Energy efficient torque distribution strategies for pure electric vehicles with independent drivetrains

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DIPARTIMENTO DI INGEGNERIA DELL’ENERGIA, DEI

SISTEMI, DEL TERRITORIO E DELLE COSTRUZIONI

Corso di Laurea Magistrale in Ingegneria Elettrica

Energy efficient torque distribution strategies for

pure electric vehicles with independent

drivetrains

Relatori Candidato

Giovanni Lutzemberger Sara Salamone

Basilio Lenzo Luca Sani

Francesco Bucchi

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Acronyms

BEV Battery Electric Vehicles

CA Controll allocation strategy

ED Even distribution strategy EM Electric motor

I-CA Implicit-Controll allocation strategy ICE Internal combustion engine

IM Induction motor

OCV Open-Circuit Voltage

PM Permanent magnet

PMSM Permanent magnet synchronous machines

SA Single axle strategy SOC State of Charge

SRM Switched reluctance motors

TV Torque Vectoring

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A Vehicle frontal area

Cd Aerodynamic drag coefficient

f_r Rolling resistance coefficient

g Gravitational acceleration

ig Gear ratio

m Vehicle mass

µ Coefficient of road adhesion

ρ Air density

rd Effective wheel radius

τsw Switching torque

V Vehicle longitudinal speed

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Chapter 1 Introduction

1.1 Motivation

Battery Electric Vehicless (BEVs) are considered a promising option to reduce pol-lution and the greenhouse effect. Today, the majority of automotive manufacturers have already developed their BEV models [1], but there are still some technological barriers that obstacle the success of electric vehicles on the market.

The driving range of BEV is still limited compared to combustion engine vehicles. Yet, there is ongoing research on how to improve performance and energy efficiency. This issue can be addressed by improving battery technology or developing opti-mal driving strategies that through advanced control on drivetrains and ancillaries can maximize energy efficiency. In this term, electric vehicles with indiependent drivetrains, can offer great potential for flexibility. In this particular configuration each wheel is driven directly, with in-wheel motor, or with indipendent drivetrain consisting of on-board motor and a gear-box. Hence, the torque for each wheel is controlled indipendently, providing great possibilities for the improvement of the vehicle handling, safety as well as efficiency [2]. Due to an actuation redundancy, the desired behavior of the vehicle can be achieved with an infinite number of dif-ferent front-to-rear wheel torque combinations, providing a freedom in the torque allocation. Therefore, the torque demand can be distributed among the multiple motors meeting various criteria, including efficiency. Especially for BEV, an opti-mized torque distribution strategy that improve vehicle economy is important since it directly contribute to driving range extension and to reduce the stress on crucial components such as the energy storage system.

Following this concept, it is of interest to examine how different driving strategies can affect the efficiency of the whole powertrain but also of its components. Such analysis requires simulation models that combine multidisciplinary domain in a

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gle environment. A dynamic model that reveals vehicle behavior and implements the torque allocation strategies, should be integrated with the elements of the elec-tric powertrain to represent the actual power usage and energy flux under different torque allocations.

The energy-efficient torque distribution scheme presented in this study, drawn from [3], investigates the potential for energy saving via wheel torque distribution through an off-line optimization that consists in the minimization of power losses. During driving, different sources of power losses occur, thus for the development of an effec-tive control strategy is important the understating of power losses and their relation with vehicle motion. In accordance with these considerations, a vehicle simulation model has been implemented in Simulink/Simscape environment, including models for electrical machine, the converter and the energy storage system, to correctly represent their efficiencies and examine energy consumption under different torque allocation scheme.

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Chapter 2 State of the Art and Previous

work

In this chapter an introduction to the subject of this thesis is given, including a focus of current electric vehicle architectures with an insight of vehicles with indipendent drivetrains, followed by a review of the research conducted on torque allocation schemes aimed at energy efficiency.

Once the background is defined, the contribution of the present study and the frame of the previous work are presented.

2.1 State of the Art and litereture review

A wide literature discusses Torque Vectoring (TV) controllers for electric vehicles with multiple drivetrains. The safety and handling benefits of TV have been widely assessed [4].In this context, several publications examine the opportunity to increase the efficiency of over-actuated electric vehicles by energy-optimal torque distribu-tions, also called control allocation schemes.

The various studies differ in many aspects: the considered sources of power losses, the drivetrain layout with in-wheel [5] [6] or on-board motors [7] [8] [3] and in some cases the proposed optimal torque distribution provide diverse configurations. For instance, [7] and [9] focus on the motor power losses, while [10] and [5] consider the motor drive, references [3] [6] [11] look at the whole powertrain, including tyre power losses. Especially regarding the drivetrain, power losses are defined through efficiency map [7], experimental measurement [12] or loss/efficiency mapping [8] [5]. These different methods will be discussed later in the next chapter.

Other papers [13] [11] point out also the importance of including tyre slip in the power loss computation. Although effective, they are based on relatively complex

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algorithms, which are quite difficult to implement in practice at the industrial level [14]. Additional aspects taken into account by studies on CA strategies include switching operations and effects on driving pleasure [15].

An interesting investigation is conducted in [7], indeed the possibility of energy saving via control allocation is extended to three different motor types. Reductions in motor power losses are achieved using an optimization scheme based on motor efficiency maps obtained by experimental measurements provided by manufactur-ers. The paper also carries out that a quadratic cost function for representing motor power losses is not an effective approach, more complex functions are needed. How-ever, simulation tests are limited to straight-line driving and a ramp steer maneuver, both at constant longitudinal acceleration, which are simple for representing a real usage.

A noteworthy contribution is provided by [3] who continued the work presented in [12], with a more detailed study based on data collected from the Range Rover Evoque prototype with four identical on-board drivetrains. The paper discussed a computationally efficient torque distribution strategies aimed at minimizing overall power loss while providing the required level of overall force and yaw moment.

With respect to the resulting torque distribution strategy, it is possible to come upon some differences in the literature. References [9] and [5] suggest that the total torque should be evenly distributed among the four motors in all conditions to maximize energy efficiency. This conclusion seems to be in contrast to those reached later by other authors [3] [7] [12] that generally at low torque demand, suggest the use of one single axle. This is likely due to many different factors: the type of motor used, the driving pattern at which tests are performed and the method used to express power losses. Specifically, in some case studies [5] [7] the driving pattern used to asses the benefit of the strategies are simple maneuvers, such as straight line at a constant speed.

Accordingly, there is a need to test the torque allocation strategies over driving cycles that reproduce a real type of usage and allow relevant consideration on ve-hicle energy consumptions. Furthermore, a sensitive analysis of the benefit of an optimal torque distibution strategy in a side-by-side comparison for different motor topologies over real driving cycles is missing.

Finally, component sizing plays an important role in performance and range economy of electric powertrains as well. Nonetheless, compared to control allocation research, fewer [16] [17] studies investigate the optimal component sizing problem for fully electric vehicles with independently actuated motors.

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2. State of the Art and Previous work

2.2 Previous work and contribution of the study

This thesis has been realized in the framework of the research presented in [3]. The main contribution of this mentioned study are:

the analytical solution of the control allocation problem for the case of identical drivetrains on the front and rear axles, based on an analytical interpolating function of the experimental drivetrain power losses. The optimal solution is parameterized as a function of the vehicle speed;

a fast and easily implementable torque distribution strategy maximizing en-ergy efficiency, based on the proposed analytical solution

the simulation-based and experimental validation of the torque distribution algorithm in cornering conditions and over driving cycles.

damping during extreme transients to provide stable vehicle response. These benefits, relevant to both human-driven and self-driven electric vehicles, have extensively been discussed and experimentally demonstrated in the literature [1–4].

Typically, a direct yaw moment controller (see Fig.1) for an electric vehicle with multiple motors consists of three main layers:

(i) A reference generator (layer 1), which determines the tar-get values of the vehicle states (such as the reference yaw rate, _w_ref, which is calculated through a multidimensional look-up table in this study, see Ref. [1]) based on the driver inputs (i.e., the steering wheel angle, d, and the accelerator and brake pedal positions,paandpb) and the

measured or estimated vehicle states (e.g., vehicle speed, V, and longitudinal acceleration, ax). Owing to the nature

of the vehicle system, in this configuration, the reference generator uses feedback signals from the plant. For exam-ple, the actual yaw rate of the car is a function ofV and ax

for a given d. Despite the significant impact of torque-vectoring control on the cornering response, it is not possible—and would not make sense—to fully compen-sate for the variation of vehicle yaw response with speed and longitudinal acceleration. As a consequence, also _w_ref has to be varied as a function ofV and ax, which can be

estimated from vehicle sensor measurements.

(ii) A high-level controller (layer 2) generating the overall traction/braking force,Fc

x, and yaw moment demands,Mcz,

to achieve the reference values of the states. The yaw moment actuation is considered here through two inde-pendent drivetrains on at least one axle (i.e., torque-vectoring differentials are not considered).

(iii) A low-level controller (i.e., the “control allocator” in Fig.

1, layer 3), which outputs the reference torques, sdi, for the

individual wheels corresponding toFc

xandMczcomputed

by the high-level controller (layer 2).

With respect to the control allocator (layer 3), special interest is on four-wheel-drive electric vehicles because of their actuation redundancy [5–10]. For example, in straight-line conditions, the overall traction/braking force demand can be generated through infinitely different front-to-rear wheel torque distributions. More-over, in the case of four independent drivetrains (i.e., one drive-train per wheel), infinite solutions are available for the generation of Mc

z actuated through different torques at the left and right

wheels. The general problem of optimally distributing the refer-ence signals to redundant actuators is referred to as control allocation [11–13].

As mentioned before, the torque-vectoring controller can signif-icantly alter the vehicle cornering response by controlling the yaw moment. The yaw moment is primarily generated by the differ-ence between the sums of wheel torque values on the left- and right-hand sides of the vehicle. In this respect, the front-to-rear wheel torque distribution on each vehicle side can have an impact on the overall yaw moment, which is normally very limited. Also,

the steering angle of the front wheels and the interaction between the longitudinal and lateral tire forces can induce a variation of the total yaw moment dependent on the adopted control allocation cri-terion. However, this influence is compensated by the feedback nature of the controller, which tends to make the vehicle follow the reference yaw rate.

For electric vehicles, an important target of the control alloca-tion algorithm is the minimizaalloca-tion of the power losses. In particu-lar, Refs. [2] and [5] analyze the different sources of power loss at the vehicle level, including the contributions associated with the electric motor drive, the transmission system, tire rolling resist-ance, and tire slip in longitudinal and lateral directions. A quasi-static model is adopted to assess the performance of different wheel torque distribution criteria. However, given the complexity of the assessed control allocation algorithms and the required state estimators, the analyses in Refs. [2] and [5] were not experimen-tally validated. Many papers, for instance [14–20], present advanced control allocation strategies for reducing the motor drive power losses, the tire slip power losses, or other tire-related per-formance indicators. They are based on relatively complex algo-rithms, which are quite difficult to implement in practice at the industrial level. Although the presented results show energy effi-ciency benefits in specific tests, the required optimal wheel torque distributions are not analyzed in detail. For real vehicle implemen-tations, there is a clear need to better understand the physics of the system and develop simple yet effective control allocation algo-rithms. This objective is consistent with the general trend toward the actual implementation of optimized energy- and vehicle-related control systems [21–25].

In the specific field of electric vehicles with multiple motors, Yuan and Wang [17] derive the analytical expressions of the power loss characteristics of electric motor drives. The conclusion is that a 50:50 front-to-rear wheel torque distribution should be the most efficient solution for the case of identical drivetrains on the two axles. Nevertheless, in practice, the same paper recom-mends to use the drivetrain/s of a single axle at low torque demands, and a 50:50 front-to-rear torque distribution at medium-high torques, without specifying the threshold for implementing the transition between the two modes. This control policy is indi-rectly confirmed by the experimental tests in Refs. [26] and [27], but these studies do not critically analyze the resulting optimal torque distribution. Dizqah et al. [28] present a simple control allocation algorithm for electric vehicles with identical power loss characteristics on the front and rear axles. However, this prelimi-nary investigation does not consider the variation of tire rolling resistance and slip ratios as a result of load transfers due to longi-tudinal and lateral accelerations.

This paper significantly extends the preliminary results of Diz-qah et al. [28] toward the implementation of simple and industri-ally viable control allocation formulations. These must: (i) be based on the results of experimental drivetrain power loss meas-urements for the whole drivetrain on rolling road facilities, rather than on complex formulations of the individual power loss contri-butions related to the inverter, motor, transmission, tires, etc.,

Fig. 1 Simplified schematic of the vehicle control system

121004-2 / Vol. 139, DECEMBER 2017 Transactions of the ASME

Downloaded From: https://dynamicsystems.asmedigitalcollection.asme.org/ on 08/14/2017 Terms of Use: http://www.asme.org/about-asme/terms-of-use

Figure 2.1: Simplified schematic of the vehicle control system proposed in [3]

In this context, power losses have been experimentally derived as the difference between the measured electrical input power at the inverter and the mechanical power at the roller of the rolling road. Therefore, the measured power loss charac-teristics include the losses in power electronics, electric motor, mechanical transmis-sion, and tyre as a whole. With the knowledge of this data, it is not possible to split all the power loss sources and investigate the relationship between component power loss, vehicle dynamics and the optimal torque distribution strategy, that potentially could lead to an overall optimization both in the component selection and in the system management [17].

In addition, to conduct the power losses measurements the electric power was provided by an external supply. Even though the battery efficiency is no directly influenced by the torque distribution, by improving the drivetrain efficiency, espe-cially the motor operating efficiency, the driving range can be extended. Thus, the weight and size of the battery can be reduced.

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In the light of the above considerations, the aim of this thesis is to investigate the link within drivetrain behaviour, vehicle performance and optimal torque distribu-tion, to make more clear the influences of torque distribution on energy consumption and efficiency. In terms of contributions, the main points are:

modeling of a vehicle longitudinal dynamic in Simulink/Simscape environment to predict performance and range, assisting component sizing and selection; integration of electric machines, converters, and the energy storage system,

particularly with respect to their efficiencies to assess the effects of different torque distribution strategies on the whole propulsion system and its compo-nents;

analysis of the relation between power losses and vehicle performances that can support the optimization of torque distribution strategies and powertrain design.

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Chapter 3 Vehicle Dynamic Model:

fundamental of propulsion and

torque distribution strategies

The present chapter deals with the vehicle dynamics modeling and the energy-efficient torque allocation strategies, which are basic concepts of this work. The understanding of vehicle dynamics is fundamental since it reveals what loads the powertrain needs to cope with during driving and the powertrain impact on the vehicle performance, whether it may be the time to accelerate, or average energy consumption per driven distance. With electric vehicles the prediction of perfor-mance and range is important. Therefore, a comprehensive vehicle model is neces-sary to simulate the vehicle response in terms of electric energy flow and efficiency, for specific speed and load conditions. Once the vheicle model is implemented, the data produced by the simulations can also have other uses in addition to predicting performance and range. For example, we will see how data about the motor torque and speed can be used to optimise both the torque distribution schemes and the compromises involved in the sizing of the motor and other subsystems.

3.1 Fundamental of vehicle propulsion

Vehicle fundamentals mathematically describe vehicle behavior based on the general principles of mechanics. A rolling vehicle, consisting of thousands of components, can be modeled with various levels of detail depending on the targeted to be studied. A large body of literature in this field already exists [18]. For the type of dynamical studies in this thesis, where powertrain load levels and energy consumption will be analyzed, it is reasonable to assume that the vehicle body is rigid. Furthermore,

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2.2.1 Rolling Resistance

The rolling resistance of tires on hard surfaces is primarily caused by hysteresis in the tire materials.Figure 2.2shows a tire at standstill, on which a force, P, is acting at its center. The pressure in the contact area between the tire and the ground is distributed symmetrically to the central line, and the resultant reaction force, Pz, is aligned to P. The deformation, z,

versus the load, P, in the loading and unloading process is shown inFigure 2.3. Due to

hys-teresis in the deformation of rubber material, the load at loading is larger than that at

unloading at the same deformation, z, as shown inFigure 2.3. When the tire is rolling, as

shown inFigure 2.4a, the leading half of the contact area is loading, and the trailing half

is unloading. Consequently, the hysteresis causes an asymmetric distribution of the ground

P

r

z Pz

FIGURE 2.2

Pressure distribution in contact area. V F_w h_w T_rf F_t hg T_rr L_b L L_a W f W r Mg_cos a Mg_sin a O Mg a FIGURE 2.1

Forces acting on a vehicle moving uphill.

Ehsani, Mehrdad, et al. Modern Electric, Hybrid Electric, and Fuel Cell Vehicles, CRC Press LLC, 2018. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/shu/detail.action?docID=5264494.

Created from shu on 2020-04-27 06:31:15.

Figure 3.1: Forces acting on a vehicle moving up to a grade [19]

the discussion of vehicle fundamentals will be restricted to , thus the longitudinal forward movement, under the assumption that vehicle stability is not under any circumstances violated. This model will focus on vehicle performance, such as speed, gradeability, acceleration, energy consumption, as well as braking performance.

The behavior of a vehicle moving along a direction is completely determined by all the forces acting on it in the specific direction. Figure 3.1 shows the forces acting on a vehicle as it travels at a given speed along a roadway with a specific grade. The tractive effort Ft, in the contact area between the tyres of the drive wheels and the

road surface propels the vehicle forward. It is produced by the power plant torque and transferred through transmission and eventually final drive to the drive wheels. The tractive effort, has to accomplish the following:

overcome the rolling resistance; overcome the aerodynamic drag;

provide the force needed to overcome the grading resistance, which is the component of the vehicle weight acting down the slope;

accelerate the vehicle, if the velocity is not constant.

Fundamental principles of mechanical systems can be used to express this relation-ship between the vehicle acceleration and the forces acting on the vehicle body as

δma = Ft− Fw− Fg− Fr (3.1)

where m is the vehicle mass, δ is the mass factor that equivalently converts the rotational inertias of rotating components into translational mass. a is the accelera-tion of the vehicle. Ft is the total tractive force acting upon the vehicle body, Fw is

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3. Vehicle Dynamic Model: fundamental of propulsion and torque distribution strategies

the aerodynamic drag force, Fg is the grading resistance force, and Fr is the rolling

resistance force.

Each of these components will be considered in turn.

3.1.1 Vehicle Resistance

As shown in Figure 3.1, vehicle resistances opposing the movement of the vehicle include rolling resistance of the tyres, appearing in Figure 3.1 as rolling resistance torques Trf and Trr, aerodynamic drag Fw, and grading resistance Fg (the term

M gsinα Figure 3.1).

All resistances and thir influence on energy consumption will be discussed in the following sections.

Rolling resistance

The rolling resistance is primarily due to the hysteresis in the tyre materials. Fig-ure 3.2 shows the deformation z, versus the load P , in the loading and unloading process. Due to hysteresis in the deformation of rubber material, the load at loading is larger than that at unloading at the same deformation.

reaction forces. The pressure in the leading half of the contact area is larger than that in the trailing half, as shown inFigure 2.4a. This phenomenon results in the ground reaction force shifting forward somewhat. This forward shifted ground reaction force, with the normal load acting on the wheel center, creates a moment that opposes rolling of the wheel. On soft surfaces, the rolling resistance is primarily caused by deformation of the ground surface, as shown inFigure 2.4b. The ground reaction force almost completely shifts to the leading half. The moment produced by the forward shift of the resultant ground reaction force is called rolling resistance moment, as shown inFigure 2.4a, and can be expressed as

Tr= Pa. (2.2)

To keep the wheel rolling, a force, F, acting on the center of the wheel is required to balance this rolling resistant moment. This force is expressed as

F=Tr rd = Pa rd = Pfr , (2.3) Deformation (z) P₂ P₁ F or ce ( P) FIGURE 2.3

Force acting on a tire versus tire deformation in loading and unloading.

Moving direction

On hard road surface On soft road surface

(a) (b) Moving direction P P F F r r r_d z z a P Pz P_x FIGURE 2.4

Tire deﬂection and rolling resistance on a (a) hard and (b) soft road surface.

Fundamentals of Vehicle Propulsion and Braking 19

Figure 3.2: Force acting on a tyre versus tire deformation in loading and unload-ing. [19]

reaction forces. The pressure in the leading half of the contact area is larger than that in the

trailing half, as shown inFigure 2.4a. This phenomenon results in the ground reaction force

shifting forward somewhat. This forward shifted ground reaction force, with the normal load acting on the wheel center, creates a moment that opposes rolling of the wheel. On soft surfaces, the rolling resistance is primarily caused by deformation of the ground surface, as

shown inFigure 2.4b. The ground reaction force almost completely shifts to the leading half.

The moment produced by the forward shift of the resultant ground reaction force is called

rolling resistance moment, as shown inFigure 2.4a, and can be expressed as

Tr = Pa. (2.2)

To keep the wheel rolling, a force, F, acting on the center of the wheel is required to balance this rolling resistant moment. This force is expressed as

F=Tr rd = Pa rd = Pfr, (2.3) Deformation (z) P2 P1 F or ce ( P) FIGURE 2.3

Force acting on a tire versus tire deformation in loading and unloading.

Moving direction

On hard road surface On soft road surface

(a) (b) Moving direction P P F F r r rd z z a P Pz Px FIGURE 2.4

Tire deﬂection and rolling resistance on a (a) hard and (b) soft road surface.

Figure 3.3: Pressure distribution at the contact patch [19].

When the tiyre rolls (Figure 3.3), as a result of the hysteresis, the normal pressure in the leading half of the contact patch is higher than that in the trailing half. The normal force due to the road is shifted from the center of the tyre in the direction of motion. This forward shifted ground reaction force, with the normal load acting on the wheel center, creates a moment that opposes rolling of the wheel. This moment is called rolling resistance moment and can be expressed as

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Tr = P a (3.2)

To keep the wheel rolling, a force F , acting on the center of the wheel is required to balance this rolling resistant moment. This force is expressed as

F = Tr rd

= P a rd

= P fr (3.3)

where rd is the effective radius of the tyre, and fr = a/rd is called the rolling

re-sistance coefficient. In this way, the rolling rere-sistance moment can be equivalently replaced by a horizontal force acting on the wheel center in the opposite move-ment direction of the wheel. This equivalent force is called rolling resistance with a magnitude of

Fr= P fr (3.4)

where P is the normal load acting on the center of the rolling wheel.

The rolling resistance coefficient fr, depends on the tyre and its working

opera-tion such as tyre temperature and tyre inflaopera-tion pressureon; the road surface, and the vheicle speed. Accordingly fr decreases with increasing pressure and tyre

tempera-ture while increases with increasing speed. Several equations for estimating rolling resistance have been developed over the years [20]. Before choosing an expression for estimating fr, the overall degree of accuracy required for the calculations should be

established. In vehicle energy consumption calculations, it is sufficient to consider the rolling resistance coefficient as a constant [20]. In [21] it has been observed that for passanger tyre the fr value ranges from 0.0065 to 0.0133. For improving energy

economy, tyre with low rolling resistance cofficient, down to 0.006 [22], has been specifically designed for electric vehicles.

Model fr

e-Up! 0.007

e-Golf 0.0065

Roadster 0.011

Table 3.1: Rolling resistance coefficient value for commercial BEVs [1][23]

Aerodynamic drag

As air travels over the body of the vehicle, it generates a force opposing vheicle motion. It mainly results from two components: shape drag and skin friction. Re-garding the shape drag, the forward motion of the vehicle pushes the air in front of it. However, the air cannot instantaneously move out of the way, and its pressure is thus increased, resulting in high air pressure. In addition, the air behind the vehicle

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2.2.2 Aerodynamic Drag

A vehicle traveling at a particular speed in air encounters a force resisting its motion. This force is referred to as aerodynamic drag. It mainly results from two components: shape drag and skin friction.

Shape drag: The forward motion of the vehicle pushes the air in front of it. However, the air cannot instantaneously move out of the way, and its pressure is thus increased, resulting in high air pressure. In addition, the air behind the vehicle cannot instan-taneouslyﬁll the space left by the forward motion of the vehicle. This creates a zone of low air pressure. The motion of the vehicle, therefore, creates two zones of pressure that oppose the motion by pushing (high pressure in front) and pulling it backward (low pressure at the back), as shown inFigure 2.5. The resulting force on the vehicle is the shape drag. The term“shape drag” comes from the fact that this drag is completely determined by the shape of the vehicle body.

Skin friction: Air close to the skin of the vehicle moves almost at the speed of the vehicle, while air away from the vehicle remains still. In between, air molecules move at a wide range of speeds. The difference in speed between two air molecules produces a friction that results in the second component of aerodynamic drag.

Aerodynamic drag is a function of vehicle speed, V, vehicle frontal area, Af, shape of the

vehicle body, and air density,ρ:

FW=1

2ρAfCD(V− Vw)

2_, ₍₂_.8)

where CDis the aerodynamic drag coefﬁcient that characterizes the shape of the vehicle

body, and Vw is a component of the wind speed in the vehicle moving direction, which

has a positive sign when this component is in the same direction of the moving vehicle and a negative sign when it is opposite to the vehicle speed. The aerodynamic drag coef ﬁ-cients for typical vehicle body shapes are shown inFigure 2.6.

2.2.3 Grading Resistance

When a vehicle goes up or down a slope, its weight produces a component that is always directed in the downward direction, as shown inFigure 2.7. This component either opposes

Moving direction

High pressure Low pressure

FIGURE 2.5 Shape drag.

Figure 3.4: Shape drag: high-pressure and low-pressure zone [19]

cannot instantaneously fill the space left by the forward motion of the vehicle, re-sulting in low air pressure. These high-prssure and low-pressure (Figure 3.4) zones act against the motion of the vehicle, generating the shape drag which is completely determined by the shape of the vehicle body. The skin friction is due to the friction between the air close to the skin of the vehicle, which moves almost at the speed of the vehicle, and the air away from the vehicle that remains still. In comparison, shape drag is much larger in magnitude than skin friction and constitutes more than 90% [24] of the total external aerodynamic drag of a vehicle.

Aerodynamic drag is a function of the effective cross sectional area of the vehicle A, and aerodynamic drag coefficient Cd, which are highly dependent on the shape

of the vehicle body

Fw =

1

2ρ A Cd(V − Vw)

2 _(3.5)

where ρ is the air density, V is the vehicle longitudinal speed, and Vw is the

compo-nent of the wind speed in the vehicle moving direction, which is positive when it is in the same direction of the moving vehicle.

The frontal area A, seldom provided by manufacturers [25], is about 1.8 m2 in a modern car [18], and the drag coefficient Cd ranges between 0.20 and 0.35 [18] [19].

Model A Cd (m2) e-Up! 2.09 0.308 e-Golf 2.19 0.281 GM EV1 - 0.19 ModelS 2.34 0.24 Roadster - 0.36

Table 3.2: Area and drag coefficient value for commercial BEVs [1] [26][23]

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de-sign. Particularly, in order not to compromise the design and comfort for passengers, most work on aerodynamical drag reduction has been focused on the Cdvalue, rather

than on the area A.

Grading resistance

As a vehicle travels up or down an incline, the gravity acting on it produces a force which is directed downward, as shown in Figure 3.1. This force opposes the forward motion during grade climbing and aids in the forward motion during grade descend-ing. In typical vehicle performance models, only uphill operation is considered as it resists the total tractive force. This grading force is usually called the grading resistance and its espression is a function of the road angle α, vehicle mass m, and the gravitational acceleration g

Fg = mgsin(α) (3.6)

As the road angle α is often small, it is generally replaced by the grade value defined as

i = H

L = tan(α) ≈ sin(α) (3.7)

The common grades on interstate highways are limited to 4 percent wherever pos-sible [20].

Effects on energy economy

Aerodynamic and rolling resistance forces are of particular interest for their effect on energy economy. Indeed, to maximise the efficiency of any vehicle the mass, aerodynamic drag and rolling resistance have to be minimised, while at the same time maximising motor and transmission efficiencies. This is particularly important in the design of BEVs in order to reduce the mass of expensive batteries required.

As seen in Equation 3.5, the aerodynamic drag is directly proportional to the drag coefficient and depends on the speed with a cubic relationship. Thus at high speeds a small increase in speed results in a large increase in vehicle power required (Figure 3.5), with an associated penalty to energy economy. In this term, a reduction of Cdwill result in a reduction of the power and the energy supplied by the battery to

overcome this resistance and the range of the vehicle will be enhanced. Eventually, in the design process an improved aerodynamics could lead to a lighter battery, with associated cost savings.

Especially in BEVs there is greater opportunity for reducing Cd. This is due

to major flexibility in the location of the components, a changed need for frontal

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Figure 3.5: Power requirement to overcome aerodynamic drag for vehicle of different frontal areas and drag coefficients for at a range of speeds [26].

Figure 3.6: Power requirements to overcome the sum of aerodynamic drag and rolling resistance at a range of speeds, with low-resistance tyres fr= 0.005 and with

ordinary tyres fr = 0.015 [26].

cooling air inlet and a more flat underbody in BEVs [26]. Table 3.3 show the Cd

value for the same vehicle model propelled by a Internal combustion engine (ICE) and by an electric power plant. However, it must be recalled that at on very low-speed vehicles the aerodynamic shape is irrelevant, while it is important for medium and high-speed vehicle.

About the rolling resistance, the force and hence the power to overcome the rolling resistance is proportional to the rolling resistance coefficient (Figure 3.6). Thus, a low resistance coefficient is desirable and the choice of tyres is important.

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Model Cd ICE Cd Electric

e-Up! 0.32 0.308

e-Golf 0.275 0.27

e500 0.362 0.311

Table 3.3: Cd values for ICE and electric motor powered BEVs [1][27].

Finally, by comparing the power requirement to overcome rolling resistance and aerodynamic drag at different speeds (Figure 3.7) it is evident that at low speeds, aerodynamics have very little influence, whereas at high speeds they are the major influence on power requirements. In term of grading resistance and rolling resistance,

Figure 3.7: Power requirements to overcome rolling resistance and aerodynamic drag at different speeds.[26].

the mass of the vehicle has also an important effect, increasing the power required to overcome this resistance force. Apart from this relation, the mass of the vehicle will also have considerable influence on vehicle performance (section 3.1.3).

3.1.2 Maximum tractive effort and tyre-ground adhesion

By rearranging Equation 3.1, we arrive at an equation that expresses the vehicle motion in the longitudinal direction:

mdV

dt = (Ftf + Ftr) − (Frf + Frr+ Fw+ Fg) (3.8) The first term on the right-hand side of Equation 3.8 is the total tractive effort, that for an all-wheel-drive vehicle is the sum of the tractive effort of the front and rear tires, Ftf and Ftr. The second term is the resistance force seen is subsection 3.1.1.

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Considering one wheel (Figure 3.8), the tractive effort on it can be expressed as:

Ft= Tw rd = Tmigηt rd (3.9)

where Tw is the torque on the drive wheel and rd is the effective wheel radius.

In turn, Tw is generated by the power plant, in this case an electric motor, and

transmitted through the transmission, a fixed gearbox with a constant gear ratio ig. Thus, the torque on the drive wheel is obtained as the torque delivered by the

motor Tm multiplied by the gear ratio ig and its efficiency ηt.

The torque on the drive wheels, transmitted from the power plant, is expressed as

Tw= igi0ηtTp, (2.27)

where igis the gear ratio of the transmission deﬁned as ig= Nin/Nout(Nin—input rotating

speed, Nout—output rotating speed), i0is the gear ratio of theﬁnal drive, ηtis the efﬁciency

of the driveline from the power plant to the drive wheels, and Tp is the torque output

from the power plant.

The tractive effort on the drive wheels, as shown inFigure 2.11, can be expressed as Ft=

Tw rd .

(2.28)

Substituting Equation 2.27 into Equation 2.28 yields the following result: Ft =

Tpigi0ηt rd .

(2.29) The friction in the gear teeth and bearings creates losses in the mechanical gear transmis-sion. The following are representative values of the mechanical efﬁciency of various components:

Clutch: 99%.

Each pair of gears: 95%–97%. Bearing and joint: 98%–99%.

The total mechanical efficiency of the transmission between the engine output shaft and drive wheels is the product of the efficiencies of all the components in the driveline. As afirst approximation, the following average values of the overall mechanical efficiency of a manual gear-shift transmission may be used:

Direct gear: 90%. Other gear: 85%.

Transmission with very high reduction ratio: 75%–80%.

T_w N_w V F_t r_d FIGURE 2.11

Tractive effort and torque on a drive wheel.

28 Modern Electric, Hybrid Electric, and Fuel Cell Vehicles

Figure 3.8: Tractive effort and torque on a drive wheel [19].

In vehicle operation, the maximum tractive effort on the drive wheels, should not exceed the maximum values determined by the tyre-ground cohesion otherwise, the wheels will spin on the ground, leading to vehicle instability. Therefore, the maximum tractive effort on the drive wheel depends on the longitudinal force that the adhesive capability between the tyre and the ground can supply, rather than the maximum torque that the motors can deliver. Accordingly, the acceleration may be limited by the adhesive capability between the tyre and the ground which sometimes is the main limitation of vehicle performance.

The maximum tractive effort that the tyre-ground contact can support is usually described by the product of the normal load and the coefficient of road adhesion µ. Equation 3.10 and equation 3.11 show the maximum tractive effort at the front and at the rear

Ft,max = µ Wf (3.10)

Fr,max= µ Wr (3.11)

where Wf and Wrare the normal load acting on the front and rear axle, respectively.

Particularly, the normal loads can be determined through the equilibrium equation for moments about the centers of the tyre-ground area. Neglecting the resistance

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forces it is possibile to obtain the following espression for the normal loads Wf = mg L Lb− hg a g (3.12) Wr = mg L La+ hg a g (3.13) The first term on the right-hand side of Equations 3.12 and 3.13 is the static load on the front and rear axles when the vehicle is at rest. The second term represent the dynamic component, due to the load tranfer from the front to the rear axle during acceleration.

inFigure 2.24b. This maximum braking force limited by the adhesive capability can be expressed as

Fb max= μW, (2.51)

where µ is the adhesive coefficient of the tire–ground contact. Similar to the traction case, the adhesive coefficient varies with the slipping of the tire, as shown in Figure 2.25. However, the slip is defined in braking as

s= 1 −rω V

× 100%, (2.52)

where V is the vehicle translational speed,ω is the wheel rotation speed, and r is the wheel radius. In this deﬁnition, when ω = 0, that is, the wheel is completely locked, s = 100%.

Figure 2.25shows the typical relationship between adhesive coefﬁcient and wheel slip. There exists a maximum value in the slip range of 15%–20% and somewhat declining at 100% slip.Table 2.3shows the average values of tractive effort coefﬁcients on various roads.2

50 0 100% Slip 15-20 Adhesive coeﬃcient ( m ) A B Longitudinal Lateral _m_s mp O FIGURE 2.25

Variation of tractive effort coefﬁcient with longitudinal slip of a tire.

TABLE 2.3

Average Values of Tractive Effort Coefﬁcient on Various Roads

Surface Peaking Values (μp) Slipping Values (μs)

Asphalt and concrete (dry) 0.8–0.9 0.75

Concrete (wet) 0.8 0.7

Asphalt (wet) 0.5–0.7 0.45–0.6

Gravel 0.6 0.55

Earth road (dry) 0.68 0.65

Earth road (wet) 0.55 0.4–0.5

Snow (hard packed) 0.2 0.15

Ice 0.1 0.07

40 Modern Electric, Hybrid Electric, and Fuel Cell Vehicles

Figure 3.9: Variation of the adhesion coefficient with longitudinal slip of a tire [19].

The coefficient of road adhesion depends nonlinearly on the longitudinal tyre slip s, which is caused by deformation of the tyre during acceleration and decelerations. The slip s, of a tyre is defined as

s = 1 − V wrd × 100% (3.14)

where V is the translational speed, w is the angular speed of the tyre, and rd is the

wheel radius.

As shown in Figure 3.9, starting from zero µ increases linearly with increasing slip. The peak tractive effort is reached at a slip of 15% - 20% where the coefficient peaks at values around 0.8 to 1, depending on type of tyre and road condition. At higher value of the slip the tractive effort coefficient falls rapidly from the peak value to the purely sliding value, resulting in an unstable condition. For normal driving, the slip of the tyre must be limited in a range of less than 15% - 20%.

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3.1.3 Vehicle Performance

Performance characteristics of a road vehicle refer to its capability to accelerate and decelerate, negotiate grades and reach a maximum speed. These characteristics are different depending on the vehicle type and size. Mass of the vehicle is of great importance to vehicle performance. By saving weight all aspects of vehicle performance would be improved [26][20], including energy consumptions which are proportional to the mass of the vehcile. The tractive and braking effort developed by the tyres and the resisting forces acting on the vehicle determine the performance potential of the vehicle (Figure 3.10). Typically, overall vehicle performance also deals with cornering ability, but as this is mainly a function of suspension geometry and vehicle design [18] it is outside the scope of this study.

Figure 3.10: Tractive effort of electric vehicle with single-speed transmission and its resistance [19].

Maximum Speed

The maximum speed of a vehicle is the highest constant cruising speed that the vehicle can achieve at full power on a level road. The maximum speed of the vehicle is determined by the maximum speed of the power plant and the gear ratio of its transmission

Vmax =

π wem,maxrd

ig

(3.15) where wem,maxis the maximum speed of the engine, in this case the electrical motors,

rd is the radius of the wheels and ig is the gear ration of the transmission, which in

this case will a fixed gearbox for each motor.

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Gradeability is usually defined as the grade or the grade angle that a vehicle can overcome at a certain constant speed. By combing Equation 3.8 and 3.9, the tractive effort and resistance equilibrium for a vehicle moving on a road with a small grade at constant speed, can be written as

Tmigηg rd = mgfr + 1 2ρ A CdV 2 _{+ mgi} _(3.16)

The gradeability of the vehicle can also be obtained from the diagram in Figure 3.10, in which the tractive effort and resistance are plotted.

Acceleration performance

The acceleration of a vehicle is a key performance indicator, indeed when high-performance vehicles are compared to one another, one of the first parameter to be reviewed is the acceleration performance. It is usually described by the acceleration time covered from zero speed to a certain high speed, 0-100 km/h for example, on level ground. Such acceleration figures are found from simulation or testing of real vehicles, that for electic vehciles are carried out at maximum torque. The maxi-mum torque of an electric motor is a fairly simple function of angular speed. In most cases, at low speeds, the maximum torque is a constant, until the motor speed reaches a critical value, the base speed, after which the torque falls in such a way that the power remains constant. Refering to Equation 3.18, the acceleration of the vehicle on level ground is

a = Tmigηg rd − mgfr − 1 2ρ A CdV 2 m (3.17)

For an electric vehicle powered by an electric motor with a single-gear transmission the acceleration with respect to vehicle speed is shown in Figure ??.

Finally, solving equation 3.18, from low speed V1 to high speed V2, it is possibile

to predict the time to accelerate the vehicle.

ta= Z V2 V1 m Tmigηg rd − mgfr − 1 2ρ A CdV2 dV (3.18)

3.1.4 Braking performance and distribution

The braking performance of a vehicle is an important factor in vehicle safety. A braking system for a vehicle must quicly reduce vehicle speed and maintaining the vehicle traveling direction stable and controllable under various road conditions. These requirements are satisfied by applying sufficient braking force on the wheels

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and properly allocating the total braking force on the front and rear wheels. One of the most important features of BEVs is their ability to recover significant amounts of braking energy, increasing the overall effciency of the vehicle. Thus, when it comes to BEVs the design and control of the braking systems have to ensure the vehicles braking performance and its ability to recover as much braking energy as possible.

The braking torque applied by the braking system results in a braking force developed on the interface between the road and the tyre. The braking force can be expressed as the ratio between the braking torque and wheel radius. The braking force increases with the braking torque untill the braking force reaches the limit of tire-road adhesion and it cannot increase any further, although the braking torque may still increase. Even in braking, the maximum braking force is limited by the adhesive capability, which varies with the slip ratio. However the slip is defined in braking as

s =1 −wrd V

× 100% (3.19)

In this definition, s=100% is reached when w = 0 and the wheel is completely locked. Hence, the maximum braking force is limited by the tire-ground adhesion and is proportional to the normal load acting on the tire. Similar to when the total tractive force was calculated, the normal load should again satisfy the equilibrium equations for moments about the front and rear tyre contact point, resulting in

Wf = mg L Lb + hg a g (3.20) Wr = mg L La− hg a g (3.21) During braking, there is a load transfer from the rear axle to the front axle.

3.1.5 Consideration on energy economy

The various parameters that contribute to the motion of a vehicle have been ex-amined individually, to be integrated in the vehicle dynamic model. Each element presented in the previuos section affect vehicle behaviour both in the the dynamic performance, but also in terms of energy economy. In BEVs the coordination be-tween element and their efficiencies is extremely important to cut energy consump-tion and extend the driving range. Various choices are available in the design process of a BEV to achieve the desired level of energy economy. In this light, the techniques that can be implemented mainly invole the following aspects:

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reduce the rolling resistance and good aerodynamic design is an important method to reduce the aerodynamic drag at high speed [20][26];

Reducing vehicle mass: saving weight as well as reducing energy consumption which are proportional to vehicle mass, directly contribute to the size of the batteries and these are both heavy and expensive. For the chassis material, both carbon fibre and aluminium are considerably more expensive than steel. However, by using these materials not only is the car lighter, but for a given range a considerable amount of expensive batteries can be saved.

Properly matched transmission: parameters of the transmission such as gear ratios, greatly affect operating condition of the motoros. In the design of the transmission, the parameters should be defined so that the motors operate in the high-efficiency region;

Advanced drivetrains: advanced electric motors drive may greatly enhance the overall efficiency of vehicles, as the motor lossed accounts for the largest portion of the total losses [8] [7].

Improving engine operation efficiency: effective control techniques on drive-trains present great potential for improving vehcile energy economy [15][5][12]. As we will see, electric vehicles with multiple motors can offer the opportu-nity of an optimal torque distribution among the four drivetrains that enhance energy efficiency, through power losses minimization.

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3.2 Torque distribution strategies

Within various BEV topologies, electric vehicles with indiependent drivetrains, have recently raised interest from both industry and academia [28] [29]. With this vehicle architecture the overall wheel torque request can be managed with different config-uration: front-wheel drive, rear-wheel drive and four wheel drive. This allow the implementation of control technique such as TV, which can improve driveability, safety as well as efficiency. Owing to the use of multiple motors, in straight-line condition, the overall traction/braking torque demand can be realized through in-finitely different front-to-rear wheel torque combinations. This provides a freedom in the torque allocation and thus, the opportunity to achieve an optimal torque distribution among the four drivetrains that enhance the energy efficiency.

The presented energy efficient torque strategies has been carried out by the au-thors in [3] in the framework of European projects iCOMPOSE. In particular, the potential for energy saving via wheel torque distribution strategies is investigated through an off-line optimization that consists in the minimization of power losses. During vehicle operation, differen sources of power loss exist such as dissipations due to the drivetrains and the ones related to the tyre. The development of enery-efficient torque distribution strategies requires the analysis of the various sources of power losses with the aim to detect the major contribution that can be mini-mized. To assess the benefit of the optimal torque allocation strategies, the energy consumptions are also analyzed when operating in a front-wheel-drive mode Single axle strategy (SA), and in a four-wheel-drive mode Even distribution strategy (ED) with constant 50:50 front-to-rear wheel torque distribution. The torque distribu-tion strategies are compared under different standardized driving cyle, to test them under different operating conditions.

In the presetn section, the energy-efficient torque distribution problem is intro-duced, followed by a review of power losses source. Then the implemented torque distribution strategies are described together with the control structure.

Problem Formulation

For an effective control strategy, an optimal torque distribution scheme, apart from minimizing power losses, must also mantain the total torque demand Ttof the

vehi-cle (Equation 3.22), ensuring that the motor torque limits are respected (Equation 3.27)

Figure 3.11 show the vehicle geometry. In particular, the front-left, front-right, rear-left and rear-right wheels are respectively specified as 1, 2, 3 and 4. As a consequence of the dristributed drivetrains, the front and rear semi-wheelbases, df

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and dr are assumed equal. Finally, the wheel radius is rd and the half track of the

vehicle is th.

Figure 3.11: Vehicle geometry.

The total torque requested at the wheel Tt can be expressed as

Tt= τw,1 + τw,2 + τw,2 + τw,4 = Fxrd (3.22)

where τw,1−4are the torque at each wheel, while Fx and rdare respectively the total

traction force and the wheel radius.

In straight-line condition, the sum of torques on the left-side wheels τw,l is equal

to the sum of torques on the right-side wheels τw,r, as

τw,l = τw,1 + τw,3 = Tt 2 (3.23a) τw,r = τw,2 + τw,4 = Tt 2 (3.23b)

As a consequence, the wheel torque allocation problem can be approached inde-pendently and for one side of the vehicle. In particular, for each side of the car, the problem will consist in the definition of the front-to-total ratio r of the side torque demand that minimize power losses.

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The front-to-total ratio of the side torque can generally be written as

r = τw,f τw,f + τw,r

= τw,f τw,s

(3.24)

where τw,f and τw,r are respectively the wheel torque at the front and at the rear,

for one side of the car. Accordingly their sum returns the wheel side torque τw,s.

As the drivetrains operate in different regions of the torque-speed space, power losses vary under the various condition. Necessarily, to determine the optimal front-to-total ratio, a function that evaluates the side power losses at different vehicle speed and front-to-total torque distribution has to be minimized. In this light, the power losses for one side of the vehicle can be expressed as

Ploss,s(V, τw,s, r) = Ploss,f(V, τw,s, r) + Ploss,r(V, τw,s, r) (3.25)

where it is specified the relation

τw,f = r τw,s

τw,r = (1 − r) r τw,s

(3.26)

Notably, in Equation 4.2 the vehicle speed V is used as the speed of each vehicle corner, which is reasonable for small yaw rate [7] [3] [12]. Besides, the power loss characteristics of the four drivetrains are assumed to be equal. In the optimization procedure the maximum torque that can be deliveder by each motor, τm,max, must

be considered into the contrain of the problem. Considening ig the total gear ratio

of the transmission, the following constains must be respected r τw,s ig < τm,max (3.27a) (1 − r) τw,s ig < τm,max (3.27b)

Finally, in order to avoid oversteering behavior [12][13], r is set to be greater that 0.5.

The optimization problem is solved in MATLAB environment. With respect to one side, for every combination of vehicle speed-side torque demand, by varying the torque distribution between the front and the rear, i.e. by varying the r value between 0.5 and 1, the side power losses are generated according to Equation 4.2. Then, using function fmin, with the interior point method adopted, the front-to-total ratio that minimize power losses is selected.

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Power Loss Sources

Understanding power loss sources during driving is the basis for developing control strategy for energy efficiency. In particular, for this study it is of interest to focus on power losses components that are influenced by the torque distribution.

Generally, vehicle power losses involve losses in the drivetrains and at the tyres. For a BEV, drivetrain power losses include the contributions of the converters, electric motors, and transmissions. Tyre contributions are instead related to rolling resis-tance and tire slip. Although the energy storage system introduce energy losses which influece the whole efficiency, this contribution is not considered as it is not direcly affected by different torque allocations.

Regarding the tyre, tyre slip losses are influenced by the wheel torque control strategy. The longitudinal tire slip power losses at the i -th wheel is given as

Ploss,slip = Fx,ivx,slip (3.28)

where Fx,i is the longitudinal traction force at the wheel and vx,slip is the slip speed.

As showed in [14] this contribution is less significant than the drivetrain losses in most driving condition and the practical implementation of these strategies requires estimator of slip velocities of each tire, which is beyond the capability of the current production vehicle state estimators.

About the rolling resistance contribution, as seen in section 3.1.1 it is linearly depen-dent on the vertical load. The load transfer in longitudinal acceleration (Equation 3.12 and 3.13) causes a linear variation of rolling resistance power loss, accordingly the power loss characteristics at each corner may vary. However, in normal driving conditions the load transfers reach low value and it is reasonable to assume that rolling resistance power losses are equal at each vehicle corner. The derived power loss characteristic, does not change the front-to-total optimal torque distribution [3]. For the given reason, tyre losses are not considered in the optimization procedure.

Turning the attention to the drivetrain, the electric power Pel provided by the

battery is converted by the electric motors to generate mechanical power Pm

avail-able at the wheel. Specifically, the power flux delivered by the battery is decreased of a certain portion depending on the efficiency of the inverter, the motor and the gearbox, as

Pm = (ηinverterηmotorηgearbox) Pel (3.29)

Power losses associated with the drivetrain represent the major contribution to ve-hicle power losses [8] [10] [30]. Besides, the operational efficiency of the drivetrain

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components is influenced by the torque allocation thus, the enery-efficient torque distribution is achieved through the analysis and the minimization of the drivetrain power loss. Every component of the drivetrain and the relation between its efficiency and vehicle operation behavior is analyzed and discussed in the following section, which deals with the whole powertrain including the battery.

Torque distribution schemes and contol structure

The torque distribution strategies implemented in the model are presented:

Even Distribution strategy (ED): the total torque demand is distributed equally between the four drivetrains. In this case, the front-to-total ratio of the total side torque is set to 0.5;

Single Axle strategy (SA): the total torque demand is supplied only by the front drivetrains. Here, the front-to-total ratio of the side torque is set to 1. The front motors are selected, instead of the rear ones, for safety reasons, preferring an under steering behavior rather than over steering one. In the case of wheel torque saturation the torque demand beyond the limit is transferred to the rear drivetrains;

Controll allocation strategy (CA): the optimal torque distribution is achieved by switching between ED and SA, depending on side torque demand and vehicle speed. The switching torque τsw between ED and SA is computed

offline by comparing, at constant speed, the power losses of the ED and SA configurations. In this case, r can assume only the value associated with the SA and the ED, respectively 1 and 0.5. In particular, the switching torque is the torque value at which the power losses of the ED are equal to the ones of the SA. It can be obtained as the solution of the following equation

Ploss,f(V, τsw) + Ploss,r(V, 0) = 2 Ploss,r(V,

τsw

2 ) (3.30)

Notably the switching torque is a function of the vehicle speed. In particular, Figure show the τsw for one side of the vehicle with respect to vehicle speed.

Focusing on one side, the CA stategy provides the following torque distribution

τw,f(V, τw,s) =    τw,s, if τw,s ≤ τsw(V ) 0.5 τw,s, if τw,s > τsw(V ) τw,r(V, τw,s) =    0, if τw,s ≤ τsw(V ) 0.5 τw,s, if τw,s > τsw(V ) (3.31)

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Figure 3.12: Switching torque of the vehicle as a function of vehicle speed.

According to Equation 4.5, if for a certain vehicle speed the side torque demand is lower then the switching torque then the SA will be the optimal solution, otherwise if the torque demand is greater than the switching torque then the ED is the optimal solution.

The online calculation of the switching torque is based on a look-up table with the vehicle speed in input.

Implicit-Controll allocation strategy (I-CA): provide the optimal torque distri-bution through a map that for every side torque-speed combinations returns the front-to-total torque ratio that minimizes power losses. The map generated through the offline optimization, allows intermediate distribution between the single axle and the even distribution (Figure 3.13). The online calculations of the I-CA are based on a 2D look-up table of the optimal front-to-total torque distribution, receiving in inputs vehicle speed and vehicle side torque demand. Figure 3.14 show a semplified vehicle control structure. The reference generator, consisting in a proportional-integral controller, through a feedback control on ve-hicle velocity, outputs the total torque demand to follow an input driving profile. The torque distribution strategy block, through the use of look-up tables (for the CA and I-CA), sets the wheel torque at each wheel for every control strategies. The use of look-up table in the on line torque allocation allows real-time operation with minimum demand on the processing hardware.

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3. Vehicle Dynamic Model: fundamental of propulsion and torque distribution strategies Vehicle speed (km/h) Side Torque (Nm) 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

Figure 3.13: I-CA: map of r as a function of vehicle speed and demanded wheel torque on the vehicle side.

Reference generator Tt

Torque distribution strategy min f =P Ploss r Fx x y 4 3 1 2

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Electrical powertrain: modeling

and requirement

The powertrain of the considered BEV consist of four independent drivetrains and the rechargeable energy storage system. Specifically, each drivetrain comprises the electric motor, the power electronic converter and the single-speed transmission.

As posed in section 3.2 the problem of the energy-efficient torque distribution allocation is addressed through the minimization of the drivetrain power losses. Specifically, the optimization procedure requires the knowledge of the relationship between the drivetrain power losses and vehicle motion parameter. In this chapter, the operational efficiency of the drivetrain components with respect to the driving performance is investigated and formulated.

The purpose is to built a comprehensive model which describes the vehicle mo-tion, allowing the implementation of the torque distribution strategies, as well as includes the elements of the electrical powertrain. The integration of these sys-tem domains supports the validation of the powertrain working condition under the different torque distribution strategies and the computation of the relative energy consumption to effectively assess which distribution method most improve the en-ergy saving.

In accordance with the above considerations in the following sections the drive-train component power losses are analyzed and the model adopted introduced. Sub-sequently, to complete the powertrain, also the energy storage system is presented. Finally, once the model has been completely defined with the vehicle motion and the powertrain elements, the power requirement and sizing criteria are discussed.

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4. Electrical powertrain: modeling and requirement

4.1 Drivetrain efficiency model

The problem of the energy efficient torque distribution has been formulated from the aspect of drivetrain power loss minimization, as they are influenced by the torque allocation during driving.

Each drivetrain consist of the electric motor, the power converter and a single-gear transmission (Figure 4.1). The Electric motor (EM) converts the electric energy from the battery into mechanical energy to propel the vehicle, or vice versa, to enable regenerative braking to generate electricity for charging the on-board energy storage. The power converter is used to supply the electric motor with proper voltage and current to produce proper torque and speed, according to the torque distribution strategy. Since the speed-torque profile on an EM is close to the ideal, a single-gear or transmission meet the vehicle performance requirement.

wm,τm

ww,τw

EM Power

Converter

Figure 4.1: Semplified drivetrain structure.

The efficiency model adopted for each drivetrain component is now presented and discussed.

4.1.1 Electric machine

In BEV, the electric machine is the core energy conversion components. The features required of an EM for traction applications include high torque for starting, at low speeds and hill climbing, and high power for high-speed cruising; wide speed range at constant power; high efficiency over wide speed-torque ranges; intermittent overload capability; reliability and robustness; low acoustic noise and low torque ripple, as well as acceptable cost. Different machine type for vehicular application have been introduced, over the past few years. According to the research conducted in [31], [1] [32], state-of-the-art drivetrains are often equipped with Induction motors (IMs) and the Permanent magnet synchronous machiness (PMSMs). In addiction, Switched

Energy efficient torque distribution strategies for pure electric vehicles with independent drivetrains

DIPARTIMENTO DI INGEGNERIA DELL’ENERGIA, DEI

SISTEMI, DEL TERRITORIO E DELLE COSTRUZIONI

Corso di Laurea Magistrale in Ingegneria Elettrica

Energy efficient torque distribution strategies for

pure electric vehicles with independent

drivetrains

Contents

Acronyms

Chapter 1

Introduction

1.1

Motivation

Chapter 2

State of the Art and Previous

work

2.1

State of the Art and litereture review

2.2

Previous work and contribution of the study

Chapter 3

Vehicle Dynamic Model:

fundamental of propulsion and

torque distribution strategies

3.1

Fundamental of vehicle propulsion

3.1.1

Vehicle Resistance

3.1.2

Maximum tractive effort and tyre-ground adhesion

3.1.3

Vehicle Performance

3.1.4

Braking performance and distribution

3.1.5

Consideration on energy economy

3.2

Torque distribution strategies

Electrical powertrain: modeling

and requirement

4.1

Drivetrain efficiency model

4.1.1

Electric machine