Eindhoven University of Technology MASTER Battery-Electric

Eindhoven University of Technology

MASTER

Battery-Electric Powertrain Design Analysis for an Efficient Passenger Vehicle

Duclos, J.

Award date:2021

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https://research.tue.nl/en/studentTheses/42037d81-24cc-4854-8480-84e8fad8a536

Battery-Electric Powertrain Design Analysisfor an Efficient Passenger Vehicle

Master thesis

J. (Julien) Duclos BSc

0815840

Master: Automotive TechnologyDepartment: Mechanical EngineeringResearch Group: Control Systems Technology (CST)

Supervisor: dr.ir. T. (Theo) Hofman

Report ID: CST2021.032

Eindhoven, June 2021

Declaration concerning the TU/e Code of Scientific Conduct for the Master’s thesis

I have read the TU/e Code of Scientific Conduct i.

I hereby declare that my Master’s thesis has been carried out in accordance with the rules of the TU/e Code of Scientific Conduct

Date: 22/06/2021

Name: J. Duclos

ID-number: 0815840

Signature:

Insert this document in your Master Thesis report (2nd page) and submit it on Sharepoint

i See: http://www.tue.nl/en/university/about-the-university/integrity/scientific-integrity/ The Netherlands Code of Conduct for Academic Practice of the VSNU can be found here also. More information about scientific integrity is published on the websites of TU/e and VSNU

Version 202007

Preface

This document includes the report of the author’s graduation project, entitled “Battery-Electric Powertrain Design Analysis for an Efficient Passenger Vehicle”. The project wasaimed at the investigation of various technological and topological powertrain design choicesin a battery-electric passenger vehicle. In particular, it focuses on understanding the influenceof the topology, machine technology and gearbox type on the vehicle’s energy consumptiongiven its performance constraints. The research was conducted at the Control Systems Tech-nology (CST) group of the Eindhoven University of Technology (TU/e), the Netherlands,under supervision of dr.ir. T. Hofman, as part of the graduation from the Automotive Tech-nology (AT) Master program. A written thesis, in the form of an 11-page (draft) publicationpaper1, is included and summarizes the main result(s) of the graduation work.

1A first version of the work has been submitted to the 2021 IEEE Vehicle Power and Propulsion Conference(IEEE VPPC 2021), held on 25-28 October, 2021 in Gijon, Spain.

ii

Summary

The flexibility of electric machines enables a broad scope of possible choices in battery-electricpowertrain design. To investigate the energetic effect of these design choices, different battery-electric powertrain configurations, varying in topology, machine technology, and transmissionarchitecture are investigated for a passenger hatchback vehicle. For each configuration, theelectric machine(s), gear ratio value(s), and controls are jointly optimized in an integrated(bi-level) fashion using the particle swarm optimization (PSO) algorithm. The combinedcomponent and topological choices lead to a large variation in the vehicle’s energy consump-tion and can significantly influence the total electric machine sizing in the powertrain. Theenergy consumption of the vehicle was lowered by: (i) using two axles (AWD) instead ofone axle (RWD) to drive the vehicle; (ii) choosing a distributed drive system, over a centraldrive system; followed by (iii) using permanent-magnet-synchronous type machine(s) insteadof asynchronous induction machines; (iv) using multiple electric machines in single-axledtopologies; and, (v) using two-speed instead of single-speed gearbox(es).

iii

Contents

Declaration TU/e Code of Scientific Conduct i

Preface ii

Summary iii

Thesis paper 6

I. Introduction 6

II. E-Powertrain system design 7A. Topology design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7B. Mathematical problem formulation: bi-level optimization . . . . . . . . . . . 7

III. Vehicle modeling 8A. Vehicle road-load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9B. Braking strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9C. Final drive transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10D. Gearbox transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10E. Electric machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10F. Battery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10G. Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11H. Model validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

IV. Optimization results 12A. Numerical implementation and solver . . . . . . . . . . . . . . . . . . . . . . 12B. Optimal design results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12C. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

V. Conclusion 14

Appendix 14

References 15

iv

Thesis paper

5

GRADUATION REPORT (4AT98), JUNE 2021 1

Battery-Electric Powertrain Design Analysisfor an Efficient Passenger Vehicle

J. (Julien) Duclos, (0815840)Dept. of Mechanical Engineering (ME), Control Systems Technology (CST) group

Eindhoven University of Technology (TU/e), P.O. Box 513, 5600 MB Eindhoven, The NetherlandsE-mail: [email protected]

Abstract—The flexibility of electric machines enables a broadscope of possible choices in battery-electric powertrain design. Toinvestigate the energetic effect of these design choices, differentbattery-electric powertrain configurations, varying in topology,machine technology, and transmission architecture are investi-gated for a passenger hatchback vehicle. For each configuration,the electric machine(s), gear ratio value(s), and controls arejointly optimized in an integrated (bi-level) fashion using theparticle swarm optimization (PSO) algorithm. The results showthat the combined component and topological choices can leadto a maximum reduction of 17.61% in the vehicle’s energy con-sumption and can significantly influence the total electric machinesizing in the powertrain. For the individual design choices, ithas been showed that the energy consumption of the vehicle islowered by: (i) using two axles (AWD) instead of one axle (RWD)to drive the vehicle (-12.0%); (ii) choosing a distributed drivesystem, over a central drive system (-6.6%); followed by (iii) usingpermanent-magnet-synchronous type machine(s) instead of asyn-chronous induction machines (-3.0%); and, (iv) using multipleelectric machines in single-axled topologies (-0.80%). Althoughthe smallest energetic impact was observed for (v) using two-speed(s) instead of single-speed(s) (-0.38%), it affected the (total)electric machine size the most (-10.3%).

I. INTRODUCTION

More recently new passenger car models equipped withbattery-electric powertrains (e.g. EQS, E-tron, Polestar, etc.)are being introduced by original equipment manufacturers(OEMs) [1]. Among these battery-electric vehicles (BEVs)announced and already on the market, there exist a largevariation in the powertrain design: the layout (topology), i.e.locations of the electric machine (front-wheel (FWD), rear-wheel (RWD), all-wheel driven (AWD), etc.), type of electricmachine(s) (e.g. permanent-magnet synchronous (PSM) orasynchronous induction machine (AIM) and how they areconnected via transmission components (back-to-back, co-axial, transverse, etc.) to the driven wheels. Given the largevariation in powertrain designs (architectures), the main ques-tion that arises is what the optimal powertrain configurationfor such a BEV is, and more fundamentally, what underlyinginfluence the possible design choices have on the vehicle’skey performance indicators (KPIs), like the vehicle efficiency(kWh/km) or driving range (km), performance (acceleration,top speed, gradeability) and costs (of ownership).

Related literature: In the context of BEVs, the powertraindesign problem and the influence of design choices thereinhas previously been addressed in literature. Common practiceherein in is to subdivide the design problem into four different

Fig. 1. Schematic overview of the battery-electric vehicle powertrain model,including a battery pack (BAT), electric machine(s) (EM) using either PSMor AIM technology, gearbox(es) (GB) having either one (1-spd) or twospeeds (2-spd) that is/are connected via an optional final drive (FD) trans-mission to the wheels (W). The parameters in gray represent the design andcontrol variables that are optimized in this study: electric machine size(s) Pm,gearbox ratio size(s) Rg, gear position γ(t), and torque split φ(t) in casemultiple EMs are present. The arrows indicate the direction of the power flowfrom component input to output in a forward fashion.

layers: (i) drivetrain topology, (ii) component technology,(iii) component sizing, and (iv) powertrain control. This isalso referred to as a system-level design (SLD) [2]. Mostpowertrain design research has only focused on one or twodesign levels at maximum. Mainly addressing, for a fixedtopology, the component specifications (e.g. optimal sizing andtechnology), investigating the effect of the transmission tech-nology [3]–[8], the type of electric machine technology [9],[10], or both at the same time [11]. Other research investigatesthe effect of different topology designs (e.g. optimal sizing perlayout) [12]–[16], in which the technology of the componentsare kept fixed most of the time. To the knowledge of the author,there is no present work that investigates the powertrain designproblem for a BEV addressing both the topology layout, andtransmission and machine technology simultaneously.

Statement of contributions: Against this backdrop, this workinvestigates both the effect of technological and topologicaldesign choices in a battery-electric powertrain for a passengervehicle. In particular, it focuses on understanding the influenceof these design choices with respect to the vehicle’s energyconsumption given its performance constraints. The mainunderlying research question is centered on how to optimaldesign the electric powertrain with respect to the selection ofthe topology, machine technology and gearbox type. Thereto,an efficient passenger vehicle, i.e., with the BMW i3 94 Ahcharacteristics (cf. Table I), is selected to simulate the effectof the design choices on a vehicle currently on the market.


To investigate the different powertrain options, a modularbackwards-facing quasi-static simulation model of the BEVpowertrain has been developed, schematically represented ina forward fashion by the overview in Fig. 1. The model in-cludes the longitudinal vehicle dynamics, electric machine(s),gearbox/final drive transmission unit(s), a battery pack and acontrol system (power-split and gear shifting). In addition, aregenerative braking strategy is developed to model the brak-ing behavior of a real braking controller. Using the developedsimulation model, a variety of powertrain configurations havebeen studied for the selected hatchback vehicle.

The powertrain design considerations that are investigatedthrough this study are related to: (1) the electric machinetechnology (PSM or AIM), (2) the gearbox type (single-/twin-speed), (3) the drive system type (central or distributed drive),(4) the number of electric machines (1, 2 or 4), and (5) thedrivetrain architecture (for RWD and AWD vehicles). The firsttwo design choices refer to the component technology usedwhereas the other three influence the topology layout of theresulting powertrain design. For each powertrain configuration,the electric machine size(s), gear ratio(s), and controls arejointly optimized to be able to compare the energetic influenceof each design choice and to extract the best configuration.

Organization: The remainder of this paper is organized asfollows. First, in Section II, the electric powertrain system de-sign problem and its mathematical formulation are presented.Secondly, in Section III, the model of the BEV is derived,including all the component modules of its powertrain sys-tem. Next, the optimization results for the different topologyconfigurations are analyzed and discussed in Section IV. Thework is concluded by Section V in which its most importantfindings are summarized.

II. E-POWERTRAIN SYSTEM DESIGN

In this section, the electric powertrain design problem is de-scribed in more detail. Thereto, the different topology designsthat are considered here are presented and a mathematical for-mulation of the battery energy minimization problem, jointlyoptimizing the electric machine size(s), the transmission units’gear ratio(s) and control(s) subjected to the component andvehicle-specific performance constraints, is given. The set ofpowertrain topologies will be discussed first.

A. Topology design

Topology is defined as the layout of the components in apowertrain, how they are connected to each other and to thewheels of the vehicle. Depending on the set of componentsused, a large number of different topology design are possible.In this work, topologies are considered that are constructedwith up to two differently sized electric machines EMk,with machine index k, attached to one gearbox GBl perdriven axle l, and the choice whether (or not) a final drivetransmission FD is used to connect to the pair of wheel(s) W.The powertrain system is powered by a single battery packBAT via integrated power electronics (not modeled sepa-rately), being the only source of (electrical) energy supply. Theresulting set of different topology designs is given in Fig. 2.

Fig. 2. Set of different powertrain topologies studied, including both cen-tral (C) and distributed (D) drive system and using either one (single) ortwo (double) axles to drive the vehicle. The first subscript (e.g. C1, D1, etc.)denotes the topology index, the second subscript after the dot (j) indicates thetotal number of gears in the gearbox(es) and the superscript (m) representsthe machine technology used in that topology. W = wheel, EMk = elec-tric machine (with power electronics), GBl = gearbox transmission unit,BAT = battery pack, FD = final drive transmission.

Here, the topologies can be categorized column-wise intocentral (C) and distributed (D) drive systems. Furthermore,the topologies can be classified row-wise into single-axle(RWD) and double-axle driven (AWD) architectures. Power-train configurations, denoted by Tmi,j (-), are considered witheither having single-speed or two-speed gearbox(es), whichare connected to either PSM or AIM machines. The subscripti ∈ C1, . . . ,C3,D1, . . . ,D3 indicates the topology index,j ∈ 1, 2 the number of gears in the (discrete) gearbox(es)and m ∈ PSM,AIM the machine technology used in thetopology configuration. Given these topological and techno-logical design variations, a set of 36 different functionalpowertrain configurations T could be created from the set of 9possible topology layouts. To be able to compare the optimaldesign of all 36 configurations, a powertrain optimizationproblem is mathematically formulated next.

B. Mathematical problem formulation: bi-level optimization

The powertrain design problem focuses on the minimizationof the supplied battery energy ∆Es (kWh), calculated fromthe integral of the internal battery power Ps(t) (W) over arepresentative drive cycle with t ∈ [t0, tend], as given by:

∆Es =

∫ tend

t0

Ps(t)dt. (1)

To obtain an optimal system design for each powertrainconfiguration Tmi,j ∈ T, the physical components and the


control algorithm should be (co-)designed in an integratedmanner [2]. Therefore, the combined powertrain design prob-lem P is formulated as a bi-level optimization, where both thepowertrain design parameters xp and time-dependent controlvariables xc(t) are optimized:

P : minxp,xc(t)

∆Es

(xp,xc(t) | xv,Λ(t),Tmi,j

)

s.t.gp(xp) ≤ 0, hp(xp) = 0,

gc (xp,xc(t)) ≤ 0, hc (xp,xc(t)) = 0.

(2)

Here, g(.) and h(.) are the inequality and equality constraints,respectively, xv the vehicle parameters, and Λ(t) the drivecycle information. Due the unidirectional coupling betweenthe plant and control variables, the problem is solved usinga nested optimization approach [2], with two individual opti-mization loops. In the outer loop, expressed by the high-levelproblem Ph, the optimal sizing parameters x?p are found by thegiven optimal control signals x?c(t), as the result of the lower-level problem Pl in the inner loop. The subscript (.)? refersto the optimal value of the variable for which the optimizationproblem is solved (minimizer).

Component sizing (high-level problem): The high-levelproblem, as expressed in Eq. (3), aims to find the opti-mal sizing parameters xp = Pm,Rg, related to the EMsize(s), Pm = Pm1 , Pm2, and the set of gear ratio value(s),rgl = rgl(1), rgl(2) of the gearboxe(s) Rg = rg1 , rg2present in the specific topology configuration Tmi,j , by solving:

Ph : minxp

∆Es

(xp | x?c(t),xv,Λ(t),Tmi,j

)

s.t. gp(xp) ≤ 0, hp(xp) = 0,(3)

subjected to the vehicle performance and component require-ments, which are captured by the expressions given by theinequality gp and equality constraints hp, as given below.

gp

gp1 : top speed ≥ vmax (km/h)gp2

: kinetic top speed ωm(vmax) ≥ vmax

rw(rpm)

gp3 : acceleration time for 0-100 (km/h) ≤ tacc (s)gp4

: low-speed gradeability q0 ≥ 30 (%)gp5

: mutual gear sizing rgl(1) ≥ rgl(2) (-)

hp

hp1

: battery energy content Eb (kWh)hp2

: final drive ratio rfl = 1 (-)

The performance constraints, as in the Appendix, are relatedto the minimum top speed (gp1

), the maximum achievabletop speed without over-speeding the E-machine(s) (gp2

), theacceleration time (gp3 ) and the minimum gradeability fromstand-still (gp4 ). The component constraints are related to thegear step design (always speed reducing during up-shifting),and the battery size (hp1

) and the final drive ratio(s) (gp2)

that are kept at a fixed value. The latter assuming that thecombined gearbox and final drive ratio is optimized in case ofa central drive topology. In order to solve for the lower-leveloptimal control problem, as expressed in Eq. (4), the givenfunctional powertrain configuration, feasible machine size(s)and gear ratio value(s), vehicle parameters and drive cycledata are provided as input from the higher-level problem.

Control optimization (low-level problem): The low-levelproblem aims to find for each feasible combination of val-ues xp, the optimal control signals xc(t) present in thepowertrain configuration Tmi,j , by solving:

Pl : minxc(t)

∆Es

(xc(t) | xp,xv,Λ(t),Tmi,j

)

s.t. gc(xc(t) | xp) ≤ 0, ξ(t) ∈ [¯ξ, ξ],

(4)

where ξ(t) (-) refers to the system’s battery state-of-charge(SoC) and gc is the set of inequality constraints that areincluded in the simulation model, given by:

gc

gc1 : max. electric machine torque ≤ τmk(Nm)

gc2 : min. electric machine torque ≤¯τmk

(Nm)gc3 : max. electric machine speed ≤ ωmk

(rad/s)gc4 : max. battery charge current ≤ Ib (A)gc5 : max. battery discharge current ≤ Ib (A)

related to the physical limitations of the components in thepowertrain, such as the maximum (gc1 ) and minimum (gc2 )torque, and maximum speed (gc3 ) of each electric machineEMk; and, the maximum charge (gc4 ) and discharge (gc5 )current of the battery pack BAT.

The control variables xc(t) consist, in case of multi-ple E-machines and 2-spd gearbox(es), of the gear positionγ(t) = γl(t) of each gearbox l ∈ 1, 2 and the torque splitφ(t) ∈ [0, 1] between the electric machines, defined as:

φ(t) =τm1(t)

τm1(t) + τm2

(t), (5)

where τm1(Nm) is the torque generated by EM1 and τm2

(Nm)is the torque generated by EM2 (Nm). The presence of theactual control variables depends on the specific topology.

III. VEHICLE MODELING

From a more generic perspective, without showing theparticular topology, the technology design variations, relatedto the electric machines (EMs: PSM or AIM), the gearboxchoice (GB: 1-spd or 2-spd) as a powertrain block model, areshown in Fig. 1. Here, each model block represents a storage,conversion or transmission loss element that is present in thepowertrain. The change in internal battery storage energy isgiven by ∆Es, and the auxiliary power by Paux. Furthermore,the power flows are indicated by arrows from componentinput to output (load) in a forward fashion, yet the powerdemands are simulated in a backwards fashion (up-stream).The final drive unit (FD) is introduced as a fixed-power-splitdevice in case of a central drive system. The backward-facingvehicle simulation model is constructed in a modular wayand incorporates the different topology designs, componenttechnologies as power-based building blocks (cf. Fig. 1). Be-low the assumptions and each of the (quasi-static) powertraincomponent models used in the BEV simulation model isdiscussed in more detail.


TABLE IVEHICLE TECHNICAL SPECIFICATIONS [17]

Parameter Symbol Value Unit

VEHICLE BODY

Curb weight (EU) mc 1320 (kg)Frontal area Af 2.38 (m2)Air drag coefficient cd 0.29 (-)Wheel base l 2.57 (m)Center of gravity height hg 0.47 (m)Static weight distribution (F/R) - 47/53 (%)

POWERTRAIN

Drivetrain configuration - RWD (-)Electric machine technology - PSM (-)Peak motor power Pm 125 (kW)Rated motor speed ωm,r 4800 (rpm)Maximum motor torque τm 250 (Nm)Maximum motor speed ωm 11400 (rpm)

TRANSMISSION

Tyre model - Ecopia EP600 (-)Tyre size - 155/70-R19 (-)Rolling resistance coefficient [18] cr 6.4 (kg/ton)Wheel radius rw 0.3498 (m)Gear ratio rg 9.665 (-)

BATTERY

Battery pack configuration - 96s1p (-)Nominal battery pack voltage Uoc 353 (V)Nominal battery pack capacity Q0 94 (V)Battery pack energy content (gross/net) Eb 33.2/27.2 (kWh)

PERFORMANCE

Top speed vmax 150 (km/h)Acceleration (0-100 km/h) tacc 7.3 (s)Energy consumption (NEDC) Ev 13.1 (kWh/100km)Electric driving range (NEDC) Dr 300 (km)

COMPONENT MASSES

Electric machine mass [19] mm,0 42 (kg)Motor inverter mass [19] mi,0 19 (kg)Gearbox/final drive mass [20] mg,0 21 (kg)Battery pack mass [21] mb,0 256 (kg)

BMW i3 94 Ah (2017)

A. Vehicle road-load

Given a reference driving profile with velocity v(t) (m/s)and road gradient α(t) () together with the vehicle parametersin Table I, the required traction force, Ft(t) (N), to propel thevehicle in longitudinal direction, considering the aerodynamicdrag, gradient resistance, and vehicle (acceleration) inertia, canbe calculated as [22]:

Ft(t) =1

2ρaAf cd v

2(t) + crmv g cos (α(t))

+mv g sin(α(t)) +mv λdv(t)

dt,

(6)

where ρa = 1.225 (kg/m3) is the air density, Af (m2) is thefrontal area, cd (-) is the air drag coefficient, cr (-) is the tirerolling friction coefficient, mv (kg) is the total mass of thevehicle, g = 9.81 (m/s) is the Earth’s gravitational accelerationand λ = 1.05 (-) [23] is the rotational inertia factor accountingfor the mass of the rotating parts. The corresponding angularspeed ωw (rad/s) and torque demand τw (Nm) at the (four)wheels of the vehicle are given by:

ωw(t) =v(t)

rw, (7)

τw(t) = Ft(t) rw, (8)

where rw (m) is radius of an individual wheel. Dependenton the drivetrain configuration of the powertrain topology, therequired torque τw (Nm) is distributed over the wheel pair ofthe front (τw1

) and/or rear axle (τw2) of the vehicle:

τw(t) = τw1(t) + τw2(t). (9)

B. Braking strategy

The dynamic braking capability of the electric motor en-ables recharging the battery when driving. To estimate theamount of regenerative braking that the EM can apply withoutdestabilizing the vehicle, the maximum braking force Fbr (N)must be determined. This maximum braking force is propor-tional to the normal load(s) acting on the vehicle and theadhesion between the tires and the road [23]:

Fbr = µ (Fz1 + Fz2) = µmv g, (10)

where µ = 0.8 (-) is the adhesion coefficient between thetires and the road, Fz1 and Fz2 , are the normal loads on thefront and rear axles in (N), respectively. In the proposed BEVmodel, the braking force Fbr(t) = (Ft(t) < 0) is distributedbetween the mechanical and regenerative brakes:

|Fbr(t)|µ = Fbr,m(t) + Fbr,r(t), (11)

where Fbr,m (N) is the mechanical braking force and Fbr,r (N)is the regenerative braking force. The latter is dependent onthe drive-train configuration of the powertrain topology:

Fbr,r(t) =

(1− β(t)) Fbr(t), if RWD,Fbr(t), if AWD.

(12)

Here, for AWD topologies, maximum regenerative brakingis possible within the limitation of the tyre-road adhesion.For RWD layouts, the actual regenerative braking that canbe applied is represented by the brake fraction 0 ≤ β ≤ 1 (-),being the ratio of the (maximum) braking force on the frontaxles, Fbr1 (N), to the total braking force of the vehicle [23]:

β(t) =Fbr1(t)

Fbr=Fz1(t)

mv g, (13)

where Fz1(t) accounts for the load transfer from the rear axleto the front axle while braking, which is dependent on thevehicle deceleration rate, av(t) < 0 (m/s2):

Fz1(t) =mv g

l

(b− hg

∣∣∣∣av(t)

g

∣∣∣∣µ

), (14)

where l (m) is the wheel base of the vehicle, b (m) representthe distance from the rear-axle to the vehicle’s center ofmass (CoM), which is calculated from the vehicle’s (F/R)weight distribution, and hg (m) is the height of the CoM.Subsequently, the total required (net) resultant force at thewheels, Fr (N), is given by:

Fr(t) = Ft(t) + Fbr,m(t). (15)


C. Final drive transmission

When a final drive (FD) is present in the powertraintopology, the final drive (bevel gear differential) is modeledwith a fixed efficiency of ηfd = 0.93 (-) [24]. In that case, theangular speed ωfl (rad/s) and torque τfl (Nm) at the input ofthe final drive unit at axle l follows from:

if FD present:

ωfl(t) = ωw(t) rfl ,

τfl(t) =τwl

(t)

rflη−sign(τw(t))fd

(16)

If no final drive is present, the wheels are directly attachedto the gearbox and the axle wheel torque is divided withoutdifferential losses over the two individual driven wheels ateach side (left/right) of the driven axle(s). Hence, the torqueand angular speed relations of Eq. (16) become:

if FD not present:

ωfl(t) = ωw(t),

τfl(t) =τwl

(t)

2 .(17)

D. Gearbox transmission

The gearbox transmission (GB), being either a single-speed(1-spd) or a twin-speed (2-spd) discrete gearbox, is modeledusing a fixed gear-dependent efficiency ηgbl

:

ηgbl= ηgl(γl(t)), (18)

where the gear efficiency ηgl(γl(t)) is determined by thenumber of gear-stages that are necessary to realize the gearratio [25]. It is assumed that one gear-stage, i.e. one pair ofgears, has a constant efficiency of ηgp = 0.985 (-) and canrealize a maximum gear ratio of four [25]. The efficiency per(selected) gear is then given by:

ηgl(γl(t)) =

ηgp if rgl (γl(t))

rfl≤ 4,

η2gp if 4 <rgl (γl(t))

rfl≤ 16,

(19)

where rgl (γ(t)) represents the gear of the gearbox unit ataxle l for which the efficiency is calculated. To account forthe presence of a final drive, the value of rf equals one fora distributed topology and seven for a central drive topology.The angular speed ωgl (rad/s) and torque τgl (Nm) at the outputof gearbox unit l are calculated as:

ωgl(t) = ωfl(t) rgl (γl(t)) , (20)

τgl(t) =τfl(t)

rl (γl(t))ηgl (γl(t))

−sign(τfl (t)) , (21)

where the corresponding gear ratio and efficiency are depen-dent on the gear position γl at any given time t.

E. Electric machine

The electric machine (EM) that is connected to the gearboxis modeled using a lookup map Mmk

of the motor/generatorefficiency. Dependent on the machines’ torque τmk

(Nm) andspeed ωmk

(rad/s) operating point, a value for the efficiencyηmk

(-) is obtained from the map:

ηmk=Mmk

(ωmk

,τmk

smk

). (22)

The parameter smk(-) is used to scale the original efficiency

map along the torque axis to account for the performance ofthe differently-sized EMs [22], according to:

smk=Pmk

Pm,0, (23)

where Pmk(W) is the peak mechanical power of the resized

electric machine and Pm,0 (W) is the peak mechanical powerof the original electric machine. Using the machine efficiency,the EM electrical input power Pelk (W) can be calculated as:

Pelk(t) = ωmk(t) τmk

(t) η−sign(τmk

(t))mk . (24)

Similarly, the maximum torque τmk(and minimum torque

¯τmk

) of the sized EM is scaled accordingly:

τmk= smk

τm. (25)

The electric machine model uses the efficiency maps ofFig. 3, representing both the PSM and AIM technology. Themaps are produced with generated data from MOTOR-CAD®,using high-fidelity finite element models (FEM), and compro-mises only the electric machine. Therefore, to account forthe motor inverter losses, an additional (fixed) efficiency ofηi = 0.95 (-) is assumed.

F. Battery

The battery pack of the vehicle is modeled witha basic equivalent circuit model [22]. Data from aSamsung SDI 94 Ah Li-ion cell is used to simulate the batterypack that is originally present in the BMW i3 94 Ah vehicle.The corresponding battery cell specifications are shown inTable II. In the battery model considered, the open-circuit

TABLE IIBATTERY PACK CELL SPECIFICATIONS [27]


Cell capacity Q0,cl 94 (Ah)Nominal cell voltage Uoc,cl 3.68 (V)Max. cell charge current Ich 72 (A)Max. cell discharge current Idi 150 (A)

Max. cell state of charge ξ 1 (-)Min. cell state of charge

¯ξ 0.2 (-)

Initial cell state of charge ξ(t0) 1 (-)

Samsung SDI 94 Ah (Li-ion)

voltage Uoc (V) and internal resistance Rb (Ω) are determinedby the configuration of the battery, i.e., the number of cellsin series nse = 96 (-), and the number of parallel branches,npa = 1 (-), in the pack, as stated in Table I:

Uoc(t) = nse Uoc,cl (ξ(t)) , (26)

Rb(t) =nsenpa

Rcl (ξ(t)) , (27)

where Uoc,cl (ξ(t)) (V) and Rcl (ξ(t)) (Ω) are the open-circuitand the internal resistance of a single cell, respectively, whichare modeled as function of the state-of-charge ξ(t), as shownin Fig. 4. The intermediate state-of-charge of the battery is


(a) Peak efficiency ηm = 0.981 (-), peak power Pm,0 = 176 (kW). (b) Peak efficiency ηm = 0.966 (-), peak power Pm = 184 (kW).

Fig. 3. MOTOR-CAD® generated efficiency map (M) of the permanent-magnet synchronous machine (PSM) (a) and asynchronous induction machine(AIM) (b) in motoring/generator mode, designed for the same peak performance targets: maximum torque τm = 350 (Nm), peak power Pm = 150 (kW)and maximum speed ωm = 12000 (rpm) [26]. These base maps are used to linearly scale the electric machine efficiency (ηm) in the torque direction withthe (peak) power (Pm).

calculated using Coulomb counting:

ξ(t) = ξ(t0)−∫ t

t0

Ib(t)

Q0dt, (28)

where Q0 (Ah) is the nominal battery (pack) capacity, whichdepends on the nominal cell capacity Q0,cl (Ah), as:

Q0 = npaQ0,cl. (29)

The internal battery current Ib (A) is determined by applyingKirchhoff’s voltage law to the equivalent battery circuit:

Ib(t) = ηcoUoc(t)−

√U2oc(t)− 4RbPb(t)

2Rb(t), (30)

Fig. 4. Open-circuit voltage (Uoc,cl) and internal resistance (Rcl) of thesingle battery cell [27].

where ηco = 0.98 (-) is the Coulombic efficiency that isincorporated when the battery is charging (Ib(t) < 0). Thepower at the battery terminals Pb (W) is computed with:

Pb =

nm∑

k=1

Pmk+ Paux, (31)

where nm (-) is the total number of electrical machines inthe specific topology. A fixed average auxiliary power Paux =800 (W) is added, that covers the devices that need to beactivated during the homologation test [28]. Subsequently, theinternal power of the battery Ps (W) follows from:

Ps(t) = Ib(t)Uoc(t). (32)

G. Mass

The change in the total mass of the vehicle, mv (kg), withthe scaled powertrain components is calculated using:

mv = m0 +mpt, (33)

where m0 (kg) is the base mass of the vehicle withoutpowertrain components, obtained by:

m0 = mc − (mm,0 +mi,0 +mg,0︸︷︷︸mpt,0

), (34)

where mc (kg) is the (original) curb mass of the vehicle (incl.75 (kg) passenger) and mpt,0 is the total mass of the individualpowertrain components that were originally present in thevehicle (cf. Table I): the electric machine mass mm,0 (kg),motor inverter mass mi,0 (kg) and gear box mass mg,0 (kg).The mass of the (scaled) powertrain, mpt (kg), including allthe powertrain components, is then given by:

mpt = mb,0 +

nm∑

k=1

(mmk+mik) +

ngb∑

l=1

mgbl, (35)


where ngb is the number of gearboxes present in the pow-ertrain topology. The mass models used for the differentcomponents are described in Table III. Dependent on themachine technology, either the mass model of the PSM orAIM is used to compute the mass of the electric machine(s):

mmk∈ mPSM,mAIM. (36)

TABLE IIIINDIVIDUAL POWERTRAIN COMPONENT MASS MODELS


Motor inverter mik 0.1(

Pmk1000

)(kg)

Gearbox transmission [24] mgl 1.723 (rlτmk )0.439 j0.219l (kg)

PSM machine mPSM 0.22(

Pm1000

)(kg)

AIM machine mAIM 0.31(

Pm1000

)(kg)

H. Model validation

The simulation model is validated for the BMW i3 hatch-back vehicle using the parameters given in Table I. Thereto,the energy usage of the original (base) powertrain config-uration (TPSM

C1.1 ) with the original component sizes, PSMmachine and single-speed transmission was computed overthe New European Driving Cycle (NEDC) and compared tothe available value from the OEM: 13.1 (kWh/100km). Thesimulated energy consumption was 13.2 (kWh/100km). Thedifference between the simulated and published NEDC valuesof ε = 1.5 (%) is found to be acceptable and, therefore, thevehicle model is assumed to be representative and sufficientaccurate for the design optimization study as discussed next.

IV. OPTIMIZATION RESULTS

This section describes the optimization results for the hatch-back vehicle with the different optimized powertrain con-figurations, varying in topology choice, machine technologyand transmission architecture. Initially, the problem is solvedsubjected to the vehicle performance constraints originatedfrom the OEM specifications (cq. Table I) to analyze theeffect of the considered powertrain design choices within

the design space of the given vehicle type. To investigatethe influence of the vehicle performance requirements on theoptimal design(s), an additional experiment is done, where theproblem is optimized without these higher-level (performance)constraints. In both cases, the representative Worldwide Har-monised Light Vehicle Test Procedure (WLTP) is used as inputfor the optimization. The energy consumption for the basevehicle with its original powertrain over the WLTP cycle isfound to be Ev,0 = 15.05 (kWh/100km), and is used as areference for comparison in the results next.

A. Numerical implementation and solver

The optimization problems Ph and Pl, together with theBEV model are numerically implemented in MATLAB®. Asolution to the higher-level problem is found using particleswarm optimization (PSO) [29]. The lower-level control prob-lem has been included in the simulation model using a localminimization method. Here, the control vectors γ(t), φ(t)are gridded (vectorized) and for each time step t the values forthe control variables are selected that minimize the objectivefunction change for that specific time step. For all powertrainconfigurations, the same PSO algorithm settings were used.

B. Optimal design results

Table IV shows the results of the optimization prob-lem (2) for the 36 powertrain configurations considered. Foreach optimized configuration, it states the optimal power-train mass m?

pt (kg), overall transmission efficiency η?t (-),overall machine efficiency η?m (-) and the total (peak) motorpower P ?m (kW) of the EMs present in the topology, and theoptimal vehicle energy consumption E?v (kWh/100km). It alsoincludes the relative energy savings δE?v (%) when changingfrom a 1-spd to a 2-spd gearbox architecture, and the relativedifference in energy consumption δEv,0 (%) with respect tothe previous calculated reference. Optimized values for thecomponent sizes can be found in the Appendix (Table V). Inboth tables, the data is sorted row-wise by the different topol-ogy layouts and gearbox type, where the first column undereach parameter contains the values for the PSM technologyand the second column the values for the AIM technology.

TABLE IVPOWERTRAIN OPTIMIZATION RESULTS

T (PSM | AIM) mpt (kg) ηt (-) ηm (-) Pm (kW) E?v (kWh/100km) δE?

v (%) δEv,0 (%)

TC1.1 347.3 358.0 0.9161 0.9160 0.9050 0.8735 126 121 15.03 15.44-0.45 -0.17

-0.17 2.56TC1.2 354.5 367.1 0.9161 0.9161 0.9148 0.8796 114 113 14.96 15.41 -0.61 2.39

TC2.1 347.3 358.0 0.9161 0.9161 0.9182 0.8888 126 121 14.91 15.33-0.04 -0.08

-0.95 1.85TC2.2 354.2 369.5 0.9161 0.9161 0.9220 0.8947 113 113 14.90 15.32 -0.98 1.78

TC3.1 370.5 380.5 0.9161 0.9161 0.9189 0.8903 125 123 13.57 14.02-0.04 -0.26

-9.87 -6.83TC3.2 384.4 399.8 0.9160 0.9161 0.9237 0.8964 110 105 13.57 13.99 -9.83 -7.08

TD1.1 368.7 379.3 0.9702 0.9702 0.9054 0.8740 122 116 14.20 14.59-1.12 -0.29

-5.67 -3.04TD1.2 383.5 392.6 0.9802 0.9764 0.9152 0.8790 110 112 14.04 14.55 -6.73 -3.32

TD2.1 367.6 379.3 0.9702 0.9702 0.9186 0.8890 112 116 14.08 14.49-0.44 -0.11

-6.46 -3.71TD2.2 381.4 392.2 0.9790 0.9702 0.9195 0.8925 109 108 14.02 14.51 -6.87 -3.61

TD3.1 396.2 402.0 0.9807 0.9812 0.9183 0.8893 122 117 12.44 12.85-0.29 -0.26

-17.37 -14.64TD3.2 415.4 421.4 0.9835 0.9817 0.9230 0.8956 101 103 12.40 12.81 -17.61 -14.87


Comparing the results from Table IV, the topology withthe lowest energy consumption is TD3, which is valid forboth the PSM and AIM technology. This topology has fourelectric machines, distributed (D) over each of the four drivenwheels (AWD), which are differently sized with respect tothe front and rear axle. The optimal topology configura-tion is equipped with PSM machines that are connected toa 2-spd gearbox (i.e. TPSM

D.2 ), and gives an overall lowestenergy consumption of E?v = 12.40 (kWh/100km), which isδEv,0 = −17.61 (%) less than with the original powertrain.Also, the total machine size for this optimal topology config-uration, is reduced by −19.9 (%). In case AIMs are used, thisreduction in machine size is smaller: −14.4 (%). The topologyTC1, which is originally used in the base vehicle, has thehighest energy consumption among the considered topologies.This topology drives the pair of wheels on the rear axle (RWD)with one centrally (C) placed electric machine. The configu-ration with a 1-spd gearbox and AIM machine (i.e. TAIM

C1.1)resulted in having the highest energy consumption overallwith a value of E?v = 15.44 (kWh/100km). Considering thistopology configuration with a PSM machine, as present in theBMW i3 vehicle, it can be observed that the optimized valuesfor the components, as shown in Table V, are similar to thevalues from the OEM (cq. Table I). The energy consumptionwith the optimized components differs only δE∗

v = −0.2 (%)from the reference value, and shows that the vehicle’s orig-inal powertrain is close to its optimal design. The relativedifference between the optimized powertrain topologies withthe highest (TC1) and lowest energy consumption (TD3) wassimilar for both the PSM and AIM technology. For the set oftopologies studied, the choice for a different topology layoutled to a variation of−17.2 (%) (PSM) and−16.8 (%) (AIM) inthe absolute energy consumption of the vehicle. This variationis only slightly increased by respectively −0.3 (%) (PSM) and−0.2 (%) (AIM) when including the gearbox choice.

Design choices in the powertrain influence the characteris-tics of the individual powertrain component blocks, and hence,the overall energy consumption of the powertrain system in thevehicle. Focusing on the local effect of each of the powertraindesign choices (1) – (5) as presented in Section I, the fol-lowing observations can be done for the BMW i3 hatchbackvehicle without compromising its performance constraints.

1) Machine technology: The PSM technology gives forall topologies on average a difference in the vehicle energyconsumption of δEv = −3.0 (%) (similar for both the 1-spdand 2-spd) compared to the AIM technology. This is directlyrelated to the (overall) efficiency of the PSM machine, whichis higher in the same order of magnitude compared to theAIM: δηm = +3.4 (%). The powertrain mass decreases for alltopologies on average with δmpt = −2.8 (%) in case PSMsare used, being a consequence of the higher power density(kW/kg) of the PSM machine. For the AIM technology, thetotal size (power) of the machines in combination with the1-spd gearbox is slightly lower. For all the topologies ingeneral, the gear ratios are larger sized for AIMs. Possibly,to compensate for the higher mass of the machine.

2) Gearbox architecture: Re-allocating the machine’s op-erating points towards higher efficiency regions using 2-spdinstead of 1-spd transmission architecture(s) gives, for boththe PSM and AIM, a relative average decrease in energy con-sumption of −0.38 (%) and −0.16 (%), respectively. Althoughthe total machine efficiency increases for both machine typeswith 0.62 (%) on average, the gain in efficiency is counteractedby the increased powertrain mass caused by the heavier twin-speed gearbox. This relative mass increase is on average+3.4 (%) for the PSM and +3.8 (%) for the AIM, explainingthe lower energy consumption gain for AIMs. Using two-speedgearboxes, the gears can be sized such that a higher (lower)gear can be realized in the gearbox unit. Due to this largertorque multiplication factor, less torque is required from theelectric machines, leading to an average decrease in the totalmachine size of −10.3 (%) and −8.36 (%) for the PSM andAIM machine(s), respectively. The largest differences werefound for topologies having one differently sized EM and a1-spd transmission. For distributed topologies, using a two-speeds was more beneficial than for central topologies. Theeffect of the two-speed gearbox was less present for topologieshaving multiple EMs; both single and double-axled.

3) Drive system type: Going from a central drive (C) to adistributed drive system (D) gives an average energy saving(for both the PSM and AIM technology) of −6.6 (%). Thisrelative difference is maximum (−8.5 (%)) for the double-axle (AWD) topology (TC3 → TD3). The powertrain massfor distributed driven topologies is on average +6.6 (%) higherthan their central counterparts because of the double gearboxesthat are required. Nevertheless, this extra mass is counteractedby the absence of the final drive differential losses, increasingthe total transmission efficiency in the powertrain. The (total)electric machine size(s) was lower for the distributed drivepowertrain systems, compared to the central drive ones.

4) Number of electric machines: Adding an additional,differently sized EM in the topology lowers the absoluteenergy consumption for all topology variations. The extra EMenables a torque split over the electric machines in such away that they operate in more efficient operating regions for aparticular torque demand, hence increasing the total machineefficiency in the topology. The relative change in the absoluteenergy when having multiple EMs, is similar for both single-axled central and distributed topologies, with average valuesequal to δE?v = −0.80 (%) (PSM) and δE?v = −0.69 (%)(AIM) when equipped with 1-spd gearbox(es). The effect isless for configurations having 2-spd gearbox(es), that alsoinfluence the location of the working points in the EM’sefficiency map. For all dual-motor configurations, one EM waslarger sized than the second EM present.

5) Drivetrain configuration: Double-axle driven topologieshave the possibility of recuperating electric energy on boththe front and rear wheels during regenerative braking, whichsubstantially decreases the vehicle’s energy consumption com-pared to single-axled driven variants. The additional com-ponents necessary increase the total powertrain mass, butcause the total machine efficiency to be higher. The machine


efficiency that increases with the same order of magnitude forboth EM types (+1.5 (%)), together with more regenerativebraling capability, results in a average lower (net) energyconsumption of δE?v = −9.3 (%) for central driven topologiesand δE?v = −12.0 (%) for distributed driven topologies,compared to TC1 and TD1, respectively. For both drive systemtypes, the total electric machine size decreases with respectto their single-axled equivalents when equipped with 2-spdgearboxes. This trend is not visible in case 1-spds are used.

Optimizing the problem solely on the driving cycle, i.e.without including the vehicle performance constraints, hadsome significant effect on the electric machine sizing of alltopologies. On average, the total machine size were decreasedwith −60 (%) for the PSM and −55 (%) for the AIM. As a re-sult, overall lower vehicle energy consumptions were obtainedas consequence of lower powertrain masses and higher (total)machine efficiencies. A possible reason for the large reductionin machine size is the absence of acceleration constraint. Being(close to) active for each powertrain configuration, it was adominant factor that bounded the optimal solution. For theseparate powertrain design choices, similar trends as in thepreviously discussed constrained case were visible.

C. Discussion

A few comments are in order. First of all, the set of optimalcomponent parameters x?p and control variables x?c(t) for eachpowertrain configuration was subject to a pre-defined grid (e.g.control steps, component bounds) and the parameters chosenfor the optimization algorithm. The settings of the optimizationalgorithm (e.g. stopping criteria) were the same for all power-train configurations, which could have an influence the optimalsolution, especially for configurations with a higher numberof design parameters. Although the PSO algorithm convergedto an optimum, it did not guarantee finding the exact globalsolution. Hence, the optimality of the computed results, andthe likely-hood of finding a global optimal solution with thePSO solver should be verified in extended research (e.g. by anexhaustive search routine). Based on that, the PSO parametersettings might need to be re-tuned. Another note is to be maderegarding the effect of the 2-spd gearbox, which was less thanexpected. Due to the high-speed, low-torque characteristicsof the EM used in the simulation model, it was able tosatisfy both the acceleration and top speed constraint witha single gear. As a result, the energetic benefits were almostcompletely counteracted by the mass increase of the additionalgear, causing the difference in energy to be relatively small.In future research, choosing another EM sizing method (e.g.based on torque) or considering another lower-speed EM typemight show more the effect of the extra gear.

V. CONCLUSION

This work investigates the effect of technological and topo-logical design choices in the battery-electric powertrain ofan efficient passenger vehicle. Thereto, a powertrain designoptimization study, centered on the energy consumption of thevehicle over the WLTP driving cycle, is performed for a groupof various powertrain configurations. In total, 36 different

powertrain configurations are investigated, constructed from9 topology layouts, for which two different machine technolo-gies (PSM/AIM) and transmission architectures (1-spd/2-spd)are considered. Results showed that, for the BMW i3 hatch-back vehicle, a maximum decrease in the energy consumptionof −17.37% can be achieved when changing to a distributed,double-axled powertrain topology, while keeping the originalPSM machine technology and 1-spd gearbox type. Amongthe topology configurations studied, also a variation of thetotal machine size of −19.9 (%) could be observed. Fromthe separate study on the design consideration, the largesteffect on the energy consumption on average was achievedby: (i) using two axles (AWD) instead of one axle (RWD) todrive the vehicle, for distributed drive systems (−9.3%) andcentral drive systems (−12.0%); (ii) choosing a distributeddrive system, over a central drive system (−6.6%); followedby (iii) using PSM type machine(s) instead of AIM (−3.0%);and, (iv) using multiple electric machines instead of a singleone in single-axled topologies, in case of the PSM (−0.80%)and the AIM (−0.69%). The smallest energetic impact wasobserved for (v) equipping the powertrain with a two-speedinstead of a single speed connected to the PSM (−0.38%) andthe AIM (−0.16%), however, it decreased the total EM size byrespectively −10.3% and −8.36%. Furthermore, it was foundthat the observed trends were also valid when the performanceconstraints were left out and that the acceleration constrainthas a significant impact on the required (peak) power of theelectric machine(s) in the topology.

APPENDIX

PERFORMANCE CONSTRAINTS

A. Maximum vehicle speed

The target for the minimum speed that the vehicle needsto achieve is evaluated based on the tractive power requiredPt(vmax) to overcome the resistance forces Fres(vmax) at thedefined top speed vmax and the (total) maximum tractive powerfrom the electric machine(s) at that speed Pt(vmax):

gp1 : Pt(vmax)− Pt(vmax) ≤ 0 (37)

The tractive power Pt(vmax) is defined as the sum of themaximum tractive power Ptk

(vmax

rw, γl

)from each elec-

tric machine k with maximum electromechanical powerPmk

(ωmax, γl) given the angular speed ωmax(vmax) and gearposition γl at which the maximum tractive power is achievedand the required kinematic speed is not violated:

Pt(vmax) = Fres(vmax) vmax (37a)

Fres(vmax) = crmvg +1

2ρaAfcdv

2max (37b)

Pt(vmax) = maxγ

nm∑

k=1

Ptk

(vmax

rw, γl

))(37c)

Ptk(ωmax, γl) = Pmk(ωmaxrgl (γl) rfl) ηgl (γl) ηfl , (37d)

with :ωmk

rgl(γl)rfl≥ ωmax


TABLE VOPTIMIZED COMPONENT SIZES

T (PSM | AIM) P ?m1

(kW) P ?m2

(kW) r?g1 (1) (-) r?g1 (2) (-) r?g2 (1) (-) r?g2 (2) (-)

TC1.1 126 121 - - 8.9 10.2 - - - - - -TC1.2 114 113 - - 10.9 12.5 5.1 8.2 - - - -

TC2.1 92 91 34 30 8.9 10.2 - - - - - -TC2.2 84 84 29 30 11.0 13.6 7.0 9.1 - - -

TC3.1 85 97 40 26 10.0 10.2 - - 8.1 10.2 - -TC3.2 83 64 28 41 14.1 15.9 5.3 9.3 10.6 15.4 3.0 8.7

TD1.1 61 58 - - 8.8 10.1 - - - - - -TD1.2 55 56 - - 11.5 11.4 4.0 4.0 - - - -

TD2.1 41 44 15 14 10.1 10.2 - - - - - -TD2.2 32 38 23 16 11.0 12 4 8.3 - - - -

TD3.1 53 53 8 5 8.7 10.2 - - 9.9 9.9 - -TD3.2 45 46 5 5 15.3 14.0 6.6 9.2 10.3 11.5 3.8 1.3

B. Maximum kinematic speed

It is required that the angular motor velocity at the minimumtop speed ωm(vmax) cannot over-speed the E-machine(s) withmaximum angular speed ωm(vmax) at defined top speed vmax:

gp2: ωm(vmax)− ωm(vmax) ≤ 0 (38)

The maximum angular velocity ωm(vmax) is determined bythe machine k with the lowest maximum machine speed inthe highest possible gear position γl:

ωm(vmax) = min(ωm1

(vmax), . . . , ωmnm(vmax)

)(38a)

ωmk(vmax) = max

γl

(rgl(γl) rfl

vmax

rw

)(38b)

C. Acceleration

The target (0-100 (km/h)) acceleration time t∗acc is evaluatedwith the actual acceleration time of the vehicle tacc:

gp3 : tacc − t∗acc ≤ 0 (39)

The acceleration time tacc is defined as the time to reach thedesired velocity tspd and the time nshifttshifts that is neededto perform the number of (additional) shifts. The former isdependent on the acceleration ai as a result of the net tractiveforce Ft(ti)−Fres(vi) acting on the vehicle at each time instantti and speed vi. The maximum traction force Ft(ti) followsfrom the maximum tractive torque τ(ti), which is definedas the sum of the maximum tractive torques τtk (γl(ti)) ofeach machine k with instant gear position γl(ti) at which themaximum torque is achieved with no machine over-speeding:

tacc = tspd + nshift tshift (39a)

tspd =

tend∑

i

vi+1 − vi(ai+1 + ai)/2

(39b)

ai =Ft(ti)− Fres(vi)

mv(39c)

Fres(vi) = crmv g +1

2ρaAf cd v

2i (39d)

Ft(ti) =τt(ti)

rw(39a)

τt(ti) = maxγ(ti)

nm∑

k=1

τtk (γl(ti))

)(39b)

τtk (γl(ti)) = rgl(γl(ti)) rfl τmk(ωmk

(γl(ti))) (39c)ηg (γl(ti)) ηf , with : ωmk

(γl(ti)) < ωmk

ωm (γl(ti)) = rg (γl(ti)) rfvirw

(39d)

D. Low-speed gradeabilityThe requirement on the minimal slope α0 ∝ q0 at which the

vehicle should be able to drive away from standstill is checkedusing the corresponding gradient resistance force Fg(α0) atthat slope and the maximum traction force Ft at the wheelsat standstill (v = 0):

gp4: Fg(α0)− Ft ≤ 0 (40)

The maximum traction force is the total torque τt from theelectric machine(s) τtk(γl) with gear position γl at which themaximum traction force is achieved:

Fg(α0) = mv g sin(α0) (40a)

Ft =τtrw

=1

rwmaxγ

nm∑

k=1

τtk (γl)

)(40b)

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