Control and Design Optimization of an Electric Vehicle Transmission Using Analytical Modeling Methods

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Control and Design Optimization of an

Electric Vehicle Transmission Using

Analytical Modeling Methods ?

Olaf Borsboom ∗Thijs de Mooy ∗Mauro Salazar ∗

Theo Hofman ∗

∗Eindhoven University of Technology, 5600 MB, Eindhoven,

The Netherlands (e-mail: o.j.t.borsboom@tue.nl,

t.mooy@student.tue.nl, {m.r.u.salazar, t.hofman}@tue.nl).

Abstract: This paper introduces a framework to systematically optimize the control and design

of an electric vehicle transmission, connecting powertrain sizing studies to detailed gearbox

design methods. To this end, we ﬁrst create analytical models of individual components:

gears, shafts, bearings, clutches, and synchronizers. Second, we construct a transmission by

systematically conﬁguring a topology with these components. Third, we place the composed

transmission within a powertrain and vehicle model, and compute the minimum-energy control

and design, employing solving algorithms that provide global optimality guarantees. Finally, we

carry out the control and design optimization of a ﬁxed- and two-gear transmission for a compact

family electric vehicle, whereby we observe that a two-gear transmission can improve the energy

consumption by 0.8 %, while also achieving requirements on gradeability and performance. We

validate our framework by recreating the transmission that is mounted in the benchmark test

case vehicle and recomputing the energy consumption over the New European Driving Cycle,

where we notice an error in energy consumption of 0.2 %, aﬃrming that our methods are suitable

for gear ratio selection in powertrain design optimization.

Keywords: Electric vehicles, transmissions, optimal design, optimal control, analytical

modeling

1. INTRODUCTION

In the past decade, we have witnessed a substantial in-

crease in battery-electric vehicle sales (IEA, 2021). How-

ever, the range and the pricing of these vehicles is still a

impediment in the large-scale transition, especially when

compared to their conventional counterparts (Paoli and

G¨ul, 2022). To improve the performance and cost of these

vehicles, we have to investigate the powertrain. Yet the

powertrain is an intricate system, leaving design engineers

with numerous choices in topology, technology, sizing, and

control. This problem requires holistic tools to quickly

explore the design space, in order to ﬁnd what powertrain

conﬁguration is the right one for the speciﬁc vehicle at

hand.

The one component that we address in this paper—the

transmission—has been extensively researched for applica-

tion within conventional vehicle powertrains. Since in gen-

eral, an electric motor (EM) can provide suﬃcient torque

over a larger speed range than an internal combustion

engine, strictly speaking, electric vehicles do not require

a multi-speed transmission, because a ﬁxed-gear transmis-

sion (FGT) suﬃces. This conﬁguration is also what is most

commonly available on the current market (EVDatabase,

2022). However, given that torque and speed are in fact

limited, designers must come to a compromise between

acceleration and gradeability on the one hand, and top

speed on the other. For this reason, more high-performance

vehicles, from the makes of, e.g., Audi and Porsche, and

?This publication is part of the project NEON (with project

number 17628 of the research programme Crossover, which is (partly)

ﬁnanced by the Dutch Research Council (NWO)).

Pac

clutch

synchr.

γfd

γ1

γ2

bearings

shafts

gear

Preq

Fig. 1. A schematic layout of the powertrain, including an

EM and a two-gear transmission. This conﬁguration

consists of three shafts, six bearings, three gear pairs,

a synchronizer (synchr.), and a clutch.

the Formula E vehicles, have been equipped with multi-

speed transmissions (K¨oller and Schmitz, 2021; Hewland,

2022). Some vehicle types focusing particularly on energy

eﬃciency have the EMs mounted directly in the wheels,

eliminating the need for a transmission altogether. Deter-

mining the suitable transmission for a speciﬁc car moti-

vates a methodology that can quickly construct, model,

and assess diﬀerent types of transmissions in the context of

a full powertrain. Conﬁgurations like the one in Fig. 1 must

be evaluated on their energy eﬃciency, whilst ensuring a

certain performance, in a modular and ﬂexible fashion.

Related literature: This research relates to three general

research streams. The ﬁrst stream deals with (hybrid-)

electric vehicle powertrain control and design optimiza-

tion from a system perspective. This is also known as

powertrain co-design, and it is typically solved with con-

vex optimization (Pourabdollah et al., 2018; Borsboom

et al., 2021) or derivative-free solvers (Ebbesen et al., 2012;

Hegazy and van Mierlo, 2010). To ensure computational

arXiv:2210.13287v2 [eess.SY] 28 Oct 2022

tractability, these methods either assume a ﬁxed trans-

mission eﬃciency or disregard losses entirely.

The second stream relates to the transmission design and

control optimization at the component level (Qin et al.,

2018; Lei et al., 2020; Patil et al., 2020). However, the opti-

mization problems usually have component-speciﬁc objec-

tives, like minimizing the noise and volume or maximizing

the strength of the gear teeth, leveraging computationally-

expensive ﬁnite-element models. Both the objectives and

the methodologies do not connect well to the holistic,

system-level perspective that is necessary in powertrain

design optimization.

The third and ﬁnal research stream pertains to more

detailed transmission models in powertrain design, striking

a balance between accurate modeling and system level

design (Machado et al., 2021). The works in Hofstetter

et al. (2018); Anselma et al. (2019) minimize the losses,

but merely for one conﬁguration. The procedure in Kr¨uger

et al. (2022) considers multiple topologies but provides

no global optimality guarantees, whereas optimality is

guaranteed in Leise et al. (2019), but ﬁxed eﬃciencies for

all transmission components are considered.

To conclude, to the best of the authors’ knowledge, there

are no methods that can model and predict the losses

of diﬀerent transmission conﬁgurations in a modular and

ﬂexible fashion, in the context of a full electric powertrain,

optimizing the design and control, whilst guaranteeing

global optimality of the solution.

Statement of contributions: To address this issue, this

paper presents a modular design and control optimization

framework of electric vehicle transmissions. Speciﬁcally, we

ﬁrst derive detailed analytical loss models of all compo-

nents that compose a transmission: the gears, the shafts,

the bearings, the clutches and the synchronizers. Second,

using engineering rules and manufacturer data, we me-

thodically design a transmission system for a powertrain

by combining the individual component models in the

desired conﬁguration. Third, we model the full drivetrain

and the car, and pose an optimization problem to minimize

the energy consumption over a drive cycle. Finally, we

showcase our framework by optimizing the design of the

two-gear transmission (2GT) of Fig. 1 and an FGT for a

compact family electric car.

Organization: This paper is organized as follows: Sec-

tion 2 presents the transmission component and shifting

loss models, the systematic design approach, and the sur-

rounding optimal design and control problem . We show-

case our optimization framework with numerical results in

Section 3. Finally, we draw the conclusions in Section 4,

together with an outlook to future research.

2. METHODOLOGY

In this section, we construct the optimization problem,

starting with the objective. Subsequently, we develop a

model of the car and the powertrain, whereby we exten-

sively elaborate on the gearbox and its components. To

increase readability, some equations are purposely omitted

from the main text. However, the interested reader can ﬁnd

these in the Appendix. Moreover, to keep our derivations

concise, we will abandon the time dependency t, when

its obvious from the context. Finally, we summarize the

problem and present the solving method.

2.1 Objective and Longitudinal Vehicle Dynamics

The objective of the optimization is to minimize the energy

provided to the EM over a drive cycle:

min Eac,(1)

where Eac is the energy consumed at the (electrical) input

of the EM. The car is modeled following a quasi-static

approach (Guzzella and Sciarretta, 2007) in time domain.

The power request at the wheels Preq equals

Preq =1

2ρacdAfv3+v(mv+mgb) (g(crcos β+ sin β) + a),

(2)

where v,aand βare the time-dependent velocity, acceler-

ation, and road inclination, respectively, provided by the

drive cycle, ρais the air density, cdis the drag coeﬃcient,

Afis the frontal area of the car, mvis the vehicle mass

without the gearbox, mgb is the gearbox mass, gis the

gravitational constant, and cris the rolling resistance co-

eﬃcient.

2.2 Transmission

In this section, we systematically design a transmission

and model the components that a transmission contains,

including its losses in an analytical fashion: the spur gear

pairs (g), the shafts (s), the bearings (b), the clutches (cl),

and the synchronizers (syn). Using these component mod-

els, we can construct multiple transmission conﬁgurations

in a ﬂexible and modular fashion, and evaluate diﬀerent

transmission types. In this paper, we focus on the FGT

and the 2GT, both designed in two stages. We describe

the modeling procedure for a 2GT, and we can construct

an FGT model by eliminating a pair of gears, a clutch,

and a synchronizer.

Primarily, we optimize the gear ratio values γj. These are

bounded by

γj∈[γmin, γmax], j ∈ {1,2,fd}(3)

where γ1is the ratio of the ﬁrst gear pair, γ2is the ratio of

the second gear pair, γfd is the ﬁnal drive ratio, and γmin

and γmax are the minimum and maximum values of the

ratios. We ensure that the gear ratios are ordered from a

high to a low ratio with the following constraint:

γ1> γ2.(4)

Furthermore, the number of gear teeth on the pinions (the

driving gears) Nt∈Nhas to be a natural number, as well

as the number of teeth on the driven gears γjNt:

γjNt∈N, j ∈ {1,2,fd}.(5)

We demand the car to be able to launch at the maximum

road inclination angle βmax in the highest gear ratio γ1,

guaranteed by the constraint

γ1γfd ≥grwmv(crcos βmax + sin βmax)

Tm,max

,(6)

which partially deﬁnes our design space, where Tm,max is

the maximum output torque of the EM and rwis the radius

of the wheels.

Initial Design We initialize the design of the gearbox

according to the procedures in Maciejczyk and Zdziennicki

(2011) and start with the shafts. In the conﬁguration of the

transmission (see Fig. 1, we consider three shafts (Ns= 3):

the input shaft (k= 1), the intermediate shaft (k= 2),

and the output shaft (k= 3), which are all assumed to be

short and therefore inﬁnitely stiﬀ, neglecting shaft torsion.

The torque limit of the gearbox is then determined by the

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ControlandDesignOptimizationofanElectricVehicleTransmissionUsingAnalyticalModelingMethods?OlafBorsboomThijsdeMooyMauroSalazarTheoHofmanEindhovenUniversityofTechnology,5600MB,Eindhoven,TheNetherlands(e-mail:o.j.t.borsboom@tue.nl,t.mooy@student.tue.nl,fm.r.u.salazar,t.hofmang@tue.nl).Abstract:Thi...

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Control and Design Optimization of an Electric Vehicle Transmission Using Analytical Modeling Methods

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