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Energy Conversion and Management xxx (xxxx) xxx
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Energy Conversion and Management
journal homepage: www.elsevier.com/locate/enconman
Optimal drivetrain design methodology for enhancing dynamic and energy
performances of dual-motor electric vehicles
Chi T.P. Nguyen a, b, Bảo-Huy Nguyễn a, c, *, João Pedro F. Trovão a, d, e, f, Minh C. Ta a, c
e-TESC Lab, University of Sherbrooke, Sherbrooke, QC J1K 2R1, Canada
Thai Nguyen University of Technology, Thai Nguyen, Viet Nam
CTI Lab. for EVs, School of Electrical and Electronic Engineering, Hanoi University of Science and Technology, Hanoi, Viet Nam
Canada Research Chair in Efficient Electric Vehicles with Hybridized Energy Storage Systems, Canada
INESC Coimbra, University of Coimbra, DEEC, Polo II 3030-290 Coimbra, Portugal
Polytechnic Institute of Coimbra, IPC-ISEC, DEE, 3030-199 Coimbra, Portugal
Electric vehicle
4WD dual-motor
Off-road vehicle
Drivetrain design optimization
Electric vehicles (EVs) are a dominant transportation trend towards a low-carbon future. However, there is a lack
of systematic studies for full-electric off-road vehicles in terms of drivetrain design and energy management
issues. This paper proposes a novel methodology to develop a drivetrain system for a four-wheel-drive (4WD)
dual-motor off-road EV. The obtained design delivers high performance in both driving performance and energy
efficiency of a desired vehicle using the two electric motors with their multi-speed gearbox. The proposed
approach is to minimize the power envelope (force-speed characteristic) difference between a targeted model
and the system under study. As a result, the objective function finds an optimal solution for a proper set of gear
ratios and constant power speed ratio (CPSR) through a two-layer hierarchical scheme. The upper layer sets the
targeted power envelope based on the expected driving measures. The lower layer minimizes the gap of the
power envelope characteristics between the targeted vehicle and the dual-motor vehicle. In parallel, two opti­
mization loops are implemented at the lower layer. The outer loop determines the proper value of CPSR,
meanwhile, the gear ratios corresponding to the selected CPSR are optimized by a particle swarm optimization
(PSO) algorithm in the inner loop. The results verify that the dynamic and energy performances of the studied
dual-motor vehicle are enhanced over critical testing scenarios. In a comparison with the single-motor singlespeed gearbox configuration, the optimized design shows that the top speed is 55% higher; the acceleration time
to 45 km/h is reduced by 55.83%; and the gradeability is improved by 162.5%. Moreover, the test under a typical
off-road cycle reveals a 1.9% overall efficiency increase compared to the dual-motor drivetrain using the same
non-optimized gear ratio. The proposed comprehensive design approach has the potential to be applied to a wide
range of vehicle areas.
gravity that is interested in off-road vehicle stability.
1. Introduction
Conventional vehicles burning gasoline and diesel fuel in an internal
combustion engine (ICE) are a major factor of the air pollution that leads
to an inevitable transition to electric vehicles (EVs). Among them,
heavy-duty off-road vehicles for mining, agriculture, and military are
increasingly popular [1–3]. EVs benefit from off-road duties thanks to
their fast response and the capability of producing high torque at low
speed. Furthermore, the mass of the battery packs usually weighs haft of
a ton and is located on the vehicle floor. This keeps a low center of
1.1. Literature review
In order to adapt to all terrains and difficult off-road conditions, e.g.,
on snow, mud road, or steep uphill driving, strong traction power should
be enforced to all the wheels simultaneously. A promising solution is to
add one more electric motor to the other axle which is known as the
dual-motor four-wheel drive (4WD) configuration [4]. This structure not
only overcomes the fault states of one motor but also improves the
robustness and reliability of the vehicle. A large number of commercial
* Corresponding author at: e-TESC Lab, University of Sherbrooke, Sherbrooke, QC J1K 2R1, Canada and CTI Lab. for EVs, Hanoi University of Science and
Technology, Hanoi, Viet Nam.
E-mail address: huy.nguyenbao@hust.edu.vn (B.-H. Nguyễn).
Received 5 October 2021; Received in revised form 10 November 2021; Accepted 21 November 2021
0196-8904/© 2021 Elsevier Ltd. All rights reserved.
Please cite this article as: Chi T.P. Nguyen, Energy Conversion and Management, https://doi.org/10.1016/j.enconman.2021.115054
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Objective function
Rotation speed
Particle position in PSO
Particle speed in PSO
Best personal solution in PSO
Best global solution in PSO
Individual particle
Gear connected to IM
Gear connected to PMSM
[13] proposes a calculation solution for the best gear ratio and shifting
procedure. The optimized vehicle drivetrain saves 14.53% fuel con­
sumption while improving the acceleration performance. Addressing an
innovative configuration of a hybrid vehicle with two in-wheel motors,
the authors of [14] developed an optimization of the powertrain design
for reducing the vehicle energy consumption.
passenger EVs have been launched with the dual-motor 4WD configu­
ration such as Audi E-Tron Sportback, BMW iX3, the design of Model S,
X, and Y from Tesla.
However, it is worth noticing that the existing EVs still keep a simple
gearbox system. Due to the torque-speed characteristic of an electric
motor that can provide high torque for a wide speed range, the
commercialized EVs are generally equipped with a single-speed gearbox
to reduce the drivetrain mass and cost. In the case of the EV required to
fulfill extreme off-road conditions, the single-speed transmission offers
only one solution as a trade-off between the acceleration ability and the
top speed. Besides, although the electric motor efficiency can reach
more than 95%, this value is changed over various operation ranges.
Consequently, the single-speed transmission cannot ensure high opera­
tion efficiency for different driving conditions. Recent investigations
have proved that multi-speed transmissions can solve the existing
drawbacks of conventional systems. A two-speed Automatic Mechanical
Transmission (AMT) integrated into EVs shows a better performance in
terms of acceleration time, maximum speed, and energy economy [5,6].
To increase the motor efficiency while keeping the dynamic perfor­
mance, the authors of [7] validate the effective operation of two-speed
Dual Clutch Transmission (DCT) and simplified Continuous Variable
Transmission (CVT). The use of a six-speed forward and three-speed
reverse transmission is beneficial for heavy-duty vehicles in [8]. In
practice, in order to climb the high slopes while carrying heavy loads,
the vehicle needs a very high torque at a certain power limitation. The
reverse gear, in addition, also supports the vehicle’s back maneuvers.
However, the larger electric system size and energy losses are asking
more concerns.
Besides the dual-motor multi-speed drivetrain configurations, how to
optimize the gearbox ratios is also an important question because an
appropriate design should offer both dynamic and energy performance
enhancement. In [9] a genetic algorithm (GA)-based gear ratio design
strategy is developed for maximizing the mean efficiency. The work [10]
investigates different combinations of proper gear ratios. The objective
function considers the acceleration time and the energy consumption
under specific working conditions of a dump truck. The articles [6,11]
address the gear ratio optimization problem based on dynamic pro­
gramming (DP) leading to the minimum energy consumption. The
simulation results demonstrate the overall energy of the two-speed
system is saved by 4.2% under the EPA75 driving cycle. Arming to
improve the acceleration time and energy economy, the ratio selection
of a two-speed gearbox is combined with the torque distribution prob­
lem between two motors in [12]. By means of optimization of the gear
ratio and shifting strategy in [7], the motor efficiency is enhanced
associated with faster acceleration and higher maximum speed where
under HWFET driving cycle, 14% battery energy is saved. The paper
1.2. Motivation and contribution
The common approach of the aforementioned articles is to apply
state-of-the-art technologies and advanced optimization methods to
maximize the dynamic quality and to minimize the energy consumption
of the vehicles under study. Hence, it is not surprising that these works
have conducted high-performance EV drivetrain designs. However,
there is a lack of a method that optimizes the vehicle drivetrain
regarding a design reference with the pre-defined desired performance.
Let us take an analogous example in automatic controller synthesis to
illustrate this issue. Given a plant, the control engineer has a set of
required performances of the closed-loop system including overshoot,
settling time, steady-state error, and gain and phase margins. The
developed control techniques allow the engineer to synthesize the
controller to achieve these pre-defined criteria. Moreover, there are
applications of optimization algorithms for controller design without the
need for human expertise when dealing with complex control systems.
This analogy raises a question of a systematic methodology to auto­
matically deduce an optimal design for the vehicle drivetrain. This
design can achieve fully pre-defined performance criteria in terms of
dynamics and efficiency. Also, this technique can be applied to all EVs.
In addition, most of the existing studies focus on passenger vehicles but
the drivetrain optimization of off-road EVs has not been paid full
attention to.
To fill this gap of knowledge, this paper proposes a novel method­
ology for the optimal drivetrain design of a dual-motor 4WD off-road
configuration to improve both the dynamic performance and energy
efficiency of the vehicle. The study may contribute to the research field
in the following points:
(i) A comprehensive methodology to optimize the drivetrain system
of the dual-motor off-road EV is introduced. The novelty of this
methodology is based on minimizing the difference between the
power envelopes of the studied vehicle and the reference design.
To the best of the authors’ knowledge, this is the first time that
the power envelope gap minimization is discussed for the drive­
train design of the EV.
(ii) By considering the power envelop in the design process, the in­
fluence of the constant power speed ratio (CPSR) on the dynamic
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performance of the vehicle is appropriately addressed. The role of
this design variable has also not been investigated in the previous
studies. Moreover, the combination of the CPSR and the motors
efficiency maps allows us to instantaneously improve both the
dynamic performance and energy efficiency of the studied
maximum efficiency of PMSM; ηtranIM and ηtranPM are the transmission
efficiency for IM and PMSM, respectively.
Besides the one-fixed ratio gear transmission, the multi-speed
structures as two-speed, three-speed or four-speed [12,15–17] are also
suggested for the dual-motor configurations. The multi-speed system
allows both dynamic and economic performance enhancement. How­
ever, its negative features are poor transmission efficiency, increased
weight, and higher costs. Therefore, it is necessary to choose a proper
drivetrain architecture.
In the studied EV, IM is connected to a fixed-speed gear, while a twospeed gear is used for PMSM for the following reasons. In terms of
driving performance, engaging two motors to two individual fixed-speed
gearboxes generally helps reduce the complexity, the energy loss, and
the price of the transmission system. However, this is a trade-off be­
tween the high-speed performance and the-high force performance. For
example, the gearbox ratios are calculated to reach the targeted
maximum traction force. Higher gear ratios allow higher traction force,
but meanwhile reduce the vehicle speed. Although the maximum force is
met, which leads to faster acceleration, the top speed of the vehicle is
limited by the dropped maximum speed of IM or PMSM. In contrast, to
hit the targeted top speed, another smaller gear ratio pair is chosen. In
this case, the lower power envelope (F − V curve) of the dual-motor
model reduces the traction force capacity, and the acceleration takes
more time. When the two-speed gear is utilized for PMSM, the benefits of
high-speed and high-force performance are obtained at the same time.
The studied model could provide a high maximum speed and satisfy the
high traction force needed for acceleration and climbing. Considering
the cost of the system and multi-speed energy loss, the fixed gear ratio is
good enough for IM.
In terms of efficiency performance, since IM is featured by high ef­
ficiency on a large speed range (Fig. 2(a)), the fixed-speed gearbox is
used to cover the whole vehicle speed range without a large efficiency
decrease. While efficiency map of PMSM presented in Fig. 2(b) shows an
unequal efficiency over its speed range. More than 90% efficiency is
expected at the speed range from 3500 to 5000 rpm. At a lower speed
level, the motor efficiency drops dramatically. In particular, under 1000
rpm of the motor speed, the efficiency value is smaller than 60%. If the
single-speed gearbox is employed and PMSM works with the full speed
range, the operating points will fall outside the high-efficiency region.
This reduces the overall efficiency, whereas the multi-speed gear moves
operating points to the higher efficiency region which is well docu­
mented in [18,19]. As a result, a two-speed gearbox is designed for
PMSM to achieve a high-performance and high-efficiency system.
By applying the proposed approach, the optimized drivetrain system
demonstrates a superior off-road capability in terms of higher maximum
speed, faster acceleration, stronger gradeability, as well as less energy
consumption. The proposed topology with a proper set of CPSR and gear
ratios is evaluated via critical testing scenarios including speed-range
test over a standard driving cycle, acceleration test up to a given
speed, and hill climbing test up to 42% of the road grade. The dynamic
and the energy performances are then examined with load change and
grade fluctuation over a real-world off-road driving cycle.
1.3. Paper organization
Section 2 of this paper presents the studied dual-motor 4WD off-road
configuration and proposes the methodology of the drivetrain system
optimization. Thereafter, Section 3 provides the optimization results
and the performance improvement validations via extensive simula­
tions. Finally, the conclusions and perspectives are given in Section 4.
2. Methodology of drivetrain design optimization for the dualmotor 4WD EV
2.1. Dual-motor 4WD configuration
The studied dual-motor EV is driven by an induction motor (IM) in
the front and a permanent magnet synchronous motor (PMSM) in the
rear axle as presented in Fig. 1. The drivetrain system in each axle
consists of one electric motor which converts the electric energy fed by a
battery to mechanical power and drives the wheels via selectable
gearbox ratios. In practice, IM is preferred for a wide speed range at low
torque, while on the contrary, PMSM offers high starting torque at low
speed. Hence, taking the benefits of both motors can extend the
maximum dynamic performance of the vehicle. The total maximum
traction power is provided by IM and PMSM given by:
Ptotmax = PIMmax .ηIMmax .ηtranIM + PPMmax .ηPMmax .ηtranPM
where PIMmax and ηIMmax are the maximum power and the maximum ef­
ficiency of IM; PPMmax and ηPMmax are the maximum power and the
Fig. 1. Studied dual-motor 4WD off-road EV platform.
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during driving and less energy consumption can extend the driving
range, but none of these measures correlate well with driving experience
in general and extreme off-road conditions in particular. Moreover,
since the objective function is built by the F − V characteristic difference
between two vehicle models, the proposed approach is not limited by
the drivetrain system or the optimization method. Another special
advantage of this optimization method using the power envelope gap
minimization is to consider CPSR in the design process. As specified by
one maximum power (Pmax ) value of the vehicle, there are numerous
targeted power envelopes. These curves are bounded by the maximum
vehicle speed and the maximum traction force which depends on the
CPSR. According to [20], CPSR is defined as a ratio of the maximum
vehicle speed (Vmax ) and the base speed (Vbase ) during the constant
power region given by:
The maximum traction force is expressed by:
Fmax =
For a certain maximum power and maximum speed, the large CPSR
(or low base speed) increases the maximum traction force value. This
means the vehicle can show a better traction ability. The chosen targeted
curve should satisfy all the required driving criteria, also it should have
the narrowest power envelope gap with the dual-motor one. In this way,
the dual-motor model gains a superior performance as the expected
model. To the best of our knowledge, this is the first study to take CPSR
solution into the gear ratio optimization.
In this problem, the gear ratios of IM and PMSM, and CPSR are
selected as the design variables. The objective function is used to
determine an optimal solution for a drivability improvement under
dynamic constraints. The optimization mathematical model is presented
Fvehtarget (CPSR, V) − Fvehpro (V)
Besides given parameters by the vehicle and electric motors, the
optimal solution is found out by minimizing the difference between two
power envelope curves based on a proper set of gear ratios and CPSR
given by:
kgearIM , kgearPM , CPSR =
(kgearIM ,kgearPM )∈Ωkgear ,CPSR∈ΩCPSR
Fig. 2. Efficiency map of the electrical drives: (a) IM; (b) PMSM.
2.2. General methodology
The main goal of this study is to deliver a driving performance of the
reference vehicle using two electric motors with their multi-speed
gearbox as close as possible. The power of the EV is generated by one
front-axle IM and one rear-axle PMSM. Each electric motor has indi­
vidual power envelope characteristics depending on its gear ratios. On
the other hand, the power envelopes of the reference vehicle are
determined by different CPSRs. This study aims to select the motor gear
ratios and CPSR for an optimal solution. As a result, we defined the
objective function to be the power envelope difference between the
targeted model and the dual-motor model, and this objective function is
minimized based on a proper set of gear ratios and CPSR.
There are some alternative methods to design gearbox ratios, for
example, energy consumption minimization or acceleration time
minimization-based approaches have already been considered in some
reviewed studies. We do not aim to outperform an existing approach in a
single dimension such as energy consumption or rate of acceleration.
Our goal is to design the drivetrain system to satisfy predetermined
criteria using the limited resources in hand (using two electric motors
and their gearbox structure). The power envelope characterizes driving
experience in different driving conditions while two other metrics do not
have this property. Better acceleration time matters in brief moments
with Ωkgear and ΩCPSR are the feasible ranges of the gear ratios and CPSR.
Fvehtarget (CPSR, V) is the power envelope of the targeted vehicle which
is based on the following equation:
∫ Vbase
∫ Vmax
Fvehtarget (CPSR, V) =
Ftargetmax +
∫ Vbase
∫ Vmax
As defined in (2), (3), the targeted maximum force is higher when
rising CPSR at a certain maximum power and the fixed top speed, while
the base speed is decreased. This large maximum force enables the
vehicle to shorten the acceleration time and climb a greater incline. This
means CPSR has significant effects on the power envelope of the targeted
vehicle, and also contributes to the objective function (J) in (4).
Fvehpro (V) is the power envelope of the studied vehicle which is driven
by IM and PMSM as:
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Fvehpro (V) = FIM (V) + FPM (V) =
FPMmax +
⎨ FIMmax = TIMmax (V)kgearIM R
= TPMmax (V)kgearPM
⎨ VIMbase = kgearIM Rwheel
⎩ VIMmax =
⎨ VPMbase = kgearPM Rwheel
⎩ VPMmax =
FIMmax +
clearly reflect the role of the motor gear ratios to the vehicle perfor­
mance as well as the objective function.
The full structure of the proposed methodology consists of two layers
described in Fig. 4. The upper layer determines the targeted power en­
velope through the steps: (i) from the desired performance re­
quirements, three main criteria including the top speed, acceleration
time, and gradeability are determined; (ii) three criteria define corre­
sponding maximum power value; (iii) the targeted power envelope is
drawn by this maximum power value as a benchmark curve. This curve
is bounded by the maximum vehicle speed and the maximum driving
force which depends on the CPSR. Then, the objective function (J) in (4)
is implemented in the lower layer to find the optimal gear ratios. Due to
the possible CPSR variation, the targeted power envelope can be
changed. Thus, CPSR is integrated into the objective function as one of
the design variables.
The gear ratio optimization inside the lower layer is addressed by
employing particle swarm optimization (PSO). Unlike traditional opti­
mization methods, PSO looks for the solutions without using the
gradient of the problem. The performance of the objective function (J) in
(4) is plotted in a three-dimension space in Fig. 5 for two different gear
ratio pair cases corresponding CPSR = 4.
As can be seen in Fig. 5a, b, the proposed objective function is
expensive to evaluate and non-smooth. This is because the function is
carried out by separated speed conditions and respecting dynamic
constraints. In such conditions, the derivative-based methods fail to
optimize the objective function or find a local solution and miss the
global one. One more reason, although PSO shares similarities with
Evolutionary Computation (EC) techniques, the PSO process does not
require genetic operators such as crossover and mutation. It is mathe­
matically less complex and converges to the solution faster. However, it
is noticed that PSO is chosen to use for the gear ratio optimization
method, but it is not the only method that can be employed. The pro­
posed methodology does not limit any optimization algorithm depend­
ing on the designer’s decision.
in which FIMmax and TIMmax are the maximum traction force and the
maximum torque of IM; FPMmax and TPMmax are the maximum traction
force and the maximum torque of PMSM; ωIMbase and ωIMmax are the based
rotational speed and the maximum rotational speed of IM; ωPMbase and
ωPMmax are the based rotational speed and the maximum rotational speed
of PMSM; kgearIM and kgearPM are the gear ratios of IM and PMSM; ηtran is
the transmission efficiency; Rwheel is the wheel radius.
Fig. 3 describes the logic used in the optimization of the objective
function under two typical cases of different gear ratios of IM and
PMSM. The power envelope of the reference model is shown in the black
curve according to the selected CPSR of 4. IM and PMSM characterize
the red power envelope and the blue one, respectively. The combination
of two electric motors creates the power envelope of the dual-motor
model plotted in purple. The proposed methodology is to minimize
the power envelope difference of the two models. In other words,
minimizing the objective function (J) in (4) is to minimize the green
area. It is noticed that the various gearbox ratios change the motor
power envelope characteristics. Using (7)–(10), the higher gear ratio
increases the maximum force but reduces the speed of the vehicle and
vice versa. The effect of motor F − V characteristics combined with the
different gear ratios shapes the purple curve. The green area which
corresponds to the mismatch between the targeted vehicle and dualmotor model is also different under each configuration. The optimal
solution is found out by minimizing the green area. Fig. 3a and b of Case
1 (kgearIM = 15, kgearPM = 10) and Case 2 (kgearIM = 10, kgearPM = 20)
2.3. Power envelope definition of the targeted vehicle
The reference vehicle offers three main driving criteria involving the
top speed, the acceleration time, and the climbing ability. A maximum
power value determined by these criteria characterizes the F − V curve
well known as the power envelope of the vehicle [20]. The computation
of the maximum power value of the targeted vehicle is presented below.
This process is also summarized in the flow chart in Fig. 6.
Fig. 3. The effect of gear ratios on the power envelope of the dual-motor vehicle as well as the objective function corresponding to CPSR = 4.
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Fig. 4. Hierarchical optimization framework.
Fig. 5. The 3D plots of the objective function performance in two different gear ratio pairs with CPSR = 4.
2.3.1. Power envelope based on maximum speed performance
A maximum power value is required to hit the maximum vehicle
speed (Vmax ) calculated by:
Ptargetmax (Vmax ) = Ftargetmax (Vmax ) .Vmax
+ Mgsin(α)
Ftargetmax (Vmax ) = Mgfroll cos(α) + ρair Cx AVmax
where M is the vehicle total mass; g the acceleration of gravity; froll the
rolling coefficient; ρair the air density; Cx A the aerodynamic standard;
and α the climbing angle of the road. In this case, it is assumed that α = 0
where Ptargetmax (Vmax ) is the maximum power at Vmax ; and Ftargetmax (Vmax ) is the
maximum force at Vmax of the targeted model.
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Fig. 6. Flow chart of maximum power definition of the targeted vehicle.
2.3.2. Power envelope based on acceleration performance
The acceleration time (tacc ) is how long it takes the vehicle to
accelerate from zero to a given speed with a full load. This time is used to
measure the vehicle’s acceleration ability. The drivetrain system
generally needs to be developed to shorten the acceleration time. The
consumed power during accelerating up to the speed (Vacc ) is:
Ptargetmax (acc) = Ftargetmax (acc) .Vacc
at Vacc ; and Ftargetmax (acc) is the maximum force at Vacc of the targeted EV,
as presented in the following:
Ftargetmax (acc) = Mgfroll cos(α) + ρair Cx AV 2 + Mgsin(α) + M
The acceleration time from zero to Vacc is deduced from (14) with α =
0 as follows:
where Vacc is the acceleration speed, Ptargetmax (acc) is the maximum power
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tacc =
Ptargetmax (acc)
Vbase Ptargetmax (acc)
⎨ vi (k + 1) = wvi (k) + c1 r1 Pbest,i (k) − xi (k) +
c2 r2 (Gbest (k) − xi (k) )
xi (k + 1) = xi (k) + vi (k + 1); i = 1 to n
) dV
− Mgfroll + 12ρair Cx AV 2
Ptargetmax (acc)
) dV
where c1 and c2 are the cognitive and social parameters, respectively,
which impact directly on the position of Pbest,i (k) and Gbest (k) ; r1 and r2
the vectors random numbers generated at every iteration (0 ≤ r ≤ 1) ; w
the inertia factor balancing global and local search; and n the swarm
size. Once the particle’s velocity and position have been calculated,
Pbest,i (k) and Gbest (k) are updated simultaneously.
Mgfroll + 12ρair Cx AV 2
) dV
Mgfroll + 12ρair Cx AV 2
The acceleration time is calculated in two separate regions as the
constant force region and the constant power region divided by the base
speed (Vbase ). Nevertheless, the base speed is changed by the CPSR factor
as expressed in (2). The previous work [20] shows that at the same
maximum power and maximum speed, higher CPSR results in a base
speed decrease and a larger maximum traction force. This can reduce the
acceleration time. In the fact of automobile applications, the CPSR value
is limited by the mechanical structure up to 10:1 [21].
To handle (15), firstly, the maximum power value (Ptargetmax (Vmax ) )
computed in (11) is recalled into (15) for an acceleration time evalua­
tion from 0 to Vacc . Compared to the desired acceleration time, if the
calculated time by (15) is longer, the maximum power value will be
increased in accordance with the result of (1). After that, the updated
Ptargetmax is used to re-calculate the acceleration time according to (15).
Because the tacc is related to the CPSR factor, at first this calculation is
applied for the case of the minimal ratio of the CPSR. A low ratio of CPSR
implies a long acceleration time period. Therefore, at this CPSR value, if
the required acceleration time is satisfied, the updated Ptargetmax is suffi­
cient for the expected acceleration. Otherwise, this action is repeated for
a new larger CPSR until the tacc is equal or less than the expected time.
2.4.1. Optimization control variables and dynamic constraints
As presented, after defining the maximum power value, which is
enough to cover all expected driving performances, the power envelope
of the targeted model is generated. This curve is limited by the targeted
maximum vehicle speed and maximum traction force based on CPSRs.
And then, the design variables including the motor gear ratios standing
for kgearIM , kgearPM1 , and kgearPM2 and CPSR are embedded in the opti­
mization problem to minimize the power envelope difference of the
targeted vehicle and the studied dual-motor vehicle. The feasible ranges
of these design variables are confirmed based on the EV dynamic con­
straints as the following. Maximum force constraint. Under the departure condition or
climbing, the cooperation of two electric motors is implemented to in­
crease the traction force. The gear ratio pair in this event would be kgearIM
and higher kgearPM1 . The maximum traction force provided by two mo­
tors is necessary to exceed the total resistance force in (16) given by:
2.3.3. Power envelope based on gradeability performance
Off-road vehicles are often featured by the greater traction force at
the low-speed range in uphill situations than on-road vehicles are. In the
case of this targeted model, the gradeability in which the vehicle can
ascend a maximum road slope (α) at the speed (Vgrade ) is supplied by the
maximum force defined by:
Ftargetmax (grade) = Mgfroll cos(α) + ρair Cx AVgrade
+ Mgsin(α)
Ptargetmax (grade) = Ftargetmax (grade) .Vgrade
+ TPMmax
≥ Ftargetmax (grade)
(19) Maximum speed constraint. The top vehicle speed is basically
limited by the maximum motor speed. Meanwhile, kgearIM and kgearPM2
can make a decision on the value of the maximum speed of IM and
PMSM (see (9),(10)). For this reason, the upper bounds of the motor gear
ratios are constrained by the maximum vehicle speed (Vmax ) given by:
ωIMmax 3.6Rwheel
⎨ kgearIM max =
ωPMmax 3.6Rwheel
⎩ kgearPM2 max =
where Vgrade is the given speed in gradeability, Ptargetmax (grade) is the
maximum power at Vgrade , Ftargetmax (grade) is the maximum force at Vgrade of
the targeted EV.
In conclusion, in the comparison of three maximum power values in
(11), (13), and (17), the highest value is selected as the best maximum
power of the targeted vehicle, which satisfies all performance re­
quirements. The targeted power envelope is then extracted by this
power value with the possible CPSRs.
2.4.2. Objective function
According to the general function in (4), two separate objective
functions are suitable for the studied drivetrain structure where IM uses
the fixed-speed gear and PMSM takes the two-speed gear. They are
purposed to optimal solutions over two different speed regions. In the
low-speed region, the objective function (J1 ) looks for the design vari­
ables as kgearIM , kgearPM1 to reach the targeted maximum traction force.
2.4. Optimal gearbox design using PSO
J1 =
PSO algorithm was introduced in 1995 [22] as a meta-heuristic
optimization method. PSO simulates the behavior of a social group,
such as a bird, fish, or ant flock. Each particle represents a point in the
search space. The current particle’s position is determined by the n
-dimension vector xi (k) . In the kth iteration, each particle’s position is
renewed by its own updated velocity vector vi (k) , where i is the indi­
vidual particle and k is the iteration number. The particles fly towards
the best personal solution Pbest,i (k) and the best global solution denoted
by Gbest (k) . The optimal solution can be represented by Gbest position.
The updated velocity and position vectors of the particle swarm are
formulated by:
On the other hand, in the high-speed region, the best combination of
kgearIM , kgearPM2 is selected by the objective function (J2 ) for a maximum
speed satisfaction given by:
Fvehtarget (CPSR, V) − Fvehpro1 (V)
J2 =
Fvehtarget (CPSR, V) − Fvehpro2 (V)
Subject to:
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Fig. 7. Flow chart of the gear ratio optimization.
accordance with different CPSRs in the outer loop, the minimum value
depicted to fopt is selected. The proper set of CPSR and gear ratios is
found out.
kgearIM min ≤ kgearIM ≤ kgearIM max
kgearPM1 min ≤ kgearPM1 ≤ kgearPM1 max
kgearPM2 min ≤ kgearPM2 ≤ kgearPM2 max
3. Results and discussion
CPSRmin ≤ CPSR ≤ CPSRmax
+ TPMmax
≥ Ftargetmax (grade)
3.1. Studied EV system
It is observed by (4), (6), the power envelope of the targeted model
can be reshaped by the CPSR range. This means CPSR also contributes to
the objective function evaluation. For this reason, the optimal gear ratio
determination should take care of the best CPSR selection within (1, 10)
interval as explained in Subsection 2.3.2. A two-loop optimization
schedule is demonstrated in the flow chart in Fig. 7.
Firstly, the minimum value of CPSR is selected in the outer loop.
Corresponding to the initial CPSR, the inner loop shown detailed in the
right-side box realizes the specified targeted power envelope for the gear
ratio design using PSO. This algorithm evaluates the objective functions
in (21) and (22) to find the best values of gear ratios. This process is
iterated on gradually increasing CPSR to its maximum value. Eventually,
in comparison of objective function results obtained by PSO in
The target vehicle is based on the off-road e-Commander of e-TESC
laboratory at the University of Sherbrooke. The original e-Commander
4WD off-road topology (T1) includes one IM driving two or four wheels
selectable by a shaft driven with a lockable rear differential. The 48 V IM
with the peak power of 22.5 kW and the continuous power of 8.2 kW
connects to the wheels by a fixed gear ratio of 20.5. This motor mounted
on the front axle fed by a three-phase inverter and supplied by a 6.9 kWh
battery pack provides the power to the studied e-Commander. The
vehicle model and the control scheme are organized and simulated using
energetic macroscopic representation (EMR) in Matlab/Simulink™. The
EMR model includes the battery model. The traction system uses the
electric motor with its gearbox transmission to transfer torque from the
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Table 2
Full-scale system parameters.
Vehicle (e-Commander)
Vehicle total mass
Aerodynamic standard
Rolling coefficient
Air density (at 20 ◦ C)
Wheel radius
Transmission efficiency
Fig. 8. Power envelopes of the T1 and T2.
Acceleration time from 0 to 45 km/h
Grade slope at 20 km/h
Induction motor
Maximal power
22.5 kW
Maximal efficiency
86 %
60 Nm
8000 rpm
Permanent magnet synchronous motor
Maximal power
16 kW
Maximal efficiency
93 %
34 Nm
Maximal speed
5000 rpm
Battery bank capacity
Battery bank resistance (at 80% SoC)
No. of cells in series
No. of cells in parallel
110 Ah
6 mΩ
Number of
200, 300}
{100, 200,
0.8, 1.0}
{0, 1, 2}
{0, 1, 2}
▪ Mean value of the best global solutions f(xG ) over the number
of iterations for the best run expressed by:
f (xG (ite))
Mean = ite=1
▪ Standard deviation (SD) value given by:
√ max
(f (xG (ite) − Mean )2
SD = ite=1
The sensitivity analysis is implemented under five scenarios as
included in Table 3. In each scenario, the tested parameter is tuned in its
value range, while the others keep recommended values.
The previous works in the literature notice that the bigger number of
particles increases the chances of searching successfully but also wastes
processing time. Danial et al. [26] suggest a selected population from 25
to 500 individuals. In some other cases, the swarm size is decided by the
design variable number. The number of times that the particles move to
a new place is called the iteration number. As an agreement of the
convergence rate and the computational cost, a potential range of the
iteration is between (100, 1000) by [26]. The effect of the previous
Table 3
PSO parameter testing scenarios.
Table 1
EV requirements.
Gearbox ratio in T2
Maximal torque
In PSO, tuning control parameters could considerably affect
computational behavior. Hence, a parameter-sensitive analysis should
be investigated to determine the impact of these parameters on the
overall PSO performance. The sensitivities of five control parameters
including the number of iterations (ite), the number of particles (n), the
inertial weight (w), the cognitive coefficient (c1), and the social coeffi­
cient (c2) are evaluated by the following criteria:
Acceleration speed
Gearbox ratio in T1
Maximal speed
3.2. Drivetrain design optimization results
Maximum speed
871.1 kg
1.3 m2
1.223 kg/m3
0.3175 m
91 %
Maximal torque
motor armature to the driven force. The interaction between the traction
force and the environment force delivers the vehicle speed in the EV
chassis system. The global EV model using EMR representation was
provided in [23,24]. After that, an on-road driving test with the tra­
jectory at the University of Sherbrooke campus conducted on this EV
prototype by the e-TESC Lab validates the vehicle’s mathematical model
described in previous studies [25].
The dynamic performance of T1 is presented in the green solid line in
Fig. 8. The gear ratio used in the T1 comes from the datasheet provided
by the manufacturer for the original e-Commander version. The top
speed is limited at 45 km/h, and the maximum force is 3500 N. The
acceleration time to reach 45 km/h is 12 s. One 16 kW (peak power)
PMSM is added on the rear axle of T1 to become the dual-motor topology
(T2). The same gear ratio of 13.3 is used for two motors in T2. Since the
gear ratios were not considered as a part of the design in the past, despite
the larger traction power capacity, the dynamic performance of dualmotor T2 was very poor.
The blue dashed line in Fig. 8 referring to the T2 performance shows
that the top vehicle speed ends at 45 km/h, and the maximum traction
force is around 3600 N. In comparison with T1, using two electric mo­
tors in T2 does not see any driving performance increase. In other words,
the dual-motor and one fixed-speed gearbox do not take full advantage
of the high-power system. Motivated from the T2 performance limita­
tion, this study is carried out to deliver the desired performance
following the maximum power capacity of the vehicle. As a result, T3 is
introduced as an improvement of T2. The drivetrain structure of T3
considering the gear ratios and CPSR is appropriately designed by
applying the proposed methodology. The T3 drivability is expected to
satisfy the high-performance requirement as given in Table 1. Full-scale
parameters of the e-Commander 4WD off-road EV are summarized in
Table 2.
70 km/h
45 km/h
5.5 s
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Fig. 9. PSO parameter selection procedure and the results.
velocity on the current velocity is denoted by the inertial weight (w). The
larger w stimulates global exploration but probably overlooks local
exploitation. The w is introduced between (0.4, 1.2) for the best PSO
actions in [27]. The acceleration coefficients (c1, c2) represent the par­
ticle moving toward the personal best and the global best, respectively.
The sum of both values should not be higher than four as the conception
of PSO in [22]. As a result of this analysis, the swarm size is examined
from 100 to 300 particles; the number of iterations is within a range of
(100, 300); the inertial weight is between (0.4, 1) and c1, c2 are varied
from zero to two.
As seen in Fig. 9, in S1, the SD values show that the more particles
covering a search space allow the more efficient performance. Also, due
to the higher number of iterations, the seeking actions in S2 are more
successful with lower SD values. Although the computation time in S1
and S2 is longer when the population size (n) and the iteration number
(ite) grow, the process just takes about 10 s or less. S3 results indicate
that the particles are more likely to explore the new space by a longer
step based on a larger w (higher SD). Conversely, a smaller w motivates
the particles to exploit the expected local area (lower SD). The selected w
value should therefore take care of both global and local search. Addi­
tionally, the coefficients c1, c2 are expected to remember the last best
own position and also share the information to the others as reflected by
the smallest SD values in S5 results. The best PSO behavior is evaluated
by the minimal values of SD as well as balancing between the global and
the local search. After the above sensitive analysis, the PSO control
parameters are used including the number of particles of 300, the iter­
ation of 300, the inertia weight between (0.4, 0.8), and the acceleration
coefficients of 1.
The optimization results are found that the best value of CPSR is 4;
the IM-connected gear ratio is 13.5; the first and the second gear ratio for
PMSM are 24.5 and 7.75, respectively. The sum of two motor power
envelopes is the power envelope of the dual-motor vehicle as plotted in
the red dot line in Fig. 10. The gear shifting of PMSM occurs at the
vehicle speed of 23.6 ± 0.1 km/h to avoid the eventual chattering
around the switching point.
3.3. Performance validations of the optimized drivetrain design
The studied dual-motor EV is modeled and controlled based on
MATLAB/SimulinkTM. The topologies T2 and T3 employ the same rulebased torque distribution strategy which is presented in our previous
study [23]. Below 800 rpm of the PMSM speed, the torque demand is
completely produced by IM. After 800 rpm, both PMSM and IM share the
same torque amount to the wheels with the torque split ratio of 0.5.
To verify the effective performance of the optimal drivetrain design,
the performance of three topologies T1, T2, and T3 is compared and
analyzed in the following driving conditions:
▪ Vehicle operation during a modified New European Driving
Cycle (NEDC) with the top speed limited at 70 km/h;
▪ Vehicle acceleration from 0 to 45 km/h on a flat road;
▪ Vehicle gradeability with 20 km/h;
▪ Vehicle operation under a typical off-road driving cycle
Fig. 10. Power envelope of the dual-motor T3 with optimized gear ratios.
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Fig. 11. Vehicle speed response under NEDC cycle.
Fig. 14. Vehicle speed response under a 0–45 km/h acceleration.
performance is executed by applying full throttle to the given speed and
then analyzing the results. The acceleration is evaluated by the accel­
eration time from 0 to 45 km/h with the maximum motor torque.
Thanks to the two-speed gear solution of PMSM, the higher gear ratio
allows a significant acceleration improvement to around 23.6 km/h. T3
provides a maximum traction force up to 4300 N followed by 3250 N
and 3070 N of T2 and T1, respectively (see Fig. 13). On the other hand,
after 23.6 km/h known as the gear switching speed of PMSM, the lower
gear ratio is then engaged. The maximum force is reduced leading to
slower acceleration. In Fig. 14, T1 accelerates from rest to 45 km/h in 12
s, while it takes only 5.3 s to accelerate T2 and T3 up to this given speed.
In short, the acceleration time from stop to 45 km/h of the proposed
design (T3) is less than 5.5 s of the required time in Table 1 and is
reduced by 6.7 s compared to T1.
Fig. 12. EV operation points under NEDC cycle.
3.3.3. Gradeability performance
In this case, it is assumed that throughout acceleration from 0 to 20
km/h, EV will be asked for further gradeability. This test validates the
maximum road slope that the vehicle can ascend respected to its dy­
namic limitations. As discussed in the previous part, smaller than 23.6
km/h, T3 shows the incredible traction force involved in the higher gear
ratio of PMSM (in red points in Fig. 15). For this reason, the dual-motor
model can climb up to 42% uphill grade. This figure is much higher than
the 30% and 16% of the maximum slope for T2 and T1 climbing,
▪ Vehicle overall efficiency under ORDC.
3.3.1. High-speed performance
The results in Fig. 11 and Fig. 12 demonstrate that the maximum
speed of T3 with optimal gear ratios is increased up to 70 km/h, whereas
T1 and T2 can only reach 45 km/h as the top vehicle speed. This is
because the motor speed range is limited by using the single-speed gear
in T1 and T2. As consequence, to deliver the possible maximum force for
acceleration and climbing, the top vehicle speed should be lower (as at
45 km/h). Furthermore, by means of sharing the same gear ratio for two
motors, T2 does not give IM a chance to hit 8000 rpm of the maximum
motor speed. The IM speed here is being restricted by the PMSM
maximum speed at 5000 rpm. However, the T3 design enables two
motors to take full their speed range.
3.3.4. Off-roading ability performance
The efficient performance of the proposed T3 is validated by the real
conditions of the off-road vehicles through a typical driving profile. The
off-roaders are actually ready to adapt to the load mass change on any
extreme trail or terrain. The dual-motor vehicle using optimized gear
ratios is operated under ORDC in Fig. 16. Over the first 500 s, the model
is driven on a flat road without load, and sometimes the downhill spots
occur. Between 500 s and 600 s, 272 kg, the full cargo capacity, of the
extra load mass is filled onto the vehicle. From 600 s to 1300 s, the eCommander is asked to climb different road slopes from 0% to about
30% with the full load. As presented in Fig. 16, the off-roading abilities
are realized by the optimal drivetrain design of T3. On smooth roads and
no-load conditions, the vehicle could run up to 70 km/h of the top speed.
The large maximum force allows the model to climb on a hill of 32%
grade at 10–20 km/h. In the higher-speed climbing conditions, at
around 30–40 km/h, after 900 s, the EV can ascend a lower gradient
under 10%. The red operation points of T3 in Fig. 17 almost cover the
working space of the force-speed map. They are located up to 70 km/h of
the top vehicle speed, and 4300 N of the peak force. These red points
also follow well the boundary of the power envelope of T3. There is no
doubt that, in some places, T3 can work outside the working space of T1
and T2. By these results, the driving performance improvement of T3 is
totally validated.
3.3.2. Acceleration performance
A common procedure to test the accelerating or climbing
Fig. 13. EV operation points under a 0–45 km/h acceleration.
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Fig. 15. EV operation points under a climbing at 20 km/h.
85%. However, the maximum speed of IM in T2 is limited by 5000 rpm,
so the high efficiency at the high-speed region is omitted. Meanwhile, T3
enables the red points to have high efficiency in the full speed range.
Seeing the blue points representing the T2 operation in the PMSM effi­
ciency map (Fig. 18b), at the motor speed lower than 2000 rpm, the
motor must work in less than 70% of efficiency conditions. In addition,
the T2 operation points are not observed in the highest efficiency area
(more than 90%) at above 3500 rpm. Compared to T2, the designed
gearbox of T3 can adjust the position of the red points into the higher
efficiency area at the high-speed region. As we can see, PMSM efficiency
can be more than 90% in the 3500–4500 rpm range. In conclusion, T3
operates in the areas that both motors are the most efficient and this can
be observed by the position of red points.
As a result, the overall efficiency is enhanced by the proper drive­
train design of T3 under the ORDC as presented in Fig. 19. The average
efficiency of T2 is 70.1% whereas it figures 72.0% for T3. The maximum
value of the T3 efficiency hits 81.1% followed by 78.8% at the highest
efficiency value of T2.
Due to the efficiency enhancement of the proposed powertrain
design, the driving range of the vehicle is also extended. To examine the
range autonomy of the topologies, we employ a repeated speed profile of
Worldwide harmonized Light vehicles Test Cycles (WLTC). The cycle has
been modified for suitable to the speed range of the studied topologies
with the maximum speed of 45 km/h. The vehicle stops when the battery
is discharged to 20% of its state-of-charge (SoC). As revealed in Fig. 20,
with the T3 topology, the vehicle can run 1.4 and 3.7 km longer than the
one with T2 and T1, respectively. Moreover, it is noteworthy that this
comparison is even not fair for the proposed T3 configuration because it
only runs up to 45 km/h (the limit of T1 and T2) whereas its capability is
70 km/h. Hence, T3 does not work at its optimal efficiency region in this
It should be noted that the comparative study conducted in this
section is not to validate the advantages of the optimization method
which is driven by the well-known PSO algorithm. The optimization is a
part of the powertrain design methodology proposed in this paper. The
output of the design process is the configuration T3 which has been
justified being superior to the conventional unoptimized topologies T1
and T2.
Fig. 16. Off-road driving cycle (ORDC).
4. Conclusion
The paper has proposed a novel comprehensive approach for the
drivetrain structure optimization of the dual-motor off-road EV. The
proposed drivetrain design for the dual-motor model produces the high
driving performance referenced by the targeted vehicle. The proposed
methodology is based on minimizing the power envelope difference
between the targeted and the dual-motor models. The optimization
process is systematically presented in two layers. The first step is to build
the targeted power envelope according to the pre-determined driving
criteria in the upper layer. It is noted that CPSR plays an important role
in the power envelope as well as the dynamic performance of the tar­
geted vehicle. Hence, CPSR needs to be considered in the design process.
Next, the minimization of the power envelope gap between the two
Fig. 17. EV operation points under ORDC.
3.3.5. Overall efficiency improvement
Fig. 18 shows the motor operation points in T2 (blue points) and T3
(red points) during ORDC. In the efficiency map of IM (Fig. 18a), T2 and
T3 operating points cover the efficiency regions which are higher than
C.T.P. Nguyen et al.
Energy Conversion and Management xxx (xxxx) xxx
Fig. 18. Operation points under ORDC in (a) IM efficiency map, (b) PMSM efficiency map.
Fig. 19. System efficiency under ORDC.
models is carried out in the lower layer. This layer finds the gear ratio
solutions for two electric motors, corresponding to the optimal CPSR
value by the PSO algorithm. The proposed design approach is applied to
the drivetrain system of the dual-motor 4WD e-Commander off-road EV
to evaluate its effectiveness. Compared to T1 and T2, the gear ratio
optimization implemented in T3 offers a significant enhancement in
multiple driving criteria. The top speed of T3 is 55% higher than T1 and
T2; acceleration time from stop to 45 km/h is reduced by 55.83% in the
comparison with T1; and the gradeability is improved 162.5% and 40%,
respectively, compared to the T1 and T2. In real-world conditions, T3
verifies the great off-roading ability, also the average efficiency
improvement to 72.0% from 70.1% of T2 overall efficiency.
This novel methodology provides the comprehensive drivetrain
design solution for high performance EV configurations. The proposed
multi-step procedure is promising for a variety of the EV applications. In
a future study, the torque distribution between two motors will be
optimized to maximize the global energy efficiency. Thereby, the dualmotor multi-speed gear vehicle is expected for the most efficient driving
performance and energy efficiency.
Fig. 20. Driving range comparison.
C.T.P. Nguyen et al.
Energy Conversion and Management xxx (xxxx) xxx
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Chi T.P. Nguyen: Investigation, Methodology, Software, Validation,
Writing – original draft. Bảảo-Huy Nguyễễn: Investigation, Software,
Validation, Writing – review & editing. João Pedro F. Trovão: Re­
sources, Conceptualization, Validation, Funding acquisition, Supervi­
sion, Writing – review & editing. Minh C. Ta: Validation, Writing –
review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to influence
the work reported in this paper.
This work was supported in part by Grant 950-230672 from Canada
Research Chairs Program, in part by Grant 2019-NC-252886 from Fonds
de recherche du Québec - Nature et Technologies, in part by FCT-Portuguese
Foundation for Science and Technology project UIDB/00308/2020, and
by the European Regional Development Fund through the COMPETE
2020 Program within project MAnAGER (POCI-01-0145-FEDER028040).
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