actuators

Article
An Analysis of and Improvements in the Gear Conditions of the
Automated Mechanical Transmission of a Battery Electric Vehicle
Considering Energy Consumption and Power Performance
Huang Xu 1,2 , Mengchen Yang 3 , Zhun Cheng 3, * and Xiaoping Su 1

1 School of Mechanical and Power Engineering, Nanjing Tech University, Nanjing 211816, China;
xuh1@niit.edu.cn (H.X.); suxiaoping@njtech.edu.cn (X.S.)
2 School of Transportation Engineering, Nanjing Vocational University of Industry Technology,
Nanjing 210023, China
3 Department of Vehicle Engineering, Nanjing Forestry University, Nanjing 210037, China; a_ymcc@163.com
* Correspondence: cz38@njfu.edu.cn

Abstract: The design of the gear quantity and transmission parameters of a vehicle has large effects
on its economical and power performance. This paper mainly researches the gear conditions (includ-
ing the gear quantity and each gear’s transmission parameters) of two-gear and three-gear AMT
(Automated Mechanical Transmission). This research uses Cruise software to build a multi-gear
simulation model of a BEV (Battery Electric Vehicle) and adopts the LHS (Latin hypercube sampling)
method to design an experiment plan and conduct a simulation experiment. This paper proposes a
systematic method for influencing factor analyses and the optimization of transmission parameters,
combining fuzzy theory, multiple regression, and particle swarm optimization. The research results
show that the gear quantity allowing for optimal overall performance is three. The highest score
obtained in the results of the simulation experiment for three-gear AMT is 11.15% higher than that
of the two-gear AMT. The optimal design plan for the two-gear AMT is a small ig1 with a big k1 , in
which case the highest score of the regression model increases by 2.67% compared with that before
modeling. The optimal design plan for the three-gear AMT is a big k1 with a big k2 , in which case the
Citation: Xu, H.; Yang, M.; Cheng, Z.; highest score of the regression model increases by 12.78% compared with that before modeling. Then,
Su, X. An Analysis of and
this research uses PSO (particle swarm optimization) to further optimize the regression models and
Improvements in the Gear Conditions
compares the difference between the highest scores in the results of the simulation experiment. The
of the Automated Mechanical
difference between the highest scores of the three-gear and two-gear AMT further increases to 21.95%
Transmission of a Battery Electric
after optimization. As shown in the results, the key factor influencing the performance of two-gear
Vehicle Considering Energy
Consumption and Power Performance.
and three-gear AMT is gear quantity.
Actuators 2024, 13, 432. https://
doi.org/10.3390/act13110432 Keywords: BEV; AMT; LHS; gear quantity; transmission plan; an influencing factor analysis; multi-
objective optimization design
Academic Editors: Seongjin Yim
and Kanghyun Nam

Received: 22 August 2024

Revised: 7 October 2024 1. Introduction
Accepted: 8 October 2024 With the continuous development of global industries, the energy waste problem
Published: 26 October 2024
and environmental concerns are becoming increasingly serious. In this regard, striving
to develop a BEV (Battery Electric Vehicle) has been considered an important way to
ameliorate energy consumption, environmental pollution, and other problems, and is an
Copyright: © 2024 by the authors.
important developmental direction in the automobile field [1]. Moreover, a BEV has great
Licensee MDPI, Basel, Switzerland. advantages over internal combustion engine vehicles in terms of carbon peaking and carbon
This article is an open access article neutrality goals, structure simplification, exhaust gas emissions, operational efficiency,
distributed under the terms and and so on. To bring these advantages into play, research on automatic transmissions is
conditions of the Creative Commons required [2].
Attribution (CC BY) license (https:// Currently, most BEVs in mass production adopt a direct-driven mode and only few
creativecommons.org/licenses/by/ high-end electric vehicles have a two-gear AMT. This paper explores the overall perfor-
4.0/). mance changes in a BEV after the configuration of a two-gear or three-gear AMT. The

Actuators 2024, 13, 432. https://doi.org/10.3390/act13110432 https://www.mdpi.com/journal/actuators

Actuators 2024, 13, 432 2 of 25

first type of BEV with a two-gear AMT belongs to the Porsche Taycan, which was mass-
produced in 2019 and which is as expensive as the BMW i8 [3]. The research in this paper
can be applied to other vehicle models, like commercial battery electric vehicles. The
optimization of the gear design and transmission parameters in this research can also serve
as a reference for time and cost reductions in terms of transportation, etc. Therefore, there
is plenty of room for research on the application of two-gear and multi-gear AMT to BEVs.
Research on the multiple gears of electric vehicles can improve the speed ratio range
and the distribution of AMT, thereby improving the vehicles’ performance, saving fuel, and
enhancing shift smoothness. Currently, studies on the multiple gears of electric vehicles
focus on a comparison of the overall vehicle performance, and most of them focus on
electric vehicles with a two-gear AMT and a fixed-gear-ratio transmission. To solve the
low working efficiency of the driving motor of a BEV with a fixed speed ratio, the authors
of [4] changed a fixed gear ratio into a two-gear AMT plan with the goal of reserving the
power battery and driving motor. The simulation analysis and comparison results show
that the two-gear AMT had a significant advantage in terms of the overall performance
improvement of the BEV. The authors of [5] first compared the overall performance of BEVs
with a fixed gear ratio and a two-gear AMT and then optimized the fixed gear ratio and
two-gear transmission ratio, respectively, using a genetic algorithm. The results show that
the BEV with the two-gear AMT had a better performance. With the goal of satisfying the
desired power of electric off-roaders, the authors of [6] optimized the transmission ratio
of a two-gear AMT with the minimum overall energy consumption as the optimization
objective and obtained the optimal transmission ratio. The authors of [7] proposed an
optimization method for the drivetrain system of a dual-motor off-road vehicle based on
the power envelope (force–speed characteristics) difference between the minimizing target
and the dual-motor model. The results show that compared to a single-motor drive system,
the overall performance shows a significant improvement in terms of power. Furthermore,
when compared to a non-optimized dual-motor system, there is a noticeable improvement
in economic efficiency.
Compared with a vehicle with a fixed gear ratio, a vehicle with a multi-gear AMT
can further improve the vehicle’s overall performance through different gear-shift control
strategies. The authors of [8] developed a gear-shift control strategy using a transmission
ratio obtained by the initial parameter matching and then adjusted the transmission ratio
and proposed a transmission ratio range optimization plan to optimize the matching of the
number of teeth, but this research adopted the same gear-shift control strategy throughout
the entire process. To solve this problem, in the optimization of the transmission ratio
of a BEV with a two-gear AMT, the authors of [9] dynamically adjusted the shift control
strategy for each group of transmission ratios obtained after optimization. The simulation
results show that the overall energy consumption reduced to some extent compared with
that in a plan that did not adjust the shift control strategy dynamically. The authors of [10]
proposed a new dual-motor four-wheel-drive electric vehicle configuration, featuring a
two-gear transmission on the rear axle, based on the characteristics of the vehicle’s driving
cycles. An energy management strategy was then developed, focusing on the optimal
low-energy torque distribution for different gears. The driving efficiency under the same
conditions but with different gears was compared, leading to the development of an optimal
economical gear-shifting control strategy. The authors of [11] proposed a transmission
system design consisting of a two-speed transmission without a clutch, paired with an open
differential on each axle. This setup resulted in a drivetrain system with eight different
gear combinations. Additionally, a “torque fill” control strategy was introduced to manage
the gear-shifting process, compensating for torque gaps during shifts. The results show a
significant reduction in energy consumption during both steady-state driving cycles and
other driving cycles.
However, three-gear AMT is rarely applied in the electric vehicle field, and their
development is currently mainly in the academic research stage. The authors of [12] con-
sidered the minimization of power loss in the power transmission process as the objective
Actuators 2024, 13, 432 3 of 25

function and evaluated the energy utilization performance of an SPHT (Serial-Parallel

Hybrid Transmission) with different numbers of gears using a dynamic planning method.
The experimental results show that the fuel consumption of the engine reduced when the
number of gears changed from 1 to 2 and from 1 to 3. The authors of [13] adopted a DOE
(Design of Experiment) method allowing for an SPHT to choose the optimal transmission
ratios of transmissions with three kinds of gear quantities and analyzed the effects on the
overall acceleration performance, climbing performance, and fuel consumption when the
hybrid power system was equipped with different transmissions. The results show that
an increase in gears could improve the overall power performance of a vehicle and could
reduce the entire vehicle’s battery discharge power and capacity, but this would reduce the
improvement in overall fuel consumption. Reference [14] proposed a three-gear variable
speed transmission to ensure the motor remains in the peak efficiency range, consider-
ing the minimum fuel consumption in a UDDS (Urban Dynamometer Driving Schedule)
driving cycle as the objective, and obtained the final shift strategy after optimizing the
transmission ratio. The energy consumption of a BEV with a three-gear AMT reduced by
9.3% compared with that of a BEV with a single-speed reducer. Reference [15] proposed a
multi-target transmission ratio optimization method considering gear shift performance.
The method first optimized the three-gear AMT’s transmission ratio to improve power
performance, economical efficiency, and comfort, improved aspects of the three-gear AMT
performance, such as shift time, clutch friction, and shift wobble, and finally made a com-
parison with two-gear and fixed-speed-ratio AMT in terms of performance. The results
showed that the three-gear AMT had a superior performance to that of fixed-speed-ratio
and two-gear AMT as a whole.
The studies above referred less to the influence of driving cycle on the reliability of
research results regarding transmission parameters, but were more focused on driving
cycle. This paper uses three driving cycles for the experiment to enhance the reliability
of the experimental data. Because the increase in driving cycles causes an increase in the
experiment groups, the paper uses an LHS (Latin Hypercube Sampling) experiment design
method to reduce the number of experiment groups while maintaining the experimental
plan’s randomness, homogeneity, and rationality [16], and then builds a comprehensive
evaluation system with the energy consumption and power performance indexes in three
driving cycles using fuzzy theory to analyze the AMT’s gear conditions (including the
gear quantity and each gear’s transmission parameters). Then, the paper builds multiple
regression models for two-gear and three-gear AMT based on the fuzzy theory evaluation
system to explore the transmission parameter design plan. Finally, the paper optimizes two-
gear and three-gear AMT with the PSO and the enumeration method, and then discusses
the key factor causing the difference in performance between two-gear and three-gear AMT.

2. Establishment of the BEV Simulation Model

2.1. The Working Principle
Figure 1 shows a schematic diagram of a two-gear AMT structure. As illustrated,
the power from the driving motor is transmitted to the planetary carrier, which drives
intermediate shaft 1. When clutch 1 is engaged and clutch 2 is disconnected, the power
continues to drive intermediate shaft 2 via the first gear’s driven gear, causing the driving
gear of the main reducer to rotate. This rotation drives the driven gear of the main reducer,
transmitting power to the main reducer, which ultimately drives the differential to output
power from both ends of the output shaft. This describes the power transmission route
for the first gear. When clutch 1 is disconnected and clutch 2 is engaged, the power flows
through the second gear’s driven gear to drive intermediate shaft 2. The subsequent power
transmission follows the same route as that of the first gear, forming the second-gear power
transmission route.
 power from both ends of the output shaft. This describes the power transmission
the first gear. When clutch 1 is disconnected and clutch13 2 is engaged, the pow
through the second gearʹs driven gear to drive intermediate shaft 2. The subseque
14
transmission follows the same route as that of the first gear, forming the sec
Actuators 2024, 13, 432 4 of 25
power transmission
16 route. 15

3 4
5 6 8
Figure 1. 2Schematic diagram of two-gear AMT. 7 (1—driving
9 motor; 2—input shaft; 3—bra
planet1 wheel; 5—planetary carrier; 6—driving gear of the 10
first gear; 7—clutch 1; 8—driving
the second gear; 9—clutch 2; 10—intermediate shaft 1; 11—driven gear of the second gea
EM main reducing gear; 13—intermediate shaft
driving gear of 11 2; 14—driven gear of main re
gear and differential gear; 15—output shaft; 16—driven gear of the 12 first gear.)

Figure 2 shows the structure schematic of a three-gear AMT, which operates sim
to the two-gear AMT described above. For BEVs, multi-gear AMT 13 offers the followi
vantages over systems with fewer gears or fixed-ratio transmissions: 14
(1) Higher efficiency. Multi-gear AMT enables a more precise match between th
16 15 positions, the electric v
tor and its operating speed. By selecting the appropriate gear
can ensure that the motor more frequently operates within its optimal range across
Figure 1. Schematic diagram of two-gear AMT. (1—driving motor; 2—input shaft; 3—brake;
ent vehicle
4—planet
speeds, thereby enhancing
wheel; 5—planetary carrier;
overall
6—drivingAMT.
performance
gear of(1—driving
and efficiency.
the first gear; 7—clutch 1; 8—driving
Figure 1. Schematic diagram of two-gear motor; 2—input shaft; 3—
gear of the second gear; 9—clutch 2; 10—intermediate shaft 1; 11—driven gear of the smoother
(2) Improved driving experience. Multi-gear AMT facilitates second gear; accele
planet wheel; 5—planetary carrier; 6—driving gear of the first gear; 7—clutch 1; 8—driv
and deceleration,
12—driving gear ofleading to a gear;
main reducing more comfortable
13—intermediate driving
shaft 2; 14—drivenexperience.
gear of main reducing
the second
gear and
gear; 9—clutch
differential gear;
2; 10—intermediate
15—output shaft; 16—driven
shaft
gear of the
1;first
11—driven
gear).
gear of the second
(3) Enhanced energy utilization. Multi-gear AMT adjusts
driving gear of main reducing gear; 13—intermediate shaft 2; 14—driven gear of main
gear positions effec
across
gear various
Figure
and driving
2 shows
differential the conditions,
structure
gear; 15—output which
schematic improves
of a three-gear
shaft; 16—driven AMT, energy
gear which efficiency
operates
of the and exten
similarly
first gear.)
rangetoofthethe battery.
two-gear AMT described above. For BEVs, multi-gear AMT offers the following
advantages over systems with fewer gears or fixed-ratio transmissions:
Figure 2 shows the structure schematic of a three-gear AMT, which operates
to the two-gear AMT described above. For BEVs, multi-gear AMT offers the follo
vantages over systems with fewer gears or fixed-ratio transmissions:
(1) Higher efficiency. Multi-gear AMT enables a more precise match betwee
tor and
EM its operating speed. By selecting the appropriate gear positions, the electr

can ensure that the motor

i1 more
i2
frequently
i3
operates within its optimal range acro
ent vehicle speeds, thereby enhancing overall performance and efficiency.
(2) Improved driving experience. Multi-gear
i0 AMT facilitates smoother acc
and deceleration, leading to a more comfortable driving experience.
(3) Enhanced energy utilization. Multi-gear AMT adjusts gear positions e
across various driving conditions, which improves energy efficiency and ex
range of the battery.
Figure 2. Schematic diagram of three-gear AMT.
Figure 2. Schematic diagram of three-gear AMT.
(1) Higher efficiency. Multi-gear AMT enables a more precise match between the
motor and its operating speed. By selecting the appropriate gear positions, the electric
Despite
vehicle canthe advantages
ensure over
that the motor AMT
more with fewer
frequently operatesgears
withinor
itsfixed-speed-ratio
optimal range across transm
plans, the three-gear
different AMT
vehicle speeds, hasenhancing
thereby limitations, including
overall a complex
performance design, a more in
and efficiency.
(2)EM
Improved driving experience. Multi-gear AMT facilitates smoother acceleration
and deceleration, leading to i1a more comfortable driving experience.
i2 i3
(3) Enhanced energy utilization. Multi-gear AMT adjusts gear positions effectively
across various driving conditions, which improves energy efficiency and extends the range
i0
of the battery.
Despite the advantages over AMT with fewer gears or fixed-speed-ratio transmission
plans, the three-gear AMT has limitations, including a complex design, a more intricate
production process, and greater difficulties in their control compared to two-gear AMT.
Therefore, the selection of an appropriate transmission system requires a comprehensive

Figure 2. Schematic diagram of three-gear AMT.

Despite the advantages over AMT with fewer gears or fixed-speed-ratio tran
Actuators 2024, 13, 432 5 of 25

consideration of the practical operating conditions and requirements of the vehicle. More-
over, the claim that multi-gear AMT outperforms AMT with fewer gears has yet to be
validated in real-world driving scenarios. This study focuses on the factors influencing
gear quantity and transmission parameters. Additionally, the research provides a refer-
ence for simulation experiments, assisting manufacturers before sample car production
and thus reducing the need for physical and financial resources during the research and
development phase.

2.2. Parameter Matching

Using a BEV with a fixed-speed-ratio transmission plan as the research object, this
paper proposes and designs two-gear and three-gear AMT plans based on this concept.
The specific parameters of the vehicle used are presented in Table 1.

Table 1. Basic parameters of the vehicle.

Parameter Value
Vehicle mass m/kg 1565
Rolling radius of wheel r/m 0.301
Air resistance coefficient Cd 0.284
Windward area A/m2 1.88
Rolling resistance coefficient f 0.015

Given the high efficiency, high reliability, wide operating range, and favorable external
characteristics of the permanent magnet synchronous motor, this study selected it as the
driving motor for the BEV.
Based on the design values of the performance indices listed in the table, calculations
were conducted to select the driving motor parameters for the model. In Formula (1),
Pmax1 represents the motor output power of an electric vehicle at a maximum speed of
150 km/h on a level road, Pmax2 denotes the motor output power at 30 km/h with a
maximum gradeability of 30%, and Pmax3 refers to the motor output power when the
vehicle accelerates from 0 to 100 km/h within 11 s [17].

Cd Au amax 2
  
u amax


 Pmax1 = 3600 ηT m g f + 21.15
Cd Au a 2
  
ua
Pmax2 = 3600 η m g f cos α max + m g sin α max + 21.15 (1)
 T
2 3
 
m g f um tm
√ m + Cd Aum tm

1
 P
max3 = 3600 tm ηt + δmu

1.5 m 2 t 52.875

where uamax denotes the maximum speed, um represents the final speed within the acceler-
ation range, δ is the rotational mass conversion coefficient, tm is the acceleration time, ua
refers to the continuous climbing speed, and αmax indicates the maximum grade ability.
The parameters listed in Table 1 were substituted into Formula (1), resulting in the
following formula:


 Pmax1 = 35kw



Pmax2 = 40kw (2)




Pmax3 = 73kw


The driving motor’s power must simultaneously satisfy the design requirements for
maximum speed, maximum gradeability, and a 0–100 km/h acceleration time. Therefore,
the motor output power Pmax should meet the condition Pmax ≥ max( Pmax1 , Pmax2 , Pmax3 ).
Considering that the motor inevitably incurs some energy losses, a margin of 10~15% is
recommended when selecting motor power. Thus, Pmax = 75 kw.
Figure 3 shows a contour map of motor efficiency.
 the motor output power Pmax should meet the condition Pmax ≥ max Pmax1 , Pmax2 , Pmax3 . ( )
Actuators 2024, 13, 432 Considering that the motor inevitably incurs some energy losses, a margin of 10~15% is
6 of 25
recommended when selecting motor power. Thus, Pmax = 75 kw.
Figure 3 shows a contour map of motor efficiency.

9
63 6
79.7 43
339

422

25
77.50

.65
81.9

86.
2

22
88
193

. 88
84.

90
88.6525
22 9
6
79.7339
77.5043

81.963
Torque (Nm)

86.4

90.8
32
75.2746

822
84.19

88.6525

90.8822
86.4229
81.9636
79.7339
77.5043

84.1932
2
882
90.
88
.6 52

86.
5
79.7339 81.9636
77.5043

422
84
.19
9
32

Figure 3.
Figure 3. Contour
Contourmap
mapofof
motor efficiency.
motor efficiency.

Based on
Based onthe
themotor
motorcharacteristics
characteristics andand
the economic
the economicperformance discussed
performance in this in this
discussed
paper, an economical shift in control strategy, commonly used in
paper, an economical shift in control strategy, commonly used in the market andthe market and by by
re-research
search institutions, is adopted. This strategy selects the intersection point of the speed–
institutions, is adopted. This strategy selects the intersection point of the speed–efficiency
efficiency curves between two adjacent gears, used as the shift point [18].
curves between two adjacent gears, used as the shift point [18].
The gear count of the AMT is closely linked to the overall performance of BEVs. In-
The the
creasing gear countofof
number theenables
gears AMT the is closely linkedtoto
driving motor the overall
operate performance
more frequently withinof BEVs.
Increasing
its high-efficiency range, significantly enhancing economic performance. However,frequently
the number of gears enables the driving motor to operate more a
within
higher its high-efficiency
gear count also raisesrange, significantly cost
the manufacturing enhancing
of AMT,economic performance.
making it essential However,
to select
aan
higher gear count
appropriate alsoofraises
number gears.the manufacturing
Additionally, it has cost
beenofnoted
AMT, making
that it essential
most BEVs have a to select
an appropriate
direct-driven number ofWhile
configuration. gears.two-gear
Additionally,
BEVs areitalready
has been noted thattheir
in production, mostoverall
BEVs have a
presence in theconfiguration.
direct-driven market remainsWhilelimited. Therefore,
two-gear this paper
BEVs primarily
are already examines thetheir
in production, per- overall
formance in
presence of BEVs equipped
the market with two-gear
remains limited. andTherefore,
three-gear AMT.
this paper primarily examines the
To betterof
performance meet theequipped
BEVs power performance requirements
with two-gear of vehicles,
and three-gear this paper designs
AMT.
the first-gear transmission ratio, which is primarily based on two
To better meet the power performance requirements of vehicles, this performance indicators:
paper designs
maximum gradeability and maximum speed [19].
the first-gear transmission ratio, which is primarily based on two performance indicators:
(1) A design is established with a stable speed ua = 30 km/h and a gradeability of 30%.
maximum gradeability and maximum speed [19].
C Au a 2and a gradeability of 30%.
r with a stable speed ua = 30 km/h
(1) A design is iestablished
⋅i ≥ ( m gf cos α + m g sin α + d ) (3)
g1 0
Tm ax ⋅ η t m ax m ax
21.15
2
ra C Au
i g1 · i0the
where Tmax represents ≥ maximum(driving
mg f costorque,
αmax +uamg
is the αmax + dclimbing
sincontinuous ) speed, (3)
Tmax · η t
and ig1 denotes the first-gear transmission ratio. 21.15
(2) The design is based on the maximum motor speed and driving power required to
where Tmax represents the maximum driving torque, ua is the continuous climbing speed,
achieve a top vehicle speed of 150 km/h.
and ig1 denotes the first-gear transmission ratio.
(2) The design is based on thei maximum ⋅ nmax speed and driving power required to
0.377 rmotor
g1
⋅ i0 ≤ (4)
achieve a top vehicle speed of 150 km/h. u amax

0.377r · nmax
i g1 · i0 ≤ (4)
uamax

r Cd Aumax 2
i g1 i0 ≥ (mg f + ) (5)
Tnmax ηt 21.15
where Tnmax represents the torque at the maximum driving motor speed, umax denotes the
maximum vehicle speed, and nmax is the peak driving speed.
The transmission ratio range for the first gear was designated as i g1 ∈ [1.89, 5.04].
The ratio between AMT gears should not exceed 1.8 [20], as higher ratios may lead to
difficulties in gear shifting. A total of 1.8 was set as the upper limit, and 1.1 as the lower limit,
which should not be too low; thus, the acceptable range for the gear ratio is [1.1,1.8]. The
Actuators 2024, 13, 432 7 of 25

first-gear transmission ratio was calculated based on the power performance requirements.
From this, using Formula (6), the second-gear transmission ratio for two-gear AMT was
determined according to the ratio range defined by k1 , and the third-gear transmission ratio
for three-gear AMT was designed according to the ratio range defined by k2 .

 k1 ∈ [1.1, 1.8]
(6)
k2 ∈ [1.1, 1.8]


By using the control variable method, all factors except the number of gears in the
transmission were standardized. For both two-speed and three-speed transmissions, the
same motor, the same range of first-gear transmission ratios, and consistent gear–gear ratio
ranges were employed. These ranges were determined using identical power performance
metrics, ensuring the reliability of our analysis concerning the number of gears.

2.3. Establishment of the Complete Vehicle Model

The vehicle model was developed using AVL-Cruise (version: 2019) software, which
is primarily employed to simulate and analyze key vehicle performance metrics such as
power and fuel economy. The modular modeling approach of the software streamlines the
entire vehicle modeling process, making it more efficient and user-friendly. Additionally, it
features an interactive interface with other software platforms like Matlab and dSPACE.
AVL-Cruise offers a variety of vehicle models, including new energy vehicles, hybrid
electric locomotives, off-road vehicles, and driving cycles. The user-friendly design of
the AVL-Cruise software has led to its widespread use in vehicle-related enterprises and
research institutes [21].
Actuators 2024, 13, x FOR PEER REVIEW 8 of 27
The electric vehicle model developed with this software is shown in Figure 4.

Figure 4.
Figure 4. Full-vehicle
Full-vehiclemodel
modelof of
a BEV.
a BEV.

(1) To
(1) To build
buildthethemodel,
model,start by by
start dragging the necessary
dragging modules
the necessary from the
modules Modules
from the Modules
library into the model construction area, establishing both mechanical and electrical con-
library into the model construction area, establishing both mechanical and electrical con-
nections between them.
nections between them.
(2) Open the parameter settings interface for each module and input the specific pa-
(2) Open
rameters thetoparameter
required settings
complete the vehicle interface for each module and input the specific
model configuration.
parameters required to complete the vehicle model configuration.
(3) After setting the parameter, connect the data buses between components to ensure
accurate data flow for inputs and outputs.
(4) Configure the simulation tasks in the Task Folder according to the research objec-
tives.
(5) Run the vehicle model, and the result manager in the software will record all ex-
perimental data and corresponding graphs from the simulation tasks.
Actuators 2024, 13, 432 8 of 25

(3) After setting the parameter, connect the data buses between components to ensure
accurate data flow for inputs and outputs.
(4) Configure the simulation tasks in the Task Folder according to the research objectives.
(5) Run the vehicle model, and the result manager in the software will record all
experimental data and corresponding graphs from the simulation tasks.
Given the large number of components in the model, the focus will be on the most
critical modules: the drive motor and the AMT module for modeling explanations.
Actuators 2024, 13, x FOR PEER REVIEW The drive motor model corresponds to the eDrive module shown in Figure 94,ofwith 27 its
Actuators 2024, 13, x FOR PEER REVIEW
main parameter settings interface illustrated in Figure 5. 9 of 27

Figure 5.
Figure 5. Drive
Drivemotor
motorparameter
parametersettings interface.
settings interface.
Figure 5. Drive motor parameter settings interface.
The required
The required input
input parameters
parameters include
include the
the motor
motor type,
type, rated
rated voltage,
voltage, peak
peak speed,
speed, and
The required inputFor
parameters include the motor type, rated voltage, peak speed,
other specifications. For instance, in Figure 5, “Type of Machine: PSM” indicates that
and other specifications. instance, in Figure 5, “Type of Machine: PSM” indicates that the
andmotor
the othertype
specifications. For instance, in Figure 5, “Type of Machine:Figure
PSM”6indicates that
motor type is aisPermanent
a Permanent Magnet Synchronous
Magnet Synchronous Motor
Motor(PMSM).
(PMSM). Figure displays the the
6 displays
the motor
interface fortype is a Permanent Magnet Synchronous Motor (PMSM). Figure 6 displays the
interface for inputting
inputtingthe theefficiency
efficiencymap map data of of
data thethe
drive
drivemotor, where
motor, (a) represents
where (a) represents
interface
speed, (b) for inputting the efficiency map data of the drive motor, where (a) represents
speed, (b) represents
representstorque,
torque,andand(c)(c)represents
represents motor
motorefficiency. Once
efficiency. the the
Once efficiency mapmap
efficiency
speed,
data are(b) represents
entered into torque,
this and (c)
interface, therepresents
drive motor
motor efficiency.
model is Once the efficiency map
complete.
data are entered into this interface, the drive motor model is complete.
data are entered into this interface, the drive motor model is complete.

Figure 6. Drive motor map data settings interface.

Figure6.6. Drive
Figure Drive motor
motor map
map data
datasettings
settingsinterface.
interface.
The AMT model corresponds to the Gear Box module shown in FIG. 4, with its main
The AMT
parameter model corresponds
configuration interface to the Gear Box
illustrated module
in Figure 7. shown in FIG. 4,parameter
After entering with its main
b,
parameter
which configuration
represents interface ratio
the transmission illustrated
of eachingear,
Figure 7. After automatically
the module entering parameter
matchesb,
which represents the transmission ratio of each gear, the module automatically
the number of teeth on the driving and driven gears for the current gear position. Addi- matches
the number
tionally, of teeth
values on the
for e and driving
f can andas
be input driven
neededgears
for for the current
research gearThis
purposes. position.
allowsAddi-
the
Actuators 2024, 13, 432 9 of 25

The AMT model corresponds to the Gear Box module shown in FIG. 4, with its main
parameter configuration interface illustrated in Figure 7. After entering parameter b, which
represents the transmission ratio of each gear, the module automatically matches the
number of teeth on the driving and driven gears for the current gear position. Additionally,
values for e and f can be input as needed for research purposes. This allows the module to
Actuators 2024, 13, x FOR PEER REVIEW
automatically adjust the ratio of teeth between the driving and driven gears, yielding the10 of 27
appropriate transmission ratio.

Figure7.7.Transmission
Figure Transmission gear
gear ratio
ratio settings
settings interface.
interface.

The
Theparameter
parameter configuration
configuration process for for
process other modules
other in the
modules inmodel is similar
the model to that
is similar to that
presented
presentedabove,
above,with each
with settings
each window
settings window offering a user-friendly,
offering highly
a user-friendly, integrated
highly integrated
visual
visualinterface that
interface facilitates
that easy
facilitates input.
easy input.
3. Experiment Plan Design and Result Analysis
3. Experiment Plan Design and Result Analysis
3.1. LHS Design Principle
3.1. LHS Design Principle
Mc Kay et al. [22] introduced the LHS method in 1979, which is a design approach
Mc Kay
for selecting et al. [22]
sample introduced
points the LHS method
in a multi-dimensional in It1979,
space. whichused
is widely is a in design
numericalapproach
for selecting
simulations and sample pointsdesign.
experiment in a multi-dimensional space. It is widely used in numerical
simulations and experiment
Unlike unrestricted randomdesign.
sampling, LHS utilizes the principle of stratified sampling.
It begins by defining
Unlike a sampling
unrestricted range,
random and basedLHS
sampling, on the domain
utilizes thexiprinciple
∈ [xi a , xi b ],ofdivides this sam-
stratified
i i i i i =i xi i,
range evenly into S sections of equal length, such that x =
pling. It begins by defining a sampling range, and abased0 on 1the domainx < x < . . . < x s −1x ∈ [x
<i x s a, xb b], di-
i vides . . ., n.
= 1, 2, this In this way, into
the sampling range is segmented n hypercubes. Next, an
into Sthat
range evenly S sections of equal length, such xia = xi0 < xi1 < … < xis−1 <
nx× S xmatrix
is = ib, i = 1,M 2,is generated through
…, n. In this way,a the
random permutation
sampling range isof segmented
sequence {1, into 2, . . .,Snn}.hypercubes.
In the
matrix,
Next, an n × S matrix M is generated through a random permutation of sequence the
each row corresponds to a small hypercube, from which a point is selected as {1, 2, …,
final sample point. Figure 8 illustrates a comparison between LHS sampling points and
n}. In the matrix, each row corresponds to a small hypercube, from which a point is se-
randomly generated sampling points.
lected as the final sample point. Figure 8 illustrates a comparison between LHS sampling
As shown in Figure 8, each partitioned interval includes LHS sample points, unlike
points and
random randomly
sampling, where generated
points may sampling points.
be unevenly distributed or clustered on one side, re-
sulting in a suboptimal range and distribution. One key advantage of LHS is that it ensures
sample points are spread across the entire range of input variable values. Additionally,
LHS rearranges the sample points for the input variables while minimizing the correlations
among the randomly sampled points for each variable.

Figure 8. A comparison of LHS sampling and random sampling.

 xis = xib, i = 1, 2, …, n. In this way, the sampling range is segmented into S
Next, an n × S matrix M is generated through a random permutation of seq
n}. In the matrix, each row corresponds to a small hypercube, from which
Actuators 2024, 13, 432 lected as the final sample point. Figure 8 illustrates a comparison
10 ofbetween
25
points and randomly generated sampling points.

Figure
Figure 8. A8.comparison
A comparison of LHS and
of LHS sampling sampling and random
random sampling. sampling.
In addition, the Monte Carlo method is a simple, effective, and widely adopted
As shown
experimental in FigureHowever,
design approach. 8, eachitspartitioned interval
ability to ensure includes
the quality of sampleLHS
pointssample
random sampling, where points may be unevenly distributed or clustere
becomes evident only with large-scale sampling. In contrast, LHS can deliver highly
accurate and reliable results even with small-scale sampling [23].
resulting in a suboptimal range and distribution. One key advantage of LH
On the other hand, in full factorial experiments, the number of experimental designs
grows exponentially as the number of input variables increases. In contrast, LHS can
maintain a relatively low number of experimental designs, even when the number of
input variables varies, while still ensuring the randomness and validity of the sample
points. Therefore, LHS offers a significant advantage in studies involving varying numbers
of variables.

3.2. LHS Experiment Table

Based on the LHS method described in Section 3.1, the experimental design table
required for this study was prepared. Table 2 presents 20 transmission schemes for two-
speed transmission, generated using the LHS method, with the specific data detailed
as follows.

Table 2. Transmission plans of two-gear AMT.

No. First-Gear Transmission Ratio/ig1 Gear–Gear Ratio k1

1 2.11 1.77
2 2.31 1.21
3 2.63 1.25
4 2.92 1.41
5 3.87 1.11
6 3.72 1.57
7 3.22 1.48
8 3.35 1.76
9 4.82 1.42
10 4.5 1.55
11 4.08 1.16
12 2.83 1.36
13 3.1 1.17
14 2 1.64
15 4.94 1.71
16 4.57 1.34
17 4.11 1.59
18 3.55 1.29
19 2.48 1.5
20 4.29 1.69
Actuators 2024, 13, 432 11 of 25

Actuators 2024, 13, x FOR PEER REVIEW 12 of 27

Figure 9 gives 20 groups of three-gear AMT plans obtained with the LHS method.
Table 3 shows specific data.

1.7

1.6

1.5

1.4

1.3

1.2

1.1
1.8 5
1.6 4
1.4 1.2 3
2
k1 ig1
Figure9.9.LHS
Figure LHSgeneration
generationfigure
figureofofthe
thetransmission
transmissionplans
plansofofthree-gear
three-gearAMT.
AMT.

Table3.3.Transmission
Table Transmissionplans
plansofofthree-gear
three-gearAMT.
AMT.

No.
No. First-Gear Transmission
First-Gear TransmissionRatio
Ratio ig
ig11 Gear–Gear
Gear–Gear Ratio
Ratio kk11 Gear–Gear Ratiok2k2
Gear–Gear Ratio
11 3.73
3.73 1.53
1.53 1.29
1.29
22 4.23
4.23 1.44
1.44 1.31
1.31
33 3.53
3.53 1.7
1.7 1.22
1.22
44 5 5 1.3
1.3 1.65
1.65
5 2.2 1.32 1.51
56 2.2
4.29
1.32
1.23
1.51
1.68
67 4.29
2.32 1.23
1.74 1.68
1.74
78 2.32
1.98 1.74
1.6 1.74
1.77
89 1.983.2 1.37
1.6 1.18
1.77
10 2.94 1.58 1.16
9 3.2 1.37 1.18
11 3.03 1.49 1.57
10
12 2.94
3.88 1.58
1.63 1.16
1.13
11
13 3.03
4.46 1.49
1.47 1.57
1.45
14
12 2.57
3.88 1.67
1.63 1.48
1.13
15
13 4.03
4.46 1.4
1.47 1.71
1.45
16 2.83 1.77 1.36
14
17 2.57
4.83 1.67
1.25 1.48
1.27
15
18 4.03
4.66 1.4
1.16 1.71
1.54
19
16 2.48
2.83 1.17
1.77 1.62
1.36
20
17 3.43
4.83 1.1
1.25 1.41
1.27
18 4.66 1.16 1.54
Using the LHS method,
19 2.48sample points are evenly and 1.17randomly distributed 1.62across the
entire
20 sampling range. Based
3.43 on this, performance experiments
1.1 are conducted
1.41on BEVs
equipped with two-gear and three-gear AMT. Compared to the combined experiments,
the number of tests
Using the LHS ismethod,
significantly
sample reduced, which
points are greatly
evenly and lowers
randomly thedistributed
time needed for
across
the
theexperiments.
entire sampling range. Based on this, performance experiments are conducted on BEVs
equipped with two-gear and three-gear AMT. Compared to the combined experiments,
3.3. Comparison and Analysis of the Experimental Results of Two-Gear and Three-Gear AMT
the number of tests is significantly reduced, which greatly lowers the time needed for the
As previously mentioned, to ensure the reliability of the experimental results, the study
experiments.
selected three driving cycles: the NEDC (new European driving cycle), the FTP-75 (Federal
Test
3.3. Procedure), andAnalysis
Comparison and the WLTC (World
of the Light Duty
Experimental Test of
Results Cycle), as shown
Two-Gear in FigureAMT
and Three-Gear 10a–c.
As previously mentioned, to ensure the reliability of the experimental results, the
study selected three driving cycles: the NEDC (new European driving cycle), the FTP-75
(Federal Test Procedure), and the WLTC (World Light Duty Test Cycle), as shown in Fig-
ure 10 (a), (b), and (c).
Actuators 2024,
Actuators 13,13,
2024, x FOR
432 PEER REVIEW 13 of
12 27
of 25

120
150
100

Speed（km/h）
80
100
60

40
50

20

0 0
0 200 400 600 800 1000 0 200 400 600 800 1000 1200 1400
Time(s) Time(s)

(a) (b)

150
Speed（km/h）

100

50

0
0 500 1000 1500
Time(s)
(c)
Figure 10.10.
Figure Driving cycles:
Driving (a)(a)
cycles: NEDC; (b)(b)
NEDC; FTP75; (c) (c)
FTP75; WLTC.
WLTC.

AA performance
performance simulation
simulation experiment
experiment was
was conducted
conducted forfor
thethe BEV
BEV model
model using
using thethe
simulation
simulation softwareCRUISE,
software CRUISE,based
basedon
onthe
the driving cycles
cyclesmentioned
mentionedabove.
above.Table
Table4 presents
4 pre-
the the
sents performance simulation
performance datadata
simulation for the
forBEV equipped
the BEV withwith
equipped a two-gear AMT,AMT,
a two-gear whilewhile
Table 5
provides
Table the corresponding
5 provides data for
the corresponding thefor
data BEV
thewith
BEVawith
three-gear AMT.AMT.
a three-gear

Table
Table 4. Performance
4. Performance experiment
experiment results
results of of two-gear
two-gear AMT.
AMT.

NDEC EnergyNDEC Energy FTP75 Energy WLTC Energy 100 km

No.
FTP75 EnergyConsumption
Consumption
Con- WLTC Energy Con- 100 kmAcceleration
Consumption
Accelera-
No. Consumption
sumption (kwh)
(kwh) (kwh) sumption (kwh)
(kwh) tion Time
Time(s)
(s)
(kwh)
1 1 1.968 1.968 3.412 3.412 4.512 4.512 11.09
11.09
2 2.049 3.498 4.626 10.85
2 2.049 3.498 4.626 10.85
3 2.066 3.503 4.712 10.53
3 4 2.066 2.061 3.503 3.482 4.712 4.707 10.53
10.49
4 5 2.061 2.193 3.482 3.677 4.707 5.052 10.49
11.59
5 6 2.193 2.091 3.677 3.518 5.052 4.787 10.43
11.59
7 2.071 3.493
6 2.091 3.518 4.787 4.737 10.42
10.43
8 2.044 3.458 4.681 10.33
7 9 2.071 2.178 3.493 3.662 4.737 5.025 10.42
12.06
8 10 2.044 2.136 3.458 3.594 4.681 4.907 10.33
10.95
9 11 2.178 2.185 3.662 3.693 5.025 5.032 12.57
12.06
10 12 2.136 2.064 3.594 3.484 4.907 4.713 10.52
10.95
13 2.107 3.577 4.815 10.88
11 14
2.185 1.994
3.693 3.42
5.032 4.526 12.57
11.23
12 15 2.064 2.135 3.484 3.591 4.713 4.908 10.52
11.01
13 16 2.107 2.18 3.577 3.664 4.815 5.032 10.88
12.13
14 17 1.994 2.108 3.42 3.549 4.526 4.835 10.43
11.23
18 2.12 3.577 4.872 10.86
15 2.135 3.591 4.908 11.01
19 2.028 3.443 4.623 10.76
16 20 2.18 2.104 3.664 3.541 5.032 4.826 12.13
10.36
17 2.108 3.549 4.835 10.43
 Actuators 2024, 13, x FOR PEER REVIEW 14 of 27

Actuators 2024, 13, 432 13 of 25

18 2.12 3.577 4.872 10.86
19 2.028 3.443 4.623 10.76
Table 5. Performance
20 experiment results of three-gear
2.104 3.541 AMT. 4.826 10.36

Table NDEC Energy

5. Performance FTP75
experiment results of three-gearWLTC
Energy AMT.Energy 100 km
No. Consumption Consumption Consumption Acceleration
NDEC Energy
(kwh) (kwh)
FTP75 Energy Con- WLTC Energy Con- 100Time
(kwh) (s)
km Accelera-
No. Consumption
sumption (kwh) sumption (kwh) tion Time (s)
1 2.046
(kwh) 3.483 4.669 10.39
2 1 2.0772.046 3.523
3.483 4.747
4.669 10.31
10.39
3 2 2.0262.077 3.453
3.523 4.576
4.747 10.36
10.31
4 3 2.0882.026 3.533
3.453 4.779
4.576 10.32
10.36
5 4 1.9492.088 3.41
3.533 4.474
4.779 11.05
10.32
6 5 2.0641.949 3.502
3.41 4.721
4.474 10.08
11.05
7 6 1.93 2.064 3.381
3.502 4.417
4.721 10.88
10.08
8 7 1.9641.93 3.407
3.381 4.435
4.417 11.24
10.88
9 8 2.0371.964 3.5
3.407 4.623
4.435 10.48
11.24
10 9 2.0222.037 3.4743.5 4.604
4.623 10.47
10.48
11 10 1.9732.022 3.409
3.474 4.52
4.604 10.48
10.47
12 11 2.081.973 3.51
3.409 4.749
4.52 10.31
10.48
13 12
2.0642.08
3.501
3.51
4.72
4.749
10.09
10.31
14 1.956 3.387 4.442 10.89
13 2.064 3.501 4.72 10.09
15 2.025 3.455 4.621 10.18
14 1.956 3.387 4.442 10.89
16 1.958 3.397 4.497 10.58
15 2.025 3.455 4.621 10.18
17 2.147 3.643 4.942 11.39
16 1.958 3.397 4.497 10.58
18 2.111 3.583 4.843 10.67
17 2.147 3.643 4.942 11.39
19 1.98 3.436 4.515 10.82
18 2.111 3.583 4.843 10.67
20 2.058 3.83 4.655 10.06
19 1.98 3.436 4.515 10.82
20 2.058 3.83 4.655 10.06
Based on the results of 20 sets of two-gear AMT simulation and 20 sets of three-gear
AMT simulation, fourBased on the results
indices of 20 sets
were used to of two-gear
create AMTplots
scatter simulation and 20 sets
to compare theof data,
three-gear
as
AMT simulation, four indices were used to create scatter plots to compare the data, as
shown in Figure 11.
shown in Figure 11.
Energy Consumption（kwh）

Energy Consumption（kwh）

Actuators 2024, 13, x FOR PEER REVIEW 15 of 27

(a) (b)

5.2 Two-gear AMT

Energy Consumption（kwh）

Three-gear AMT

5
Time（s）

4.8

4.6

4.4

4.2
5 10 15 20
Experiment number
(c) (d)
Figure 11. Comparison diagrams of experimental results: (a) NEDC; (b) FTP75; (c) WLTC; (d) 100
Figure 11. Comparison diagrams of experimental results: (a) NEDC; (b) FTP75; (c) WLTC; (d) 100 km
km acceleration time.
acceleration time.
Clearly, both energy consumption and the 0-100 km/h acceleration time benefit from
being minimized, as lower values lead to a better performance in both areas. As shown in
the four pictures, most red points are positioned below the blue points when comparing
both energy consumption and 100 km acceleration time across three driving cycles, sug-
gesting that the simulation results for the three-gear AMT are generally superior to those
of the two-gear AMT.
Actuators 2024, 13, 432 14 of 25

Clearly, both energy consumption and the 0-100 km/h acceleration time benefit from
being minimized, as lower values lead to a better performance in both areas. As shown in
the four pictures, most red points are positioned below the blue points when comparing
both energy consumption and 100 km acceleration time across three driving cycles, sug-
gesting that the simulation results for the three-gear AMT are generally superior to those
of the two-gear AMT.
However, differences in driving cycles, economic performance metrics, and power per-
formance metrics make it challenging to determine which AMT offers superior performance.
To address this, a comprehensive evaluation system was developed, employing fuzzy theory
to assess and identify the optimal gear quantity by considering four key metrics.
The fuzzy comprehensive evaluation method is an integrated evaluation approach
based on fuzzy mathematic theory [24]. It converts subjective assessments into objec-
tive quantitative indices using membership degree theory and establishes a multi-factor
evaluation system to objectively assess complex systems. The specific steps are as follows:
(1) To establish a factor set and a rating set, this paper selects energy consumption
in the NEDC, FTP75, and WLTC driving cycles, along with the 100 km acceleration time,
as the four performance indicators, denoted as eNEDC , eFTP75 , eWLTC , t. These indicators
are the factor set for evaluating the pros and cons of different gear counts in AMT, i.e.,
U = {eNEDC , eFTP75 , eWLTC , t}. Subsequently, each factor is categorized into four levels—
great, good, moderate, and weak—with corresponding scores of s1 = 100, s2 = 75, s3 = 50,
and s4 = 25, forming the rating set S = {s1, s2, s3, s4} = {100, 75, 50, 25}.
(2) Construct index performance evaluation sections. As previously mentioned, for
the four performance indices, lower values indicate better performance. Based on the
experimental results of two-gear and three-gear AMT shown in Tables 4 and 5, the minimum
and maximum values from 40 sets of data for each performance index were used as the
standards for “great” and “weak” grades, respectively. Intermediate values were divided
according to an equidistant partitioning principle, resulting in the index performance
evaluation sections outlined in Table 6.

Table 6. Index performance evaluation sections.

Performance Index
Comment
Grade eNEDC eFTP75 eWLTC
T (s)
(kwh) (kwh) (kwh)
Great <1.93 <3.381 <4.417 <10.06
Good 1.93~2.0615 3.381~3.6055 4.417~4.7345 10.06~11.315
Moderate 2.0615~2.193 3.6055~3.83 4.7345~5.052 11.315~12.57
Weak >2.193 >3.83 >5.052 >12.57

(3) Select the membership degree function and construct a fuzzy evaluation matrix.
Three common types of membership degree function are the triangular, trapezoidal, and
normal distribution functions. Compared to the triangular function, both the normal
distribution and trapezoidal functions are more efficient at filtering low-value information.
Moreover, the normal distribution function not only offers effective filtering but also
possesses smoothness and gradual transition characteristics, making it more precise in
describing and modeling complex change processes. Therefore, this paper utilized the
normal distribution function to construct the membership degree functions.
The paper describes experiment group i as Xi = { xi,1 , xi,2 , xi,3 , xi,4 } (where the first
20 groups represent two-gear AMT experiments and the last 20 represent three-gear AMT
experiments, i.e., i = 1, 2, . . .,20), in which xi,1 , . . ., xi,4 are the values of the four performance
indices obtained from experiment group i. Let ri,j (s1 ) denote the membership degree of
xi,j (the j-th index value in experiment group i) to evaluation grade vk . Using the normal
distribution membership function of the great grade as an example, we present the general
form of the normal distribution membership degree function [25]:
Actuators 2024, 13, 432 15 of 25




 1 xi,j < µ j,1



µ j,1 ≤ xi,j ≤ µ j,2

 2
− ( xi,j − µ j,1 ) / 2σj,1
ri, j (s1 ) = e (7)




xi,j > µ j,2



 0

where µj,k represents the expected value of the normal distribution membership function
for index j with respect to evaluation grade Skwhile , σ denotes the standard deviation of
the normal distribution membership function, which must satisfy the requirements of the
membership normalization and is defined as σj,k = (µ j,k+1 − µ j,k−1 )/6.
Taking the first performance index, eNEDC , as an example, the expected value µ and
standard deviation σ are calculated and presented in Table 7.
Actuators 2024, 13, x FOR PEER REVIEW 17 of 27
Table 7. Membership degree function parameters of performance index eNEDC .

k
1 2 3 4
Parameter
1.86425 ri , j ( v1.99575
) 2.12725 2.25875
ri′, j ( v k ) =
µ k
, j = 1, 2, 3, 4
0.0438 4 0.0438 0.0438 0.0438 (8)
 ri , j ( v k )
σ
k =1

The parameters from Table 7 were applied to the normal distribution membership
where ri′,and
function, (v ) denotes
j k using the the membership
specific degree
expressions, theofmembership
performance index j to
function evaluation
curves for thegrade
four
grades were obtained, as shown in Figure 12.
vk after normalization in experiment group i.
Membership degree

Figure 12.
Figure 12. Energy
Energy consumption
consumptionelevation
elevationgrade
grademembership
membershipdegree
degreefunction inin
function the NEDC
the driving
NEDC driv-
cycle.
ing cycle.

Similarly, theshows
Formula (9) normal
thedistribution
fuzzy evaluation membershipmatrix:functions for the other three indexes
were obtained using this method.
After obtaining the membership ri′,1 (v1degrees
) ri′,1 (v2of ri′,1 (vevaluation
) the 3
) ri′,1 (v4 ) grades for each index in
 
experiment group i using the membership r′ (v ) r′ degree function, normalization  was required
( v ) r ′ ( v ) r ′ ( v )
to form the fuzzy evaluation Ri = 
matrix i ,2 1
for i ,2
experiment2 i ,2 3
groupi ,2
i. 4

The normalization of the
(9)
membership degrees is illustrated rin
′i ,3 (v1 ) ri′,3 (v2 ) ri′,3 (v3 ) ri′,3 (v4 )
Formula (8): 
 
 
r′ (v )ri,jri(′,4v(kv)2 ) ri′,4 (v3 ) ri′,4 (v4 )
r ′ i,j (vk)i ,4= 1 , j = 1, 2, 3, 4 (8)
4
The fuzzy comprehensive elevation matrices ∑ r i,j ( v k ) for 40 sets of experimental data were
k =1
obtained using the aforementioned method.
where(4) r ′ i,j
To (vcalculate
k ) denotes the membership
fuzzy degree
comprehensive of performance
evaluation index
vectors, it j to evaluation
is necessary to first grade
deter-
vmine
k after
thenormalization
weights of thein experiment group i.indices. The research presented in this paper
four performance
emphasizes economic performance while also considering power performance. Com-
pared to the NEDC driving cycle, the WLTC driving cycle provides a more realistic as-
sessment, while the FTP75 driving cycle is specific to urban conditions in American cities
and thus differs from urban driving cycles elsewhere. Based on these considerations, the
weights were allocated as W = W , W , W , W  = 0.2,0.2,0.4,0.2 .
Actuators 2024, 13, 432 16 of 25

Formula (9) shows the fuzzy evaluation matrix:

r ′ i,1 (v1 ) r ′ i,1 (v2 ) r ′ i,1 (v3 ) r ′ i,1 (v4 )

 
 
 ′
 r i,2 (v1 ) r ′ i,2 (v2 ) r ′ i,2 (v3 ) r ′ i,2 (v4 )


 
Ri =   (9)
 r ′ ( v1 ) r ′ ( v2 ) r ′ ( v3 ) r ′ ( v4 )
 
 i,3 i,3 i,3 i,3


 
r ′ i,4 (v1 ) r ′ i,4 (v2 ) r ′ i,4 (v3 ) r ′ i,4 (v4 )

The fuzzy comprehensive elevation matrices for 40 sets of experimental data were
obtained using the aforementioned method.
(4) To calculate fuzzy comprehensive evaluation vectors, it is necessary to first deter-
mine the weights of the four performance indices. The research presented in this paper
emphasizes economic performance while also considering power performance. Compared
to the NEDC driving cycle, the WLTC driving cycle provides a more realistic assessment,
while the FTP75 driving cycle is specific to urban conditions in American cities and thus
differs from urban driving cycles elsewhere. Based on these considerations, the weights
were allocated as W = [WeNEDC , WeFTP75 , WeWLTC , Wt ] = [0.2, 0.2, 0.4, 0.2].
Bi is defined as the fuzzy comprehension evaluation vector for experiment group i,
with the calculation given by Formula (10):

Bi = W ◦ Ri (10)

where ◦ is a fuzzy operator.

There are four common fuzzy operators: M (∧, ∨) , M (∧, ⊕), M (·, ∨) and M(·, +).
The first two fuzzy operators may produce calculation errors that are indeterminable in the
evaluation results, while M (·, ∨) tends to be unstable [26]. Therefore, this paper selected
the product-sum fuzzy operator M (·, +), which is a weighted mean operator.
Following the fuzzy calculations described above, 40 sets of comprehensive evaluation
vectors were obtained, as presented in Tables 8 and 9.

Table 8. Comprehensive evaluation vectors of two-gear AMT.

Bi Vector Bi Vector
1 [0.0820, 0.8840, 0.0340, 0.0000] 11 [0.0000, 0.0060, 0.6760, 0.3180]
2 [0.0020, 0.9080, 0.0900, 0.0000] 12 [0.0120, 0.7380, 0.2500, 0.0000]
3 [0.0100, 0.7300, 0.2600, 0.0000] 13 [0.0000, 0.3880, 0.6120, 0.0000]
4 [0.0140, 0.7580, 0.2280, 0.0000] 14 [0.0560, 0.8720, 0.0720, 0.0000]
5 [0.0000, 0.0360, 0.6640, 0.3000] 15 [0.0000, 0.3140, 0.6740, 0.0120]
6 [0.0140, 0.4800, 0.5060, 0.0000] 16 [0.0000, 0.0060, 0.6760, 0.3180]
7 [0.0160, 0.6420, 0.3420, 0.0000] 17 [0.0120, 0.7380, 0.2500, 0.0000]
8 [0.0360, 0.8460, 0.1180, 0.0000] 18 [0.0000, 0.3880, 0.6120, 0.0000]
9 [0.0000, 0.0180, 0.7960, 0.1860] 19 [0.0560, 0.8720, 0.0720, 0.0000]
10 [0.0000, 0.3140, 0.6740, 0.0120] 20 [0.0000, 0.3140, 0.6740, 0.0120]

Table 9. Comprehensive evaluation vectors of three-gear AMT.

Bi Vector Bi Vector
1 [0.0220, 0.8680, 0.1100, 0.0000] 11 [0.0900, 0.9100, 0.0000, 0.0000]
2 [0.0280, 0.5800, 0.3920, 0.0000] 12 [0.0300, 0.5700, 0.4000, 0.0000]
3 [0.0360, 0.9440, 0.0200, 0.0000] 13 [0.0920, 0.6380, 0.2700, 0.0000]
4 [0.0260, 0.4800, 0.4940, 0.0000] 14 [0.2480, 0.7420, 0.0100, 0.0000]
5 [0.1540, 0.8200, 0.0260, 0.0000] 15 [0.0700, 0.8980, 0.0320, 0.0000]
6 [0.0940, 0.6300, 0.2760, 0.0000] 16 [0.1360, 0.8640, 0.0000, 0.0000]
7 [0.3960, 0.5960, 0.0080, 0.0000] 17 [0.0000, 0.1100, 0.8620, 0.0280]
8 [0.2180, 0.7080, 0.0740, 0.0000] 18 [0.0020, 0.3660, 0.6320, 0.0000]
9 [0.0120, 0.9380, 0.0500, 0.0000] 19 [0.0500, 0.9440, 0.0060, 0.0000]
10 [0.0160, 0.9600, 0.0240, 0.0000] 20 [0.1000, 0.5720, 0.2280, 0.1000]
 5 [0.1540, 0.8200, 0.0260, 0.0000] 15 [0.0700, 0.8980, 0.0320, 0.0000]
6 [0.0940, 0.6300, 0.2760, 0.0000] 16 [0.1360, 0.8640, 0.0000, 0.0000]
7 [0.3960, 0.5960, 0.0080, 0.0000] 17 [0.0000, 0.1100, 0.8620, 0.0280]
8 [0.2180, 0.7080, 0.0740, 0.0000] 18 [0.0020, 0.3660, 0.6320, 0.0000]
Actuators 2024, 13, 432 9 [0.0120, 0.9380, 0.0500, 0.0000] 19 [0.0500, 0.9440, 0.0060, 0.0000]
17 of 25
10 [0.0160, 0.9600, 0.0240, 0.0000] 20 [0.1000, 0.5720, 0.2280, 0.1000]

(5)
(5) To
Toperform
performdefuzzification,
defuzzification, the
the fuzzy
fuzzy comprehensive
comprehensive evaluation
evaluation vectors
vectors for
for 40
40
groups
groupsof ofexperimental
experimentalresults
resultswere
wereobtained
obtainedininStep
Step4.
4. However,
However, comparing
comparingthese
these vectors
vectors
directly
directly does
does not
not provide
provide aa clear
clear visual
visual representation.
representation. ToTo better
better observe
observe the
the outcomes,
outcomes,
defuzzification applied to the 40 groups of evaluation vectors, yielding g
defuzzification was applied to the 40 groups of evaluation vectors, yielding gi i, the specific
was , the specific
score
scorefor
for each
each experimental
experimentalgroup.
group. The
The defuzzification
defuzzification calculations
calculations were
were conducted
conducted based
based
on
on the
the comment
commentset setSSestablished
establishedininstep
step1.1.The
Thecalculation
calculationformula
formulaisisas
asfollows:
follows:
T
g S=·SB⋅iBT
gi =
i i
(11)
(11)

Based
Based on
on Formula
Formula (11),
(11), 20
20 sets
sets of
of test
test results
results were
were obtained
obtained forfor both
both the
the two-speed
two-speed
and
and three-speed transmissions. The detailed scoring results are illustrated inFigure
three-speed transmissions. The detailed scoring results are illustrated in Figure13.
13.
85

80

75

70

65
Score

60

55 Experiment score of two-gear AMT

Experiment score of three-gear AMT
50

45

40
0 5 10 15 20
Experiment number

Figure 13. Comparison of the experimental; result scores of two-gear and three-gear AMT.

According to Figure 13, in the high score range of 75 and above, there are nine groups
of experimental results for the three-gear AMT, which is eight more than those for the two-
gear AMT. In the low score range of 50 and below, there are four groups of experimental
results for the two-gear AMT, while none are observed for the three-gear AMT. In the
intermediate range of 75–50, there are 15 groups of experimental results for the two-gear
AMT and 11 groups for the three-gear AMT. It is evident that most experimental scores
for the two-gear AMT fall within the intermediate range. The average calculated scores
are 65.82 and 66.88, respectively, indicating that there is minimal difference in overall
performance between the two types of AMT in this section.
In general, when designing gear configurations, a three-gear AMT is preferable for
achieving high performance, without considering design complexity and manufactur-
ing challenges. Conversely, a two-gear AMT is more suitable when prioritizing cost-
effectiveness and reducing design complexity and manufacturing difficulty while main-
taining good performance.
Certainly, when considering the broader transmission sector, the design complexity
and manufacturing process of three-speed transmission are not necessarily more challeng-
ing than those of two-speed transmission. Different transmission types and configurations
have distinct characteristics, such as high transmission efficiency, lightweight construction,
simplicity, and compactness. This paper examines the AMT introduced in Section 2.1. For
this type of AMT, the three-speed transmission incorporates an additional gear pair and
clutch compared to the two-speed version. These modifications have a relatively minor
impact on vehicle quality and the complexity of processing and manufacturing.

4. Analysis and Optimization of Gear Conditions

4.1. Multiple Regression Model of the Vehicle Performances
In the study, each dependent variable, such as NEDC energy consumption or FTP75
energy consumption, is associated with at least two independent variables, including ig1 ,
Actuators 2024, 13, 432 18 of 25

k1 , and others. To analyze the relationship between multiple independent and dependent
variables, a multiple regression model was employed.
To conduct a more comprehensive analysis, we defined the experimental result score
of the two-gear AMT obtained using the fuzzy evaluation system as y5 , while the four
indices eNEDC , eFTP75 , eWLTC , t were denoted as y1 , y2 , y3 , and y4 , respectively. Based on the
collected scatter point data, multiple regression models were developed for ig1 and k1 with
the aforementioned five indices. Figure 14a illustrates the 3D plot and contour map of the
regression model for eNEDC with ig1 and k1 . Similarly, regression models for eFTP75
Actuators 2024, 13, x FOR PEER REVIEW
, eWLTC ,
20 of 27
t, and y5 with ig1 and k1 were also established.

1.8

1.7

1.6

1.5

k1
1.4

1.3

1.2

1.1
2 2.5 3 3.5 4 4.5 5
ig1
(a)
k1

(b)
k1

(c)
Figure 14. Regression models of energy consumption with ig1 and k1. (a) NEDC; (b) FTP75; (c)
Figure 14. Regression models of energy consumption with ig1 and k1 . (a) NEDC; (b) FTP75; (c) WLTC.
WLTC.

Table 10 presents the 14,

Adjusted-R 2 values for five regression models. The models for
In Figure it is evident that the energy consumption across the three driving cycles
y1 , y2 , y3 , and yfollows
5 exhibit high
a similar levels
trend. of fit,
For any while
given valuethe
of ig1model for
within its t, (i.e.,
range, y4 ), demonstrates
the energy consumption a
in all three
lower, yet acceptable, Adjusted-R 2 value.
cycles decreases as k1 increases, exhibiting comparable downward trends, al-
beit at different rates. Similarly, for any selected value of k1 within its range, energy con-
sumption across the three cycles declines as ig1 decreases. Notably, when ig1 decreases
Table 10. Fitting while
precision of the multiple
k1 increases, regression
there is a more model
pronounced (of yi with
downward ig1 in
trend and k1 ) of
energy two-gear AMT.
consumption.
Figure 15 illustrates the regression model for the 100 km acceleration time based on
Multiple Regression
ig1 and k1.Model
The fitting accuracy
1 is 0.89. The 2paper focuses 3on economic performance
4 while
5
(of yi with ig1 and k1 )
R2 0.99109 0.99530 0.99392 0.89923 0.98389
Actuators 2024, 13, 432 19 of 25

In Figure 14, it is evident that the energy consumption across the three driving cycles
follows a similar trend. For any given value of ig1 within its range, the energy consumption
Actuators 2024, 13, x FOR PEER REVIEW
in all three cycles decreases as k1 increases, exhibiting comparable downward trends, 21 of 27
albeit at different rates. Similarly, for any selected value of k1 within its range, energy
consumption across the three cycles declines as ig1 decreases. Notably, when ig1 decreases
while alsothere
k increases, considering power
is a more performance,
pronounced ensuring that
downward theinfitting
trend accuracy
energy remains within an
consumption.
Actuators 2024, 13, x FOR1PEER REVIEW 21 of 27
acceptable reference range.
Figure 15 illustrates the regression model for the 100 km acceleration time based on
ig1 and k1 . The fitting accuracy is 0.89. The paper focuses on economic performance while
also consideringalsopower performance, ensuring that the fitting accuracy remains within an
considering power performance, ensuring that the fitting accuracy remains within an
acceptable reference range.
acceptable reference range.

k1
k1
Figure 15. Regression models of 100 km acceleration time with ig1 and k1.

In the model, 100 km acceleration time t exhibits a trend distinct from that of energy
consumption. When ig1 is within the range of 3~4 and k1 is between 1.5 and 1.8, t is mini-
mized.
Figure
Figure 15. Regression 15. Regression
models of 100 kmmodels of 100 kmtime
acceleration acceleration
with ig1time
andwith
k1 . ig1 and k1.
Figure 16 presents the multiple regression model derived from the scores of 20 ex-
In the model, In the
perimental
100 kmmodel,
results 100
for akm
acceleration acceleration
two-gear
time AMT, time ta
exhibits
obtained
t exhibits usingadistinct
trend atrend
fuzzy distinct
from from
evaluation that
thatsystem.
of en-of energy
As de-
consumption.
fined in the When
comment ig
ergy consumption. When ig1 is within the range of 3~4 and k1 is between 1.5 and 1.8, t mini-
1 is
set S within
of the the
fuzzy range of
system, 3~4 and
better k 1 is between
experimental 1.5 and
outcomes 1.8, t
yield is
higher
mized.
is minimized. scores.
Figure
It can be16observed
presentsthatthe the
multiple regression
experimental model
group scoresderived fromexhibit
generally the scores of 20 ex-
an increasing
Figure 16 presents the multiple regression model derived from the scores of 20 experi-
perimental
trend, which results for a two-gear
corresponds inversely AMT, obtained
to the usingtrend
decreasing a fuzzy evaluation
in energy system. AsSpe-
consumption. de-
mental results for a two-gear
fined in the
cifically, when
AMT, set
comment obtained
ig1 decreases the using
S ofand kfuzzy
a fuzzy
system,
1 increases,
evaluation
better
the
system.
experimental
experimental group
As defined
outcomes yield ahigher
scores show more
set S of the fuzzy
in the comment pronounced
scores. upward system, better experimental
trend. Through the model, the outcomes
highest scoreyieldachieved
higher scores.
is 78.2309.
It can be observed that the experimental group scores generally exhibit an increasing
trend, which corresponds inversely to the decreasing trend in energy consumption. Spe-
cifically, when ig1 decreases and k1 increases, the experimental group scores show a more
pronounced upward trend. Through the model, the highest score achieved is 78.2309.
k1
k1

Figure 16. Regression models of AMT experiment results with ig1 and k1.
Figure 16. Regression models of AMT experiment results with ig1 and k1 .
When designing a two-gear AMT, if prioritizing economic performance, a design can
It can be observed that the experimental group scores generally exhibit an increasing
be developed within the specified range by referring to the models of y1, y2, and y3. If pri-
trend, which corresponds inversely to the decreasing trend in energy consumption. Specif-
oritizing
Figure 16. power performance,
Regression models of AMTthe model of y4results
experiment shouldwith
be used
ig1 andask1a. reference. For a design
ically, when ig1 that
decreases and k1 increases,
focuses on economic
the experimental group scores show a more
performance while also considering power performance, the
pronounced upward model trend.
When Through
of y5designing
can the model,
serve aastwo-gear
a guide AMT,theifthe
within highest
designscore
prioritizing achieved
economic
value is 78.2309.a design can
range. performance,
When designing a two-gear
be developed
In the experiment AMT,
within the if prioritizing
specifiedthree-gear
involving economic
range by AMT,referring
eachperformance,
todependent a design
the modelsvariable
of iscan
y1, y2y, iand y3. If pri-
associated
be developed within
oritizing
with the
three specified
power range
performance,
independent by referring
the model
variables ig1, k1, of
andto the models
y4 should be used
k2. Unlike of y , y , and
as 1a reference.
two-gear AMT,
2 y 3 . Ifa design
For
three-gear AMT
prioritizing powerthatperformance,
introduces the model
focusesanonadditional
economic of y4 should
performance
independent be used
while
variable, also as a itreference.
considering
making power
impossible For adirectly
designobserve
toperformance, the
model of y can serve as a guide within the design
that focuses on economic performance while also considering power performance, the
5 value range.
model of y5 can serve In as
theaexperiment
guide withininvolving three-gear
the design value AMT, each dependent variable yi is associated
range.
with three independent variables ig1, k1, and k2. Unlike two-gear AMT, three-gear AMT
introduces an additional independent variable, making it impossible to directly observe
Actuators 2024, 13, 432 20 of 25

Actuators 2024, 13, x FOR In theREVIEW

PEER involving three-gear AMT, each dependent variable yi is associated
experiment 2
with three independent variables ig1 , k1 , and k2 . Unlike two-gear AMT, three-gear AMT
introduces an additional independent variable, making it impossible to directly observe
all data variations using a 3D diagram. To address this, for each yi , one of the three
all data variations
independent variables using a 3D diagram.
was fixed sequentially To address
while keeping this,two
the other for unchanged.
each yi, one For
of the three
pendent variables was fixed sequentially while keeping the other
the fixed factor, three values—small, medium, and large—were selected from its range two unchanged. F
and substitutedfixed
intofactor, three values—small,
the multiple medium,
regression model, and large—were
as presented in Table 11.selected from its rang
This method
enables dimensionality reductions, allowing us to observe the overall change trends. This metho
substituted into the multiple regression model, as presented in Table 11.
ables dimensionality reductions, allowing us to observe the overall change trends.
Table 11. Fixed values of independent variables of three-gear AMT.
Table 11. Fixed values of independent variables of three-gear AMT.
Factor ig1 1 k 2 k
Factor ig1 k1 k2
Classification Small Medium Big Small Small
Classification Medium
Medium Big SmallMedium
Big Small Medium Big Medium
Big Small
Value 1.98 3.43 5 1.1 1.44 1.77 1.13 1.45 1.77
Value 1.98 3.43 5 1.1 1.44 1.77 1.13 1.45

Due to the largeDuenumber

to the large numbergenerated
of images of imagesaftergenerated after the dimensionality
the dimensionality reduction,reductio
paper
this paper focuses onfocuses on the regression
the regression models of models of the experimental
the experimental result scores, result
whichscores,
are which a
the main concern.
mainFor the regression
concern. models ofmodels
For the regression indicesofy1indices
, y2 , y3 ,yand
1, y2,yy43,, only
and ythe most
4, only the most
representative sentative
images were selected
images werefor discussion
selected and analysis.
for discussion and analysis.
We analyzed the Weregression
analyzed models of threemodels
the regression economic performance
of three economic indices, y1 , y2 , and
performance indices, y1, y
y3 . Figure 17ayillustrates
3. Figure 17a the eWLTC multiple
illustrates the eWLTCregression model, which
multiple regression model, showswhich anshows
overall an overall
trend in economic performance similar to the performance of the two-gear
in economic performance similar to the performance of the two-gear AMT. When AMT. When
k1 increases and ig1 decreases,
creases energy consumption
and ig1 decreases, is significantly
energy consumption reduced. The
is significantly trend The
reduced. in trend
the 100 km acceleration time (t) is also largely consistent with the trend observed
100 km acceleration time (t) is also largely consistent with the trend observed in the in the
two-gear AMT. However,
gear upon further
AMT. However, uponexamination, it is observed
further examination, it isthat when kthat
observed 1 reaches
when k1 reach
its maximum,maximum,
the trend differs
the trendslightly;
differsin this case,
slightly; t decreases
in this as k2 and
case, t decreases asig increase.
k21 and ig1 increase. T
Therefore, when designing a three-gear AMT with a primary focus on
fore, when designing a three-gear AMT with a primary focus on power performa power performance,
a design featuring
designa large transmission
featuring ratio and gear–gear
a large transmission ratio and ratio may be ratio
gear–gear considered.
may be considered.

12

4.8
10
y4

4.7

4.6
8
4.5 2
5
4.4 4 3
1.8
3 4 1.6
1.8 1.6 1.4
1.4 1.2 2 1.2
1 ig1 ig1 5 1
k1 k2
(a) (b)
Figure
Figure 17. Multiple 17. Multiple
regression modelregression model of the
of the experimental experimental
results results
of three-gear AMT.of three-gear AMT.
(a) eWLTC ; (b) t. (a) eWLTC

For ease of analysis andofdiscussion,

For ease thediscussion,
analysis and max–min values from (a)values
the max–min to (i) are
frompresented
(a) to (i) are pres
based on Figure 18. The
based corresponding
on Figure data are summarized
18. The corresponding data areinsummarized
Table 12. in Table 12.

Table 12. Max–min corresponded

Table 12. Max–mintocorresponded
the fixed value
toof each
the factor.
fixed value of each factor.

Factor Factorig1 ig1 k1 k1 k2 k2

Value 1.98
Value3.43 5
1.98 3.43
1.1
5
1.44
1.1
1.77
1.441.13 1.77 1.45 1.13 1.77 1.45 1.
Max–min 12.6674 Max–min
25.1476 12.6674
55.5793 25.1476
45.6714 55.5793
32.5267 45.6714
37.283 32.5267
44.2613 37.283 44.261335.4028
32.4745 32.4745 35.4
Actuators2024,
Actuators 2024,13,
13,432
x FOR PEER REVIEW 23
21 of 27
of 25

85
100

80 80

75 60

1
1
1.2
1.4
1.5 1.8
1.8 k1 1.6 1.6
k1 1.2 1.4 1.6
1.2
1.4
1 k2 1 k2

(a) (b) (c)

80

70
y5

60

50
1
1.2 2
1.4 3
k2 1.6 4 ig1
5

(d) (e) (f)

80
100
80
70
70 80
60

y5
50 60 60
2
50
1 1 1.2 3
2 1.2 2
3 1.4 3 1.4 4 ig1
k1 1.5 4 ig1 k1 1.6 4 ig1 k1 1.6
5 5 1.8 5

(g) (h) (i)

Figure 18.
Figure 18. Multiple
Multipleregression
regressionmodel
model of of
experimental
experimental results of three-gear
results AMT.
of three-gear (a) ig(a)
AMT. 1 = 1.98; (b) ig1
ig1 = 1.98;
= 3.43; (c) ig1 = 5; (d) k1 = 1.1; (e) k1 = 1.44; (f) k1 = 1.77; (g) k2 = 1.13; (h) k2 = 1.45; (i) k2 = 1.77.
(b) ig1 = 3.43; (c) ig1 = 5; (d) k1 = 1.1; (e) k1 = 1.44; (f) k1 = 1.77; (g) k2 = 1.13; (h) k2 = 1.45; (i) k2 = 1.77.

Whenigig11isistreated
When treatedasasa afixed
fixed parameter
parameter andand gradually
gradually increased,
increased, the the
model’smodel’s
max-min max-
min value
value shows shows a significant
a significant rise, indicating
rise, indicating that when
that when ig1 reaches
ig1 reaches its maximum,
its maximum, i.e., igi.e., ig1
1 = 5,
it= greatly
5, it greatly
impacts impacts the model.
the model. UnderUnder this condition,
this condition, the highest the highest
observed observed
score isscore95.528, is
95.528, which is the peak among the nine models illustrated
which is the peak among the nine models illustrated in Figure 18a–i. Thus, when ig1 is at itsin Figure 18a–i. Thus, when
ig1 is at its maximum,
maximum, the careful
the careful design of k1 design
and k2 is ofessential.
k1 and k2 As is essential.
indicatedAs by indicated
the trend in bythethemodel
trend
in the model in Figure 18c, selecting larger values for k and
in Figure 18c, selecting larger values for k1 and k2 can yield a transmission plan with higher
1 k 2 can yield a transmission
plan with
scores. higherwhen
However, scores. ig1However, when ig
= 3.43, as shown = 3.43, 18b,
in1 Figure as shown
the model in Figure 18b, thegreater
demonstrates model
demonstrates
stability compared greater stability compared
to scenarios with larger to ig
scenarios
1 values. with
In larger
this ig
situation,1 values.
k 1 andIn this
k 2 situa-
have a
tion, k1 and
broader design k2 have
range, a broader design range,
and the highest and the highest
score achieved is 92.6383, score
justachieved
three points is 92.6383,
lower than just
three
the pointsscore
highest lowerfor than
thethe highest score
maximum for the maximum
ig1 , showing only a slight ig1,difference.
showing only a slightthis
Therefore, dif-
ference.
model is Therefore,
recommended this asmodel is recommended
a reference. When ig1 =as1.98, a reference.
the max-min When ig1 =is1.98,
value onlythe max-
12.6674,
minsmallest
the value is amongonly 12.6674,
the nine themodels
smallest inamong
Figure the nine
18a–i, models inthe
suggesting Figure
model 18a–i, suggesting
is most stable
under
the modelthis is condition.
most stable However,
under this thecondition.
highest score is onlythe
However, 84.8852,
highestwhich score is onlysignificantly
84.8852,
lower
whichthan the highest lower
is significantly scoresthan
observed when ig
the highest 1 takes
scores on medium
observed when orig high values.
1 takes on medium
When
or high values.k1 and k2 are treated as fixed values, the comparisons between Figure 18d,g,
Figure When k1and
18e,h, andFigure
k2 are18f,i,
treatedalong
as with
fixed the overall
values, themax–min
comparisons analysis of k1 and
between Figurek2 , reveal
18d,g,
that the influence
Figure 18e,h, and Figure of k on the model and the trend in the changes in
1 18f,i, along with the overall max–min analysis of k1 and k2, reveal score are similar to
those of kinfluence
that the 2 . The highestof k1 onscore
theamong
model the andtwothe groups,
trend incomprising
the changesa in total
scoreof six
aremodels,
similar is to
observed
those of kwhen2. The khighest
1 = 1.77.score
Therefore,
amongthe analysis
the primarily
two groups, focuses on
comprising casesofwhere
a total k1 takes
six models, is
on large, medium,
observed when k1 =and small
1.77. values.the analysis primarily focuses on cases where k1 takes
Therefore,
When
on large, the fixed
medium, andvalue k1 = 1.1 and k1 = 1.44, the highest scores for the model
smallofvalues.
are 85.6201
When the fixed value of k1 = 1.1 both
and 81.3867, respectively, and khigher
1 = 1.44,thanthethe max–min
highest scores value
for the ig1 = 3.43.
for model are
However,
85.6201 and 81.3867, respectively, both higher than the max–min value for ig1 = 3.43. is
there is a significant difference between the highest scores, so this case not
How-
considered for the time being. With k1 = 1.77, the highest score reaches 94.2959. Although
 Actuators 2024, 13, x FOR PEER REVIEW 24 o

Actuators 2024, 13, 432 22 of 25

ever, there is a significant difference between the highest scores, so this case is not con
ered for the time being. With k1 = 1.77, the highest score reaches 94.2959. Although
max–min remains relatively
the max–min remains relatively large,
large, it has it has significantly
significantly decreased to
decreased compared compared
the case to the c
where ig
where ig1 = 5. This substantially enhances the stability of the model and broadensbroadens
1 = 5. This substantially enhances the stability of the model and the the
design rangesign
for range
ig1 and fork2igin
1 and
the khigh-score
2 in the high-score
segment.segment. Additionally,
Additionally, the highest
the highest score isscore is o
1.2321than
only 1.2321 lower lower than
that thatscenario
in the in the scenario
where igwhere
1 = 5. ig 1 = 5. Therefore,
Therefore, for both for both
models, models,
it is it is
recommended ommended
to use thetomodel
use the model
with fixedwith fixed
value k1 =value k1 =a 1.77
1.77 as as a reference.
reference.
ConsideringConsidering
model stability model stability and comprehensive
and comprehensive performance, the performance, the multiple reg
multiple regression
sion model of the experimental result scores with fixed
model of the experimental result scores with fixed values of ig1 = 3.43 and k1 3.43 values of ig1 = and k1 = 1.7
= 1.77
recommended
is recommended for designingfor designing a transmission
a transmission plan. As plan.
shownAs shown
by theby the score
score trendstrends
in in Fig
Figure 18b,f,18b and 18f,aachieving
achieving higher score a higher scorethat
requires requires that the
the design design
values of kvalues of k1ig
1 , k2 , and , k12, and ig1
increased.
are increased.

4.2. Analysis 4.2.

of the Optimization
Analysis Results of Two-Gear
of the Optimization Results and Three-Gear
of Two-Gear andAMT
Three-Gear AMT
In Section 4.1,
Infavorable results
Section 4.1, were results
favorable achieved forachieved
were the transmission plans of two-gear
for the transmission plans of two-g
and three-gear
andAMT. However,
three-gear AMT. there might still
However, therebemight
betterstill
solutions that
be better have notthat
solutions yethave
beennot yet b
identified within the model. Therefore, a PSO (Particle Swarm Optimization)
identified within the model. Therefore, a PSO (Particle Swarm Optimization)method [27] method
was introduced to optimize the multiple regression models of the experimental
was introduced to optimize the multiple regression models of the experimental scores for scores
the two-gearthe
andtwo-gear
three-gear
and three-gear AMT, which are the main focus of this study. an
AMT, which are the main focus of this study. Subsequently, Subsequen
optimal solution was derived
an optimal solutionusingwasthe enumeration
derived method.
using the Finally, the
enumeration optimal
method. solutions
Finally, the optimal
obtained from both obtained
lutions the PSO and
from the enumeration
both the PSO and method were compared.
the enumeration method were compared.
During the optimization process using PSO, the algorithm
During the optimization process using PSO, the algorithm records eachrecords
improvedeach impro
solution to replace
solutionthe
toprevious second-best
replace the solution. Formula
previous second-best (12)Formula
solution. represents
(12)the velocity the velo
represents
and position update formula of PSO [28]:
and position update formula of PSO [28]:
(t+1) (t) (t+1) (t) (t) (t) (t) (t) (t) (t) (t)(t)
Vid = ω · Vid V id· ( P=idω ⋅ V−id +Xc1id⋅ rand
+ c1 · rand ) ⋅+( P id c−
2 ·X
rand
id
(t+1)
) + ·c2(⋅P
rand ⋅ ( P
gd −gd XX
(t)
−
gdgd)
)
(t+1) (12)
(t+1=) X id + V
+ X
(t+1) (t)
Xid = Xid Vidid id

where V (t)id represents the velocity of particle i when exploring a d-dimensional space a
(t)
where Vid represents the velocity of particle
(t)
i when exploring a d-dimensional space after
(t) while Kid denotes the position of particle i in the d-dimensional sp
the t-th iteration,
the t-th iteration, while Kid denotes the position of particle i in the d-dimensional space
(t) (t) (t) (t)
at the same at the samePiteration.
iteration. id and Pgd Pidcorrespond
and Pgd correspond
to the localtoandthe local
globaland global values,
optimal optimal values
respectively. spectively.
ω is the inertia
ω weight, c
is the inertia
1 and c are
weight,
2 the individual and global learning factors,
c 1 and c 2 are the individual and global learn
and rand is afactors,
randomand number within the range (0,1).
rand is a random number within the range (0,1).
Figure 19 presents
Figure the convergence
19 presents plot of the PSO
the convergence plot optimization results for the
of the PSO optimization ex- for the
results
perimental score multiple regression model. Due to the excessive number of enumeration
perimental score multiple regression model. Due to the excessive number of enumera
schemes, visual observation
schemes, visualthrough figures
observation is challenging.
through figures is Therefore, the optimal
challenging. solutions
Therefore, the optimal s
obtained via the enumeration method are listed in the following table.
tions obtained via the enumeration method are listed in the following table.

78.462

78.46
(11,78.461815)
78.458
Score

78.456

78.454

78.452

78.45

0 20 40 60 80 100
Iteration number
(a) (b)
Figure 19. Optimization
Figure 19. Optimization results’map:
results’ convergence convergence map:AMT;
(a) two-gear (a) two-gear AMT; (b)
(b) three-gear three-gear AMT.
AMT.

As shown in Table 13, the parameter optimization results obtained from the two
optimization methods are largely consistent. The results were rounded to two decimal
places, yielding the highest scores for two-gear and three-gear AMT within their respective
Actuators 2024, 13, 432 23 of 25

value ranges, at 78.46 and 95.68, respectively. These scores represent increases of 2.97%
and 12.96% compared to the highest scores in the fuzzy system evaluation described in
Section 3.

Table 13. Optimal solutions of the enumeration method and PSO.

Type of AMT Two-Gear AMT Three-Gear AMT

Transmission Parameter ig1 k1 ig1 k1 k2
Value (PSO/Enumeration Method) 1.89/1.89 1.428/1.43 4.716/4.72 1.8/1.8 1.8/1.8
Optimal Solution (PSO/Enumeration Method) 78.461815/78.461723 95.676017/95.675989

In the optimization of transmission parameters, the two-gear AMT shows limited

improvements, reflecting a low ceiling for optimization; while the three-gear AMT demon-
strates significant optimization potential, indicating a higher optimization limit. Based
on the comprehensive evaluation index y5 , the highest score for the optimized three-gear
AMT was 21.95% greater than that of the optimized two-gear AMT. Prior to optimization,
as indicated by the fuzzy evaluation scores in Section 3, the highest score for the three-gear
AMT was 11.15% higher than that of the two-gear AMT. These results show a substantial
difference in the comprehensive scores of two-gear and three-gear AMT both before and
after optimization. Furthermore, the difference increases post-optimization, suggesting that
a higher gear count effectively enhances parameter optimization outcomes. Therefore, the
key factor in improving the overall performance of BEVs is the adjustment of gear quantity.

5. Conclusions
The study designed an experimental plan using the LHS method, enhancing the ran-
domness, uniformity, and rationality of the plan while reducing the number of experiments.
After conducting simulation experiments, scores for two-gear and three-gear AMT were
obtained through a fuzzy evaluation system. Based on these scores, multiple regression
models were developed, and the optimal transmission plans and scores for two-gear and
three-gear AMT were determined using the PSO and enumeration methods. The following
conclusions were drawn:
(1) The score results from the fuzzy evaluation system were discussed, and design
plans for gear quantity were proposed. To achieve a high performance, the three-gear option
is preferable; for cost reduction while maintaining performance, the two-gear option is more
suitable. Additionally, the fuzzy evaluation system integrates different evaluation factors
within a unified framework, ensuring comprehensive and rational evaluation results.
(2) Multiple regression models for two-gear and three-gear AMT were analyzed, and
high-score recommended design plans for transmission parameters were provided. For the
two-gear AMT, a design with a small ig1 and a high gear–gear ratio is recommended; for
the three-gear AMT, a design with a large ig1 and a high gear–gear ratio is preferred.
(3) The transmission parameters of two-gear and three-gear AMT were optimized
using the PSO and enumeration methods. The optimization results showed improvements
of 2.97% and 12.96%, respectively. Additionally, the analysis of the highest scores ob-
tained compared to pre-optimization scores revealed that the key factor for enhancing BEV
performance is adjusting the gear quantity. Both optimization methods yielded nearly
identical results; however, Particle Swarm Optimization significantly reduced the number
of calculations, thereby enhancing research efficiency.

Author Contributions: Methodology, H.X., Z.C., and M.Y.; software, M.Y., H.X., and Z.C.; validation,
H.X. and M.Y.; investigation, H.X. and Z.C.; resources, H.X. and Z.C.; writing—original draft prepara-
tion, M.Y. and Z.C.; writing—review and editing, H.X. and Z.C.; supervision, H.X., Z.C., and X.S.;
and project administration, H.X., Z.C., and X.S. All authors have read and agreed to the published
version of the manuscript.
Actuators 2024, 13, 432 24 of 25

Funding: This research was funded by the Jiangsu Province Vocational College Young Teacher
Enterprise Practice Training Project (grant number: 2023QYSJ058), and the 2024 Innovation Training
Program for College Students at the School of Transportation Engineering/Nanjing Vocational
University of Industry Technology.
Data Availability Statement: The data presented in this study are available on demand from the
corresponding author or first author at (cz38@njfu.edu.cn or xuh1@niit.edu.cn).
Acknowledgments: The authors thank the Jiangsu Province Vocational College Young Teacher
Enterprise Practice Training Project (grant number: 2023QYSJ058), and the 2024 Innovation Training
Program for College Students at the School of Transportation Engineering/Nanjing Vocational
University of Industry Technology for funding. We also thank the anonymous reviewers for providing
critical comments and suggestions that improved the manuscript.
Conflicts of Interest: The authors declare no conflicts of interest.

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