Abstract
The electric vehicle has become one of the most promising technical pathways for decarbonising road transport. Although the electric vehicle produces zero tailpipe emissions, several issues relating to driving range, energy efficiency, battery cost and control complexity still restrict the large-scale deployment of battery electric vehicles. This study focuses on an electric vehicle equipped with a two-speed wet dual-clutch transmission, with the aim of improving both powertrain parameter matching and regenerative braking control through a combination of vehicle dynamic modelling, multi-objective optimisation and intelligent fuzzy control. The research begins by deriving the required motor power, motor torque, battery capacity and transmission gear ratios from the longitudinal vehicle dynamics equation. A high-fidelity electric vehicle simulation model is then constructed in MATLAB, including a second-order RC battery model, an inverter loss model, a design-oriented permanent magnet synchronous motor model, a multi-loss dual-clutch transmission model and a complete longitudinal vehicle model. The electric vehicle powertrain parameters are optimised with the non-dominated sorting genetic algorithm II, in which the energy consumption of the electric vehicle over the WLTP driving cycle and the total powertrain cost are considered as two conflicting objectives. For the regenerative braking control problem, a fuzzy logic controller is developed to distribute the front-axle braking torque between the electric motor and the hydraulic friction brake. The fuzzy controller receives the required braking intensity, vehicle speed and battery state of charge, and it generates an electric braking distribution coefficient. To overcome the dependence on expert knowledge in the design of the membership functions, the genetic algorithm is applied to tune the fuzzy membership parameters. The optimised electric vehicle control strategy is evaluated under both the NEDC and WLTP cycles. The simulation results demonstrate that the optimised fuzzy regenerative braking control strategy noticeably reduces the battery state-of-charge drop and increases the recovered braking energy. After the optimisation, the two-speed DCT electric vehicle achieves an energy consumption of about 12.50 kWh per 100 km under the WLTP cycle, while maintaining satisfactory acceleration and gradeability. This work provides a practical framework for electric vehicle powertrain design and energy management.
Keywords: electric vehicle; two-speed wet dual-clutch transmission; genetic algorithm; fuzzy control; regenerative braking; powertrain optimisation; brake energy recovery
1. Introduction
In recent decades, the global automotive industry has faced increasingly severe pressure from climate change, urban air pollution and the depletion of fossil fuel resources. Electric vehicles have received worldwide attention as an important alternative to conventional internal combustion engine vehicles. The market penetration of the electric vehicle has grown rapidly, especially in China, where annual sales of new energy vehicles exceeded 12.8 million units in 2024. According to many transportation roadmaps, the electric vehicle will continue to expand its market share, and the share of electric vehicles in total vehicle sales is expected to reach around 40% in the mid-2020s.
Even though modern traction batteries have improved considerably, the energy density and cost of the battery are still fundamental bottlenecks for the electric vehicle. Unlike conventional vehicles, the electric vehicle can recover part of its kinetic energy during deceleration by operating the traction motor as a generator. Regenerative braking therefore offers an effective means of extending the driving range without increasing the battery capacity. Nevertheless, the braking energy recovery of an electric vehicle is not straightforward. During braking, the demanded braking torque must be shared between the motor and the friction braking system, while maintaining directional stability and satisfying legal braking requirements. In city driving, the average braking intensity of an electric vehicle is often below 0.2 g, and a large portion of the total energy supplied by the battery may be dissipated during repeated acceleration and deceleration events. A well-tuned regenerative braking control strategy can make a substantial contribution to the energy economy of the electric vehicle.
Another important issue for the electric vehicle is the design of the powertrain. Many current electric vehicles use a fixed single-speed reduction gear. This layout is simple and inexpensive, but a single gear ratio often forces the drive motor to operate away from its optimal efficiency region because the motor speed is rigidly linked to the vehicle speed. Multi-speed transmissions, such as a two-speed dual-clutch transmission, can improve motor operating points during normal driving and therefore reduce energy consumption. The dual-clutch transmission also avoids torque interruption during gear shifting, which improves both driving comfort and longitudinal controllability. However, the addition of a multi-speed gearbox introduces the need to carefully match the gear ratios with the motor characteristics and with the battery voltage level.
In the literature, several studies have attempted to optimise either the electric vehicle powertrain components or the regenerative braking controller. Some research groups have used multi-objective genetic algorithms to determine the optimum motor power and gear ratio of an electric vehicle. Others have designed fuzzy-logic controllers to determine the share of regenerative braking. Nevertheless, most earlier investigations either neglected the variable efficiency of the motor, inverter and transmission, or simplified the battery as a fixed voltage source. Such simplifications can produce misleading results because the efficiency maps of the electric drive system vary strongly with torque and speed. Moreover, the regenerative braking controller is often designed manually, based on experience rather than on systematic optimisation.
In this study, I carry out a comprehensive optimisation of an electric vehicle powertrain that incorporates a two-speed wet dual-clutch transmission. The work is divided into two main parts. In the first part, I complete the preliminary parameter matching of the drive motor, the battery and the two gear ratios, followed by the construction of a detailed electric vehicle simulation platform. The simulation platform is composed of a second-order RC battery model, an inverter loss model, an electromagnetic motor design model, a dual-clutch transmission loss model and a longitudinal vehicle dynamic model. In the second part, I formulate a multi-objective optimisation problem for the electric vehicle powertrain using energy consumption and cost as the objective functions. A genetic algorithm is run on the simulation platform to obtain a Pareto frontier. In the braking control part, a regenerative braking fuzzy controller is designed based on the ideal braking force distribution, the ECE R13 legal boundary and the constraint of the front-wheel-drive architecture. The genetic algorithm is then adopted again to optimise the membership functions of this fuzzy controller. The resulting electric vehicle control strategy is verified under both the NEDC and WLTP driving cycles.
2. Powertrain Architecture and Component Parameter Matching
2.1 Vehicle and Powertrain Configuration
The studied electric vehicle is a front-wheel-drive passenger car equipped with a two-speed wet dual-clutch transmission. The powertrain consists of a permanent magnet synchronous motor, a dual-clutch module, a two-speed gear set, a final drive, an inverter and a high-voltage lithium-ion battery pack. The motor is connected to the two-speed DCT, and the DCT uses two concentrically arranged shafts to transmit torque to the differential. The structure enables gear shifts without interrupting the driving torque. A controller chooses the appropriate gear according to the vehicle speed and driver demand.
The main vehicle parameters are given in the following table. The table also contains the target performance requirements, which are used later as constraints in the matching and optimisation process.
| Parameter | Symbol | Value | Unit |
| Vehicle kerb mass | \(m_v\) | 1455 | kg |
| Frontal area | \(A\) | 2.593 | m² |
| Tyre dynamic radius | \(R_D\) | 0.273 | m |
| Aerodynamic drag coefficient | \(C_D\) | 0.456 | – |
| Rolling resistance coefficient | \(f\) | 0.012 | – |
| Front and rear static axle load ratio | – | 0.50/0.50 | – |
| Wheelbase | \(L\) | 2490 | mm |
| Centre of gravity height | \(h_g\) | 510 | mm |
| Final drive ratio | \(i_0\) | 4.058 | – |
| Maximum speed target | – | 150 | km/h |
| Gradeability target 1 | – | 30% at 5 km/h | – |
| Gradeability target 2 | – | 20% at 30 km/h | – |
| 0–100 km/h acceleration target | – | 10–12 | s |
| Constant-speed driving range target | – | 150 | km |
2.2 Drive Motor Matching
The longitudinal motion of an electric vehicle is governed by the equilibrium between the tractive force and the total resistance. The total resistance consists of rolling resistance, aerodynamic drag, gradient resistance and acceleration resistance. The vehicle dynamic equation can be written as:
$$
F_t=F_f+F_w+F_i+F_j
$$
Expanding the four resistance components gives the standard vehicle driving equation:
$$
\frac{T_m i_g i_0 \eta_T}{R_D}=m_v g f\cos\alpha+\frac{C_D A}{21.15}u^2+m_v g\sin\alpha+\delta m_v\frac{\mathrm{d}u}{\mathrm{d}t}
$$
where \(T_m\) is the motor torque, \(i_g\) is the selected gear ratio, \(i_0\) is the final drive ratio, \(\eta_T\) is the transmission efficiency, \(u\) is the vehicle speed in km/h, \(\alpha\) is the road gradient, and \(\delta\) is the rotational mass conversion factor, which is taken as 1.07.
The required motor peak power must simultaneously satisfy the maximum-speed condition, the maximum-grade condition and the acceleration condition. The corresponding power requirements can be expressed as follows:
$$
P_{u_{\max}}=\frac{u_{\max}}{3600\eta_T}\left(m_v g f+\frac{C_D A u_{\max}^2}{21.15}\right)
$$
$$
P_{\alpha_{\max}}=\frac{u_\alpha}{3600\eta_T}\left(m_v g f\cos\alpha_{\max}+m_v g\sin\alpha_{\max}+\frac{C_D A u_\alpha^2}{21.15}\right)
$$
$$
P_{a,\max}=\frac{1}{3600\eta_T}\max_{t}\left[u(t)\left(m_v g f+\frac{C_D A u(t)^2}{21.15}\right)+\delta m_v a_v(t)\left(\frac{u(t)}{3.6}\right)\right]
$$
In these formulas, \(u_\alpha\) is the vehicle speed during the climbing test, \(a_v\) is the longitudinal acceleration in m/s², and the factor 3600 arises from the conversion between kW and N·m/s when the vehicle speed is expressed in km/h. According to the vehicle data and the target performance indices, I calculated the required power values. The peak motor power was then taken as 15% larger than the largest calculated value to provide a power reserve. The rated motor power was set to approximately half of the peak power so that the motor can be operated with an overload factor of about two during short-duration high-power demands.
The maximum motor speed is determined by the maximum vehicle speed and the overall driveline ratio. For the maximum-speed condition, the required motor speed is:
$$
n_{\max}\ge \frac{u_{\max} i_g i_0}{0.377 R_D}
$$
In this study, the ratio between the peak speed and the rated speed is selected as two. The motor peak torque and rated torque are then obtained from the power-speed relationship:
$$
T_{\max}=9550\frac{P_{\max}}{n_{\max}}, \qquad T_{\mathrm{rated}}=9550\frac{P_{\mathrm{rated}}}{n_{\mathrm{rated}}}
$$
The initial parameter matching of the electric vehicle drive motor is summarised as follows: peak power 100 kW, rated power 50 kW, peak speed 8000 rpm, rated speed 4000 rpm, peak torque 238 N·m and rated torque 120 N·m. The drive motor selected in this work is an interior permanent magnet synchronous motor because of its high power density, wide field-weakening range and excellent efficiency.
2.3 Battery Matching
The traction battery of an electric vehicle is one of the most important and expensive components. In this research, the battery is a lithium nickel manganese cobalt oxide pack composed of 21700 cylindrical cells. The nominal terminal voltage of each cell is 3.7 V, the nominal capacity is 5 Ah, and the internal resistance is 1.5 mΩ.
To satisfy the target range under the constant-speed test method, the total battery capacity must be large enough to supply the required power over the entire range. The available capacity can be estimated from the vehicle energy balance. In the preliminary matching stage, the battery capacity is calculated using the 60 km/h constant-speed condition. The electric power consumed by the vehicle depends on the rolling resistance and aerodynamic drag. The total energy used by the battery can be written as:
$$
C_{\mathrm{bat}} \ge \frac{S}{3600 U_{\mathrm{bat}} \eta_m \eta_T (1-\eta_b) F_j}\left(m_v g f+\frac{C_D A}{21.15}u^2\right)
$$
where \(S\) is the target driving range, \(U_{\mathrm{bat}}\) is the battery terminal voltage, \(\eta_m\) is the motor efficiency, \(\eta_T\) is the transmission efficiency, \(\eta_b\) is the auxiliary-system energy consumption factor, and \(F_j\) is the actual-to-rated discharge ratio of the battery, which typically lies between 0.8 and 0.95.
The battery must also be capable of supplying the peak motor power during acceleration and overtaking. The corresponding minimum capacity is:
$$
C_{\mathrm{discharge}}\ge \frac{1}{k U_{\mathrm{bat}}}\left(\frac{P_{\max}}{\eta_m}+P_{\mathrm{aux}}\right)
$$
where \(k\) is the maximum allowable discharge C-rate and \(P_{\mathrm{aux}}\) is the power consumed by vehicle accessories. Taking both the range and the power requirement into account, the total capacity of the battery pack in the present electric vehicle is determined as 115 Ah. The rated voltage is chosen as 320 V according to the voltage specification of the motor and the national standard for electric vehicle high-voltage systems.
The number of cells connected in series is determined by the battery pack voltage, while the number of cells in parallel is determined by the total capacity. Since the nominal cell voltage is 3.7 V, the number of series-connected cells is:
$$
N_s=\frac{U_{\mathrm{bat}}}{U_{\mathrm{cell}}}=\frac{320}{3.7}\approx 87
$$
The number of parallel branches is obtained from the capacity ratio:
$$
N_p=\frac{C_{\mathrm{bat}}}{C_{\mathrm{cell}}}=\frac{115}{5}=23
$$
The total number of cells in the battery pack is therefore 87 × 23 = 2001 cells.
2.4 Two-Speed DCT Gear Ratio Matching
The gear ratios of the two-speed transmission must be selected so that the electric vehicle can achieve the required acceleration and climbing performance in first gear and the required maximum speed in second gear. The maximum first gear ratio is constrained by the peak motor torque and by the maximum force that can be transmitted before wheel slip occurs. The first gear must provide enough wheel torque to balance the vehicle resistance during the acceleration test and during the climbing test.
For the first gear, the lower bound of the ratio can be expressed by the following torque balance relation:
$$
i_1\ge \frac{R_D\left[\left(m_v g f\cos\alpha+m_v g\sin\alpha\right)+\dfrac{C_D A}{21.15}u^2+\delta m_v\dfrac{\mathrm{d}u}{\mathrm{d}t}\right]}{T_{\max}\eta_T i_0}
$$
The wheel-slip constraint requires that the maximum motor torque transmitted to the wheels should not exceed the torque corresponding to the peak adhesion coefficient. Thus, the upper bound of the first gear ratio is:
$$
i_1\le \frac{\mu_{\max} m_v g \frac{L_2}{L}+\mu_{\max} m_v g \frac{h_g}{L}}{T_{\max}\eta_T i_0}R_D
$$
For the second gear, the ratio must be sufficiently small to allow the electric vehicle to reach its maximum design speed. At the same time, the second gear ratio must be large enough to enable the motor to deliver sufficient torque at high speed. Therefore, the second gear ratio is obtained from both the maximum speed condition and the maximum-torque-at-maximum-speed condition.
The initial matching result gives a first gear ratio of 1.8 and a second gear ratio of 1.3. The DSG shift schedule used in the simulation is based on vehicle speed only. The upshift point is set at 50 km/h and the downshift point is set at 42 km/h to avoid excessive gear hunting.
3. System Modelling of the Electric Vehicle
3.1 Battery Equivalent Circuit Model
In this work, the dynamic behaviour of the lithium-ion battery is represented by a second-order RC equivalent circuit. The model consists of an open-circuit voltage source \(U_{OC}\), a series Ohmic resistance \(R_0\), and two parallel RC networks. The two RC branches represent the electrochemical polarisation and concentration polarisation effects, respectively. The circuit model is sufficiently accurate for electric vehicle simulation while preserving a moderate computational cost.
The dynamic equations of the second-order RC battery model are:
$$
U_b=U_{OC}-IR_0-U_1-U_2
$$
$$
\frac{\mathrm{d}U_1}{\mathrm{d}t}=-\frac{U_1}{R_1C_1}+\frac{I}{C_1}
$$
$$
\frac{\mathrm{d}U_2}{\mathrm{d}t}=-\frac{U_2}{R_2C_2}+\frac{I}{C_2}
$$
The battery state of charge is obtained by Coulomb counting:
$$
\mathrm{SOC}(t)=\mathrm{SOC}(0)-\frac{1}{3600 C_{\mathrm{cell}}}\int_0^t I(\tau)\,\mathrm{d}\tau
$$
The parameters of the equivalent circuit are identified from open-circuit voltage tests and hybrid pulse power characterisation tests. In the HPPC test, the battery is subjected to a sequence of discharge current pulses, and the terminal voltage response is recorded. The Ohmic resistance is calculated from the instantaneous voltage jump at the beginning and end of the current pulse:
$$
R_0=\frac{(U_B-U_C)+(U_E-U_D)}{2I}
$$
After the current is interrupted, the voltage recovery curve is fitted by an exponential function. The fitted curve is used to extract the polarisation resistances and capacitances. The same identification procedure is repeated at multiple SOC values, so that the battery model can describe the parameter dependence on SOC. The identified parameters are shown in the following set of plots. The maximum error between the simulated battery voltage and the experimental voltage is 0.31 V, while the average error is only 6.7 mV, which confirms the good fidelity of the battery model.
3.2 Inverter Model
The inverter is treated as a voltage-source DC-AC converter built with insulated-gate bipolar transistors and freewheeling diodes. The motor-side inverter converts the DC power from the battery into three-phase AC power for the permanent magnet motor. During this process, power is dissipated in the switching devices and the freewheeling diodes. The main loss components are the IGBT conduction loss, the IGBT switching loss and the diode reverse-recovery loss.
The instantaneous inverter loss can be represented as:
$$
P_{\mathrm{inv}}=P_{\mathrm{cond}}+P_{\mathrm{sw}}+P_{\mathrm{rr}}
$$
where \(P_{\mathrm{cond}}\) is the conduction loss, \(P_{\mathrm{sw}}\) is the switching loss and \(P_{\mathrm{rr}}\) is the diode recovery loss. The inverter efficiency is defined as:
$$
\eta_{\mathrm{inv}}=\frac{P_{\mathrm{motor}}}{P_{\mathrm{motor}}+P_{\mathrm{inv}}}
$$
The inverter parameters used in this study correspond to a typical automotive IGBT module with a switching frequency of 16 kHz. By including the inverter loss model, the electric vehicle simulation can reflect the fact that the motor efficiency map is not the only energy conversion loss in the electric powertrain.
3.3 Motor Model and Efficiency Map
The drive motor is modelled using a design-oriented electromagnetic procedure. The geometry of the permanent magnet synchronous motor is estimated from the input rated power, rated speed, rated voltage and several predefined geometric parameters. The motor equivalent circuit parameters, including the d-axis inductance \(L_d\), q-axis inductance \(L_q\), armature resistance \(R_a\) and iron-loss equivalent resistance \(R_c\), are then computed. The torque equation of the interior permanent magnet motor is:
$$
T_m=p\left[\psi_f i_q+\left(L_d-L_q\right)i_d i_q\right]
$$
where \(p\) is the number of pole pairs, \(\psi_f\) is the permanent magnet flux linkage, and \(i_d\) and \(i_q\) are the d-axis and q-axis currents, respectively.
The motor efficiency is calculated by separating the total losses into copper loss, iron loss and mechanical loss. The motor output power is the product of the torque and the angular speed:
$$
\eta_m=\frac{T_m \omega_m}{T_m \omega_m+P_{\mathrm{Cu}}+P_{\mathrm{Fe}}+P_{\mathrm{mec}}}
$$
The copper loss is computed from the d-q axis currents:
$$
P_{\mathrm{Cu}}=R_a\left(i_{od}^2+i_{oq}^2\right)
$$
The iron loss is computed from the magnetising or core-loss currents:
$$
P_{\mathrm{Fe}}=R_c\left(i_{cd}^2+i_{cq}^2\right)
$$
From the equivalent circuit model, the full-load torque-speed characteristic and the motor efficiency map are generated over the complete operating region. The motor efficiency map is then used as a look-up table in the electric vehicle simulation. Because the motor design tool is based on electromagnetic principles, it is possible to change the motor design parameters during the optimisation process without relying on experimental data. This is important for the later multi-objective optimisation of the electric vehicle.
3.4 Dual-Clutch Transmission Loss Model
The transmission model of the two-speed wet DCT is not treated as a constant-efficiency element. Instead, the instantaneous efficiency is calculated by subtracting all relevant power losses from the input power. The main losses inside the DCT are caused by the open wet clutch, gear meshing, bearings, shaft oil shear and gear windage or oil churning.
The total transmission loss torque can be summarised as:
$$
T_{\mathrm{loss}}=T_{\mathrm{clutch}}+T_{\mathrm{mesh}}+T_{\mathrm{bearing}}+T_{\mathrm{shaft}}+T_{\mathrm{windage}}
$$
and the transmission efficiency is:
$$
\eta_T=1-\frac{T_{\mathrm{loss}}}{T_{\mathrm{in}}}
$$
The open clutch drag torque is a complicated function of the oil film distribution between the friction plates. At low relative speeds, the oil film completely fills the clutch clearance and the drag torque is relatively high. At high speeds, the centrifugal effect contracts the oil film radius, which reduces the effective radius and changes the drag torque. The clutch drag torque can be expressed by the following integral relation:
$$
T_{\mathrm{clutch}}=\frac{N_c \mu_c \omega}{2 h_c}\int_{R_i}^{R_o}r^3\,\mathrm{d}r
$$
where \(N_c\) is the number of friction surfaces, \(\mu_c\) is the oil dynamic viscosity, \(h_c\) is the oil film thickness, and \(R_o\) and \(R_i\) are the outer and inner radii of the friction plate. The effective outer radius is modified according to the oil supply flow rate and the rotational speed.
Gear meshing losses are determined from the gear sliding and rolling friction. The tooth friction coefficient is calculated using a semi-empirical formula that depends on the lubricant viscosity, the normal load and the pitch-line velocity. Bearing losses are calculated according to the national standard method by separating the load-dependent friction torque and the lubricant viscous friction torque. Oil seal losses are represented as a linear function of the shaft diameter.
Because the DCT loss calculation is based on the instantaneous input torque and input speed, the transmission efficiency varies over the entire driving range. The figure below shows the efficiency characteristics of the first and second gear ratios over the complete torque-speed domain. This variable-efficiency DCT model is essential for the fair optimisation of the electric vehicle because a constant transmission efficiency may incorrectly weight the candidate gear ratios.
3.5 Longitudinal Vehicle Model
In the longitudinal vehicle model, the electric vehicle is treated as a rigid body moving along a straight level road or an inclined road. The required wheel torque is determined by the longitudinal dynamics equation presented in Section 2. The dynamic performance indices such as the 0–100 km/h acceleration time, the maximum speed and the maximum grade angle are calculated using the full-load torque curve of the motor and the actual gear shifting logic. The energy consumption of the electric vehicle over a complete driving cycle is then calculated by integrating the instantaneous power drawn from the battery.
The energy consumption per 100 km under the designated driving cycle is one of the main evaluation indicators. In this study, the WLTP driving cycle is employed as the standard test cycle. The formula for the battery energy consumed over the cycle is:
$$
E_{\mathrm{EC}}=\frac{100}{L_c}\int_0^{t_f} \frac{P_{\mathrm{batt}}(t)}{\eta_{\mathrm{inv}}(t)\eta_m(t)\eta_T(t)}\,\mathrm{d}t
$$
where \(L_c\) is the total cycle distance, \(P_{\mathrm{batt}}\) is the battery output power, and the three efficiency terms represent the instantaneous inverter, motor and transmission efficiencies. In practice, the simulation model obtains the battery power by considering the power balance between the wheel demand and the motor output, and then applying the loss models of the inverter and the battery.
4. Multi-Objective Optimisation of the Electric Vehicle Powertrain
4.1 Problem Formulation
The design of the electric vehicle powertrain involves several conflicting objective functions. For example, a large motor power and a large battery capacity may improve driving performance and range, but they increase the purchase cost and the vehicle mass. A two-speed transmission can improve energy economy, but it adds additional mechanical loss and cost. I therefore formulate the powertrain parameter selection as a multi-objective optimisation problem.
A general multi-objective optimisation problem can be written as:
$$
\min_{x\in S} F(x)=\left[f_1(x),f_2(x),\ldots,f_n(x)\right]^T
$$
subject to equality and inequality constraints. Because no single solution can generally minimise all objectives simultaneously, the solution of a multi-objective optimisation problem is represented by a set of Pareto-optimal solutions. A solution is Pareto-optimal if no other solution can improve one objective without worsening at least one other objective.
4.2 Optimisation Variables
Five main design parameters are selected as the optimisation variables. These variables determine the properties of the electric motor and the two-speed DCT. The optimisation vector is:
$$
X=\left[P_{\mathrm{em}},n_{\mathrm{em}},U_{\mathrm{em}},i_1,i_2\right]
$$
where \(P_{\mathrm{em}}\) is the rated motor power, \(n_{\mathrm{em}}\) is the rated motor speed, \(U_{\mathrm{em}}\) is the rated voltage, and \(i_1\) and \(i_2\) are the first and second gear ratios of the DCT. A change in the rated voltage directly influences the number of battery cells connected in series, while the gear ratios determine how the motor operating point is mapped to the vehicle speed.
4.3 Objective Functions
Two objective functions are used in the optimisation. The first objective is the energy consumption of the electric vehicle over the WLTP cycle. This objective is calculated by integrating the instantaneous battery power throughout the cycle and converting the result to the energy consumption per 100 km. The second objective is the total cost of the electric vehicle powertrain. The cost model includes the cost of the electric motor, the battery pack, the dual-clutch transmission, the battery management system and the additional cooling hardware.
The objective functions are described as follows:
$$
f_1(X)=E_{\mathrm{WLTP}}(X)
$$
$$
f_2(X)=C_{\mathrm{motor}}(X)+C_{\mathrm{battery}}(X)+C_{\mathrm{DCT}}
$$
In the cost model, the motor cost is represented as an increasing linear function of the rated power, and the battery cost is calculated from the total number of cells. The DCT cost is fixed because the same transmission hardware architecture is assumed for every candidate design.
4.4 Constraints
The candidate powertrain parameters must satisfy the design constraints of the electric vehicle. The constraints include:
1. The maximum speed must be higher than 150 km/h;
2. The maximum grade angle must be greater than 30% at a low vehicle speed and greater than 20% at 30 km/h;
3. The 0–100 km/h acceleration time must satisfy either 12 s or 10 s, depending on the selected performance target;
4. The wheel torque must not exceed the adhesion limit on a high-friction road, otherwise wheel slip may occur.
These constraints are checked for every newly generated individual in the optimisation algorithm. If a candidate point violates any of the constraints, it is penalised in the selection process so that the optimisation front is pushed towards feasible designs.
4.5 Multi-Objective Genetic Algorithm
The optimisation is performed with the non-dominated sorting genetic algorithm II. This algorithm is widely used for engineering multi-objective optimisation because of its ability to maintain a well-spread Pareto front. The population size in this study is 200 and the maximum number of generations is 40. The crossover probability is set to 0.8 and the mutation probability is 0.2.
The optimisation procedure can be summarised as follows. First, an initial population is generated randomly within the predefined bounds of each variable. Every individual in the population is evaluated on the electric vehicle simulation platform, and the two objective functions are computed. The algorithm then performs fast non-dominated sorting, ranks the individuals according to their dominance level, and calculates the crowding distance to maintain diversity. After the non-dominated sorting step, genetic operators such as tournament selection, simulated binary crossover and polynomial mutation are used to create an offspring population. The parent and offspring populations are combined, and the best-ranked individuals are selected for the next generation. This process is repeated until the maximum generation number is reached.
4.6 Pareto Optimisation Results
The simulations are performed for two acceleration constraints, namely 12 s and 10 s, and for two different assumptions on the transmission efficiency model. The variable-efficiency model calculates the instantaneous DCT efficiency based on the loss model described in Section 3. The constant-efficiency model simply fixes the transmission efficiency at 0.96 for both gear ratios. Comparing the two approaches allows the importance of detailed transmission loss modelling in the electric vehicle optimisation process to be evaluated.
The Pareto frontiers for the two cases reveal a clear trade-off between the total powertrain cost and the WLTP energy consumption. In the left part of the Pareto front, increasing the cost by a small amount significantly reduces the energy consumption. In the right part of the Pareto front, adding more cost provides only a marginal energy benefit. Therefore, the knee point at the boundary between these two regions represents a balanced electric vehicle powertrain design. For all the simulations, the electric vehicle with the variable-efficiency transmission model obtains a lower energy consumption than the electric vehicle optimised with the constant-efficiency model. This result confirms that the transmission is not simply a fixed loss element; a lower gear ratio can reduce the DCT churning and bearing losses and therefore improve the total energy economy.
The selected optimal parameters for each case are summarised in the following table.
| Transmission efficiency model | Acceleration target | Energy consumption (kWh/100 km) | Powertrain cost (CNY) | Rated motor power (kW) | Rated motor speed (rpm) | Voltage (V) | First gear ratio | Second gear ratio |
| Constant | 12 s | 12.96 | 29040 | 40 | 6000 | 300 | 3.44 | 1.72 |
| Variable | 12 s | 12.48 | 29048 | 41 | 6000 | 300 | 1.75 | 1.17 |
| Constant | 10 s | 12.89 | 29149 | 51 | 6000 | 300 | 3.53 | 1.72 |
| Variable | 10 s | 12.48 | 29121 | 48 | 6000 | 300 | 1.75 | 1.17 |
Several important observations can be made from the optimisation results. First, the rated motor speed obtains the upper bound of its allowable range in all optimum solutions, which suggests that a higher rated speed can shift the motor efficiency region towards lower torque and higher speed. Second, the optimal voltage remains at the lower end of the allowed range, which indicates that reducing the number of series battery cells and the battery mass is economically beneficial under the present battery cost assumptions. Third, the optimised gear ratios under the variable-efficiency DCT model are considerably lower than those produced by the constant-efficiency model. This confirms that the gear ratios not only adjust the motor operating point, but also affect the mechanical efficiency of the transmission itself.
5. Regenerative Braking Control Strategy
5.1 Braking Force Distribution Fundamentals
For a front-wheel-drive electric vehicle, the total braking force is applied to both the front and rear axles. The distribution of the braking force between the two axles determines the longitudinal stability and the utilisation of the road adhesion coefficient. In an ideal braking process, the front and rear wheels would lock simultaneously. The locus of the front and rear braking forces that produces simultaneous wheel lock on a road with a given adhesion coefficient is known as the I-curve.
For the studied electric vehicle, the ideal braking force relationship can be written as:
$$
F_{\mu f}=m_v g\varphi \frac{L_2+h_g\varphi}{L}
$$
$$
F_{\mu r}=m_v g\varphi \frac{L_1-h_g\varphi}{L}
$$
where \(\varphi\) is the road adhesion coefficient, \(L_1\) and \(L_2\) are the horizontal distances from the centre of gravity to the front and rear axles, respectively, and \(h_g\) is the centre-of-gravity height. For a given vehicle deceleration \(z\), the complete front and rear braking forces satisfy:
$$
F_{bf}+F_{br}=m_v g z
$$
If the front wheels lock before the rear wheels, the vehicle loses steering ability but remains directionally stable. If the rear wheels lock first, the vehicle may spin. Therefore, the actual braking force distribution must always remain below the I-curve at high braking intensities and as close to it as possible.
The United Nations Economic Commission for Europe R13 braking regulation also provides a boundary for the braking force distribution of passenger cars. For adhesion coefficients between 0.2 and 0.8, the braking intensity must satisfy:
$$
z\ge 0.1+0.85(\varphi-0.2)
$$
and for braking intensities between 0.3 and 0.4, the front axle adhesion utilisation must be lower than the rear axle adhesion utilisation. Combining these conditions with the vehicle geometry gives the ECE R13 regulation boundary curve, usually denoted as the M-curve. In the \(F_{bf}\)-\(F_{br}\) plane, the area between the I-curve, the M-curve and the axle load line defines the feasible region for a stable braking force distribution.
5.2 Piecewise Distribution Strategy
For the regenerative braking control design, I divide the vehicle braking process into several regions according to the required braking intensity \(z\). When the braking intensity is low, the vehicle can be decelerated entirely by the front axle without violating the ECE regulation. Since the electric vehicle is front-wheel driven, this condition is very favourable for energy recovery. Therefore, at braking intensities lower than 0.22, all of the required braking force is delivered by the front axle and the distribution curve follows the segment AB in the braking force diagram.
As the braking intensity increases, the rear axle must supply part of the total braking force to maintain stability. In the region where \(0.22 \le z \le 0.53\), the braking distribution follows the ECE regulation line, which is often denoted as the BC segment. This line guarantees that the electric vehicle remains within the legal stability boundary while maximising the front-axle braking force and therefore allowing more regenerative braking energy to be recovered. The front braking force in this region can be obtained from the M-curve equation:
$$
F_{bf}=\frac{m_v g L_2 z + m_v g h_g(z+0.07)}{0.85 L}
$$
When the braking intensity exceeds about 0.53 but remains below the typical dry-road adhesion coefficient of 0.7, the vehicle is in a high-intensity braking condition. In this case, the braking force distribution is chosen along the F-line corresponding to \(\varphi=0.7\). This line represents the braking relationship after the front wheels have locked, and it produces a stable but less energy-efficient braking process. The corresponding distribution is described by:
$$
F_{br}=K_{CD}F_{bf}+B_{CD}
$$
Finally, when the braking intensity is above 0.7, the vehicle is considered to be in an emergency braking condition. To guarantee the safety of the passengers, the regenerative braking system is disabled and the hydraulic friction braking system provides the entire braking torque. The braking force distribution follows the ideal I-curve so that the front and rear wheels lock simultaneously:
$$
F_{bf}=m_v g\varphi\frac{L_2+\varphi h_g}{L}
$$
$$
F_{br}=m_v g\varphi\frac{L_1-\varphi h_g}{L}
$$
This multi-segment braking strategy provides a balance between brake safety and regenerative energy recovery.
5.3 Fuzzy Regenerative Braking Controller
The regenerative braking force on the front axle depends not only on the braking intensity but also on the battery state of charge, the vehicle speed and the maximum feasible motor braking torque. To determine the regenerative braking ratio in real time, I design a fuzzy logic controller. The fuzzy controller has three inputs and one output. The three inputs are:
1. The required braking intensity \(z\);
2. The instantaneous vehicle speed \(u\);
3. The battery state of charge \(\mathrm{SOC}\).
The output of the fuzzy controller is the regenerative braking force distribution coefficient \(K\), which ranges from 0 to 1. If the coefficient is equal to 1, the entire front-axle braking demand is satisfied by the electric motor. If the coefficient is lower than 1, the hydraulic braking system compensates for the remaining front braking torque.
The design of the fuzzy rules is based on the following physical considerations. When the battery state of charge is very high, the regenerative braking power must be limited to avoid overcharging the lithium-ion battery. Therefore, a smaller value of \(K\) is selected at high SOC. When the vehicle speed is extremely low, the motor back-electromotive force becomes too small to generate an effective braking torque. Therefore, the coefficient \(K\) is reduced at very low vehicle speed. When the required braking intensity is high, the vehicle requires a large and quickly responding braking torque, so the regenerative braking ratio \(K\) is reduced and the hydraulic braking system takes a larger share.
In this study, the membership functions of the fuzzy controller use a combination of Gaussian, triangular and trapezoidal functions. The input braking intensity \(z\) covers the normalised domain \([0,1]\). The vehicle speed covers the domain \([0,150]\) km/h, while the battery SOC covers \([0,1]\). The output coefficient \(K\) is defined over the interval \([0,1]\) with five linguistic levels, corresponding to very low, low, medium, high and very high regenerative braking ratios. A Mamdani fuzzy inference engine is selected because it is intuitive to design and easy to tune.
The complete fuzzy rule table consists of 27 rules that are developed from the basic engineering principles summarized above.
| Rule number | Required intensity z | Vehicle speed v | Battery SOC | Output K |
| 1 | Low | Low | High | Low |
| 2 | Low | Medium | High | Low |
| 3 | Low | High | High | Low |
| 4 | Medium | Low | High | Low |
| 5 | Medium | Medium | High | Low |
| 6 | Medium | High | High | Very high |
| 7 | High | Low | High | Medium |
| 8 | High | Medium | High | High |
| 9 | High | High | High | High |
| 10 | Low | Low | Medium | Very low |
| 11 | Low | Medium | Medium | Very low |
| 12 | Low | High | Medium | High |
| 13 | Medium | Low | Medium | Medium |
| 14 | Medium | Medium | Medium | High |
| 15 | Medium | High | Medium | High |
| 16 | High | Low | Medium | Very high |
| 17 | High | Medium | Medium | Medium |
| 18 | High | High | Medium | High |
| 19 | Low | Low | Low | Very low |
| 20 | Low | Medium | Low | Medium |
| 21 | Low | High | Low | High |
| 22 | Medium | Low | Low | Low |
| 23 | Medium | Medium | Low | Very high |
| 24 | Medium | High | Low | High |
| 25 | High | Low | Low | Medium |
| 26 | High | Medium | Low | Medium |
| 27 | High | High | Low | Medium |
The fuzzy inference system uses the centre-of-gravity method to defuzzify the output. The resulting crisp coefficient \(K\) can be expressed as:
$$
K=\frac{\displaystyle \sum_{i=1}^{n} w_i K_i}{\displaystyle \sum_{i=1}^{n} w_i}
$$
where \(K_i\) is the rule output value and \(w_i\) is the firing strength of the \(i\)-th fuzzy rule.
5.4 Genetic Optimisation of the Fuzzy Controller
One of the main drawbacks of a conventional fuzzy controller is that the membership functions are usually designed by trial and error or by expert experience. To obtain a more objective and robust regenerative braking controller, the genetic algorithm is applied to optimise the membership parameters. The optimisation parameters are the critical corner points, peak positions and boundary values of each membership function. Because the first input has three Gaussian-type membership functions, six parameters are required to describe its membership distribution. The vehicle speed and battery SOC inputs use three trapezoidal membership functions each, contributing another 24 parameters. The output membership function combines triangular and trapezoidal functions and therefore requires 17 parameters. In total, the genetic algorithm tunes 47 parameters of the fuzzy controller.
The objective of the genetic fuzzy optimisation is to maximise the total braking energy recovered by the electric vehicle during a standard driving cycle. This can be equivalently stated as maximising the expected regenerated energy, or alternatively as minimising the battery energy drawn from the grid. The fitness function is therefore related to the cumulative regenerative braking energy. During the evolution, every candidate fuzzy controller is inserted into the electric vehicle simulation model and evaluated over a selected driving cycle. The controller with higher recovered energy receives a higher fitness value. After the genetic search terminates, the best individual is selected as the optimised fuzzy controller.
Compared with the manually designed fuzzy controller, the genetically tuned controller achieves smoother transitions between different braking regions and avoids abrupt changes in the regenerative braking torque. The optimised membership functions keep a relatively high value of \(K\) during light braking at low battery charge states, while strongly suppressing regeneration when the battery is close to fully charged.
6. Simulation Results and Verification
6.1 Simulation Setup and Evaluation Metrics
The simulation models of the electric vehicle, including the DCT gearshift controller, the powertrain controller and the regenerative braking fuzzy controller, are implemented in MATLAB/Simulink. A driver model is used to track the target speed profiles of the NEDC and WLTP driving cycles. The driver model consists of a PI controller that compares the target speed and the actual vehicle speed and outputs the accelerator or brake pedal signal. The pedal signal ranges from \(-1\) to \(1\), where positive values represent acceleration and negative values represent braking.
The evaluation metrics for the regenerative braking strategy are the battery SOC drop and the recovered braking energy. The battery SOC drop reflects the net energy consumption of the electric vehicle over the drive cycle, while the recovered braking energy directly measures the amount of energy returned to the battery during regenerative braking operations.
6.2 NEDC Cycle Results
The NEDC cycle consists of four repeated urban driving segments and one extra-urban driving segment. The total cycle duration is 1180 seconds and the total distance is approximately 11 km. The urban segments contain a large number of stop-and-go events, which are favourable for the evaluation of regenerative braking. In this study, the simulation is extended to 1200 seconds in order to complete the full NEDC schedule and to allow the final braking event to be included.
The actual vehicle speed closely follows the target NEDC speed profile. The optimised fuzzy controller reduces the speed-tracking error compared with the unoptimised controller. The mean speed-tracking accuracy is approximately 98.6% for the optimised electric vehicle control strategy, which confirms that adding the regenerative braking controller does not degrade the longitudinal drivability of the electric vehicle.
The simulation results are compared for three different control strategies: the ideal braking-force distribution strategy, the manually designed fuzzy controller, and the genetically optimised fuzzy controller. The initial battery SOC is set to 0.95 and the terminal simulation condition is a battery SOC of 0.1.
The SOC trajectories show that the control strategy has a significant influence on the net energy consumption of the electric vehicle. When the ideal braking force distribution strategy is used, the battery SOC decreases relatively quickly and ends at approximately 0.81. With the manually designed fuzzy controller, the terminal SOC is about 0.851. When the genetically optimised fuzzy control strategy is applied, the terminal SOC reaches 0.889. The improvement in the terminal SOC therefore equals about 4.46 percentage points over the manually designed fuzzy controller and 9.75 percentage points over the ideal braking distribution strategy.
The corresponding recovered braking energy data also confirm the benefit of the optimised electric vehicle braking strategy. Compared with the ideal distribution strategy, the optimised fuzzy controller increases the total recovered braking energy by 27.71%. Compared with the unoptimised fuzzy controller, the improvement is still about 2.41 percentage points. This improvement is mainly due to the better coordination between the hydraulic brake torque and the regenerative braking torque during changing braking demands.
| Strategy | Initial SOC | Final SOC (NEDC) | Recovered braking energy gain vs ideal strategy |
| Ideal distribution | 0.95 | 0.810 | – |
| Conventional fuzzy | 0.95 | 0.851 | +24.72% |
| GA-fuzzy | 0.95 | 0.889 | +27.71% |
6.3 WLTP Cycle Results
The WLTP driving cycle is a more dynamic and more demanding test procedure than NEDC. It consists of low, medium, high and very-high speed phases, with a total duration of 1800 seconds and a total distance of about 23.25 km. The WLTP profile includes stronger accelerations and higher braking intensities than the NEDC profile, which makes regenerative braking control more challenging.
The optimised electric vehicle controller is first evaluated by comparing the actual vehicle speed with the WLTP target speed. The tracked speed error remains within ±0.3 m/s during most of the cycle, and the speed-tracking accuracy improves by about 1.5 percentage points compared with the unoptimised fuzzy controller. The slight improvement is caused by the better coordination between the hydraulic brake and the motor brake, which reduces the disturbance of the brake torque on the longitudinal vehicle motion.
The battery SOC comparison under the WLTP cycle shows the same trend as the NEDC cycle. Starting from 0.95, the battery SOC decreases to about 0.704 with the ideal braking distribution strategy, 0.756 with the conventional fuzzy controller, and 0.818 with the genetically optimised fuzzy controller. Thus, the optimised fuzzy controller raises the terminal SOC by 8.24% compared to the conventional fuzzy strategy and by 16.12% compared with the ideal braking distribution strategy. The recovered braking energy under the optimised fuzzy strategy is 7.1% higher than the ideal distribution strategy and about 6.45 percentage points higher than the conventional fuzzy strategy.
| Strategy | Final SOC (WLTP) | Recovered energy gain vs conventional fuzzy | Percentage SOC improvement |
| Ideal distribution | 0.704 | – | – |
| Conventional fuzzy | 0.756 | – | – |
| GA-fuzzy | 0.818 | +6.45% | +8.24 % vs conventional fuzzy; +16.12% vs ideal |
The simulation results show that the genetically optimised fuzzy brake-feedback controller is especially effective under the strongly transient WLTP conditions. The main reason is that the genetic algorithm adjusts the membership functions to give a higher priority to motor regeneration during medium-speed deceleration events, while still preventing battery overcharge and excessive motor torque. Consequently, the electric vehicle can recover more kinetic energy without violating the safety constraints imposed by the ideal braking curve and the ECE R13 regulation.
6.4 Overall Electric Vehicle Performance after Optimisation
The complete electric vehicle model with the optimised two-speed DCT is also evaluated in terms of its dynamic performance and energy consumption. The results are summarised in the following table.
| Indicator | Simulation result | Target |
| Maximum vehicle speed | 163.7 km/h | ≥150 km/h |
| Gradeability at 5 km/h | 35.45% | ≥30% |
| Gradeability at 30 km/h | 24.88% | ≥20% |
| 0–100 km/h acceleration time | 9.54 s | <12 s or <10 s depending on variant |
| Range by 60 km/h constant-speed method | 175.4 km | ≥150 km |
| WLTP energy consumption | 12.50 kWh/100 km | Improved over single-speed DCT |
The simulation confirms that the parameter matching procedure described in Section 2 provides a sound starting point for the electric vehicle powertrain, and the optimisation procedure improves the energy economy without compromising the required driving performance. The dual-clutch transmission allows the electric vehicle to operate more frequently in the high-efficiency zone of the motor, which is particularly valuable under the dynamic WLTP cycle.
7. Conclusion and Future Work
This research has presented a complete methodology for the parameter matching, multi-objective optimisation and regenerative braking control of an electric vehicle with a two-speed wet dual-clutch transmission. The main contributions are summarised as follows.
First, a detailed electric vehicle powertrain model was established, including a second-order RC battery model, an inverter loss model, an electromagnetic motor model, a loss-based dual-clutch transmission model and a longitudinal vehicle dynamics model. The model was used to evaluate the dynamic performance and the energy consumption of the electric vehicle under standard driving cycles. The simulation results showed that the selected motor, battery and DCT parameters satisfy the required maximum speed, gradeability, acceleration and driving range targets.
Second, a multi-objective optimisation framework was proposed for the electric vehicle powertrain. The rated motor power, rated motor speed, motor voltage, first gear ratio and second gear ratio were chosen as design variables, while the WLTP energy consumption and the total powertrain cost were used as objective functions. The multi-objective genetic algorithm generated a Pareto front that clearly describes the trade-off between energy economy and cost. A significant new finding is that the variable-efficiency model of the two-speed DCT changes the optimal gear ratio selection. The optimal first gear ratio is found at approximately 1.75 and the optimal second gear ratio at approximately 1.17 when the transmission losses are considered, whereas a fixed-efficiency model predicts larger gear ratios. Therefore, the detailed modelling of the electric vehicle transmission efficiency is important for obtaining correct powertrain design conclusions.
Third, a regenerative braking control strategy based on fuzzy logic was developed for the front-wheel-drive electric vehicle. The strategy uses a piecewise braking-force boundary that satisfies the I-curve, the ECE R13 regulation and the road adhesion limit. A fuzzy controller is used to calculate the dynamic distribution coefficient between motor braking torque and hydraulic friction torque. The genetic algorithm was then introduced to optimise the membership functions of the fuzzy controller. Under the NEDC cycle, the optimised electric vehicle regenerative braking strategy improved the final battery SOC by 4.46% over the conventional fuzzy strategy and by 9.75% over the ideal braking distribution strategy. Under the WLTP cycle, the improvements were even more obvious, confirming the practical benefit of the proposed controller in dynamic driving conditions.
In future work, the optimisation of the electric vehicle could be extended to consider battery ageing, thermal management and different driving styles. The fuzzy controller could also be replaced by a reinforcement learning agent that continuously adapts the electric vehicle braking strategy to the actual traffic environment. Finally, the simulation environment should be further validated with experimental tests on a full electric vehicle powertrain test bench or through on-road driving tests. In summary, the present study provides a systematic and practical design framework for improving the energy efficiency of the electric vehicle through the combined optimisation of the electric powertrain architecture and the regenerative braking energy management system.