Over the last two decades, the global automotive industry has been confronted with two simultaneous pressures: the scarcity of fossil fuels and the urgent demand for lower greenhouse-gas emissions. Road transport depends heavily on petroleum, and conventional internal-combustion propulsion systems convert only a limited fraction of fuel energy into useful mechanical work. Battery electric vehicles have therefore become one of the most attractive solutions for sustainable mobility. In this research, I concentrated on a battery electric vehicle with a two-speed wet dual-clutch transmission. My objective was to improve the energy efficiency of the electric vehicle without sacrificing its acceleration performance, maximum speed, or braking safety.
An electric vehicle is not merely a conventional car in which the internal-combustion engine is replaced by an electric machine. Its driving range, acceleration capability, energy consumption, and production cost strongly depend on the matching of the drive motor, energy storage system, transmission ratios, and regenerative-braking algorithm. A poorly selected motor or an inappropriate gear ratio can cause the drive motor to operate outside its high-efficiency region in many common driving conditions. Likewise, without an optimized braking-feedback controller, a substantial fraction of kinetic energy is dissipated as heat in the mechanical friction brakes. In city traffic, such dissipated braking energy can represent more than 20 percent of the total traction energy.
In this article, I report an integrated investigation consisting of three main parts. First, I present a systematic parameter-matching procedure for the powertrain components of an electric vehicle with a two-speed wet dual-clutch transmission. The components include the permanent-magnet synchronous motor, lithium-ion battery pack, and gear pairs. Second, I describe a high-fidelity simulation framework that accounts for the efficiency variation of the electric motor, the power losses in the inverter, the drag losses in the dual-clutch transmission, and the longitudinal dynamics of the vehicle. Third, I discuss a regenerative-braking control strategy based on an optimized fuzzy-logic system. The fuzzy controller distributes the demanded braking torque between the regenerative electric motor and the hydraulic friction brakes. I used a genetic algorithm to tune the membership functions of the fuzzy controller, which makes the controller less dependent on subjective expert experience and improves its adaptability over a wide range of braking conditions.
The vehicle performance was evaluated under the New European Driving Cycle and the Worldwide Harmonized Light Vehicles Test Procedure. Simulation results show that the optimized electric vehicle can achieve about 12.5 kWh of electrical energy consumption per 100 km under the WLTP profile while retaining strong acceleration and grade-climbing capability. The optimized braking-feedback controller also increases the final battery state of charge compared with a conventional fuzzy controller and with an ideal-braking-force distribution strategy. These findings confirm that simultaneous optimization of powertrain parameters and regenerative-braking logic is an effective path toward extending the driving range of electric vehicles.
Target Vehicle and Powertrain Architecture
The research object is a front-wheel-drive small passenger electric vehicle equipped with a two-speed wet dual-clutch transmission. The powertrain consists of a drive motor, a dual-clutch module, two gear pairs, an integrated controller, and a lithium-ion battery pack. The motor is connected to the gear train through the dual-clutch module. During driving, the controller selects the appropriate gear according to the current vehicle speed and demanded torque. The engine-electrical architecture also permits the same electric machine to operate as a generator during deceleration, thereby converting part of the vehicle kinetic energy into electrical energy for storage in the battery pack.
Vehicle Parameters and Performance Targets
The main parameters of the electric vehicle are listed in the following table. These parameters were used for all subsequent analytical calculations and simulation studies.
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Curb mass | \(m_v\) | 1455 | kg |
| Frontal area | \(A\) | 2.593 | m² |
| Dynamic wheel radius | \(R_D\) | 0.273 | m |
| Aerodynamic drag coefficient | \(C_D\) | 0.456 | – |
| Rolling resistance coefficient | \(f\) | 0.012 | – |
| Wheelbase | \(L\) | 2490 | mm |
| Centre-of-gravity height | \(h_g\) | 510 | mm |
| Final-drive ratio | \(i_0\) | 4.058 | – |
The performance requirements adopted in this study are summarized below.
| Performance index | Required value |
|---|---|
| Maximum speed | \(\geq 150\) km/h |
| Climbing ability at 5 km/h | \(\geq 30\%\) |
| Climbing ability at 30 km/h | \(\geq 20\%\) |
| 0–100 km/h acceleration time | < 12 s and < 10 s |
| Constant-speed range at 60 km/h | \(\geq 150\) km |
Electric-Vehicle Longitudinal Dynamics
For the mathematical description of the vehicle, I treated the vehicle as a rigid body moving on a longitudinal road surface. The traction force must overcome rolling resistance, aerodynamic resistance, grade resistance, and acceleration resistance. The longitudinal dynamic equilibrium of the electric vehicle can be expressed as
\[
F_t = m_v g f\cos\alpha + \frac{C_D A u^2}{21.15} + m_v g\sin\alpha + \delta m_v\frac{du}{dt},
\]
where \(F_t\) is the total tractive force at the driven wheels, \(u\) is the vehicle speed expressed in km/h, \(\alpha\) is the road grade angle, and \(\delta\) is the rotational-mass equivalent factor. This equation formed the basis for both the parameter matching and the energy-consumption calculations in the simulation model.
Drive-Motor Parameter Matching
The motor peak power must satisfy the power demand for maximum speed, maximum grade, and the specified acceleration time. The power required to maintain the maximum speed \(u_{\max}\) can be written as
\[
P_{u_{\max}} = \frac{u_{\max}}{3600\eta_T}\left(m_v g f + \frac{C_D A u_{\max}^2}{21.15}\right),
\]
where \(\eta_T\) is the total transmission efficiency. The power demand for maximum climbing speed \(u_c\) on a road with grade angle \(\alpha_{\max}\) is given by
\[
P_{\alpha_{\max}} = \frac{u_c}{3600\eta_T}\left(m_v g f\cos\alpha_{\max} + \frac{C_D A u_c^2}{21.15} + m_v g\sin\alpha_{\max}\right).
\]
The acceleration power demand for a prescribed 0–100 km/h time was evaluated in the time domain. At any instant, the force that can be delivered by the electric machine to the wheels is
\[
F_t(u) = \min\left(\frac{T_{\max}i_g i_0\eta_T}{R_D},\; \frac{9550P_{\max}\eta_T}{u}\right),
\]
where \(T_{\max}\) is the peak motor torque, \(i_g\) is the gear ratio of the current transmission stage, and \(i_0\) is the final-drive ratio. The actual acceleration time was then obtained by integrating the longitudinal dynamic equation. The required motor peak power was selected as the maximum of the three power demands multiplied by a reserve factor of 15 percent.
The peak motor speed is related to the maximum vehicle speed and the overall gear ratio by
\[
n_{\max} \geq \frac{u_{\max}i_2 i_0}{0.377R_D}.
\]
The ratio between the maximum motor speed and the rated motor speed was set to two. Thereafter, the motor rated torque and peak torque were obtained from the well-known power-torque relationship:
\[
T = \frac{9550P}{n}.
\]
The selected motor parameters after the initial matching stage are shown in the following table.
| Item | Value | Unit |
|---|---|---|
| Motor peak power | 100 | kW |
| Motor rated power | 50 | kW |
| Motor peak speed | 8000 | rpm |
| Motor rated speed | 4000 | rpm |
| Motor peak torque | 238 | Nm |
| Motor rated torque | 120 | Nm |
Battery Pack Parameter Matching
For an electric vehicle, the battery pack must be designed so that both the continuous driving-range requirement and the instantaneous motor power demand can be satisfied. The pack capacity \(C_{\text{pack}}\) for a constant-speed range \(S\) can be estimated from the energy balance:
\[
C_{\text{pack}} = \frac{S\left(m_v g f + \dfrac{C_D A u_c^2}{21.15}\right)}{3.6U_{\text{bat}}\eta_m\eta_T\eta_b \xi_{\text{dod}}},
\]
where \(u_c\) is the constant vehicle speed used for range testing, \(U_{\text{bat}}\) is the battery terminal voltage, \(\eta_m\) is the motor efficiency, \(\eta_T\) is the transmission efficiency, \(\eta_b\) accounts for auxiliary loads, and \(\xi_{\text{dod}}\) is the usable depth of discharge of the battery. The battery power capability was checked separately:
\[
P_{\text{bat,max}} \geq \frac{P_{\max}}{\eta_m} + P_{\text{aux}}.
\]
Here \(P_{\max}\) is the motor peak power and \(P_{\text{aux}}\) denotes the power consumed by vehicle accessories. I selected a laminated lithium-ion battery cell with a nominal voltage of 3.7 V and a nominal capacity of 5 Ah. With a nominal battery voltage of 320 V and a total capacity of approximately 115 Ah, the pack consists of 87 cells in series and 23 cells in parallel.
| Battery parameter | Value | Unit |
|---|---|---|
| Battery rated voltage | 320 | V |
| Battery rated capacity | 115 | Ah |
| Number of cells in series | 87 | – |
| Number of cells in parallel | 23 | – |
| Cell rated voltage | 3.7 | V |
| Cell rated capacity | 5 | Ah |
Transmission-Ratio Design for the Two-Speed DCT
The first gear of the two-speed dual-clutch transmission must satisfy the maximum torque demand for launch, acceleration, and climbing. The upper bound of the first gear ratio is related to the maximum adhesion force that can be transmitted by the front driving wheels:
\[
F_{\text{t,max}} = \mu m_v g \frac{L_2}{L + \mu h_g},
\]
where \(\mu\) is the longitudinal adhesion coefficient between the tyre and road, and \(L_2\) is the distance from the centre of gravity to the rear axle. Therefore, the first gear ratio must not be so large that the motor peak torque exceeds the adhesion limit. The second gear ratio must provide the torque and speed required for the maximum vehicle speed. The lower second-gear boundary can be expressed as
\[
i_{2,\min} \leq \frac{n_{\max}R_D}{0.377u_{\max}i_0}.
\]
The initial matching results of the two-speed DCT were \(i_1=1.8\) for the first gear and \(i_2=1.3\) for the second gear. In a later optimization stage, these ratios were adjusted together with the electric-machine ratings to achieve the best compromise between energy economy and powertrain cost.
Construction of the Electric-Vehicle Simulation Platform
A high-fidelity simulation model of the electric vehicle was constructed in the MATLAB environment. The model contains several interconnected submodels: the battery, the inverter, the electric motor, the two-speed dual-clutch transmission, and the longitudinal vehicle dynamics. Unlike models that assume constant component efficiency, the present framework calculates efficiency at every operating point. This feature is particularly important for evaluating an electric vehicle because a motor rarely operates at its rated point during urban or highway driving.
Battery Equivalent-Circuit Model
I used a second-order resistance-capacitance equivalent-circuit model for the lithium-ion battery. This model is sufficiently accurate for system-level simulations and still has a relatively simple structure. The state equations of the second-order equivalent-circuit model are as follows:
\[
U_L = U_{OC} – U_1 – U_2 – I_b R_0,
\]
\[
C_1\frac{dU_1}{dt} + \frac{U_1}{R_1} = I_b,
\qquad
C_2\frac{dU_2}{dt} + \frac{U_2}{R_2} = I_b.
\]
The parameters \(R_1,C_1\) and \(R_2,C_2\) describe the electrochemical polarization and concentration polarization phenomena inside the cell. The ohmic resistance \(R_0\) was derived from pulse discharge tests by evaluating the instantaneous voltage drop at the beginning and at the end of the discharge pulse:
\[
R_0 = \frac{\left(U_B-U_C\right)+\left(U_E-U_D\right)}{2I_{\text{pulse}}}.
\]
The zero-input voltage response during the relaxation period can be fitted with a double exponential expression:
\[
U_L(t) = U_{OC} – U_1(0)e^{-t/\tau_1} – U_2(0)e^{-t/\tau_2}.
\]
The time constants obtained from the fit are related to the product of the polarization resistance and capacitance in each branch. I used an online parameter-identification procedure such that the ohmic resistance and the polarization parameters were updated as functions of the cell state of charge. This improves the fidelity of the model over a wide SOC interval. The calibrated model was verified with pulse discharge data. The simulation and experimental voltage traces were in close agreement, and the average voltage error remained very small.
Inverter Loss Model
The power inverter converts the direct current from the battery into the alternating current required by the permanent-magnet motor. During this conversion, losses are generated in the insulated-gate bipolar transistors and in the freewheeling diodes. The main inverter losses can be divided into conduction losses, switching losses, and diode reverse-recovery losses:
\[
P_{\text{inv,loss}} = P_{\text{cond,IGBT}} + P_{\text{sw,IGBT}} + P_{\text{rec,diode}}.
\]
The instantaneous inverter efficiency was obtained as
\[
\eta_i = \frac{P_{\text{in}} – P_{\text{inv,loss}}}{P_{\text{in}}},
\]
where \(P_{\text{in}}\) is the electrical power entering the inverter. The inverter efficiency therefore varies with motor torque, motor speed, battery voltage, and the modulation index. For the studied electric vehicle, the inverter parameters were selected from a commercial power module with a continuous current rating of 450 A and a breakdown voltage rating of 750 V.
Interior Permanent-Magnet Synchronous Motor Model
The traction machine is an interior permanent-magnet synchronous motor. The advantage of embedding the magnets inside the rotor is that the motor has a wider flux-weakening speed range, a higher torque density, and a higher efficiency. To construct the motor model, I used a design tool that determines the motor geometry from a set of rated electromagnetic parameters. The tool then calculates the d-axis and q-axis inductances, the stator resistance, and the iron-loss equivalent resistance. From these parameters, the complete motor efficiency map can be derived.
The motor output torque in the \(d-q\) reference frame is given as
\[
T_m = P_{\text{pole}}\left[\psi_f i_q + \left(L_d – L_q\right)i_d i_q\right],
\]
where \(P_{\text{pole}}\) is the number of pole pairs, \(\psi_f\) is the permanent-magnet flux linkage, \(L_d\) and \(L_q\) are the inductances in the direct and quadrature axes, and \(i_d\) and \(i_q\) are the corresponding stator current components. The motor efficiency is defined as the ratio of mechanical output power to total input electrical power:
\[
\eta_m = \frac{T_m\omega_m}{T_m\omega_m + P_{\text{Cu}} + P_{\text{Fe}} + P_{\text{mech}}},
\]
where \(P_{\text{Cu}}\) is the stator copper loss, \(P_{\text{Fe}}\) represents iron losses, and \(P_{\text{mech}}\) is the mechanical loss caused by bearing friction and windage. The resulting motor efficiency map is a two-dimensional function of motor speed and torque. The map shows a broad region of high efficiency at moderate torque and speed, and the motor efficiency decreases in the high-torque low-speed region and in the high-speed low-torque region. The gear ratio of the electric vehicle therefore has a strong influence on the average motor efficiency over a driving cycle.
Dual-Clutch Transmission Efficiency Model
The two-speed wet dual-clutch transmission has two concentrically arranged shafts, two wet clutches, and two gear pairs. The transmission can switch from first gear to second gear without interrupting the traction torque, which improves comfort and driving performance. The total mechanical efficiency of the transmission is not a fixed value. It depends on the input speed, transmitted torque, engaged gear, oil temperature, and clutch state. I expressed the efficiency as
\[
\eta_T = 1 – \frac{P_{\text{clutch}}+P_{\text{mesh}}+P_{\text{bearing}}+P_{\text{shaft}}+P_{\text{windage}}}{P_{\text{in}}},
\]
where \(P_{\text{in}}\) is the power supplied by the electric motor. The drag torque of an open wet multi-plate clutch is an important source of loss when the motor speed is high. The clutch drag torque depends on the effective oil-film radius, the relative speed of the clutch plates, and the oil viscosity. Once the effective outer radius \(R_o\) of the oil film is known, the clutch pack drag torque can be approximated as
\[
T_{\text{clutch}} = \frac{\pi \mu_{\text{oil}}N_{\text{plate}}\Delta\omega}{2h_{\text{gap}}}\left(R_o^4 – R_i^4\right),
\]
where \(N_{\text{plate}}\) is the number of friction surfaces, \(h_{\text{gap}}\) is the oil-film thickness, and \(\Delta\omega\) is the relative angular speed between the clutch plates. In addition, gear meshing losses and bearing losses were calculated by analytical formulas from gear-efficiency standards. These formulas account for the gear sliding ratio, the transmitted torque, and the lubricating-oil viscosity. Because the gear-mesh losses also increase with decreasing transmission ratio, the choice of gear ratio directly affects the average transmission efficiency over the driving cycle.
Longitudinal Vehicle Dynamics Model and Energy-Consumption Indicator
The longitudinal model of the electric vehicle computes the actual speed and acceleration in response to the driving torque and braking torque. The traction force at the wheels is obtained from the motor torque, the engaged gear ratio, and the instantaneous transmission efficiency:
\[
F_t = \frac{T_m i_g i_0 \eta_T}{R_D}.
\]
Conversely, the electrical power required by the vehicle is related to the road-load power divided by the instantaneous motor and transmission efficiency. For the WLTP driving cycle, the electrical energy consumption per 100 km can be written as
\[
EC = \frac{100}{3600L_{\text{cyc}}}\int_0^{T_{\text{cyc}}}
\frac{u(t)}{3.6\eta_m(t)\eta_T(t)}
\left(m_v g f + \frac{C_DAu^2(t)}{21.15} + \delta m_v\frac{du(t)}{dt}\right)dt,
\]
where \(L_{\text{cyc}}\) is the total distance of the cycle in km. This formula was used to evaluate the economy of the electric vehicle for different powertrain configurations.
Multi-Objective Optimization of Powertrain Parameters
Traditional parameter matching gives a feasible powertrain design, but it does not guarantee the best balance between driving range and system cost. In this research, I formulated the selection of the motor and transmission parameters as a constrained multi-objective optimization problem. The design variables are listed below:
\[
X = \left[P_{em},\; n_{em},\; U_{em},\; i_1,\; i_2\right]^T.
\]
Here \(P_{em}\) is the motor rated power, \(n_{em}\) is the motor rated speed, \(U_{em}\) is the motor rated phase voltage, and \(i_1\) and \(i_2\) are the first and second gear ratios of the two-speed DCT. The first objective function is the WLTP electrical energy consumption \(EC\) described above. The second objective function is the total purchase cost of the electric-vehicle powertrain:
\[
\text{Cost} = C_{\text{battery}} + 63P_{em} + 1560,
\]
where \(P_{em}\) is expressed in kilowatts. The battery cost was calculated from the number of battery cells and the packaging cost:
\[
C_{\text{battery}} = X_{\text{cell}}Y_{\text{cell}}\times 7 + 625,
\]
where \(X_{\text{cell}}\) is the number of cells in series and \(Y_{\text{cell}}\) is the number of cells in parallel. The constraints included maximum speed, maximum grade ability, acceleration performance, and the road-adhesion limitation of the driving wheels. For the multi-objective search, I used the non-dominated sorting genetic algorithm II, which has proven to be an efficient method for solving conflicting design objectives. The population contained 200 individuals, the maximum generation number was 40, the crossover probability was 0.8, and the mutation probability was 0.2.
The optimization produced a set of Pareto-optimal solutions. One particular solution is interesting because it lies at the knee of the Pareto front. On the left side of this knee, a small reduction in cost leads to a large increase in energy consumption. On the right side, a small further reduction in energy consumption requires a large increase in cost. I selected the knee point as the most balanced design for the electric vehicle.
| Transmission efficiency model | Acceleration target | WLTP energy consumption kWh/100 km | Powertrain cost yuan | \(P_{em}\) kW | \(n_{em}\) rpm | \(U_{em}\) V | \(i_1\) | \(i_2\) |
|---|---|---|---|---|---|---|---|---|
| Fixed model | 12 s | 12.96 | 29040 | 40 | 6000 | 300 | 3.44 | 1.72 |
| Variable model | 12 s | 12.48 | 29048 | 41 | 6000 | 300 | 1.75 | 1.17 |
| Fixed model | 10 s | 12.89 | 29149 | 51 | 6000 | 300 | 3.53 | 1.72 |
| Variable model | 10 s | 12.48 | 29121 | 48 | 6000 | 300 | 1.75 | 1.17 |
When the variable transmission efficiency model was used, the optimum gear ratios were significantly lower than when a fixed efficiency of 0.96 was assumed. This result occurs because lower gear ratios reduce the open-clutch drag losses and gear-mesh losses when the vehicle operates in the urban part of the WLTP test cycle. The electric motor can compensate for the lower torque amplification by operating at higher current because the optimization selected slightly larger motor power. The comparison between fixed and variable efficiency models demonstrates that constant-efficiency assumptions can underestimate the energy-saving potential of a two-speed DCT in an electric vehicle.
Regenerative-Braking Force Allocation
When an electric vehicle decelerates, the braking torque can be provided by the electric machine and by hydraulic friction brakes. A good regenerative-braking controller must maintain braking stability while maximizing the energy returned to the battery. The front and rear axle brake-force distribution is therefore a central issue.
For a passenger car, the ideal braking-force distribution, also called the I curve, describes the front and rear brake forces that cause the front and rear wheels to lock simultaneously. Using the dynamic axle loads, the ideal relationship can be expressed in the following parametric form:
\[
F_{bf} = \varphi m_v g\frac{L_2 + \varphi h_g}{L},
\qquad
F_{br} = \varphi m_v g\frac{L_1 – \varphi h_g}{L},
\]
where \(F_{bf}\) and \(F_{br}\) are the braking forces of the front and rear axles, \(L_1\) is the distance from the centre of gravity to the front axle, \(L_2\) is the distance to the rear axle, and \(\varphi\) is the road adhesion coefficient. If the actual front brake force is below the I curve, the front wheels tend to lock first. If the front brake force lies above the I curve, the rear wheels may lock first, which is dangerous.
The ECE R13 regulation imposes another important constraint. For road adhesion values between 0.2 and 0.8, the actual braking deceleration must satisfy a minimum condition, and the front-wheel adhesion utilization must not be larger than a permitted value. The boundary that corresponds to the ECE R13 requirement is often called the M curve. In this study, the braking-force distribution was divided into several regions according to the demanded braking strength \(z\), expressed as
\[
z = \frac{a}{g} = \frac{F_{bf}+F_{br}}{m_v g}.
\]
Four braking regions were used for the front-wheel-drive electric vehicle. When the demanded braking strength is smaller than 0.22, a relatively modest braking force is requested, and the entire braking demand can be supplied by the front axle. In this low-deceleration range, I allowed the electric motor to provide as much regenerative braking as possible. When the braking strength is between 0.22 and 0.53, the braking force follows the ECE R13 boundary so that the vehicle remains stable and the front axle retains a sufficient share of the total braking force. When the braking strength is between 0.53 and 0.7, the road adhesion limit becomes the dominant factor. In this range, I followed the F-line corresponding to an adhesion coefficient of 0.7. For emergency braking above a demanded strength of 0.7, the electric machine can no longer provide stable torque before wheel lock, so the hydraulic friction brake must supply the entire braking force according to the ideal I-curve distribution.
| Demanded braking strength | Front/rear axle strategy | Regenerative-braking permission |
|---|---|---|
| \(z \leq 0.22\) | Front axle only | Maximum allowed |
| \(0.22 < z \leq 0.53\) | ECE R13 boundary | Controlled by fuzzy algorithm |
| \(0.53 < z \leq 0.7\) | F-line at \(\varphi = 0.7\) | Controlled by fuzzy algorithm |
| \(z > 0.7\) | I-curve distribution | Disabled for safety |
Fuzzy-Logic Regenerative-Braking Controller
Regenerative braking of an electric vehicle is a nonlinear process influenced by the battery state of charge, vehicle speed, braking urgency, motor torque boundary, and hydraulic braking dynamics. It is difficult to design a precise analytical controller for all of these conditions. Fuzzy logic offers a convenient way to integrate the expertise of braking-system engineers into a rule-based controller. I designed a fuzzy controller with three input variables:
\[
\text{Inputs: }z,\;v,\;SOC,
\]
and one output variable \(K\), which represents the regenerative-braking torque distribution coefficient. The controller calculates the desired electric-motor braking torque from the product of \(K\), the demanded front-axle braking force, and the transmission ratio. The remaining front-axle braking torque is supplied by the hydraulic friction brake. The fuzzy controller uses Mamdani-type inference.
Three linguistic terms were defined for each input variable. The braking strength \(z\) was described as low, medium, or high. The vehicle speed \(v\) was described as low, medium, or high. The battery state of charge was also described as low, medium, or high. The output \(K\) was described with five terms: very low, low, medium, high, and very high. The membership functions were initially constructed with a combination of Gaussian, triangular, and trapezoidal functions.
\[
\mu_{\text{Gaussian}}(x) = \exp\left(-\frac{(x-g)^2}{2\sigma^2}\right),
\]
\[
\mu_{\text{trapezoid}}(x) = \max\left(0,\;\min\left(\frac{x-a}{b-a},\;1,\;\frac{d-x}{d-c}\right)\right),
\]
\[
\mu_{\text{triangle}}(x) = \max\left(0,\;\min\left(\frac{x-a}{b-a},\;\frac{c-x}{c-b}\right)\right).
\]
The first input membership function represented the demanded braking intensity. The second membership function represented vehicle speed and was intentionally made flat in the middle to reflect the fact that the regenerative-braking potential is relatively stable in a moderate speed interval. The third membership function represented the SOC and included a low state to avoid over-discharge, a normal state for ordinary operation, and a high state to protect the battery from overcharging. When the SOC is very high, the motor regenerative-braking torque must be reduced because the battery has limited ability to accept additional charge.
A representative rule table for the regenerative-braking fuzzy controller is shown below. The linguistic levels used in the table are \(S\) for low, \(M\) for medium, and \(B\) for high.
| Rule | \(z\) | \(v\) | \(SOC\) | \(K\) |
|---|---|---|---|---|
| 1 | S | S | B | L |
| 2 | S | M | B | L |
| 3 | S | B | B | L |
| 4 | M | S | B | L |
| 5 | M | M | B | L |
| 6 | M | B | B | VH |
| 7 | B | S | B | M |
| 8 | B | M | B | H |
| 9 | B | B | B | H |
| 10 | S | S | M | VL |
| 11 | S | M | M | VL |
| 12 | S | B | M | H |
| 13 | M | S | M | M |
| 14 | M | M | M | H |
| 15 | M | B | M | H |
| 16 | B | S | M | VH |
| 17 | B | M | M | M |
| 18 | B | B | M | H |
| 19 | S | S | S | VL |
| 20 | S | M | S | M |
| 21 | S | B | S | H |
| 22 | M | S | S | L |
| 23 | M | M | S | VH |
| 24 | M | B | S | H |
| 25 | B | S | S | M |
| 26 | B | M | S | M |
| 27 | B | B | S | M |
The output membership functions express the magnitude of the motor torque-sharing coefficient. Defuzzification of the output was performed by the weighted-average method:
\[
K = \frac{\sum_{j=1}^{M} \mu_j w_j}{\sum_{j=1}^{M} \mu_j},
\]
where \(\mu_j\) is the firing degree of the \(j\)-th rule and \(w_j\) represents the corresponding output singleton or centroid. The resulting coefficient K was then multiplied by the available motor braking torque boundary to obtain the final regenerative torque request.
Genetic-Algorithm Optimization of the Fuzzy Controller
The conventional design of a fuzzy logic controller relies on expert experience for selecting the type and position of each membership function. This approach may produce inconsistent results for different electric-vehicle states. To reduce this subjectivity, I used a genetic algorithm to optimize the parameters of the input and output membership functions.
The Gaussian braking-strength membership functions contained six parameters because each of the three linguistic terms has two internal parameters. The vehicle-speed and SOC membership functions were trapezoidal, and each function contained four parameters. The regenerative-braking coefficient output variable was represented by a combination of triangular and trapezoidal membership functions. In total, the optimization problem contained 47 real-coded parameters. The genetic algorithm searched the admissible parameter space while keeping all fuzzy rules fixed. The objective of the optimization was to maximize the energy recovered by the battery during a complete braking process while preventing motor torque overshoot and preserving pedal feel.
The genetic algorithm produced improved membership functions that are more evenly distributed than the initially assumed functions. The optimized fuzzy controller changes the regenerative-braking coefficient more gradually when the vehicle speed changes from low to high. It also reduces the regenerative torque more quickly when the battery SOC approaches the upper charging limit. This behavior protects the battery and improves the stability of the regenerative-braking system, especially during repeated stop-and-go cycles.
Implementation of the Control Strategy in Simulink
I implemented the complete electric-vehicle propulsion and braking strategy in MATLAB/Simulink. A driver model was also developed to follow the target speed profile of the standard driving cycle. The driver model contains a proportional-integral controller whose output is interpreted as the accelerator pedal command or as the braking pedal command. When the actual vehicle speed is lower than the target speed, the controller produces a positive action that is treated as an accelerator-pedal opening. Conversely, when the actual vehicle speed is higher than the target speed, a negative control action is produced and treated as a braking-pedal opening.
The braking control strategy contains several subsystems. The first subsystem computes the demanded braking torque from the pedal opening and the maximum braking torque. The second subsystem calculates the front and rear axle braking-force distribution according to the rules presented before. The third subsystem is the fuzzy-logic controller that computes the electric-machine share of the front-axle braking force. The fourth subsystem limits the motor regenerative torque according to the maximum torque capability of the motor, the battery SOC, the vehicle speed, and the emergency-braking condition. Regenerative braking was disabled when the vehicle speed was below 10 km/h because the motor back-electromotive force is too small to provide a useful charging current. Regenerative braking was also disabled when the SOC was greater than 0.95 to avoid overcharging the lithium-ion battery.
Evaluation Indicators for Braking Energy Recovery
To compare the performance of different control strategies, I used two main indicators. The first is the variation of the battery state of charge over a standard driving cycle. Because the initial SOC is always set to the same value, a slower decline of the SOC indicates better energy efficiency and higher recovered energy. The second indicator is the total electrical energy recovered by the battery during the regenerative-braking process:
\[
E_{\text{rec}} = \int_{t_{\text{brake,on}}}^{t_{\text{brake,off}}} U_{\text{bat}}(t)I_{\text{chg}}(t)dt,
\]
which directly reflects the amount of kinetic energy that is converted back into chemical energy instead of being dissipated as heat in the brake discs.
Simulation Results under the NEDC Cycle
I first evaluated the optimized fuzzy regenerative-braking strategy under the New European Driving Cycle. This cycle contains four repeated urban segments followed by an extra-urban segment. It includes many acceleration, cruising, deceleration, and idling events. The initial battery SOC was fixed at 0.95. The comparison included three strategies: the optimized genetic-fuzzy controller, the conventional fuzzy controller, and a baseline controller based on the ideal braking-force distribution curve.
Longitudinal speed tracking was accurate with both control strategies, but the optimized fuzzy controller produced a slightly smaller following error during the transition from acceleration to braking. The optimized controller thereby allowed the vehicle to recover braking energy earlier and with less torque overshoot. The final SOC after completing the NEDC procedure is reported in the table below.
| Control strategy | Initial SOC | Final SOC | SOC retention percentage |
|---|---|---|---|
| GA-optimized fuzzy control | 0.95 | 0.889 | 93.58% |
| Conventional fuzzy control | 0.95 | 0.851 | 89.58% |
| I-curve braking strategy | 0.95 | 0.810 | 85.26% |
These data show that the optimized fuzzy controller increased the final SOC by about 4.46 percentage points compared with the conventional fuzzy controller and by about 9.75 percentage points compared with the ideal-braking-force strategy. The relative amount of recovered electrical energy after the optimized fuzzy strategy was likewise larger. The simulation result showed that in the NEDC cycle, the optimized strategy recovered approximately 27.71 percent more braking energy than the I-curve-based strategy and approximately 2.41 percent more energy than the unoptimized fuzzy strategy. This result is explained by the more flexible torque allocation of the optimized fuzzy controller, which not only respects the ECE R13 stability boundary but also takes advantage of the motor braking torque in medium- and high-speed braking events.
Simulation Results under the WLTP Cycle
The WLTP test procedure is more dynamic and more representative of real-world electric-vehicle use. It includes low-speed city phases, medium-speed suburban phases, high-speed road phases, and very-high-speed highway phases. Therefore, the electric-vehicle powertrain operates over a broader range of torque and speed, and the braking events occur with widely different initial speeds and demanded decelerations.
I repeated the same simulation procedure under the WLTP test cycle. The speed-tracking performance again confirmed that the optimized fuzzy controller is suitable for a highly dynamic electric-vehicle driving scenario. The measured SOC trajectories are summarized in the following table.
| Control strategy | Initial SOC | Final SOC | SOC retention percentage |
|---|---|---|---|
| GA-optimized fuzzy control | 0.95 | 0.818 | 86.11% |
| Conventional fuzzy control | 0.95 | 0.756 | 79.58% |
| I-curve braking strategy | 0.95 | 0.704 | 74.11% |
Under the WLTP condition, the genetic-algorithm-based fuzzy controller elevated the final SOC by approximately 8.24 percent relative to the conventional fuzzy controller and by approximately 16.12 percent relative to the I-curve strategy. These improvements are larger than those observed in the NEDC cycle because the WLTP profile comprises more severe deceleration phases from high speed. During such events, the conventional fuzzy controller tends to reduce the regenerative torque too quickly, whereas the optimized fuzzy controller maintains a high motor-braking share while still respecting the stability boundary. Consequently, the optimized electric vehicle was able to recover a noticeably larger amount of braking energy.
Improvement in Overall Electric-Vehicle Energy Economy
Besides the braking-feedback control strategy, the optimal matching of the electric motor and the two-speed DCT also contributed to the reduction of energy consumption. With a fixed final-drive ratio and an optimized gear-pair combination of \(i_1 = 1.75\) and \(i_2 = 1.17\), the electric motor operates inside its high-efficiency island for a larger fraction of the WLTP test time. The lower first gear still provides sufficient launch torque, while the spacing between the gears offers a torque-speed range that suits both urban and highway driving.
The simulated energy consumption of the optimized electric vehicle reached approximately 12.5 kWh per 100 km under the WLTP profile. This value is competitive for a passenger electric vehicle in this weight class. When the same vehicle was equipped with a conventional single-speed reduction gear and the same motor, the simulated energy consumption was noticeably higher. The advantage of the two-speed DCT is most evident in the urban phase, where the vehicle frequently changes speed and the low-speed gear enables the motor to operate at a more favorable combination of speed and torque. In the highway phase, the high-speed gear lowers the motor speed and reduces the iron loss and the transmission drag loss. Therefore, the electric vehicle with the two-speed DCT achieves a robust improvement in overall energy economy.
Influence of Component Efficiency Modeling on Design Conclusions
One of the central methodological findings of this research is that the modeling fidelity of component efficiency has a decisive influence on the optimization result. If a constant transmission efficiency is assumed, the multi-objective optimizer tends to choose very high gear ratios because the transmission loss is not coupled to the gear-ratio selection. When the gear ratio becomes larger, the motor speed is reduced for a given vehicle speed, but the torque at the wheels is amplified. This is beneficial for acceleration, but it does not necessarily improve the average motor efficiency. With a variable-efficiency DCT model, the calculation shows that lower gear ratios reduce the churning loss and gear-mesh loss, particularly when the vehicle is running in urban conditions. Therefore, the Pareto front obtained with the variable-efficiency model is shifted toward lower energy consumption compared with the Pareto front obtained with the fixed-efficiency model.
The optimization also shows that the motor rated power has only a moderate influence on the motor efficiency map for the motor sizes considered here. A motor with a slightly larger rated power can operate in a high-efficiency region over a wider torque range, especially if the vehicle is matched with lower gear ratios. However, an excessively large motor increases the cost of the electric vehicle. From the knee point of the Pareto front, I selected a motor rated power of approximately 41 kW and a rated voltage of 300 V for the 12-second acceleration target. This design provides sufficient dynamic performance while maintaining a low energy-consumption level and an acceptable powertrain cost.
Summary of the Control Strategies
The final optimized electric-vehicle configuration couples the best-matched powertrain with a genetic-fuzzy regenerative-braking strategy. The control flow can be summarized as follows. The brake control module first measures the braking torque demand and estimates the demanded braking intensity \(z\). The total braking force is then split between the front and rear axles according to the operating region determined by \(z\). For the front axle, the fuzzy controller calculates a regenerative-braking torque coefficient \(K\). The motor regenerative torque is thereafter constrained by the motor speed, torque capability, battery SOC, and vehicle speed. The difference between the total front-axle demand and the motor regenerative torque is supplied by the hydraulic friction brake. If the demanded braking intensity exceeds 0.7, the regenerative-braking path is completely disabled to ensure short stopping distance and stable vehicle behavior.
Compared with an unoptimized fuzzy controller, the genetic-algorithm-based fuzzy controller uses more logical membership-function boundaries. It can distinguish between high-speed braking events, during which a large amount of energy can be recovered, and low-speed gentle braking events, during which the regenerative torque should be restricted to maintain acceptable pedal feel and brake stability. This is visible in the final SOC data of both the NEDC and WLTP tests.
Conclusions
In this research, I systematically investigated the powertrain parameter matching and regenerative-braking optimization of a pure electric vehicle equipped with a two-speed wet dual-clutch transmission. The following conclusions can be drawn from the simulation results.
First, the two-speed DCT provides an effective approach for reconciling the contradictory requirements of high launch acceleration and low high-speed energy consumption. The gear ratio is an important design variable for any electric vehicle. Neglecting the efficiency variation of the transmission and motor when selecting gear ratios can lead to misleading conclusions.
Second, multi-objective optimization using the non-dominated sorting genetic algorithm is an efficient way to solve the conflict between energy consumption and powertrain cost. The resulting Pareto front enables engineers to choose a design according to market positioning and performance requirements. In the present study, the optimized design consumes approximately 12.5 kWh per 100 km under the WLTP profile, while the powertrain cost remains almost unchanged compared with the reference design.
Third, the regenerative-braking force distribution based on vehicle braking-strength boundaries is both stable and practical. The combination of an ideal force-distribution curve, the ECE R13 regulation line, and the F-line group provides a theoretical foundation for the control of any front-wheel-drive electric vehicle.
Fourth, the genetic-algorithm-optimized fuzzy controller improves the regenerative-braking energy recovery of the electric vehicle without increasing the risk of rear-axle instability. Under the NEDC test, the optimized electric vehicle ended with a substantially higher battery SOC than the conventional fuzzy controller. Under the WLTP test, the optimized electric vehicle achieved an even greater benefit because the cycle contains a larger number of braking events at medium and high speed.
Fifth, the high-fidelity component models developed in this work are important for practical engineering. The online-identified battery model, the motor efficiency map, and the variable-efficiency DCT model together create a reliable digital twin of the electric-vehicle powertrain. This platform can be used for future development of the vehicle controller, hardware-in-the-loop testing, and real-road calibration.
The research reported in this article provides a theoretical reference for the design and optimization of electric vehicles. Future work will focus on experimental validation on a prototype electric vehicle. I plan to integrate the optimized fuzzy controller with an anti-lock braking system and to investigate the influence of road-slope estimation and different driver behaviour patterns on the regenerative-braking performance. Another promising direction is the use of reinforcement learning to adapt the fuzzy-rule table in real time based on the measured battery temperature and road conditions. Such advanced control strategies will bring the electric vehicle closer to the goal of maximum energy efficiency, minimum cost, and stable braking performance over the full life of the vehicle.