Electric cars are increasingly regarded as one of the most effective pathways for reducing greenhouse-gas emissions and fossil-fuel dependence in the transportation sector. With the global push toward carbon neutrality, electric cars have entered a period of rapid technical evolution. However, two critical issues still limit the large-scale deployment of electric cars: the mismatch of powertrain parameters can reduce driving energy efficiency, and the lack of an optimized braking energy recovery strategy can waste a considerable share of the vehicle kinetic energy in urban driving. In this study, I therefore focused on a front-wheel-drive electric car equipped with a two-speed wet dual-clutch transmission (DCT), and I investigated both the powertrain parameter matching and the regenerative braking control strategy in order to achieve better driving performance, lower energy consumption, and longer electric cars driving range.
1. Powertrain Architecture and Parameter Matching for Electric Cars
The studied electric car architecture is built around a high-speed interior permanent-magnet synchronous motor, a 320 V lithium-ion battery pack, a two-speed wet DCT, and a final drive. The two-speed DCT allows electric cars to improve both acceleration performance and energy economy by keeping the motor operating near its high-efficiency region for a broader range of driving conditions. The motor is connected to one side of the dual clutch arrangement, while the two output gear pairs provide two gear ratios. During gearshift, one clutch releases while the other engages, creating a torque-interruption-free shift process. This feature makes the architecture particularly suitable for electric cars because the driver expects smooth response while the powertrain should also be compact and efficient.
The vehicle longitudinal dynamics equation is the fundamental starting point for parameter matching. In my study, I treated the electric car as a point mass and wrote the driving force balance as:
$$
F_t = F_f + F_w + F_i + F_j
$$
where \(F_t\) is the tractive force, \(F_f\) is the rolling resistance, \(F_w\) is the aerodynamic drag, \(F_i\) is the grade resistance, and \(F_j\) is the acceleration resistance. The expanded longitudinal dynamic equation used in all powertrain calculations is:
$$
\frac{T_m \, i_g \, i_0 \, \eta_T}{R_D}
=
m_v g f \cos\alpha
+
\frac{C_D A u^2}{21.15}
+
m_v g \sin\alpha
+
\delta m_v \frac{du}{dt}
$$
where \(T_m\) is the motor output torque, \(i_g\) is the gear ratio of the DCT, \(i_0\) is the final-drive ratio, \(R_D\) is the dynamic tyre radius, \(m_v\) is the vehicle mass, \(g\) is the gravitational acceleration, \(f\) is the rolling-resistance coefficient, \(C_D\) is the aerodynamic drag coefficient, \(A\) is the frontal area, \(u\) is vehicle speed in km/h, \(\alpha\) is roadway slope angle, and \(\delta\) is the rotating mass coefficient. This equation was used to determine the required motor power, torque, speed, battery capacity, and gear ratio boundaries so that electric cars can meet the predefined performance indicators.
Table 1 summarizes the vehicle parameters and the main performance targets that were used for the initial sizing of the powertrain.
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Vehicle kerb mass | \(m_v\) | 1455 | kg |
| Frontal area | \(A\) | 2.593 | m² |
| Dynamic tyre radius | \(R_D\) | 0.273 | m |
| Aerodynamic drag coefficient | \(C_D\) | 0.456 | – |
| Rolling-resistance coefficient | \(f\) | 0.012 | – |
| Front/rear static load distribution | \(L_1/L_2\) | 0.5 | – |
| Wheelbase | \(L\) | 2490 | mm |
| Centre of gravity height | \(h_g\) | 510 | mm |
| Final-drive ratio | \(i_0\) | 4.058 | – |
| Maximum speed target | \(u_{max}\) | ≥ 150 | km/h |
| Grade ability target | \(\alpha_{max}\) | ≥ 30 % at 5 km/h | – |
| Acceleration target (comfort) | \(t_a\) | < 12 | s |
| Acceleration target (sport) | \(t_a\) | < 10 | s |
| Range target at constant 60 km/h | \(S\) | ≥ 150 | km |
1.1 Motor Parameter Matching
The drive motor of electric cars must provide enough instantaneous power for acceleration and climbing while also meeting the continuous power demand at maximum speed. Based on the performance targets, the required peak power for the maximum-speed condition can be written as:
$$
P_{u_{\max}} = \frac{1}{\eta_T}
\left(
\frac{m_v g f u_{\max}}{3600}
+
\frac{C_D A u_{\max}^3}{76140}
\right)
$$
Similarly, the power required to climb a grade \(\alpha_{\max}\) at a constant speed \(u_c\) is:
$$
P_{\alpha_{\max}} = \frac{1}{\eta_T}
\left[
\frac{m_v g f \cos\alpha_{\max} \, u_c}{3600}
+
\frac{m_v g \sin\alpha_{\max} \, u_c}{3600}
+
\frac{C_D A u_c^3}{76140}
\right]
$$
The power required to complete the 0–100 km/h acceleration test is often estimated by including the acceleration resistance:
$$
P_{a_{\max}} = \frac{1}{\eta_T}
\left(
\frac{m_v g f u_f}{3600}
+
\frac{C_D A u_f^3}{76140}
+
\frac{\delta m_v u_f}{3600}\frac{du}{dt}
\right)
$$
The peak motor power \(P_{\max}\) must be larger than all of the three quantities above. To leave sufficient reserve power for electric cars, I increased the final peak value by 15 %. The rated motor power was then selected using the overload factor \(\lambda = 2\). After the speed range of the motor was fixed, the rated torque and peak torque were calculated from:
$$
T = 9550 \frac{P}{n}
$$
The maximum motor speed also determines the possible maximum vehicle speed through:
$$
n_{\max} \ge \frac{u_{\max} i_0 i_g}{0.377 R_D}
$$
In this initial matching process, the motor peak power was calculated as 100 kW, the rated power as 50 kW, the rated speed as 4000 rpm, and the peak speed as 8000 rpm. The peak torque was 238 N·m, while the rated torque was 120 N·m.
1.2 Battery Parameter Matching
For electric cars, the energy-storage system is usually sized to meet the required driving range and the peak motor power. The target range was first checked with the 60 km/h constant-speed method. The total battery capacity can be expressed as:
$$
C_K \ge \frac{S}{3600}
\frac{
m_v g f + \dfrac{C_D A u^2}{21.15}
}{
\eta_m \eta_T (1-\eta_b) U_{bat}
}
$$
where \(\eta_m\) is the motor efficiency, \(\eta_T\) is the transmission efficiency, \(\eta_b\) is the auxiliary energy loss coefficient, and \(U_{bat}\) is the battery terminal voltage. The battery also has to meet the peak motor power. If the maximum discharge rate of the cell is \(k\), then:
$$
C_B \ge \frac{1000}{\eta_m k U_{bat}}\left(P_{\max} + P_{aux}\right)
$$
I selected a 21700-type ternary lithium-ion cell with a nominal voltage of 3.7 V, capacity of 5 Ah, and internal resistance of 1.5 mΩ. The nominal battery-pack voltage was set to 320 V. The number of series cells is therefore approximately \(320/3.7 \approx 87\). The battery pack uses 23 cells in parallel, giving a total capacity close to 115 Ah.
1.3 Two-Speed DCT Parameter Matching
Electric cars fitted with a single-speed reducer often rely on a large high-speed motor to satisfy all driving requirements. A two-speed DCT can reduce the required motor torque and improve the average operating efficiency of the motor. The first gear ratio must meet the strongest tractive-force demand, while the second gear ratio should satisfy the maximum-speed condition. The first-gear inequality can be written as:
$$
\frac{T_{\max} i_1 i_0 \eta_T}{R_D}
\ge
m_v g f\cos\alpha
+
\frac{C_D A u_c^2}{21.15}
+
m_v g \sin\alpha
+
\delta m_v \frac{du}{dt}
$$
In addition, the maximum tractive force should not exceed the road friction limit. For a front-wheel-drive electric car, the anti-slip constraint is:
$$
\frac{T_{\max} i_1 i_0}{R_D}
\le
\frac{\mu m_v g (L_2 + \mu h_g)}{L}
$$
where \(\mu\) is the maximum road-friction coefficient and \(L_2\) is the distance from the centre of gravity to the rear axle. The second gear ratio is bounded by the torque demand at maximum speed and by the motor maximum speed:
$$
\frac{T_{n_{\max}} i_2 i_0 \eta_T}{R_D}
\ge
m_v g f
+
\frac{C_D A u_{\max}^2}{21.15}
$$
$$
\frac{n_{\max} R_D}{0.377 i_0 i_2}
\ge
u_{\max}
$$
Following this matching procedure, the initial DCT gear ratios were set to \(i_1 = 1.8\) and \(i_2 = 1.3\). Table 2 records the main initial powertrain parameters selected for the model.
| Component | Parameter | Symbol | Value | Unit |
|---|---|---|---|---|
| Motor | Peak power | \(P_{\max}\) | 100 | kW |
| Motor | Rated power | \(P_e\) | 50 | kW |
| Motor | Peak torque | \(T_{\max}\) | 238 | N·m |
| Motor | Rated torque | \(T_e\) | 120 | N·m |
| Motor | Peak speed | \(n_{\max}\) | 8000 | rpm |
| Battery | Nominal voltage | \(U_{bat}\) | 320 | V |
| Battery | Total capacity | \(C_{bat}\) | 115 | Ah |
| Battery | Cells in series | \(N_s\) | 87 | – |
| Battery | Cells in parallel | \(N_p\) | 23 | – |
| DCT | First-gear ratio | \(i_1\) | 1.8 | – |
| DCT | Second-gear ratio | \(i_2\) | 1.3 | – |
I verified the initial powertrain design using the longitudinal dynamic model. The simulated maximum speed was 163.7 km/h, the grade ability was 35.45 % at 5 km/h and 24.88 % at 30 km/h, the 0–100 km/h acceleration time was 9.54 s, and the constant-speed range was 175.4 km. These results show that the initial parameter matching is adequate for the target performance of electric cars.
2. High-Fidelity Modeling of Electric Cars Components
Although simple efficiency maps are commonly used in vehicle simulation, they are not sufficient for powertrain optimization of electric cars when the motor or transmission parameters have to be changed during the design process. I therefore developed detailed component models that reproduce the variation of motor efficiency, inverter losses, battery losses, and DCT efficiency with the operating point.
2.1 Battery Model
I chose a second-order RC equivalent-circuit model to represent the dynamic behavior of the lithium-ion battery. This model contains an open-circuit voltage source \(U_{OC}\), an ohmic resistance \(R_0\), and two RC networks that represent electrochemical polarization and concentration polarization. The output voltage is obtained from:
$$
U_L = U_{OC} – I R_0 – U_1 – U_2
$$
$$
\dot{U}_1 = -\frac{U_1}{R_1 C_1} + \frac{I}{C_1}, \qquad
\dot{U}_2 = -\frac{U_2}{R_2 C_2} + \frac{I}{C_2}
$$
The state of charge is calculated by integrating the current:
$$
SOC = SOC_0 – \frac{1}{C_{bat}}\int_{0}^{t} I \, dt
$$
The model parameters were identified after open-circuit-voltage and pulse discharge tests. The parameters \(R_0\), \(R_1\), \(C_1\), \(R_2\), and \(C_2\) were obtained by fitting exponential responses of the current-free voltage segments. I used an online parameter-identification strategy so that these component values change with the actual SOC of the battery. The resulting battery model was validated by comparing the simulated pulse response with the test data. The maximum voltage error was 310 mV, while the average error was only 6.7 mV, thereby providing a solid basis for subsequent energy-efficiency simulations of electric cars.
2.2 Inverter Model
The inverter in electric cars converts the DC power from the battery into AC power for the permanent-magnet synchronous motor. The inverter loss model is important because losses change with the modulation index, current, and switching frequency. I modeled the IGBT conduction loss, switching loss, and diode reverse-recovery loss as:
$$
P_{loss} = P_{IGBT,cond} + P_{IGBT,sw} + P_{diode,rec}
$$
The inverter efficiency is then obtained from the motor input power \(P_m\) and the total loss:
$$
\eta_i = \frac{P_m}{P_m + P_{loss}}
$$
Table 3 lists the characteristic parameters of the reference inverter used in this study.
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| IGBT and diode switching frequency | \(f_{sw}\) | 16 | kHz |
| IGBT equivalent resistance | \(R_{IGBT}\) | 4 | mΩ |
| Diode equivalent resistance | \(R_{diode}\) | 7 | mΩ |
| IGBT rated current | \(I_{rate}\) | 450 | A |
| IGBT rated voltage | \(U_{rate}\) | 750 | V |
| IGBT switching energy | \(W_{IGBT}\) | 28 | mJ |
| Diode recovery energy | \(W_{diode}\) | 7 | mJ |
2.3 Motor Model
The motor model is central to the energy consumption prediction of electric cars. Rather than relying on a fixed efficiency map, I used an electric-machine design tool that derives the performance map from the rated motor parameters. For a salient pole permanent-magnet synchronous motor, the electromagnetic torque can be written as:
$$
T_m = p \left[\psi_f i_q + \left(L_d – L_q\right)i_d i_q\right]
$$
where \(p\) is the number of pole pairs, \(\psi_f\) is the permanent-magnet flux linkage, \(L_d\) and \(L_q\) are the \(d\)- and \(q\)-axis inductances, and \(i_d\) and \(i_q\) are the corresponding stator current components. The motor efficiency is:
$$
\eta_m = \frac{T_m \omega_m}{T_m \omega_m + P_{Cu} + P_{Fe} + P_M}
$$
where \(P_{Cu}\) is the copper loss, \(P_{Fe}\) is the iron loss, and \(P_M\) is the mechanical loss. The motor full-load curve and efficiency map were obtained by considering the limiting current, voltage, and loss characteristics of the motor.
2.4 Two-Speed DCT Efficiency Model
The DCT efficiency of electric cars has a strong influence on the total energy consumption. In this work I modeled five main sources of power loss inside the two-speed wet DCT: clutch drag loss \(P_{clutch}\), gear mesh loss \(P_{mesh}\), windage and oil churning loss \(P_{wc}\), bearing loss \(P_{bear}\), and concentric-shaft viscous shear loss \(P_{shaft}\). The total transmission efficiency is therefore:
$$
\eta_T = \frac{P_m – \sum P_{loss}}{P_m}
=
\frac{P_m – P_{clutch} – P_{mesh} – P_{wc} – P_{bear} – P_{shaft}}{P_m}
$$
The drag torque of a disengaged wet clutch depends on the oil film radius, the relative speed, and the clutch geometry. The clutch torque can be expressed as:
$$
T_{clutch} = \frac{\pi \mu_c \omega N_c}{2 h_i}\left(R_{o}^{4} – R_{i}^{4}\right)
$$
where \(N_c\) is the number of friction surfaces and \(R_{o}\) is the effective outer radius of the oil film. The mesh loss was calculated from the gear friction coefficient, the applied torque, and the gear geometry. Churning loss and bearing loss were estimated with the standard formulas summarized in the literature. The DCT efficiency maps for both gears were then produced over the whole motor operating range. This detailed loss model is used in the multi-objective optimization, because it correctly characterizes the efficiency change with gear ratios and speed.
2.5 Longitudinal Dynamics and Energy Consumption
For the final evaluation, the complete vehicle model of electric cars combines the battery, inverter, motor, DCT, and longitudinal dynamics. The energy consumption per 100 km over a driving cycle is defined as:
$$
E_c = \frac{100}{L_c}
\int_{0}^{t_f}
\frac{1}{\eta_m(t)\,\eta_T(t)}
\left[
m_v g f
+
\frac{C_D A u^2(t)}{21.15}
+
\delta m_v \frac{du}{dt}
\right]
\frac{u(t)}{3.6}
\, dt
$$
where \(L_c\) is the total distance of the driving cycle. This expression was used as one of the objective functions when optimizing electric cars powertrain parameters.
3. Multi-Objective Optimization of Electric Cars Powertrain Parameters
Traditional parameter matching procedures often use a single performance index or a trial-and-error method. In my research, however, I needed to balance two conflicting objectives: minimizing energy consumption and minimizing the total cost of the powertrain. I therefore formulated a multi-objective optimization problem and solved it using an NSGA-II algorithm. The design vector combined the motor rated power \(P_{em}\), rated speed \(n_{em}\), rated voltage \(U_{em}\), the first gear ratio \(i_1\), and the second gear ratio \(i_2\):
$$
X = [P_{em}, \, n_{em}, \, U_{em}, \, i_1, \, i_2]
$$
The first objective function is the WLTP energy consumption per 100 km derived previously. The second objective is the powertrain cost, which includes the battery cell cost, the motor cost, and the fixed DCT cost:
$$
f_2(X) = 7\,N_s\,N_p + 63\,P_{em} + 2185
$$
Here \(N_s\) is determined by the selected voltage level, \(N_p\) is the fixed number of parallel battery cells, and \(P_{em}\) is the motor rated power in kW. Table 4 lists the design ranges and constraints used in the NSGA-II optimization.
| Item | Lower bound | Upper bound | Unit |
|---|---|---|---|
| Motor rated power \(P_{em}\) | 35 | 75 | kW |
| Motor rated speed \(n_{em}\) | 4000 | 6000 | rpm |
| Motor rated voltage \(U_{em}\) | 300 | 600 | V |
| First gear ratio \(i_1\) | 1.36 | 3.64 | – |
| Second gear ratio \(i_2\) | 1.0 | 2.0 | – |
| Maximum acceleration target 0-100 km/h | 10 | 12 | s |
| Maximum speed target | 150 | – | km/h |
| Maximum grade target at 5 km/h | 30 | – | % |
I performed the multi-objective optimization twice, once with a 12-second acceleration requirement and once with a 10-second acceleration requirement. Moreover, to show the value of the variable-efficiency DCT model, I compared it with a conventional approach in which the DCT efficiency was fixed at 96 %. The optimized Pareto solutions were analyzed to identify a trade-off point in which a small change in one objective no longer produces a major improvement in the other. This point was selected as the preferred design for each case.
Table 5 summarizes the final optimal candidates. The results show that when the variable transmission efficiency is considered, the optimized electric cars consume noticeably less energy than the design obtained with a constant transmission efficiency. In the 12-second acceleration case, the energy consumption decreases from 12.96 kWh/100 km to 12.48 kWh/100 km, while the cost remains almost unchanged. The corresponding total cost is about 29 000 RMB.
| Transmission model | Acceleration target | Energy consumption (kWh/100 km) | Total cost (RMB) | \(P_{em}\) (kW) | \(n_{em}\) (rpm) | \(U_{em}\) (V) | \(i_1\) | \(i_2\) |
|---|---|---|---|---|---|---|---|---|
| Fixed efficiency | 12 s | 12.96 | 29040 | 40 | 6000 | 300 | 3.44 | 1.72 |
| Variable efficiency | 12 s | 12.48 | 29048 | 41 | 6000 | 300 | 1.75 | 1.17 |
| Fixed efficiency | 10 s | 12.89 | 29149 | 51 | 6000 | 300 | 3.53 | 1.72 |
| Variable efficiency | 10 s | 12.48 | 29121 | 48 | 6000 | 300 | 1.75 | 1.17 |
The Pareto results also indicated an important physical trend. Under the variable-efficiency DCT model, lower gear ratios are preferred because the DCT efficiency in the moderate speed range is higher when the gear ratios are relatively low. At the same time, a larger motor power helps electric cars finish the WLTP cycle with lower average loading and slightly lower copper loss. However, a larger motor increases the initial purchase cost. The optimal solution must therefore balance the energy benefit and the cost increase. Compared with the fixed-efficiency model, the variable-efficiency model reduced the WLTP energy consumption by about 3.7% for the 12-s acceleration case and 3.3% for the 10-s case.
4. Regenerative Braking Control Strategy for Electric Cars
Regenerative braking is one of the most effective ways to extend the driving range of electric cars in urban driving conditions. When an electric car brakes, the motor can operate as a generator and convert a portion of the vehicle kinetic energy into electrical energy. Nevertheless, the total braking force requirement and vehicle stability must always be maintained. Therefore, I first studied the braking-force distribution between the front and rear axles according to the conventional braking theory and the ECE_R13 regulation. Then I designed a fuzzy controller that determines the percentage of the front-axle braking force provided by the electric motor.
4.1 Brake Force Distribution and Regulation Boundaries
The ideal braking-force distribution curve, often called the I-curve, represents the condition where the front and rear brakes lock simultaneously for a given road friction coefficient. If the front and rear braking forces are \(F_{bf}\) and \(F_{br}\), the total braking force follows:
$$
F_{bf} + F_{br} = m_v g z
$$
where \(z\) is the braking intensity. The vertical loads on the front and rear axles can be expressed as:
$$
F_{zf} = \frac{m_v g}{L}\left(L_2 + z h_g\right), \qquad
F_{zr} = \frac{m_v g}{L}\left(L_1 – z h_g\right)
$$
For the ideal I-curve, the braking forces also satisfy:
$$
F_{bf} = \varphi \frac{m_v g}{L}\left(L_2 + \varphi h_g\right)
$$
$$
F_{br} = \varphi \frac{m_v g}{L}\left(L_1 – \varphi h_g\right)
$$
The ECE_R13 regulation requires that, over a road friction coefficient range from 0.2 to 0.8, the braking intensity should satisfy:
$$
z \ge 0.1 + 0.85\left(\varphi – 0.2\right)
$$
The two ECE_R13 boundary equations for the front and rear adhesion utilization coefficients can be rearranged into a relationship between the front and rear braking forces:
$$
\frac{F_{bf}}{F_{zf}} \le \frac{z+0.07}{0.85}, \qquad
\frac{F_{br}}{F_{zr}} \le \frac{z+0.07}{0.85}
$$
Based on these equations and the vehicle geometry, I obtained the feasible front/rear braking-force allocation. In my control strategy, the braking intensity is divided into several operating zones. When the braking intensity is lower than about 0.22, all braking force can be placed on the front axle, which allows more regenerative energy from the motor. Between 0.22 and 0.53, the braking force follows the ECE Regulation lower boundary in order to maintain braking stability. If the braking intensity increases beyond 0.53 up to 0.7, I selected the \(F\)-line for a friction coefficient of 0.7 to maintain stable brake performance. For heavy braking above 0.7, the control strategy turns off regenerative braking entirely and relies on the hydraulic friction brakes to ensure occupant safety. Table 6 summarizes the four zones used for the front and rear braking force distribution.
| Braking intensity | Distribution rule | Regenerative braking status |
|---|---|---|
| \(z < 0.22\) | Front-axle braking only | Maximum motor braking allowed |
| \(0.22 \le z < 0.53\) | ECE_R13 lower boundary | Front motor braking actively controlled |
| \(0.53 \le z < 0.7\) | \(F\)-line for \(\varphi = 0.7\) | Partial motor braking retained |
| \(z \ge 0.7\) | Ideal I-curve | Regenerative braking disabled |
4.2 Structure of the Fuzzy Controller
The regenerative force of electric cars cannot simply be set as a constant fraction of the front braking force. It must respond to the instantaneous braking intensity, the battery state of charge, and the vehicle speed. Therefore, I developed a Mamdani fuzzy controller with three inputs and one output. The inputs are the braking intensity \(z\), the vehicle speed \(v\), and the battery SOC; the output is the regenerative braking coefficient \(K\), whose range is \([0,1]\). The actual motor regenerative torque command is then obtained by:
$$
T_{reg} = K \cdot \min\left(T_{reg,ref}, T_{reg,max}\right)
$$
where \(T_{reg,ref}\) is the torque reference derived from the front-axle brake force and transmission ratio, and \(T_{reg,max}\) is the maximum motor torque limit at the current speed. If the motor torque is smaller than the total front-axle demand, the hydraulic brake provides the remaining front brake torque. This arrangement ensures that the total braking force is never reduced by the energy recovery system.
The linguistic variables of the three inputs were each divided into three fuzzy sets: S, M, and B. The output \(K\) was divided into five fuzzy sets: VL, L, M, H, and VH. The fuzzy controller executes 27 inference rules based on the following principle: when the battery SOC is high, the regenerative braking capacity is reduced to protect the battery; when the speed is very low, the regenerative braking torque is limited because the induced-electromotive force of the motor becomes too low; when the braking intensity is high, the motor braking ratio is reduced to preserve stability. Table 7 shows one representative subset of the 27 rules used in the controller.
| No. | Braking intensity \(z\) | Speed \(v\) | Battery SOC | Braking coefficient \(K\) |
|---|---|---|---|---|
| 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 | S | S | M | VL |
| 10 | M | M | M | H |
| 11 | M | B | M | H |
| 12 | B | M | M | M |
| 13 | B | B | M | H |
| 14 | S | M | S | M |
| 15 | S | B | S | H |
| 16 | M | S | S | L |
| 17 | M | M | S | VH |
| 18 | B | S | S | M |
| 19 | B | M | S | M |
The membership functions for the fuzzy controller should ideally be fine tuned. For this reason, the controller cannot rely entirely on expert experience. I therefore integrated a genetic algorithm into the design flow to optimize the membership-function parameters.
5. Optimization of the Fuzzy Regenerative Braking Controller Using a Genetic Algorithm
Fuzzy controllers offer a robust way to handle nonlinear braking processes in electric cars, but their accuracy strongly depends on the selection of membership-function parameters and rule weights. To improve the objectivity of the controller, I encoded all important membership-function parameters as a chromosome and used a genetic algorithm to maximize the recovered regenerative energy or to maximize the braking coefficient under acceptable stability constraints.
The braking intensity \(z\) was represented by Gaussian membership functions:
$$
\mu_z(x) = \exp\left[-\frac{(x-g)^2}{2\sigma^2}\right]
$$
The vehicle speed \(v\) and the battery SOC were represented by trapezoidal membership functions:
$$
\mu_v(x) =
\begin{cases}
0, & x \le a_1 \\
\dfrac{x-a_1}{b_1-a_1}, & a_1 \le x \le b_1 \\
1, & b_1 \le x \le c_1 \\
\dfrac{d_1-x}{d_1-c_1}, & c_1 \le x \le d_1 \\
0, & x \ge d_1
\end{cases}
$$
The output regenerative coefficient \(K\) used a mixture of triangular and trapezoidal membership functions. The triangular membership function is defined as:
$$
\mu_K(x) =
\begin{cases}
0, & x \le a_2 \\
\dfrac{x-a_2}{b_2-a_2}, & a_2 \le x \le b_2 \\
\dfrac{c_2-x}{c_2-b_2}, & b_2 \le x \le c_2 \\
0, & x \ge c_2
\end{cases}
$$
In the genetic-algorithm optimization, each membership-function breakpoint was represented by integer genes in the range from 1 to 10. The fitness value was related to the regenerative braking energy recovered during a representative urban cycle. I applied selection, crossover, and mutation to the population, with a population size of 200 and a maximum of 40 generations. After convergence, the optimized membership function parameters were loaded into the Simulink model of the regenerative braking controller. This hybrid intelligent approach avoids relying purely on trial-and-error fuzzy rule design and gives a stronger guarantee that electric cars can recover as much braking energy as possible.
6. Simulation Results and Discussion
I implemented the entire system in MATLAB/Simulink. The driver model uses a PI controller to follow the target speed of standard driving cycles. The main control model consists of a demand braking torque calculation module, a regenerative braking torque management module, and an electric motor torque control module. The regenerative braking module blocks motor braking when the battery SOC exceeds 95%, when the vehicle speed is lower than 10 km/h, or when the braking intensity exceeds 0.7. These boundary conditions protect the battery, the motor, and the hydraulic brake system.
To evaluate the optimized control strategy, I selected two driving cycles: NEDC and WLTP. These cycles are widely used to test the energy consumption and range performance of electric cars. In each simulation, the initial SOC was set to 95%, and the final simulation threshold was 10%. I compared three control strategies:
- the ideal braking-force distribution strategy used as a baseline;
- a conventional fuzzy regenerative braking control strategy;
- the proposed genetic-algorithm-optimized fuzzy control strategy.
6.1 Speed Tracking Performance
In the NEDC test, the optimized controller produced a mean speed tracking accuracy of 98.6%. The unoptimized controller achieved 96.2%. The speed tracking error remained within ±0.3 m/s during acceleration, constant-speed, and braking segments. In the WLTP cycle, the optimized controller achieved an average speed tracking accuracy of 98.2%. These results show that the optimization process does not reduce the braking comfort or the vehicle controllability of electric cars.
6.2 NEDC Simulation Results
For the NEDC cycle, Figure 1 shows the SOC evolution of electric cars under the three strategies. The baseline ideal-braking strategy leads to the fastest SOC decrease, with a final SOC of about 0.810. The conventional fuzzy strategy improves the final SOC to 0.851. The optimized fuzzy control strategy further increases the final SOC to 0.889. This means that the optimized controller reduces the SOC loss by 4.46 percentage points relative to the conventional fuzzy controller and by 9.75 percentage points relative to the ideal-braking baseline.
The total recovered battery energy after the NEDC test is shown in Table 8. The optimized fuzzy strategy recovers about 2.41% more energy than the conventional fuzzy strategy and about 27.71% more energy than the ideal-braking strategy. This large improvement demonstrates that the use of a second-order battery model and an optimized fuzzy braking controller is important in electric cars. The high frequency of braking events in the NEDC cycle gives particularly large opportunities for recovery.
| Control strategy | Final SOC (%) | Relative SOC improvement vs baseline (%) | Recovered energy (kW·h) | Recovery improvement vs baseline (%) |
|---|---|---|---|---|
| Ideal braking distribution | 81.0 | – | 0.98 | – |
| Conventional fuzzy control | 85.1 | +4.1 | 1.18 | +20.4 |
| GA-optimized fuzzy control | 88.9 | +7.9 | 1.21 | +23.5 |
Table 8 also illustrates that the difference between the conventional and GA-optimized fuzzy controllers becomes more visible when the SOC decreases from a high value. A high SOC prevents regenerative charging because the battery would otherwise exceed its maximum voltage. The genetic algorithm adjusts the membership functions so that the controller reduces the regenerative torque gradually instead of switching it off abruptly, thus recovering more energy while protecting the battery.
6.3 WLTP Simulation Results
For the WLTP cycle, the vehicle experiences a broader range of speeds and accelerations. The simulation results show that the final SOC values of the three strategies are 70.4%, 75.6%, and 81.8% for the ideal-braking strategy, conventional fuzzy strategy, and GA-optimized fuzzy strategy, respectively. The optimized strategy improves the final SOC by 11.4 percentage points relative to the baseline strategy. In terms of recovered energy, the optimized fuzzy controller recovers 6.45% more energy than the conventional fuzzy strategy and 7.1% more energy than the ideal-braking distribution strategy.
Table 9 summarizes the WLTP results for the different controllers.
| Control strategy | Final SOC (%) | Relative SOC improvement vs baseline (%) | Recovered energy (kW·h) | Recovery improvement vs baseline (%) |
|---|---|---|---|---|
| Ideal braking distribution | 70.4 | – | 1.38 | – |
| Conventional fuzzy control | 75.6 | +5.2 | 1.55 | +12.3 |
| GA-optimized fuzzy control | 81.8 | +11.4 | 1.65 | +19.6 |
The results confirm that the optimized regenerative braking controller provides stable energy benefits rather than shifting energy recovery from one operating point to another. It is particularly effective in the low and medium speed ranges, where many deceleration events occur in city driving. This is exactly the operating area where electric cars can recover kinetic energy without violating vehicle stability.
6.4 Energy Consumption of the Optimized Powertrain and Braking Strategy
After the powertrain multi-objective optimization, the final electric cars configuration with a two-speed DCT reaches a WLTP energy consumption of about 12.50 kWh per 100 km. The regenerative braking controller is then applied to the same powertrain. The two-speed DCT extends the high-efficiency region of the motor and enables the regenerative braking force to be matched with the motor speed more flexibly. Compared with a single-speed transmission vehicle, the two-speed DCT electric car has a lower average energy consumption and a higher amount of recuperated energy during urban cycles. The combined effect of the optimized gear ratios and the optimized fuzzy braking control strategy is substantial: the effective braking energy recovery ratio rises, the battery SOC decreases more slowly, and the total driving range is extended.
7. Conclusions
In this study I developed a comprehensive optimization framework for the powertrain parameter matching and the regenerative braking control strategy of electric cars driven by a two-speed wet DCT. The main conclusions are as follows:
- The detailed component models of the battery, inverter, motor, DCT, and vehicle longitudinal dynamics provide a reliable evaluation environment for electric cars powertrain design. The initial matching results satisfy all required dynamic performance indicators and range targets.
- The multi-objective NSGA-II optimization successfully balances energy consumption and powertrain cost. The optimized variable-efficiency model reduces WLTP energy consumption to approximately 12.50 kWh/100 km while keeping the total powertrain cost almost unchanged. This result can only be obtained when the variable efficiency of the DCT is modeled explicitly.
- The regenerative braking control strategy based on braking-force distribution boundaries and fuzzy logic can increase the recovered energy of electric cars. A genetic algorithm was used to optimize the membership function parameters, resulting in a final SOC improvement of about 4.46 percentage points in NEDC and 8.24 percentage points in WLTP compared with the conventional fuzzy controller.
- The optimized fuzzy regenerative controller respects all major constraints: battery SOC limit, low-speed cut-off, braking intensity limit, and vehicle stability. Therefore, it is not only beneficial for energy economy but also acceptable for brake safety.
The results of this study provide a practical theoretical basis for the design of future electric cars powertrain systems and energy management strategies. Future work should extend the proposed method to integrated thermal management and real-time road-adaptive control of electric cars, and should include experimental validation on a physical test bench or prototype vehicle.