Electric cars are now regarded as one of the most promising ways to reduce the dependence of road transport on fossil fuels, and their market share has been growing rapidly in recent years. However, two engineering problems still prevent electric cars from achieving the same user acceptance as conventional vehicles. The first is the difficulty of matching the motor, battery, transmission and final drive in a way that satisfies acceleration, gradeability and energy consumption simultaneously. The second is the fact that a large part of the kinetic energy dissipated during braking can be recovered if the regenerative braking system is controlled intelligently. The present article is concerned with both of these issues, and it describes the work that I carried out on a front-wheel-drive electric car equipped with a two-speed wet dual-clutch transmission, hereafter referred to as a two-speed DCT.
Regenerative braking has received much attention in the context of urban driving, where frequent stops and starts create many opportunities for energy recovery. If the electric machine operates as a generator during deceleration, the braking energy can be converted into electrical energy and stored in the battery. Nevertheless, the amount of recovered energy is strongly affected by the way in which the total braking demand is divided between the front and rear axles, and by the way in which the front-axle braking demand is divided between the electric motor and the hydraulic friction brake. I therefore developed a composite control framework that combines fuzzy logic with a genetic algorithm. A dual-layer fuzzy controller was designed with the demanded braking intensity, the battery state of charge and the vehicle speed as inputs. The genetic algorithm was then used to optimize the parameters of the membership functions, so that the final control strategy no longer relied entirely on subjective expert experience.
Throughout this article, I will refer to the vehicle under study as an electric car with a two-speed DCT. Although some of the numerical data are specific to this vehicle, the methods can be transferred to other electric cars with different mass, aerodynamic drag and motor ratings. The aim is to demonstrate how an integrated simulation environment can support the simultaneous optimization of powertrain parameters and braking energy management.
Background and Research Status
The global transition towards low-carbon mobility has made electric cars a central element of the automotive industry. In many countries, government incentive schemes have been introduced not only for private customers but also for manufacturers, with the purpose of accelerating the deployment of battery electric cars, plug-in hybrid cars and fuel-cell cars. Among these alternatives, battery electric cars have expanded most quickly because their powertrains are relatively simple and because their well-to-wheel efficiency can be high when the electricity is generated from low-carbon sources.
In the Chinese market, the importance of electric cars has increased significantly. National plans have set clear targets for the average electricity consumption of pure electric passenger cars, and a target of about 12.0 kWh per 100 km was proposed for pure electric passenger cars by 2025. Although the latest battery technology has improved energy density, the range of affordable electric cars is still limited by battery cost and weight. As a result, many investigations have focused on reducing the energy consumption of electric cars through better vehicle design, more efficient components and more intelligent control strategies.
Previous studies on powertrain parameter matching have considered both single-speed and multi-speed transmissions. Single-speed configurations are simple and economical, but they often require a high-performance motor to satisfy the conflicting demands of low-speed hill climbing and high-speed cruising. Multi-speed transmissions allow the motor operating point to be shifted into a more efficient region, and they reduce the peak torque that the motor must deliver. Researchers have optimized gear ratios and motor parameters using methods such as dynamic programming, particle swarm optimization and non-dominated sorting genetic algorithms. In several of these studies, the efficiency of the motor and the transmission was assumed to be constant. I considered this assumption questionable because both the motor efficiency and the transmission efficiency vary strongly with torque, speed and engaged gear ratio.
In the field of regenerative braking, earlier work has shown that urban driving can waste a substantial percentage of the total traction energy during braking. Many control strategies have been proposed in the literature, including rule-based strategies, fuzzy logic strategies, model-predictive strategies and strategies based on multi-objective optimization. Several studies have considered the importance of the ECE_R13 braking regulation, the ideal braking distribution curve and the maximum torque capability of the motor. However, conventional fuzzy controllers often depend on membership functions that are chosen by trial and error. A genetic algorithm can be used to search for a more adequate set of membership-function parameters, which was the approach that I adopted in this work.
Vehicle Configuration and Design Targets
The vehicle selected for this research is a small urban electric car with front-wheel drive. Its driveline consists of a permanent-magnet synchronous motor, a two-speed wet DCT, a final drive and the front wheels. The main reason for choosing this architecture is that the two-speed DCT can enlarge the high-efficiency operating region of the motor, improve the acceleration performance and increase the driving range without making the vehicle cost prohibitive.
The main vehicle parameters and performance targets used in the matching study are presented in the table below.
| Parameter | Symbol | Value | Unit |
|---|---|---|---|
| Vehicle 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 | \(L\) | 2.490 | m |
| Height of center of gravity | \(h_g\) | 0.510 | m |
| Final drive ratio | \(i_0\) | 4.058 | – |
| Maximum speed | \(u_{max}\) | ≥150 | km/h |
| Gradeability at 5 km/h | – | ≥30 | % |
| Gradeability at 30 km/h | – | ≥20 | % |
| 0–100 km/h acceleration time | \(t_a\) | <12 | s |
| Driving range at 60 km/h | – | ≥150 | km |
Powertrain Parameter Matching
The first part of the matching procedure was the selection of the motor. The motor has to satisfy the maximum speed condition, the gradeability condition and the acceleration condition. The road load equation used for all three conditions can be written as
\[
F_t=F_f+F_w+F_i+F_j,
\]
where \(F_f\) is the rolling resistance, \(F_w\) is the aerodynamic resistance, \(F_i\) is the grade resistance and \(F_j\) is the acceleration resistance. For a vehicle driven through a transmission with total ratio \(i_g i_0\), the traction force at the tyre contact patch is
\[
F_t=\frac{T_m i_g i_0 \eta_T}{R_D}.
\]
The resulting longitudinal dynamics equation can be expressed as
\[
\frac{T_m i_g i_0 \eta_T}{R_D}
=
m_v g f\cos\alpha
+\frac{C_D A}{21.15}u^2
+m_v g\sin\alpha
+\delta m_v\frac{\mathrm{d}u}{\mathrm{d}t},
\]
where \(u\) is the vehicle speed expressed in km/h, \(\alpha\) is the road slope angle, \(g\) is the gravitational acceleration and \(\delta\) is the rotational mass coefficient. The rotational mass coefficient was taken as 1.07 in the calculations.
For the maximum-speed condition, the required motor power is
\[
P_{u\max}
=
\frac{1}{\eta_T}
\left(
\frac{m_v g f u_{\max}}{3600}
+
\frac{C_D A u_{\max}^3}{76140}
\right).
\]
For the gradeability condition at a steady speed \(u_c\), the required power is
\[
P_{\alpha\max}
=
\frac{u_c}{3600\eta_T}
\left(
m_v g f\cos\alpha_{\max}
+
m_v g\sin\alpha_{\max}
+
\frac{C_D A u_c^2}{21.15}
\right).
\]
For the acceleration condition within a time \(t_a\), the power is obtained by integrating the instantaneous road-load power during the acceleration event. A convenient expression for the initial design stage is
\[
P_{\mathrm{acc}}
=
\frac{1}{\eta_T}
\left[
\frac{\delta m_v}{2t_a}
\left(u_f^2+u_0^2\right)
+
\frac{2}{3}m_v g f u_f
+
\rho_a C_D A \frac{u_f^3}{6}
\right],
\]
where \(u_f\) is the final speed of the acceleration test, \(u_0\) is the initial speed and \(\rho_a\) is the air density. The motor peak power \(P_{\max}\) must be greater than or equal to the maximum of the three power demands, with a safety margin. In the present matching exercise I added a 15% margin to the maximum required value. The mathematical expression is
\[
P_{\max} \ge 1.15 \max\left(P_{u\max},P_{\alpha\max},P_{\mathrm{acc}}\right).
\]
The rated motor power is related to the peak power by the overload factor \(\lambda\). A value of \(\lambda=2\) was used here, so the rated power became
\[
P_e=\frac{P_{\max}}{\lambda}.
\]
In addition, the rated power has to be sufficient to sustain the maximum cruising speed, so the final selection must obey \(P_e\ge P_{u\max}\).
The maximum motor speed was calculated from the maximum vehicle speed and the total ratio of the highest gear:
\[
n_{\max}\ge \frac{u_{\max} i_1 i_0}{0.377 R_D}.
\]
Because the selected motor has a wide constant-power region, the rated motor speed was taken as one half of the peak motor speed:
\[
n_e=0.5 n_{\max}.
\]
The peak motor torque was obtained from the conventional power-torque relationship:
\[
T_{\max}=9550\frac{P_{\max}}{n_{\max}}.
\]
Battery Sizing
The battery of an electric car must satisfy two basic constraints. First, the stored energy has to provide the required range. Second, the instantaneous power capability has to be sufficient for the peak power demand of the motor and the auxiliary systems. Based on the 60 km/h constant-speed range target, the battery capacity can be estimated as
\[
C_{\mathrm{batt}}
\ge
\frac{\displaystyle
\left(m_v g f+\frac{C_D A u^2}{21.15}\right)S
}{
3600\,\eta_m\eta_T\eta_b \Delta\mathrm{SOC}
},
\]
where \(S\) is the required driving range, \(\eta_m\) is the motor efficiency, \(\eta_T\) is the transmission efficiency, \(\eta_b\) is an auxiliary-consumption coefficient and \(\Delta\mathrm{SOC}\) is the usable depth of discharge. The auxiliary coefficient was set to 0.2 in this work.
The power capability at the rated voltage \(U_{\mathrm{batt}}\) can be written as
\[
C_{\mathrm{power}}
\ge
\frac{1}{k U_{\mathrm{batt}}}
\left(
\frac{P_{\max}}{\eta_m}+P_{\mathrm{aux}}
\right),
\]
where \(k\) is the maximum C-rate of the cell and \(P_{\mathrm{aux}}\) is the auxiliary power demand. The final battery capacity is
\[
C_{\mathrm{pack}}=\max\left(C_{\mathrm{batt}},C_{\mathrm{power}}\right).
\]
Lithium-ion cells with a nominal voltage of 3.7 V and a nominal capacity of 5 Ah were selected for the pack. The number of series cells is
\[
N_s=\frac{U_{\mathrm{batt}}}{U_{\mathrm{cell}}},
\]
and the number of parallel branches is
\[
N_p=\frac{C_{\mathrm{pack}}}{C_{\mathrm{cell}}}.
\]
Using the above equations, the required rated voltage of the battery pack was 320 V, the capacity was about 115 Ah, the number of series cells was 87 and the number of parallel branches was 23.
Transmission Ratio Matching
The first gear ratio of a two-speed DCT must provide sufficient torque for hill climbing and acceleration, while the second gear ratio must provide sufficient speed for the maximum-vehicle-speed condition. The first-gear ratio can be determined from
\[
i_1\ge
\frac{R_D\left[
m_v g(f\cos\alpha_{\max}+\sin\alpha_{\max})
+\dfrac{C_D A u_{\alpha}^2}{21.15}
\right]}
{T_{\max} i_0\eta_T},
\]
where \(u_{\alpha}\) is the speed during the hill-climbing test. The anti-slip condition must also be considered. To avoid excessive wheel slip, the driving torque at the front wheels should not exceed the maximum torque that the road can support. This condition can be written as
\[
\frac{T_{\max} i_1 i_0 \eta_T}{R_D}
\le
\varphi_{\max} m_v g \frac{L_2+\varphi_{\max} h_g}{L},
\]
where \(\varphi_{\max}\) is the maximum adhesion coefficient and \(L_2\) is the distance from the center of gravity to the rear axle.
The second gear ratio was determined by considering the maximum-speed driving condition and the desired cruising speed. The ratio has to be small enough to permit a high vehicle speed at the motor peak speed, but it also has to be large enough to avoid an excessive torque demand at high speed. The selected ratios were further refined by the multi-objective optimization described in the next section.
The results of the initial parameter matching are shown in the following table.
| Component | Parameter | 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 | N m |
| Motor | Rated torque | 120 | N m |
| DCT | First gear ratio | 1.80 | – |
| DCT | Second gear ratio | 1.30 | – |
| Battery | Rated voltage | 320 | V |
| Battery | Rated capacity | 115 | Ah |
| Battery | Series cells | 87 | – |
| Battery | Parallel branches | 23 | – |
Modelling of the Powertrain and the Vehicle
To evaluate the powertrain candidates in a realistic manner, I built a complete simulation environment in MATLAB and Simulink. The model includes the battery, the inverter, the motor, the two-speed DCT and the longitudinal vehicle dynamics. The efficiency of each component is not constant; it is calculated as a function of the operating point.
Battery model
I used a second-order resistance–capacitance equivalent circuit model for the lithium-ion battery pack. The model consists of an open-circuit voltage source, an ohmic resistance \(R_0\) and two parallel RC branches that represent electrochemical polarization and concentration polarization. The governing equations are
\[
\begin{aligned}
U_t&=U_{oc}-U_1-U_2-I_bR_0,\\
\frac{\mathrm{d}U_1}{\mathrm{d}t}&=-\frac{U_1}{R_1C_1}+\frac{I_b}{C_1},\\
\frac{\mathrm{d}U_2}{\mathrm{d}t}&=-\frac{U_2}{R_2C_2}+\frac{I_b}{C_2}.
\end{aligned}
\]
The state of charge is obtained by Coulomb counting:
\[
\mathrm{SOC}(t)=\mathrm{SOC}_0-\frac{1}{3600C_{\mathrm{cell}}}\int_{0}^{t}\eta_b I_b(\tau)\,\mathrm{d}\tau.
\]
I identified the model parameters from pulse discharge data. The ohmic resistance was obtained from the instantaneous voltage jump at the beginning and at the end of the discharge pulse. The remaining RC parameters were extracted by fitting the zero-input voltage response after the pulse with an exponential function. This procedure was repeated at different initial SOC levels, so that the battery model could represent the variation of internal resistance with the state of charge.
Inverter model
The inverter was represented by a power-loss model composed of conduction losses, switching losses and diode recovery losses. The inverter output power was therefore smaller than the DC input power by an amount \(\Delta P_{\mathrm{inv}}\). The total inverter loss can be expressed as
\[
P_{\mathrm{inv}}=P_{\mathrm{cond}}+P_{\mathrm{sw}}+P_{\mathrm{rec}}.
\]
Using the rated module data of a commercial IGBT module, I calculated the inverter efficiency as
\[
\eta_i=\frac{P_m}{P_m+P_{\mathrm{inv}}},
\]
where \(P_m\) is the AC power supplied to the motor. The inverter switching frequency was 16 kHz, the rated current was 450 A and the rated voltage was 750 V.
Motor model
The motor was modelled with a \(d\)-\(q\) equivalent circuit. The voltage equations in the rotating reference frame are
\[
\begin{aligned}
v_d&=R_a i_d-\omega_e L_q i_q,\\
v_q&=R_a i_q+\omega_e\left(L_d i_d+\psi_{\mathrm{PM}}\right),
\end{aligned}
\]
where \(R_a\) is the stator resistance, \(L_d\) and \(L_q\) are the \(d\)- and \(q\)-axis inductances, \(\omega_e\) is the electrical angular speed and \(\psi_{\mathrm{PM}}\) is the permanent-magnet flux linkage. The electromagnetic torque is
\[
T_m=\frac{3}{2}p\left[\psi_{\mathrm{PM}}i_q+\left(L_d-L_q\right)i_di_q\right].
\]
The motor efficiency was calculated from the output power and the total losses:
\[
\eta_m=\frac{T_m\omega_m}{T_m\omega_m+P_{\mathrm{Cu}}+P_{\mathrm{Fe}}+P_{\mathrm{mech}}}.
\]
The copper loss and iron loss were computed from the equivalent currents in the \(d\)-\(q\) frame:
\[
\begin{aligned}
P_{\mathrm{Cu}}&=R_a\left(i_d^2+i_q^2\right),\\
P_{\mathrm{Fe}}&=R_c\left(i_{cd}^2+i_{cq}^2\right).
\end{aligned}
\]
A complete motor efficiency map was generated for each candidate motor design by varying the motor design parameters. This approach is more reliable than a simple look-up table because it allows the motor efficiency map to change when the rated power, rated voltage or rated speed of the motor is changed during optimization.
Transmission model
The DCT model accounts for five types of power loss: drag losses in the open wet clutch, gear mesh losses, windage and oil churning losses, bearing and seal losses, and viscous shear losses between the concentric shafts. The overall transmission efficiency is
\[
\eta_T=1-\frac{\sum P_{\mathrm{loss}}}{P_{\mathrm{in}}}.
\]
For a wet multi-plate clutch, the drag torque is highly dependent on the oil film distribution between the separator plates. At low speeds, the film covers the entire friction surface, while at high speeds centrifugal forces reduce the effective outer radius of the oil film. The drag torque of an open clutch is
\[
T_{\mathrm{cl}}
=
\int_{R_{\mathrm{in}}}^{R_{\mathrm{out}}}
2\pi r^2 \tau_c\,\mathrm{d}r,
\]
where \(\tau_c\) is the shear stress in the oil film. The decrease of the effective oil-film radius was described by a flow-balance model, which I used to obtain the clutch drag loss at each motor speed.
The gear mesh loss was calculated from the gear mesh efficiency. For a helical gear pair, the meshing loss coefficient depends on the friction coefficient and the sliding velocity. The friction coefficient was calculated with an empirical formula that is valid in the loaded and lubricated operating range of the gearbox.
The windage and churning losses were calculated according to standard gear-testing practice. The model includes the losses generated by the shaft surfaces, the gear faces and the gear teeth. The bearing losses were divided into a load-dependent component and a speed-dependent component. In addition, the relative motion of the two concentric shafts produces a viscous shear torque that is proportional to the differential speed and the oil dynamic viscosity:
\[
T_{\mathrm{shaft}}
=
\frac{4\pi^2\mu_c h_c r_{\mathrm{co}}^2 r_{\mathrm{ci}}^2}{\Delta h}\Delta\omega.
\]
Because the DCT has two different gear ratios, the transmission model was solved separately for first-gear operation and second-gear operation. The resulting efficiency map is a function of motor speed and input torque, which was then used in the vehicle-level longitudinal dynamics model.
Longitudinal vehicle dynamics
The complete electric car was represented by the longitudinal force balance. The acceleration performance was evaluated by integrating the longitudinal acceleration from the launch speed to 100 km/h. The gear shift point affects the acceleration time, so the shift schedule was included in the simulation. In this work, the upshift point from the first gear to the second gear was set at 50 km/h, and the downshift point was set at 42 km/h in order to prevent frequent shifting.
The initial matching result was verified with the simulation platform. The simulation produced a maximum vehicle speed of 163.69 km/h, a gradeability of 35.45% at 5 km/h and 24.88% at 30 km/h, and an acceleration time of 9.54 s from 0 to 100 km/h. The range calculated by the 60 km/h constant-speed method was 175.41 km. All of these values satisfy the design targets listed earlier in this section.
Multi-Objective Optimization of Powertrain Parameters
The conventional matching procedure described above gives a feasible parameter set, but not necessarily an optimal one. In a multi-objective framework, the energy consumption and the powertrain cost are conflicting objectives. I therefore formulated the following multi-objective problem:
\[
\begin{aligned}
&\min \quad \bigl(f_1(\mathbf{x}), f_2(\mathbf{x})\bigr),\\
&f_1(\mathbf{x})=E_C(\text{WLTP}),\\
&f_2(\mathbf{x})=C_{\mathrm{pack}}+C_{\mathrm{motor}}+C_{\mathrm{trans}},
\end{aligned}
\]
where \(\mathbf{x}\) is the design vector:
\[
\mathbf{x}=[P_{\mathrm{em}},n_{\mathrm{em}},U_{\mathrm{em}},i_1,i_2],
\]
and where \(P_{\mathrm{em}}\) is the motor rated power, \(n_{\mathrm{em}}\) is the motor rated speed, \(U_{\mathrm{em}}\) is the motor rated voltage, \(i_1\) is the first-gear ratio and \(i_2\) is the second-gear ratio of the DCT.
The first objective \(f_1\) is the energy consumed over the WLTP driving cycle, expressed in kWh per 100 km. The second objective \(f_2\) is the total cost of the battery, motor and transmission. According to the information collected from suppliers, the battery cell cost was taken as 7 RMB per cell and the motor cost was taken as 63 RMB per kW of rated power. The transmission cost was assumed to be 1560 RMB because it is not strongly affected by the design variables considered in this study.
The optimization constraints include the maximum vehicle speed, the gradeability, the acceleration time and the wheel-slip condition:
\[
\begin{aligned}
u_{\max}&\ge150~\text{km/h},\\
\alpha_{\max}&\ge30\%,\\
t_{0-100}&\le12~\text{s},\\
\frac{T_{\max}i_1i_0\eta_T}{R_D}&\le \varphi_{\max}m_vg\frac{L_2+\varphi_{\max}h_g}{L}.
\end{aligned}
\]
The multi-objective genetic algorithm used here is based on the NSGA-II procedure. The population size was 200, the number of generations was 40, the crossover probability was 0.8 and the mutation probability was 0.2. During each function evaluation, the motor design tool generated an efficiency map for the proposed motor, the DCT model generated an efficiency map for the proposed gear ratios, and the longitudinal vehicle model calculated the energy consumption over the WLTP cycle.
To show the influence of transmission efficiency modelling, I performed the optimization in two ways. In the first case, the transmission efficiency was fixed at 0.96. In the second case, the transmission efficiency was calculated with the detailed loss model described earlier. The Pareto solutions obtained in these two cases were markedly different. When the variable-efficiency model was used, the optimizer selected lower gear ratios because low gear ratios reduce the transmission loss at low and moderate vehicle speeds. The energy consumption at the knee point of the Pareto front was reduced by about 3.7% for the 12 s acceleration target and by about 3.3% for the 10 s acceleration target compared with the fixed-efficiency optimization. This shows that a constant-efficiency assumption can underestimate the benefit of a multi-speed transmission for electric cars.
The selected optimal design candidates are presented in the table below.
| Transmission efficiency model | Acceleration target (s) | Energy consumption (kWh/100 km) | Powertrain cost (RMB) | Rated power (kW) | Rated speed (rpm) | Voltage (V) | \(i_1\) | \(i_2\) |
|---|---|---|---|---|---|---|---|---|
| Fixed efficiency | 12 | 12.96 | 29040 | 40 | 6000 | 300 | 3.44 | 1.72 |
| Variable efficiency | 12 | 12.48 | 29048 | 41 | 6000 | 300 | 1.75 | 1.17 |
| Fixed efficiency | 10 | 12.89 | 29149 | 51 | 6000 | 300 | 3.53 | 1.72 |
| Variable efficiency | 10 | 12.48 | 29121 | 48 | 6000 | 300 | 1.75 | 1.17 |
The results show that the powertrain cost is sensitive to the acceleration target. Reducing the 0–100 km/h acceleration time from 12 s to 10 s increases the cost by a small amount, while the energy consumption remains almost unchanged. More importantly, the best design obtained with the variable-efficiency transmission model consumes less energy than the best design obtained with the fixed-efficiency model, without a significant increase in cost. The optimized electric car with the two-speed DCT therefore provides a better balance between economy and dynamic performance.
Regenerative Braking Control Strategy
The braking energy recovery potential of electric cars depends strongly on the brake force distribution. In this research I first analyzed the ideal brake force distribution curve, the ECE_R13 regulation curve, and the constant-adhesion-coefficient F and R line groups. The purpose of this analysis was to define a feasible region in which the front-axle braking force and rear-axle braking force must lie so that the vehicle remains stable during braking.
When the front axle and rear axle are locked simultaneously on a road with a given adhesion coefficient \(\varphi\), the temporary braking forces can be expressed as
\[
\begin{aligned}
F_{bf}&=\varphi m_v g\frac{L_2+\varphi h_g}{L},\\
F_{br}&=\varphi m_v g\frac{L_1-\varphi h_g}{L}.
\end{aligned}
\]
Eliminating \(\varphi\) from the above equations yields the ideal braking force distribution curve, usually called the I-curve:
\[
F_{br}
=
\frac{m_v g}{2h_g}
\left[
\sqrt{L^2+\frac{4h_g L F_{bf}}{m_v g}}
-\frac{L}{h_g}
-\frac{2F_{bf}h_g}{m_v gL}
\right].
\]
The ECE_R13 regulation requires that the adhesion utilization of the front axle should not be larger than a value related to the braking intensity \(z\). Using the definition of the front-axle adhesion coefficient,
\[
\varphi_f
=
\frac{F_{bf}L}{m_v g(L_2+h_g z)},
\]
and the regulation condition
\[
\varphi_f\le \frac{z+0.07}{0.85},
\]
I obtained the following boundary:
\[
F_{bf}
\le
\frac{m_v g(L_2+h_g z)}{L}
\frac{z+0.07}{0.85}.
\]
This inequality represents the lower boundary of the total braking force that is allowed by the regulation. In the graphical representation of the braking force distribution diagram, this boundary intersects the horizontal axis at a braking intensity of approximately \(z=0.22\). For an electric car with front-wheel drive, the front-axle braking force can therefore be provided entirely by the motor during very light braking. When the braking intensity is between 0.22 and 0.53, the front-axle braking force follows the ECE_R13 boundary curve. When the braking intensity is higher than 0.53, the braking force distribution follows the F-line corresponding to a dry asphalt road. At braking intensities greater than 0.7, the regenerative braking system is disabled so that the hydraulic brakes can provide a rapid and stable emergency braking response.
The front-wheel braking force of the studied car is composed of a regenerative component and a hydraulic friction component. To allocate these two components, I designed a fuzzy controller with three inputs and one output. The three inputs are:
- the demanded braking intensity \(z\),
- the vehicle speed \(v\),
- the battery state of charge \(\mathrm{SOC}\).
The output is the regenerative braking distribution coefficient \(K\), which represents the fraction of the front-axle braking force that is supplied by the electric motor. The total front-axle regenerative torque is therefore
\[
T_{\mathrm{reg}}=K T_{\mathrm{front}},
\]
and the remaining front-axle torque is supplied by the friction brake:
\[
T_{\mathrm{fric}}=(1-K)T_{\mathrm{front}}.
\]
The regenerative torque command is also limited by the maximum regenerative torque of the motor and by the maximum battery charging power.
The membership functions of the input and output variables were defined over appropriately chosen domains. The braking intensity \(z\) was normalized over [0,1] and divided into three fuzzy sets, namely small, medium and large. The vehicle speed was divided into three fuzzy sets, namely slow, medium and fast. The battery SOC was also divided into three fuzzy sets, representing low, normal and high SOC. The output coefficient \(K\) was divided into five fuzzy sets representing extremely low, low, medium, high and extremely high regenerative braking ratios.
The fuzzy decision table is given below. A total of 27 rules were defined. The first column is the rule number, and the following columns represent the linguistic terms of the inputs and the output.
| Rule | z | v | SOC | K | Rule | z | v | SOC | K |
|---|---|---|---|---|---|---|---|---|---|
| 1 | S | S | B | L | 15 | M | M | B | H |
| 2 | S | M | B | L | 16 | B | S | M | VH |
| 3 | S | B | B | L | 17 | B | M | M | M |
| 4 | M | S | B | L | 18 | B | B | M | H |
| 5 | M | M | B | L | 19 | S | S | M | VL |
| 6 | M | B | B | VH | 20 | S | M | S | M |
| 7 | B | S | B | M | 21 | S | B | S | H |
| 8 | B | M | B | H | 22 | M | S | S | L |
| 9 | B | B | B | H | 23 | M | M | S | VH |
| 10 | S | S | M | VL | 24 | M | B | S | H |
| 11 | S | M | M | VL | 25 | B | S | S | M |
| 12 | S | B | M | H | 26 | B | M | S | M |
| 13 | M | S | M | M | 27 | B | B | S | M |
| 14 | M | M | M | H | – | – | – | – | – |
The fuzzy inference was carried out with the Mamdani method, and the output was defuzzified by the weighted-average method:
\[
K=\frac{\displaystyle\sum_{i=1}^{n} w_i K_i}{\displaystyle\sum_{i=1}^{n} w_i}.
\]
One difficulty with fuzzy controllers of this kind is that the membership-function parameters are often defined manually. To reduce the effect of subjective experience, I applied a genetic algorithm to optimize the parameters of the membership functions. The design variables were the boundary points and peak positions of the Gaussian, triangular and trapezoidal membership functions. All variables were normalized between 1 and 10 before optimization. The objective of the genetic search was to maximize the regenerative braking energy recovered over a representative braking pattern. After the genetic optimization, the optimized membership functions were stored in the fuzzy controller and used in the subsequent vehicle-level simulations.
Simulation Platform and Evaluation Criteria
The regenerative braking controller was implemented in Simulink and connected to the vehicle dynamics model built in the earlier stage. A driver model was included in the simulation platform. The driver model compares the target speed profile with the actual vehicle speed and produces either an accelerator-pedal command or a brake-pedal command. The controller of the driver model is a PI controller:
\[
A(t)=k_p u_e(t)+k_i\int_{0}^{t}u_e(\tau)\,\mathrm{d}\tau,
\]
where \(u_e\) is the speed error, \(k_p\) is the proportional gain and \(k_i\) is the integral gain. The output is normalized in the range from \(-1\) to \(1\), where positive values represent accelerator commands and negative values represent brake commands.
The main control system contains a demanded-braking-torque calculation module, a motor-regenerative-torque management module and a motor-torque control module. The regenerative torque management module disables regenerative braking when any of the following conditions is true:
- the battery SOC is above 0.95;
- the vehicle speed is below 10 km/h;
- the demanded braking intensity is above 0.7;
- the regenerative torque command exceeds the maximum allowable motor torque.
These conditions protect the battery, the motor and the vehicle stability.
To quantify the benefit of the proposed strategy, I selected two evaluation indicators. The first indicator is the variation of the battery SOC during a complete driving cycle. The second indicator is the recovered braking energy at the end of the cycle. These indicators avoid the errors that may occur when the mechanical braking energy is converted through several intermediate power paths.
Simulation Results and Discussion
The simulations were carried out over the NEDC and WLTP driving cycles. The initial battery SOC was set to 0.95 in all tests. Three control strategies were compared:
- the ideal I-curve braking distribution strategy;
- the basic fuzzy logic regenerative braking strategy;
- the genetic-algorithm-optimized fuzzy regenerative braking strategy.
NEDC results
In the NEDC test, the optimized electric car followed the target speed profile with an average error that was about 2.4 percentage points smaller than that of the basic fuzzy controller. The speed-tracking error was kept within ±0.3 m/s. The final battery SOC was 0.81 for the I-curve strategy, 0.851 for the basic fuzzy strategy and 0.889 for the optimized fuzzy strategy. The optimized fuzzy strategy therefore increased the final battery SOC by a large amount when compared with the basic fuzzy strategy and by a much larger amount when compared with the I-curve strategy. The recovered braking energy also increased. Compared with the basic fuzzy strategy, the optimized strategy increased the recovered energy by 2.41%; compared with the ideal I-curve strategy, the improvement was 27.71%.
WLTP results
In the WLTP test, the optimized fuzzy strategy again provided a good speed-tracking performance, with a maximum speed-tracking error smaller than 0.3 m/s for the low-speed, medium-speed, high-speed and very-high-speed phases. The final SOC values were 0.704 for the I-curve strategy, 0.756 for the basic fuzzy strategy and 0.818 for the optimized fuzzy strategy. Relative to the basic fuzzy strategy, the final SOC of the optimized electric car was about 6.24% higher, and the recovered braking energy was about 6.45% higher. Relative to the I-curve strategy, the final SOC was about 16.2% higher and the recovered energy was about 7.1% higher. These results confirm that the genetic-algorithm-optimized fuzzy controller can recover more braking energy without degrading the braking safety of the vehicle.
The table below summarizes the simulation results for the two driving cycles and the three control strategies.
| Cycle | Strategy | Final SOC (–) | Recovered energy improvement vs basic fuzzy (%) | Recovered energy improvement vs I-curve (%) |
|---|---|---|---|---|
| NEDC | I-curve | 0.810 | – | – |
| NEDC | Basic fuzzy | 0.851 | – | – |
| NEDC | Optimized fuzzy | 0.889 | 2.41 | 27.71 |
| WLTP | I-curve | 0.704 | – | – |
| WLTP | Basic fuzzy | 0.756 | – | – |
| WLTP | Optimized fuzzy | 0.818 | 6.45 | 7.10 |
The simulation results clearly show that the optimized fuzzy controller is superior to both the basic fuzzy controller and the ideal I-curve strategy in terms of energy recovery. The improvement is particularly visible in NEDC, where many braking events occur at low speeds. The reason is that the optimized membership functions enable the controller to keep the regenerative braking active in a larger portion of the low-speed deceleration range while still avoiding wheel slip and battery overcharge.
Conclusion and Outlook
In this research I have investigated the powertrain parameter matching and the regenerative braking control strategy of an electric car equipped with a two-speed wet DCT. The work can be summarized by the following conclusions.
First, the traditional matching method based on the longitudinal dynamics equation gave a feasible powertrain design, but the use of a genetic-algorithm-based multi-objective optimization method further improved the result. By optimizing the rated motor power, the rated motor speed, the motor voltage and the two DCT gear ratios simultaneously, the energy consumption of the electric car over the WLTP cycle was reduced to about 12.48 kWh per 100 km. This value is comparable to that of state-of-the-art small electric cars and confirms the advantage of a two-speed DCT for widening the high-efficiency motor operating region.
Second, the use of a variable-efficiency transmission model changed the optimal solution significantly. A constant-efficiency assumption led to higher optimal gear ratios, while the detailed loss model selected lower gear ratios because they reduce the churning and meshing losses in the low-speed and medium-speed range. The difference in the predicted energy consumption was about 3% to 4% of the total energy consumption, which should not be ignored in future electric car optimization studies.
Third, the composite fuzzy-logic and genetic-algorithm approach proposed for the regenerative braking control offers an effective solution for electric cars. The optimized fuzzy controller dynamically determines the regenerative braking torque according to the demanded braking intensity, the vehicle speed and the battery state of charge. Simulation over the NEDC and WLTP cycles shows that the optimized strategy increases the final battery SOC and the recovered braking energy compared with both a basic fuzzy strategy and an ideal I-curve braking distribution strategy. The improvement in recovered energy over the I-curve strategy was particularly large in the NEDC cycle, where frequent urban braking provides many opportunities for recovery.
Fourth, the simulation platform developed in MATLAB and Simulink has been shown to be suitable for powertrain design and energy management optimization. It can calculate the acceleration performance, the gradeability, the maximum speed, the driving range, the battery SOC trajectory and the recovered braking energy of electric cars under standard driving cycles. Such a platform can be used in the early stage of vehicle development to reduce cost and shorten the design cycle.
There are several directions in which this work can be extended. A first direction is the integration of battery thermal management with powertrain optimization, because both driving range and regenerative braking power are influenced by temperature. A second direction is the application of reinforcement learning to the regenerative braking control problem in order to adapt the strategy to real traffic conditions. A third direction is the validation of the proposed strategies by experiments on a physical test bench or on a real electric car. In such tests, the brake durability, the driver comfort and the functional safety of the regenerative braking system should also be evaluated.
Overall, the methods and results described in this article provide a useful reference for the parameter design and control-strategy development of electric cars. They show that combining modern optimization algorithms with detailed component models can improve the driving range and the energy economy of electric cars without compromising their dynamic performance or safety.