A novel NSGA-based co-optimization design framework of dual-motor parameters and energy management for electric cargo vehicles

By virtue of the high overall efficiency and enhanced power performance, the dual-motor system is widely used in electric vehicles. However, it remains a significant challenge to determine the optimal dual-motor power level before energy management, and it is difficult to simultaneously equilibrate the power performance and economy. Therefore, a novel co-optimization design framework that integrates the instantaneous optimal energy consumption strategy into the non-dominated sorting genetic algorithm-II is proposed for a dual-motor electric cargo vehicle. First, based on the vehicle power performance index, an explicit power performance objective function is designed by calculating the reserve power. Second, an improved energy management strategy is developed to optimize the power distribution coefficient between dual motors, and a dual-motor average efficiency for one driving cycle is obtained simultaneously, which is defined as the outcome of economic objective function. Then, the appropriate dual-motor power level and corresponding parameters are determined based on Pareto-optimality and vehicle usage scenarios. Finally, in order to verify the superiority of the co-optimization design framework, a model predictive control-based energy management strategy is implemented for comparison. The results indicate that, the proposed methodology decreases the 50 km/h acceleration time by 1.8s, and reduces energy consumption by 10.93% over one typical driving cycle. Furthermore, by analyzing the motor operating points, it can be observed that the proposed method increases the dual-motor operating points in the efficiency region above 90% by approximately 2%–3%, while reducing the points in the efficiency region below 80% by about 0.2%.

1 Introduction

With the rapid development of vehicle electrification and energy-saving technology, electric vehicles (EVs) have increasingly shown the ability to address the global energy crisis and emission issues. EVs with low noise, low cost per mile, good acceleration performance, and other merits have been extensively studied worldwide (Feng et al., 2025), [错误!未找到引用源。]. However, the innate disadvantages of EVs over a range of parametric issues, long charging times, and inconvenient charging facilities limit their large-scale application (Kwon et al., 2020; Ruan et al., 2023). Single-motor powertrains are associated with insufficient energy savings because of their small high-efficiency regions and limited use (Machado et al., 2021; Yang et al., 2023; Xu et al., 2021; Louback et al., 2024). Recently, multi-motor EVs have become popular because of their relatively high average energy efficiency and enhanced power performance. Dual-motor power systems, which are representative of multi-motor powertrains, have attracted research attention (Wang et al., 2024; Nguyen et al., 2023a; Cao et al., 2024). However, the powertrain configurations of new types of dual-motor systems are much more complex, and dual-motor power level selection prior to the development of energy management strategies (EMSs) has become increasingly difficult. Wang et al. (2023a) reported large differences in the motor power and torque requirements of EVs between special driving conditions and general driving cycles. Therefore, it is imperative to co-optimize the motor power level and EMSs to simultaneously equilibrate the power performance and economy of dual-motor system.

The research on motor parameter determination for EVs has been widely conducted. Zhang et al. (2018) analyzed the characteristics of different base-speed dual-motor systems and presented a dual-motor parameter matching method based on a direct drive structure. Hong et al. (2021) used a reference motor map and considered dynamic performance requirements to scale up motor parameters. Nguyen et al. (2023b) identified motor parameters by considering design requirements and built a semi-static model corresponding to efficiency maps. These studies only addressed ego-vehicle configuration and design objectives but ignored the influence of driving cycles and use scenarios. Hu et al. (2019) proposed a parameter matching method based on a statistical analysis of the demand power of driving cycles and dynamic performance restraints. Wang et al. (2023a), who aimed to meet vehicle dynamic settings or economic needs under different driving conditions, calculated the parameters for dual-motor driving configurations. In another study, a shrinkage factor is integrated into particle swarm optimization to find the optimal powertrain sizing solution (Cui et al., 2023). These studies successfully identified motor parameters by statistically analyzing driving cycles and vehicle design objectives. However, determining optimal motor parameters remains challenges due to the inappropriate EMSs.

The EMSs of dual-motor systems have also been extensively investigated. Tian et al. (2023) proposed a novel dual-motor multi-mode coupling powertrain based on the instantaneous power optimal principle and devised a mode switching strategy for driving and regenerative braking conditions. Ganesan et al. (2023) identified a real-time EMS for multi-motor EVs based on mixed-integer model predictive control. Cui et al. (2023) proposed an intelligent EMS for a specific dual-motor four-wheel drive battery electric vehicle to reduce energy consumption under unknown traffic conditions. He et al. (2022) proposed a model predictive control (MPC) based longitudinal control strategy considering energy consumption. However, these studies primarily focus on optimizing the EMSs to achieve energy savings for the system, while overlooking whether the motor power level meets the vehicle’s requirements. Therefore, the co-optimization design framework (CODF) must be designed to fully optimize both dual-motor system and EMSs.

Co-optimization technology has been widely investigated in the automotive field (Zou et al., 2024; Lei et al., 2023; Wu et al., 2024; Cha et al., 2024), including the co-optimization technology of dual-motor systems. Wang and Sun (2014) proposed a global optimization method to determine the power system parameters and introduced a quantum genetic algorithm to optimize matching in dual-motor hybrid power systems; their approach could satisfy the dynamic performance and economic requirements. A past study (Wang et al., 2020) also presented the integration of a genetic algorithm into a global intelligent algorithm to obtain the optimal vital parameters of a power transmission system, and the vehicle dynamic performance and economic performance are both improved. Wang et al. (2023b) proposed a co-optimization EMS for dual-motor electric tractors to simultaneously achieve working efficiency and smoothness under typical conditions, their scheme is based on nonlinear PID control and a modified snake optimizer algorithm. However, the objective functions defined by these co-optimization algorithms, such as motor price and volume, fail to accurately reflect the performance of the dual-motor system. Co-optimization is a muti-objective optimization problem, and various methods have been developed to address such problems, including the grey wolf optimizer (GWO) (Wang et al., 2022), particle swarm optimization (PSO) (Zhou et al., 2021), and genetic algorithm (GA) (Mazouzi et al., 2024). As a representative of multi-objective genetic algorithms, the non-dominated sorting genetic algorithm (NSGA) and its variants are widely applied in powertrain parameter determination and energy management (Mosammam et al., 2024; Li et al., 2021; Castro et al., 2021). By virtue of the NSGA-Ⅱ advantages (Li et al., 2024), the dual-motor parameter selection and EMS of the electric cargo vehicle (ECV) are co-optimization to enhance the comprehensive performance.

In this research, a novel CODF that integrates the instantaneous optimal energy consumption strategy (IOECS) into the NSGA-II is proposed. Based on the power performance index, an explicit power performance objective function is proposed by calculating the reserve power. Then, for the economic objective function to be optimized, an improved EMS is designed to calculate the average dual-motor efficiency based on the China heavy-duty commercial vehicle test cycle-truck (CHTC-HT) cycle. Finally, on the basis of the objective functions used to determine the Pareto-optimal solutions, an iterative optimal algorithm is employed to obtain the final dual-motor parameters. In order to verify the superiority of the CODF, one MPC-based EMS is conducted as the comparison. The key contributions of this research are as follows:

1. A CODF is proposed to support dual-motor parameter identification and EMS, thereby establishing a connection between them.

2. The explicit power performance and economic objectives are designed to enhance the accuracy of the co-optimization algorithm.

3. An improved IOECS is integrated into the NSGA-II algorithm, allowing the motor parameters and power distribution coefficient to be co-optimized based on the objective functions.

The remaining sections of this paper are arranged as follows: Section II describes the powertrain configuration and calculations. Section III introduces the process for establishing the CODF and presents the optimal dual-motor parameters. Section IV discusses the simulation experiment platform designed to validate the superiority of this study, followed by an analysis of the simulation results. Section V presents the conclusions of this research.

2 Powertrain configuration and calculations

2.1 ECV configuration and parameters

Many new powertrain configurations have emerged as a result of dual-motor systems improving their average energy efficiency and dynamic performance (Huang et al., 2020; Gao et al., 2022). However, despite improvements in system efficiency, most configurations have complicated powertrain systems that constrain control algorithms and manufacturing. Here, a simple dual-motor system (DMS) is proposed for ECVs to simplify the powertrain configuration and reduce manufacturing costs. Figure 1 shows the proposed dual-motor torque coupling configuration. The rear axle provides the driving force, which is composed of a motor/generator 1 (MG1) and a motor/generator 2 (MG2), two joint sleeves (sleeves 1 and 2, respectively), a gear reduction, and a differential.

Figure 1

Figure 1. DMS configuration.

The three operating modes of the DMS including the MG1 single-motor drive mode, the MG2 single-motor drive mode, and the dual-motor drive mode. Table 1 shows the operating conditions of each system component under different working modes, S1 and S2 represent the working states of sleeves 1 and 2, respectively. This structure offers high transmission efficiency due to the limited number of assembly units. Its design indicates relatively simple control, a short development cycle, and the potential for reduced manufacturing costs.

Table 1

Table 1. Operating conditions of each component.

The three modes of the DMS are defined as 1-SM, 2-SM, and 3-DM. The power flow and constraints for each mode are shown in Table 2. The mode shifts are determined by only two joint sleeves. The motor torque and speed are determined by considering the driving cycle requirements, and the results are integrated with the reconstructed motor efficiency maps. On this basis, the energy consumption is calculated at each simulation step.

Table 2

Table 2. Power flow and constraints of each mode.

The ECV parameters and design requirements are shown in Table 3. The performance parameters of the ECV are as follows: a maximum speed of 100 km/h, a maximum gradeability of 20%, a climbing speed exceeding 20 km/h, and an acceleration time of 15 s from 0 to 50 km/h. These parameters are utilized to define the vehicle’s power requirements.

Table 3

Table 3. Parameters and design requirements of the ECV.

2.2 Power requirements calculation

The power requirements for the ECV to reach the maximum speed on the straight road is calculated as Equation 1:

$P_{umax}=\frac{u_{\mathit{max}}}{3600\eta }\left(mgf+\frac{C_{d}A{u}_{\mathit{max}}^2}{21.15}\right)(1)$

where $P_{uamx}$ is the power requirement under maximum speed, kW; $u_{\mathit{max}}$ is the maximum speed, km/h; $\eta$ is the efficiency of the transmission system; $m$ is the gross weight of the ECV, kg; $g$ is the acceleration of gravity, which is selected as 9.8 m/s²; $f$ is the friction coefficient of the tire; $C_d$ is the drag coefficient; and $A$ is the front area of the ECV, m².

The power requirement of the vehicle when climbing a maximum gradient at a steady speed is shown as Equation 2:

$P_{imax}=\frac{u_i}{3600\eta }\left(mgsin\alpha +mgfcos\alpha +\frac{C_{d}A{u}_i^2}{21.15}\right)(2)$

where $P_{imax}$ is the power requirement under the maximum gradient, kW; $\alpha$ is the maximum gradient, 20%; and $u_i$ is the climbing speed, 20 km/h.

According to the empirical formula (Hu et al., 2018), for a car accelerating from 0 to 50 km/h, the vehicle speed with a full accelerator pedal can be fitted as a function of time, as shown in Equation 3:

$u\left(t\right)=u_f{\left(\frac{t}{t_f}\right)}^{\chi }(3)$

where $u_f$ is the final speed, km/h; $t_f$ is the acceleration time from 0 to 50 km/h, s; and $\chi$ is the fitting coefficient, which is empirically selected as 0.5.

The power requirement satisfying the acceleration demand is expressed as Equation 4:

$P_{amax}=\frac{u_f}{3600\eta }\left\{mgf+\frac{C_d{Au}_f^2}{21.15}+\frac{\delta m{u}_f}{3.6\Delta \mathrm{t}}\left[1-{\left(\frac{t_f-\Delta t}{t_f}\right)}^{\chi }\right]\right\}(4)$

where $P_{amax}$ is the power requirement under acceleration demand, kW; $\delta$ is the rotating mass conversion; and $\Delta t$ is the iteration time step.

The power requirements of the ECV calculated according to power performance indicators are shown in Table 4.

Table 4

Table 4. Power requirements of the ECV.

The peak power of DMS must satisfy the above mentioned demand shown in Equation 5:

$P_{MG1\mathit{max}}+P_{MG2\mathit{max}}\geqq \max \left(P_{uamx},P_{imax},P_{amax}\right)(5)$

where $P_{MG1\mathit{max}}$ is the peak power of the electric machine MG1, kW, and $P_{MG2\mathit{max}}$ is the peak power of the electric machine MG2, kW.

The maximum values calculated are used to determine the vehicle’s maximum power requirement as shown in Table 4. Considering system power redundancy, the peak power demand is set at 120 kW.

3 Establishment of the CODF

The first challenge in designing an EMS is to match the optimal parameters of the two motors. Conversely, the dual-motor parameters influence the power distribution of the EMS. Therefore, the dual-motor parameters and EMS should be co-optimized to determine the parametric values. The CODF is shown in Figure 2. The framework consists of three parts: a power performance objective function, an economic objective function, and an NSGA-II-based optimization algorithm. The power performance objective function is calculated using vehicle power performance evaluation indices and dual-motor peak power variables. The economic objective function is determined by the average efficiency loss of the dual-motor system during the driving cycle operation. Specifically, the motor efficiency maps of population individuals can be constructed based on parameters, while the average efficiency loss of each individual is calculated using the IOECS and vehicle models. On the basis of these two objective functions, Pareto-optimal solutions representing the dual-motor optimal power parameters can be obtained. The proposed CODF combines the NSGA-II optimization algorithm and the IOECS algorithm to simultaneously optimize the dual-motor parameters and EMS.

Figure 2

Figure 2. CODF based on NSGS-II.

3.1 Design of the power performance objective function

According to automobile dynamics theory, the power performance of a vehicle can be evaluated by three indices: maximum speed, maximum gradeability, and acceleration time. In designing the power performance objective function, the power level of dual-motor should satisfy the basic power performance requirements of the ECV, and increasing values indicate enhanced power performance. Therefore, matrix P, which is initially calculated as the difference between the overall dual-motor power and the requisite power of power performance indices, is shown in Equation 6:

$P=\left[\begin{array}{l}{P}_{MG1\mathit{max}}+P_{MG2\mathit{max}}-P_{uamx}\\ P_{MG1\mathit{max}}+P_{MG2\mathit{max}}-P_{imax}\\ P_{MG1\mathit{max}}+P_{MG2\mathit{max}}-P_{amax}\end{array}\right](6)$

where $P$ is the dynamic performance power difference matrix.

In the vehicle design process, different usage scenarios involve great discrepancies in terms of power requirement indices. A weight matrix can be defined to balance the proportions of different design requirements and control the fluctuation of the objective function, leading to reasonable ranges. The weight matrix is given by Equation 7:

$w=\left[w_1,w_2,w_3\right](7)$

where $w_1$, $w_2$, and $w_3$ represent the weight of the maximum speed index, the weight of the maximum gradeability index, and the acceleration time index, respectively. In this research, there are no special requirements for the vehicle’s power performance indicators based on the design objectives. Meanwhile, in order to scale the counting range of the power performance objective function, the values of w are thus defined as [0.033, 0.033, 0.034].

Subsequently, the power performance objective function can be expressed as shown in Equation 8:

${\mathit{min}J}_1=\min \left(-wP\right)(8)$

With the decrease in the power performance objective function, the overall power of the dual motor increases. This setting indicates improved power performance of the vehicle.

3.2 Design of the economic objective function

The design process of the economic objective function differs from that of the power performance objective function because the values of the former are obtained via a co-optimization process. As shown in Figure 3, each NSGA-II individual generates a grope of dual-motor power values, which are used to calculate other motor parameters. Subsequently, two motor efficiency maps are produced using the linear scaling function in PSAT (Borhan et al., 2012). Then, the dual-motor power distribution coefficient k, which represents the proportional of MG1 output power at each simulation step, is determined for IOECS based on the vehicle model and driving cycle. Finally, a simulation model that combined dual-motor efficiency maps, the IOECS, a vehicle model, and a driving cycle is established. This simulation model is used to calculate the economic objective function of the CODF.

Figure 3

Figure 3. Design process of the economic objective function.

3.2.1 Calculation of the dual-motor parameters

The maximum speed requirement of the motor based on the maximum design speed of the vehicle is given by:

$n_{\mathit{max}}=\frac{u_{\mathit{max}}{i}_0}{0.377r}(9)$

where $n_{\mathit{max}}$ is the maximum speed requirement of the motor, rpm; $i_0$ is the reducing gear ratio; and $r$ is the wheel radius, m. The maximum speed requirement of the motor is 3,679 rpm.

The two motors of the DMS have the same working speed because of the torque coupling under this configuration. The motor of the DMS needs to satisfy certain characteristics, including a large torque at low speed, a wide speed range, and a strong overload capacity. Therefore, considering the maximum speed requirement of the motor, the peak speed of MG1 is determined to be 4,000 rpm, and the peak speed of MG2 is determined to be 5,000 rpm. The ratio of the motor peak speed to the rated speed represents the expanding constant power region coefficient, which is selected as 2.5 according to engineering experience. The calculated rated speeds of MG1 and MG2 are 1,600 and 2000 rpm, respectively.

In this study, the peak powers of the dual motor represent the individuals of the CODF. The ratio of the motor peak power to the rated power is the overload coefficient, which is determined to be 2 according to engineering experience.

The peak torque of the motor is obtained based on the motor peak power and rated speed as follows:

$T_{MGimax}=\frac{{9550P}_{MGimax}}{n_{MGie}}(10)$

where $T_{MGimax}$ is the peak torque of the ith motor, Nm; $P_{MGimax}$ is the peak power of the ith motor, kW; and $n_{MGie}$ is the rated speed of the ith motor, rpm.

The PSTA from Argonne National Laboratory can generate scaled maps that integrate quasi-steady models according to the original machine (Borhan et al., 2012). The dual-motor crucial parameters are calculated based on Equations 9, 10, and then delivered to the PSTA to generated the corresponding motor efficiency map.

3.2.2 Power distribution of IOECS

The system efficiency is closely related to the working efficiency of the motor in the EVs. The driving efficiency varies under the same power requirements but with different motor speeds and torques, leading to fluctuations in motor energy consumption. Therefore, the goal of IOECS is to calculate the instantaneous optimal energy consumption at each simulation step.

The feasible operating regions of the dual-motor speed and torque are obtained based on the dual-motor efficiency maps. The constraints are expressed as follows:

$\left\{\begin{array}{l}0<n_{MGi}<n_{MGimax}\\ 0<T_{MGi}<T_{MGimax}\left(n_{MGi}\right)\end{array}\right.(11)$

where $n_{MGi}$ is the speed of the ith motor, rpm; $n_{MGimax}$ is the peak speed of the ith motor, rpm; $T_{MGi}$ is the torque of the ith motor, Nm; and $T_{MGimax}$ is the peak torque of the ith motor, Nm.

When the two motors are coupled to drive, the two motors providing demand power for the vehicle to minimize instantaneous energy consumption become imbalanced. Therefore, in this study, the power distribution coefficient k is defined. This coefficient represents the proportion of the total demand power supplied by the MG1 motor and is expressed as:

$k=\frac{P_{MG1}}{P_{req}}(12)$

where $P_{MG1}$ is the MG1 motor power, kW, and $P_{req}$ is the vehicle total demanding power, kW.

For the power distribution process of the dual motor, the total power demand of each operating point can be calculated according to Equation 18. The constraint of the dual-motor power distribution is given by:

$P_{req=}{P}_{MG1}+P_{MG2}(13)$

where $P_{MG2}$ is the MG2 motor power, kW.

Second, $k$ varies from 0 to 1, from which a series of MG1 motor power values can be obtained. Considering the constraints given by Equations 11–13, the MG1 motor torque and MG2 motor torque can be calculated as follows:

$T_{MG1}=9550\frac{P_{req}k}{n_{MG1}}(14)$

$T_{MG2}=9550\frac{P_{req}\left(1-k\right)}{n_{MG2}}(15)$

Third, linear interpolation is used to obtain the motor efficiency at each particular point based on Equation 16, which denotes the table lookup function for the efficiency of the two motors. Then, the energy consumption of the two motors is calculated according to Equation 17. On this basis, a sequence of energy consumption values of the dual motors is obtained.

${\eta }_i=f\left(n_{MGi},T_{MGi}\right)(16)$

$E_{MGi}=\frac{T_{MGi}{n}_{MGi}\Delta t}{{9550\eta }_{MGi}{\eta }_0}(17)$

where ${\eta }_{MGi}$ is the ith motor efficiency under the point (${\eta }_{MGi}$, $T_{MGi}$); $E_{MGi}$ is the energy consumption of the ith motor, kJ; and $\Delta t$ is the simulation time step, s.

Finally, the total energy consumption sequence of the dual motors at each operating point is compared. The point with the lowest energy consumption is selected, and the k value corresponding to the minimum energy consumption representing the power distribution between dual motors is determined.

3.2.3 Simulation model

The simulation model, which consisted of a driving cycle, an ECV model, and the IOECS, served as the foundation for supporting the co-optimization of CODF. Data on urban cycle conditions, which represent the most recent national standards of China, are selected to simulate the actual conditions of urban roads. Considering the gross weight of the ECV, CHTC-HT is selected as the research driving cycle of this study, the driving cycle is shown in Figure 4.

Figure 4

Figure 4. Driving cycle adopted in the CODF.

The CHTC-HT-based power requirements of the ECV are calculated as follows:

$P_{req}=\frac{1}{{3600\eta }_T}\left(mgf+\frac{C_{d}A{u}^2}{21.15}+\delta m\frac{dv}{dt}\right)u(18)$

where $P_{req}$ is the power requirement of each simulation step, kW, and $u$ is the vehicle velocity of each simulation step, km/h.

3.2.4 Economic objective function

The motor average efficiency during the driving cycle is an important parameter for representing dual-motor economic performance and for calculating the objective function. Given the average efficiency obtained at each simulation step, the dual-motor energy consumption is selected for calculation. The total energy consumption of the dual-motor system is calculated as follows:

$\left\{\begin{array}{l}{E}_{MG1total}={\displaystyle \sum _{t=0}^n}{E}_{MG1}\left(t\right)\\ E_{MG2total}={\displaystyle \sum _{t=0}^n}{E}_{MG2}\left(t\right)\end{array}\right.(19)$

where $E_{MG1total}$ is the motor total energy consumption of one driving cycle of MG1, kJ, and $E_{MG2total}$ is the motor total energy consumption of one driving cycle of MG2, kJ.

Combining Equations 14, 15, 17, 19 gives the total energy consumption, which can be expressed as Equation 20:

$\left\{\begin{array}{l}{E}_{MG1total}={\displaystyle \sum _{t=0}^n}\frac{P_{req}\left(t\right)k\left(t\right)\Delta t}{{\eta }_{MG1}\left(t\right){\eta }_0}\\ E_{MG2total}={\displaystyle \sum _{t=0}^n}\frac{P_{req}\left(t\right)\left[1-k\left(t\right)\right]\Delta t}{{\eta }_{MG2}\left(t\right){\eta }_0}\end{array}\right.(20)$

Subsequently, the two-motor average efficiency can be expressed as Equation 21:

$\left\{\begin{array}{l}{\eta }_{MG1average}={\displaystyle \sum _{t=0}^n}\frac{P_{req}\left(t\right)k\left(t\right)\Delta t}{E_{MG1total}{\eta }_0\eta }\\ {\eta }_{MG2average}={\displaystyle \sum _{t=0}^n}\frac{P_{req}\left(t\right)\left[1-k\left(t\right)\right]\Delta t}{E_{MG2total}{\eta }_0\eta }\end{array}\right.(21)$

where ${\eta }_{MG1average}$ is the average efficiency of the MG1 motor at one driving cycle, and ${\eta }_{MG2average}$ is the average efficiency of the MG2 motor at one driving cycle.

The matrix E is defined based on the motor average efficiency as Equation 22:

$E=\left[\begin{array}{l}{\eta }_{MG1average}\\ {\eta }_{MG2average}\end{array}\right](22)$

where $E$ is the motor average efficiency matrix.

In the process of dual-motor system parameter determination, the power level requirements of the motors vary due to differing design objectives and corresponding operational scenarios. The weight matrix of average efficiency must be defined to meet the different design requirements. The weight matrix can also control the fluctuation of the objective function result within a reasonable range. The weight matrix is expressed as Equation 23:

$b=\left[b_1{b}_2\right](23)$

where $b_1$ and $b_2$ represent the weights of the average efficiency of the MG1 and MG2, respectively.

The two motors in this study are designed with distinct performance characteristics: MG1 is configured as a power-focused motor, while MG2 is optimized for economic performance. At this stage, the economic cruising capability of MG2 required more attention than that of MG1 during the calculation. Thus, the proportion of $b_2$ in the weight matrix is larger than that of $b_1$. In this research, the values of $b$ are defined as [0.4, 0.6].

The motor average efficiency loss based on the NSGA-II algorithm requirement is regarded as the optimization target. The economic objective function is designed as Equation 24:

$\mathit{min}{J}_2=\mathit{min}\left(1-bE\right)(24)$

The objective functions of the CODF are shown in Equation 25, and the P_MG1max and P_MG2max are the independent variables for the NSGA-II algorithm.

$\left\{\begin{array}{l}\mathit{min}{J}_1=\mathit{min}\left(-wP\right)\\ \mathit{min}{J}_2=\min \left(1-bE\right)\\ P_{MG1\mathit{max}},P_{MG2\mathit{max}}ϵ\left(0,120kW\right)\end{array}\right.(25)$

3.3 NSGA-II-based co-optimization algorithm

The dual-motor peak power is selected as the independent variable of the algorithm, and the objective functions J1 and J2 are used to calculate the results corresponding to the independent variables. Table 5 shows the basic parameters of the algorithm.

Table 5

Table 5. Basic parameter definitions.

4 Results and discussion

4.1 CODF results and analysis

As depicted by the CODF mentioned above, the CODF results after reaching maximum number of iterations are shown in Figure 1. The Figure 5a is the Pareto-optimal front, and the X-axis represents the results of the power performance objective function, while the Y-axis represents the results of the economic objective function. The Figure 5b shows the independent variable distribution corresponding to the Pareto-optimal solutions. The value of each point on this image represents the optimal grope of MG1 peak power and MG2 peak power. Given the peak power variation of the dual motors, the two objective functions cannot diminish simultaneously. A decrease in the power performance objective function will increase the economic objective function, in other words, an increase in motor power will lead to a decrease in economic function during the driving cycle. The higher the dual-motor average efficiency and the greater the economy are, the lower the vehicle power performance is. Theoretically, each point of Figure 5b represents the optimal power parameter representing the grope of the dual motor, and the values are acceptable under no other restrictions. Furthermore, with increasing dual-motor power level (red points in Figures 5a,b), the value of the power performance objective function decreases, and the value of the economic objective function increases. This trend means improved vehicle power performance and reduced economic function, and a contrasting scenario is improved vehicle economy and reduced power performance.

Figure 5

Figure 5. The CODF results. (a) Pareto-optimal front. (b) The independent variable distribution corresponding to the Pareto-optimal solutions.

Considering the user scenarios and commercial vehicle features of ECVs, the focus of this research is to improve the economic objective of ECVs as much as possible under the premise of ensuring power performance. Therefore, in this research, the marked point in Figure 5b is selected as the power level of the dual motor, and the corresponding Pareto-optimal solution in Figure 5a is also identified. The optimal peak powers of MG1 and MG2 are 70 and 58 kW after rounding, respectively. Based on the PSTA, the parameters of the dual motors are shown in Table 6 and the motor efficiency maps of MG1 and MG2 are shown in Figures 6a,b.

Table 6

Table 6. Parameters of the dual-motor.

Figure 6

Figure 6. Dual-motor maps. (a) MG1 efficiency map. (b) MG2 efficiency map.

After selecting the dual-motor parameters, the corresponding power distribution coefficient is determined. The vehicle braking conditions for the power distributions are calculated in the same manner as the process of determining the driving conditions based on the IOECS. The power distribution coefficients at each working point are shown in Figure 7a. When the wheel torque requirement is small (approximately 0–200 Nm), the power distribution coefficient k is close to 0, which means that only the MG2 motor outputs power. A good explanation is that high-efficiency regions of the MG2 motor corresponded to relatively small wheel torque demand. With increasing wheel torque (approximately 200–400 Nm), the power distribution coefficient k is mainly 1, which means that only the MG1 motor could be operated. As MG1 had a relatively large peak torque, its high-efficiency regions could satisfy the relatively high torque demand. When the wheel torque exceeded 400 Nm, the DMS entered dual-motor driving mode, and the power distribution coefficient fluctuated between 0 and 1. Then, 50 km/h is taken as the dividing line. The MG1 motor involved much greater power allocation in the region less than 50 km/h, whereas the other region had a greater proportion of power allocated by the MG2 motor. This phenomenon can be attributed to the parameters of the two motors. Furthermore, their high-efficiency regions presented an overt discrepancy between different power level motors. The power distribution coefficients of the dual motor are the optimal rules for the system parameter configuration, which are given by the IOECS. Then, these rules are employed in MATLAB/Simulink simulations. The driving cycle, IOECS, and vehicle models are combined to validate the effectiveness of the CODF.

Figure 7

Figure 7. Control rules. (a) Power distribution coefficients. (b) Mode switching rules.

The working stations of the two motors at each operating point are determined based on the power distribution rules of the dual motor. The mode switching rules are shown in Figure 7b. Given the simple structure of the DMS, the system frequently operates in dual-motor mode, which can minimize the instantaneous energy consumption.

4.2 Comparison of the simulation results

A simulation platform is designed to illustrate the effectiveness and superiority of the CODF algorithm on the power performance and energy saving. The MPC-based EMS is implemented as the comparison, and the same motor parameters determined by Pareto-optimality are adopted.

The comparative vehicle acceleration performance is shown in Figure 8. Apparently, the power performance of the CODF-based model is greater than the MPC-based model. The acceleration time from rest to 50 km/h is 12.9s with the CODF-based model, compared to 14.7s with the MPC-based model, decreases the acceleration time by 1.8s. The reason is that for a full pedal acceleration cycle, the MPC-based strategy could not distinguish the cycle requirement and only guaranteed the minimum energy consumption, and this optimal objective restricts the maximum motor output torque. With the methodology proposed in this study, the CODF-based model could provide sufficient torque to satisfy vehicle requirements.

Figure 8

Figure 8. Comparison results of vehicle acceleration performance.

An energy consumption comparative simulation based on the CHTC-HT cycle is conducted in MATLAB/Simulink. The changes in SOC during the simulation are shown in Figure 9a. The CODF-based model demonstrates the best energy-saving performance after a single driving cycle. Since IOECS can find the optimal power distribution ratio between the two motors that minimizes energy consumption for each system operating point, it achieves a globally optimal optimization effect. Therefore, compared to the MPC-based model, the CODF-based model offers better economic performance. A noticeable discrepancy between the two models is observed, with terminal SOCs of 0.4951 and 0.4577, respectively. In comparison to the MPC-based model, the CODF-based model results in a 10.93% reduction in energy consumption over one driving cycle.

Figure 9

Figure 9. Comparison results. (a) SOC changes. (b) The average efficiency.

The average efficiency comparison of the two motors is shown in Figure 9b. The average efficiency of the two motors exhibited the same variation trend as the changes in the SOC. These results indicate the apparent advantage of the CODF-based model over the MPC-based model in terms of energy savings. The dual-motor working point distributions of the CODF-based model are shown in Figures 10a,b, and the dual-motor working point distributions of the MPC-based model are shown in Figures 10c,d. Most of the CODF-based motor working points clusters in the high-efficiency areas of the motor maps, whereas more working points of the MPC-based motor are distributed in regions representing inefficiency. The working point efficiency statistics are shown in Table 7. It can be observed that, compared with MPC-based model, the proposed method increases the dual-motor operating points in the efficiency region above 90% by approximately 2%–3%, while reducing the points in the efficiency region below 80% by about 0.2%.

Figure 10

Figure 10. Motor working point distribution within the driving regions according to the CHTC-HT cycle. (a) MG1 working points of CODF. (b) MG2 working points of CODF. (c) MG1 working points of MPC. (d) MG2 working points of MPC.

Table 7

Table 7. Proportion of working point distributions.

5 Conclusion

With the aim of enhancing power performance and economic goals in the process of dual-motor system design, a CODF that could integrate the IOECS into the NSGA-II optimization algorithm is proposed in this study. First, an explicit power performance objective function is designed by calculating the reserve power based on the power performance index. Then, an economic objective function is designed, which is calculated by combining the vehicle model and driving cycle. Subsequently, the power distribution coefficient is integrated into the IOECS for co-optimization, and the optimal power distribution coefficients are confirmed based on the minimum energy consumption. Finally, a comparison simulation is conducted to verify the effectiveness and superiority of CODF-based method. The research results demonstrate that, compared to the MPC-based model, the CODF-based model reduces the acceleration time from standstill to 50 km/h by 1.8 s, while achieving a 10.93% reduction in the energy consumption over one driving cycle. Furthermore, statistical analysis of the motor operating points for both models indicates that the CODF-based model increases the proportion of operation within the high-efficiency region (≥90%) by 2%–3%, while concurrently reducing the number of operating points in the low-efficiency region (≤80%).

The CODF proposed in this study provides a novel approach for determining the power levels in dual-motor systems. It enables targeted optimization of the EMSs, facilitating the simultaneous optimization of both power performance and energy economy. However, the vehicle model and driving cycle employed in this research are simplified for the sake of computational feasibility. In future research, we will develop more sophisticated and realistic vehicle and system models that incorporate the aforementioned complex factors, thus achieving a more comprehensive optimization of dual-motor systems. This method can be adapted for parameter optimization in other dual-motor systems, offering scientific theoretical guidance for the determination of motor power levels.

Data availability statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

Author contributions

KH: Formal Analysis, Funding acquisition, Project administration, Writing – original draft. CZ: Investigation, Resources, Writing – original draft, Writing – review and editing. NS: Investigation, Resources, Writing – original draft. GZ: Project administration, Supervision, Writing – original draft. ZG: Project administration, Supervision, Writing – original draft. LC: Conceptualization, Methodology, Software, Validation, Writing – original draft, Writing – review and editing. CL: Data curation, Writing – original draft, Writing – review and editing. MG: Writing – original draft, Writing – review and editing.

Funding

The author(s) declare that financial support was received for the research and/or publication of this article. This work is supported by the National Science Foundation of China (52272394), and the Free exploration project of Natural Science Foundation of Jilin Province (YDZJ202101ZYTS159).

Conflict of interest

Authors KH, NS, GZ, and ZG were employed by The State Key Laboratory of Engine and Powertrain System, Weichai Power Co., Ltd.

The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Generative AI statement

The author(s) declare that no Generative AI was used in the creation of this manuscript.

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References

Borhan, H., Vahidi, A., Phillips, A., Kuang, M., Kolmanovsky, I., and Cairano, S. (2012). MPC-Based energy management of a power-split hybrid electric vehicle. IEEE Trans. Control Syst. Technol. 20, 593–603. doi:10.1109/TCST.2011.2134852

CrossRef Full Text | Google Scholar

Cao, K., Hu, M., Chen, L., Zhao, J., and Xiao, Z. (2024). Research on optimal economic dynamic torque distribution strategy for dual-motor four-wheel-drive considering voltage variation of power battery. J. Energy Storage 86, 111006. doi:10.1016/j.est.2024.111006

CrossRef Full Text | Google Scholar

Castro, M., Mukundan, S., Lopes Filho, C., Byczynski, G., Minaker, B., and Tjong, J. (2021). “Non–dominated sorting genetic algorithm based determination of optimal torque–split ratio for a dual–motor electric vehicle,” in IECON 2021–47th annual conference of the IEEE industrial electronics society, 1–6. doi:10.1109/IECON48115.2021.9589555

CrossRef Full Text | Google Scholar

Cha, J., Han, C., Song, S., Kang, S., Lee, D., Chang, M., et al. (2024). Hierarchical co-optimization of EV scheduling considering customer and system in distribution networks. Sustain. Energy Grids Netw. 38, 101361. doi:10.1016/j.segan.2024.101361

CrossRef Full Text | Google Scholar

Cui, H., Ruan, J., Wu, C., Zhang, K., and Li, T. (2023). Advanced deep deterministic policy gradient based energy management strategy design for dual-motor four-wheel-drive electric vehicle. Mech. Mach. Theory 179, 105119. doi:10.1016/j.mechmachtheory.2022.105119

CrossRef Full Text | Google Scholar

Feng, R., Hua, Z., Yu, J., Zhao, Z., Dan, Y., Zhai, H., et al. (2025). A comparative investigation on the energy flow of pure battery electric vehicle under different driving conditions. Appl. Therm. Eng. 269, 126035. doi:10.1016/j.applthermaleng.2025.126035

CrossRef Full Text | Google Scholar

Ganesan, A., Murgovski, N., Yang, D., and Gros, S. (2023). “Real-time mixed-integer energy management strategy for multi-motor electric vehicles,” in 2023 IEEE transportation electrification conference, 1–6. doi:10.1109/ITEC55900.2023.10186957

CrossRef Full Text | Google Scholar

Gao, B., Meng, D., Shi, W., Cai, W., Dong, S., Zhang, Y., et al. (2022). Topology optimization and the evolution trends of two-speed transmission of EVs. Renew. Sustain. Energy Rev. 161, 112390. doi:10.1016/j.rser.2022.112390

CrossRef Full Text | Google Scholar

He, H., Han, M., Liu, W., Cao, J., Shi, M., and Zhou, N. (2022). MPC-based longitudinal control strategy considering energy consumption for a dual-motor electric vehicle. Energy 253, 124004. doi:10.1016/j.energy.2022.124004

CrossRef Full Text | Google Scholar

Hong, X., Wu, J., and Zhang, N. (2021). “Parameters design and energy efficiency optimization of a new Simpson gearset based dual motor powertrain for electric vehicles,” in 2021 60th annual conference of the society of instrument and control engineers of Japan (SICE), 685–660. doi:10.13334/scie.20216586

CrossRef Full Text | Google Scholar

Hu, M., Chen, S., and Zeng, J. (2018). Control strategy for the mode-switch of a novel dual-motor coupling powertrain. IEEE Trans. Veh. Technol. 67, 2001–2013. doi:10.1109/TVT.2017.2769127

CrossRef Full Text | Google Scholar

Hu, J., Zu, G., Jia, M., and Niu, X. (2019). Parameter matching and optimal energy management for a novel dual-motor multi-modes powertrain system. Mech. Syst. Signal Process. 116, 113–128. doi:10.1016/j.ymssp.2018.06.036

CrossRef Full Text | Google Scholar

Huang, W., Huang, J., and Yin, C. (2020). Optimal design and control of a two speed planetary gear automatic transmission for electric vehicle. Appl. Sci. 10, 6612. doi:10.3390/app10186612

CrossRef Full Text | Google Scholar

Kwon, K., Seo, M., and Min, S. (2020). Efficient multi-objective optimization of gear ratios and motor torque distribution for electric vehicles with two-motor and two-speed powertrain system. Appl. Energy 259, 114190. doi:10.1016/j.apenergy.2019.114190

CrossRef Full Text | Google Scholar

Lei, N., Zhang, H., Wang, H., and Wang, Z. (2023). An improved co-optimization of component sizing and energy management for hybrid powertrains interacting with high-fidelity model. IEEE Trans. Veh. Technol. 72, 1–12. doi:10.1109/TVT.2023.3296114

CrossRef Full Text | Google Scholar

Li, Y., Wang, S., Duan, X., Liu, S., Liu, J., and Hu, S. (2021). Multi-objective energy management for Atkinson cycle engine and series hybrid electric vehicle based on evolutionary NSGA-II algorithm using digital twins. Energy Convers. Manage. 230, 113788. doi:10.1016/j.enconman.2020.113788

CrossRef Full Text | Google Scholar

Li, X., Dong, W., Wang, L., Xu, Z., and Xiao, S. (2024). NSGA-II-based hierarchical irregular subarray tiling for large-scale phased array design. IEEE Antennas Wirel. Propag. Lett. 23, 3148–3152. doi:10.1109/LAWP.2024.3427846

CrossRef Full Text | Google Scholar

Louback, E., Biswas, A., Machado, F., and Emadi, A. (2024). A review of the design process of energy management systems for dual-motor battery electric vehicles. Renew. Sustain. Energy Rev. 193, 114293. doi:10.1016/j.rser.2024.114293

CrossRef Full Text | Google Scholar

Machado, F., Kollmeyer, P., Barroso, D., and Emadi, A. (2021). Multi-speed gearboxes for battery electric vehicles: current status and future trends. IEEE Open J. Veh. Technol. 2, 419–435. doi:10.1109/OJVT.2021.3124411

CrossRef Full Text | Google Scholar

Mazouzi, A., Hadroug, N., Alayed, W., Hafaifa, A., Iratni, A., and Kouzou, A. (2024). Comprehensive optimization of fuzzy logic-based energy management system for fuel-cell hybrid electric vehicle using genetic algorithm. Int. J. Hydrogen Energy 81, 889–905. doi:10.1016/j.ijhydene.2024.07.237

CrossRef Full Text | Google Scholar

Mosammam, Z., Ahmadi, P., and Houshfar, E. (2024). Multi-objective optimization-driven machine learning for charging and V2G pattern for plug-in hybrid vehicles: balancing battery aging and power management. J. Power Sources 608, 234639. doi:10.1016/j.jpowsour.2024.234639

CrossRef Full Text | Google Scholar

Nguyen, C., Nguyễn, B., Trovão, J., and Ta, M. (2023a). Torque distribution optimization for a dual-motor electric vehicle using adaptive network-based fuzzy inference system. IEEE Trans. Energy Convers. 38, 2784–2795. doi:10.1109/tec.2023.3285225

CrossRef Full Text | Google Scholar

Nguyen, C., Nguyễn, B., Trovão, J., and Ta, M. (2023b). Optimal energy management strategy based on driving pattern recognition for a dual-motor dual-source electric vehicle. IEEE Trans. Veh. Technol. 73, 4554–4566. doi:10.1109/TVT.2023.3343704

CrossRef Full Text | Google Scholar

Ruan, J., Wu, C., Cui, H., Li, W., and Sauer, D. (2023). Delayed deep deterministic policy gradient-based energy management strategy for overall energy consumption optimization of dual motor electrified powertrain. IEEE Trans. Veh. Technol. 72, 11415–11427. doi:10.1109/TVT.2023.3265073

CrossRef Full Text | Google Scholar

Tian, Y., Zhang, Y., Li, H., Gao, J., Swen, A., and Wen, G. (2023). Optimal sizing and energy management of a novel dual-motor powertrain for electric vehicles. Energy 275, 127315. doi:10.1016/j.energy.2023.127315

CrossRef Full Text | Google Scholar

Wang, Y., and Sun, D. (2014). Powertrain matching and optimization of dual-motor hybrid driving system for electric vehicle based on quantum genetic intelligent algorithm. Discrete Dyn. Nat. Soc., 956521. doi:10.1155/2014/956521

CrossRef Full Text | Google Scholar

Wang, Y., Cai, H., Liao, Y., and Gao, J. (2020). Study on global parameters optimization of dual-drive powertrain system of pure electric vehicle based on multiple condition computer simulation. Complexity, 1–10. doi:10.1155/2020/6057870

CrossRef Full Text | Google Scholar

Wang, E., Xia, J., Li, J., Sun, X., and Li, H. (2022). Parameters exploration of SOFC for dynamic simulation using adaptive chaotic grey wolf optimization algorithm. Energy 261, 125146. doi:10.1016/j.energy.2022.125146

CrossRef Full Text | Google Scholar

Wang, J., Zhang, C., Guo, D., Yang, F., Zhang, Z., and Zhao, M. (2023a). Drive-cycle-based configuration design and energy efficiency analysis of dual-motor 4WD system with two-speed transmission for electric vehicles. IEEE Trans. Transp. Electrific. 10, 1887–1899. doi:10.1109/TTE.2023.3277479

CrossRef Full Text | Google Scholar

Wang, S., Wu, X., Zhao, X., Wang, S., Xie, B., Song, Z., et al. (2023b). Co-optimization energy management strategy for a novel dual-motor drive system of electric tractor considering efficiency and stability. Energy 281, 128074. doi:10.1016/j.energy.2023.128074

CrossRef Full Text | Google Scholar

Wang, J., Zhang, Z., Guo, D., Ni, J., Guan, C., and Zheng, T. (2024). Torque vectoring and multi-mode driving of electric vehicles with a novel dual-motor coupling electric drive system. Automot. Innov. 7, 236–247. doi:10.1007/s42154-023-00280-x

CrossRef Full Text | Google Scholar

Wu, S., Chen, Z., Shen, S., Liu, G., Liu, Y., and Zhang, Y. (2024). Co-optimization of velocity planning and energy management for intelligent plug-in hybrid electric vehicles based on adaptive dynamic programming. IEEE Trans. Veh. Technol. 73, 9812–9824. doi:10.1109/TVT.2024.3384017

CrossRef Full Text | Google Scholar

Xu, X., Liang, J., Hao, Q., Dong, P., Wang, S., Guo, W., et al. (2021). A novel electric dual motor transmission for heavy commercial vehicles. Automot. Innov. 4, 34–43. doi:10.1007/s42154-020-00129-7

CrossRef Full Text | Google Scholar

Yang, M., Nicholls, R., Moghimi, M., and Griffiths, L. (2023). Performance management of EV battery coupled with latent heat jacket at cell level. J. Power Sources 558, 232618. doi:10.1016/j.jpowsour.2022.232618

CrossRef Full Text | Google Scholar

Zhang, Z., Kuang, Z., Wang, T., and Cui, S. (2018). The analysis of dual-motor matching with different base speed for electrical vehicle. IEEE Tran. Electrific. 3, 2693–3586. doi:10.1109/ITEC-AP.2018.8433275

CrossRef Full Text | Google Scholar

Zhou, Q., Guo, S., Xu, L., Guo, X., Williams, H., and Xu, H. (2021). Global optimization of the hydraulic-electromagnetic energy-harvesting shock absorber for road vehicles with human-knowledge-integrated particle swarm optimization scheme. IEEE/ASME Trans. Mechatron. 26, 1225–1235. doi:10.1109/TMECH.2021.3055815

CrossRef Full Text | Google Scholar

Zou, Y., Yang, Y., Zhang, Y., and Liu, C. (2024). Aging-aware co-optimization of topology, parameter and control for multi-mode input-and output-split hybrid electric powertrains. J. Power Sources 624, 235564. doi:10.1016/j.jpowsour.2024.235564

CrossRef Full Text | Google Scholar