Energy Conversion and Management xxx (xxxx) xxx Contents lists available at ScienceDirect Energy Conversion and Management journal homepage: www.elsevier.com/locate/enconman Optimal drivetrain design methodology for enhancing dynamic and energy performances of dual-motor electric vehicles Chi T.P. Nguyen a, b, Bảo-Huy Nguyễn a, c, *, João Pedro F. Trovão a, d, e, f, Minh C. Ta a, c a e-TESC Lab, University of Sherbrooke, Sherbrooke, QC J1K 2R1, Canada Thai Nguyen University of Technology, Thai Nguyen, Viet Nam c CTI Lab. for EVs, School of Electrical and Electronic Engineering, Hanoi University of Science and Technology, Hanoi, Viet Nam d Canada Research Chair in Efficient Electric Vehicles with Hybridized Energy Storage Systems, Canada e INESC Coimbra, University of Coimbra, DEEC, Polo II 3030-290 Coimbra, Portugal f Polytechnic Institute of Coimbra, IPC-ISEC, DEE, 3030-199 Coimbra, Portugal b A R T I C L E I N F O A B S T R A C T Keywords: Electric vehicle 4WD dual-motor Off-road vehicle Drivetrain design optimization Electric vehicles (EVs) are a dominant transportation trend towards a low-carbon future. However, there is a lack of systematic studies for full-electric off-road vehicles in terms of drivetrain design and energy management issues. This paper proposes a novel methodology to develop a drivetrain system for a four-wheel-drive (4WD) dual-motor off-road EV. The obtained design delivers high performance in both driving performance and energy efficiency of a desired vehicle using the two electric motors with their multi-speed gearbox. The proposed approach is to minimize the power envelope (force-speed characteristic) difference between a targeted model and the system under study. As a result, the objective function finds an optimal solution for a proper set of gear ratios and constant power speed ratio (CPSR) through a two-layer hierarchical scheme. The upper layer sets the targeted power envelope based on the expected driving measures. The lower layer minimizes the gap of the power envelope characteristics between the targeted vehicle and the dual-motor vehicle. In parallel, two opti­ mization loops are implemented at the lower layer. The outer loop determines the proper value of CPSR, meanwhile, the gear ratios corresponding to the selected CPSR are optimized by a particle swarm optimization (PSO) algorithm in the inner loop. The results verify that the dynamic and energy performances of the studied dual-motor vehicle are enhanced over critical testing scenarios. In a comparison with the single-motor singlespeed gearbox configuration, the optimized design shows that the top speed is 55% higher; the acceleration time to 45 km/h is reduced by 55.83%; and the gradeability is improved by 162.5%. Moreover, the test under a typical off-road cycle reveals a 1.9% overall efficiency increase compared to the dual-motor drivetrain using the same non-optimized gear ratio. The proposed comprehensive design approach has the potential to be applied to a wide range of vehicle areas. gravity that is interested in off-road vehicle stability. 1. Introduction Conventional vehicles burning gasoline and diesel fuel in an internal combustion engine (ICE) are a major factor of the air pollution that leads to an inevitable transition to electric vehicles (EVs). Among them, heavy-duty off-road vehicles for mining, agriculture, and military are increasingly popular [1–3]. EVs benefit from off-road duties thanks to their fast response and the capability of producing high torque at low speed. Furthermore, the mass of the battery packs usually weighs haft of a ton and is located on the vehicle floor. This keeps a low center of 1.1. Literature review In order to adapt to all terrains and difficult off-road conditions, e.g., on snow, mud road, or steep uphill driving, strong traction power should be enforced to all the wheels simultaneously. A promising solution is to add one more electric motor to the other axle which is known as the dual-motor four-wheel drive (4WD) configuration [4]. This structure not only overcomes the fault states of one motor but also improves the robustness and reliability of the vehicle. A large number of commercial * Corresponding author at: e-TESC Lab, University of Sherbrooke, Sherbrooke, QC J1K 2R1, Canada and CTI Lab. for EVs, Hanoi University of Science and Technology, Hanoi, Viet Nam. E-mail address: huy.nguyenbao@hust.edu.vn (B.-H. Nguyễn). https://doi.org/10.1016/j.enconman.2021.115054 Received 5 October 2021; Received in revised form 10 November 2021; Accepted 21 November 2021 0196-8904/© 2021 Elsevier Ltd. All rights reserved. Please cite this article as: Chi T.P. Nguyen, Energy Conversion and Management, https://doi.org/10.1016/j.enconman.2021.115054 C.T.P. Nguyen et al. Energy Conversion and Management xxx (xxxx) xxx Nomenclature Variables J P T F k V ω x v η Pbest Gbest Subscripts _max _base _tran _tot _gear _ori _veh _acc _pro _target _i _gearIM _gearPM _opt _bat Objective function Power Torque Force Coefficient Velocity Rotation speed Particle position in PSO Particle speed in PSO Efficiency Best personal solution in PSO Best global solution in PSO Maximum Base Transmission Total Gearbox Original Vehicle Acceleration Proposed Targeted Individual particle Gear connected to IM Gear connected to PMSM Optimal Battery [13] proposes a calculation solution for the best gear ratio and shifting procedure. The optimized vehicle drivetrain saves 14.53% fuel con­ sumption while improving the acceleration performance. Addressing an innovative configuration of a hybrid vehicle with two in-wheel motors, the authors of [14] developed an optimization of the powertrain design for reducing the vehicle energy consumption. passenger EVs have been launched with the dual-motor 4WD configu­ ration such as Audi E-Tron Sportback, BMW iX3, the design of Model S, X, and Y from Tesla. However, it is worth noticing that the existing EVs still keep a simple gearbox system. Due to the torque-speed characteristic of an electric motor that can provide high torque for a wide speed range, the commercialized EVs are generally equipped with a single-speed gearbox to reduce the drivetrain mass and cost. In the case of the EV required to fulfill extreme off-road conditions, the single-speed transmission offers only one solution as a trade-off between the acceleration ability and the top speed. Besides, although the electric motor efficiency can reach more than 95%, this value is changed over various operation ranges. Consequently, the single-speed transmission cannot ensure high opera­ tion efficiency for different driving conditions. Recent investigations have proved that multi-speed transmissions can solve the existing drawbacks of conventional systems. A two-speed Automatic Mechanical Transmission (AMT) integrated into EVs shows a better performance in terms of acceleration time, maximum speed, and energy economy [5,6]. To increase the motor efficiency while keeping the dynamic perfor­ mance, the authors of [7] validate the effective operation of two-speed Dual Clutch Transmission (DCT) and simplified Continuous Variable Transmission (CVT). The use of a six-speed forward and three-speed reverse transmission is beneficial for heavy-duty vehicles in [8]. In practice, in order to climb the high slopes while carrying heavy loads, the vehicle needs a very high torque at a certain power limitation. The reverse gear, in addition, also supports the vehicle’s back maneuvers. However, the larger electric system size and energy losses are asking more concerns. Besides the dual-motor multi-speed drivetrain configurations, how to optimize the gearbox ratios is also an important question because an appropriate design should offer both dynamic and energy performance enhancement. In [9] a genetic algorithm (GA)-based gear ratio design strategy is developed for maximizing the mean efficiency. The work [10] investigates different combinations of proper gear ratios. The objective function considers the acceleration time and the energy consumption under specific working conditions of a dump truck. The articles [6,11] address the gear ratio optimization problem based on dynamic pro­ gramming (DP) leading to the minimum energy consumption. The simulation results demonstrate the overall energy of the two-speed system is saved by 4.2% under the EPA75 driving cycle. Arming to improve the acceleration time and energy economy, the ratio selection of a two-speed gearbox is combined with the torque distribution prob­ lem between two motors in [12]. By means of optimization of the gear ratio and shifting strategy in [7], the motor efficiency is enhanced associated with faster acceleration and higher maximum speed where under HWFET driving cycle, 14% battery energy is saved. The paper 1.2. Motivation and contribution The common approach of the aforementioned articles is to apply state-of-the-art technologies and advanced optimization methods to maximize the dynamic quality and to minimize the energy consumption of the vehicles under study. Hence, it is not surprising that these works have conducted high-performance EV drivetrain designs. However, there is a lack of a method that optimizes the vehicle drivetrain regarding a design reference with the pre-defined desired performance. Let us take an analogous example in automatic controller synthesis to illustrate this issue. Given a plant, the control engineer has a set of required performances of the closed-loop system including overshoot, settling time, steady-state error, and gain and phase margins. The developed control techniques allow the engineer to synthesize the controller to achieve these pre-defined criteria. Moreover, there are applications of optimization algorithms for controller design without the need for human expertise when dealing with complex control systems. This analogy raises a question of a systematic methodology to auto­ matically deduce an optimal design for the vehicle drivetrain. This design can achieve fully pre-defined performance criteria in terms of dynamics and efficiency. Also, this technique can be applied to all EVs. In addition, most of the existing studies focus on passenger vehicles but the drivetrain optimization of off-road EVs has not been paid full attention to. To fill this gap of knowledge, this paper proposes a novel method­ ology for the optimal drivetrain design of a dual-motor 4WD off-road configuration to improve both the dynamic performance and energy efficiency of the vehicle. The study may contribute to the research field in the following points: (i) A comprehensive methodology to optimize the drivetrain system of the dual-motor off-road EV is introduced. The novelty of this methodology is based on minimizing the difference between the power envelopes of the studied vehicle and the reference design. To the best of the authors’ knowledge, this is the first time that the power envelope gap minimization is discussed for the drive­ train design of the EV. (ii) By considering the power envelop in the design process, the in­ fluence of the constant power speed ratio (CPSR) on the dynamic 2 C.T.P. Nguyen et al. Energy Conversion and Management xxx (xxxx) xxx performance of the vehicle is appropriately addressed. The role of this design variable has also not been investigated in the previous studies. Moreover, the combination of the CPSR and the motors efficiency maps allows us to instantaneously improve both the dynamic performance and energy efficiency of the studied vehicle. maximum efficiency of PMSM; ηtranIM and ηtranPM are the transmission efficiency for IM and PMSM, respectively. Besides the one-fixed ratio gear transmission, the multi-speed structures as two-speed, three-speed or four-speed [12,15–17] are also suggested for the dual-motor configurations. The multi-speed system allows both dynamic and economic performance enhancement. How­ ever, its negative features are poor transmission efficiency, increased weight, and higher costs. Therefore, it is necessary to choose a proper drivetrain architecture. In the studied EV, IM is connected to a fixed-speed gear, while a twospeed gear is used for PMSM for the following reasons. In terms of driving performance, engaging two motors to two individual fixed-speed gearboxes generally helps reduce the complexity, the energy loss, and the price of the transmission system. However, this is a trade-off be­ tween the high-speed performance and the-high force performance. For example, the gearbox ratios are calculated to reach the targeted maximum traction force. Higher gear ratios allow higher traction force, but meanwhile reduce the vehicle speed. Although the maximum force is met, which leads to faster acceleration, the top speed of the vehicle is limited by the dropped maximum speed of IM or PMSM. In contrast, to hit the targeted top speed, another smaller gear ratio pair is chosen. In this case, the lower power envelope (F − V curve) of the dual-motor model reduces the traction force capacity, and the acceleration takes more time. When the two-speed gear is utilized for PMSM, the benefits of high-speed and high-force performance are obtained at the same time. The studied model could provide a high maximum speed and satisfy the high traction force needed for acceleration and climbing. Considering the cost of the system and multi-speed energy loss, the fixed gear ratio is good enough for IM. In terms of efficiency performance, since IM is featured by high ef­ ficiency on a large speed range (Fig. 2(a)), the fixed-speed gearbox is used to cover the whole vehicle speed range without a large efficiency decrease. While efficiency map of PMSM presented in Fig. 2(b) shows an unequal efficiency over its speed range. More than 90% efficiency is expected at the speed range from 3500 to 5000 rpm. At a lower speed level, the motor efficiency drops dramatically. In particular, under 1000 rpm of the motor speed, the efficiency value is smaller than 60%. If the single-speed gearbox is employed and PMSM works with the full speed range, the operating points will fall outside the high-efficiency region. This reduces the overall efficiency, whereas the multi-speed gear moves operating points to the higher efficiency region which is well docu­ mented in [18,19]. As a result, a two-speed gearbox is designed for PMSM to achieve a high-performance and high-efficiency system. By applying the proposed approach, the optimized drivetrain system demonstrates a superior off-road capability in terms of higher maximum speed, faster acceleration, stronger gradeability, as well as less energy consumption. The proposed topology with a proper set of CPSR and gear ratios is evaluated via critical testing scenarios including speed-range test over a standard driving cycle, acceleration test up to a given speed, and hill climbing test up to 42% of the road grade. The dynamic and the energy performances are then examined with load change and grade fluctuation over a real-world off-road driving cycle. 1.3. Paper organization Section 2 of this paper presents the studied dual-motor 4WD off-road configuration and proposes the methodology of the drivetrain system optimization. Thereafter, Section 3 provides the optimization results and the performance improvement validations via extensive simula­ tions. Finally, the conclusions and perspectives are given in Section 4. 2. Methodology of drivetrain design optimization for the dualmotor 4WD EV 2.1. Dual-motor 4WD configuration The studied dual-motor EV is driven by an induction motor (IM) in the front and a permanent magnet synchronous motor (PMSM) in the rear axle as presented in Fig. 1. The drivetrain system in each axle consists of one electric motor which converts the electric energy fed by a battery to mechanical power and drives the wheels via selectable gearbox ratios. In practice, IM is preferred for a wide speed range at low torque, while on the contrary, PMSM offers high starting torque at low speed. Hence, taking the benefits of both motors can extend the maximum dynamic performance of the vehicle. The total maximum traction power is provided by IM and PMSM given by: Ptotmax = PIMmax .ηIMmax .ηtranIM + PPMmax .ηPMmax .ηtranPM (1) where PIMmax and ηIMmax are the maximum power and the maximum ef­ ficiency of IM; PPMmax and ηPMmax are the maximum power and the Fig. 1. Studied dual-motor 4WD off-road EV platform. 3 C.T.P. Nguyen et al. Energy Conversion and Management xxx (xxxx) xxx during driving and less energy consumption can extend the driving range, but none of these measures correlate well with driving experience in general and extreme off-road conditions in particular. Moreover, since the objective function is built by the F − V characteristic difference between two vehicle models, the proposed approach is not limited by the drivetrain system or the optimization method. Another special advantage of this optimization method using the power envelope gap minimization is to consider CPSR in the design process. As specified by one maximum power (Pmax ) value of the vehicle, there are numerous targeted power envelopes. These curves are bounded by the maximum vehicle speed and the maximum traction force which depends on the CPSR. According to [20], CPSR is defined as a ratio of the maximum vehicle speed (Vmax ) and the base speed (Vbase ) during the constant power region given by: CPSR = Vmax Vbase (2) The maximum traction force is expressed by: Fmax = Pmax Vbase (3) For a certain maximum power and maximum speed, the large CPSR (or low base speed) increases the maximum traction force value. This means the vehicle can show a better traction ability. The chosen targeted curve should satisfy all the required driving criteria, also it should have the narrowest power envelope gap with the dual-motor one. In this way, the dual-motor model gains a superior performance as the expected model. To the best of our knowledge, this is the first study to take CPSR solution into the gear ratio optimization. In this problem, the gear ratios of IM and PMSM, and CPSR are selected as the design variables. The objective function is used to determine an optimal solution for a drivability improvement under dynamic constraints. The optimization mathematical model is presented by: J= Vmax ∑ ( Fvehtarget (CPSR, V) − Fvehpro (V) ) (4) V=0 Besides given parameters by the vehicle and electric motors, the optimal solution is found out by minimizing the difference between two power envelope curves based on a proper set of gear ratios and CPSR given by: ( ) kgearIM , kgearPM , CPSR = argmin(J) (5) (kgearIM ,kgearPM )∈Ωkgear ,CPSR∈ΩCPSR Fig. 2. Efficiency map of the electrical drives: (a) IM; (b) PMSM. 2.2. General methodology The main goal of this study is to deliver a driving performance of the reference vehicle using two electric motors with their multi-speed gearbox as close as possible. The power of the EV is generated by one front-axle IM and one rear-axle PMSM. Each electric motor has indi­ vidual power envelope characteristics depending on its gear ratios. On the other hand, the power envelopes of the reference vehicle are determined by different CPSRs. This study aims to select the motor gear ratios and CPSR for an optimal solution. As a result, we defined the objective function to be the power envelope difference between the targeted model and the dual-motor model, and this objective function is minimized based on a proper set of gear ratios and CPSR. There are some alternative methods to design gearbox ratios, for example, energy consumption minimization or acceleration time minimization-based approaches have already been considered in some reviewed studies. We do not aim to outperform an existing approach in a single dimension such as energy consumption or rate of acceleration. Our goal is to design the drivetrain system to satisfy predetermined criteria using the limited resources in hand (using two electric motors and their gearbox structure). The power envelope characterizes driving experience in different driving conditions while two other metrics do not have this property. Better acceleration time matters in brief moments with Ωkgear and ΩCPSR are the feasible ranges of the gear ratios and CPSR. Fvehtarget (CPSR, V) is the power envelope of the targeted vehicle which is based on the following equation: ∫ Vbase ∫ Vmax Ptargetmax Fvehtarget (CPSR, V) = Ftargetmax + V 0 Vbase ∫ Vbase ∫ Vmax Ptargetmax Ptargetmax = + (6) Vbase V 0 Vbase As defined in (2), (3), the targeted maximum force is higher when rising CPSR at a certain maximum power and the fixed top speed, while the base speed is decreased. This large maximum force enables the vehicle to shorten the acceleration time and climb a greater incline. This means CPSR has significant effects on the power envelope of the targeted vehicle, and also contributes to the objective function (J) in (4). Fvehpro (V) is the power envelope of the studied vehicle which is driven by IM and PMSM as: 4 C.T.P. Nguyen et al. Energy Conversion and Management xxx (xxxx) xxx (∫ Fvehpro (V) = FIM (V) + FPM (V) = (∫ + VPMbase ∫ VPMmax FPMmax + 0 VPMbase ) PPMmax V ⎧ ηtran ⎪ ⎨ FIMmax = TIMmax (V)kgearIM R wheel ⎪ ⎩ FPM max = TPMmax (V)kgearPM ⎧ ωIMbase ⎪ ⎪ ⎨ VIMbase = kgearIM Rwheel ωIMmax ⎪ ⎪ ⎩ VIMmax = Rwheel kgearIM ⎧ ωPMbase ⎪ ⎪ ⎨ VPMbase = kgearPM Rwheel ωPMmax ⎪ ⎪ ⎩ VPMmax = Rwheel kgearPM ηtran 0 VIMbase ∫ VIMmax FIMmax + VIMbase ) clearly reflect the role of the motor gear ratios to the vehicle perfor­ mance as well as the objective function. The full structure of the proposed methodology consists of two layers described in Fig. 4. The upper layer determines the targeted power en­ velope through the steps: (i) from the desired performance re­ quirements, three main criteria including the top speed, acceleration time, and gradeability are determined; (ii) three criteria define corre­ sponding maximum power value; (iii) the targeted power envelope is drawn by this maximum power value as a benchmark curve. This curve is bounded by the maximum vehicle speed and the maximum driving force which depends on the CPSR. Then, the objective function (J) in (4) is implemented in the lower layer to find the optimal gear ratios. Due to the possible CPSR variation, the targeted power envelope can be changed. Thus, CPSR is integrated into the objective function as one of the design variables. The gear ratio optimization inside the lower layer is addressed by employing particle swarm optimization (PSO). Unlike traditional opti­ mization methods, PSO looks for the solutions without using the gradient of the problem. The performance of the objective function (J) in (4) is plotted in a three-dimension space in Fig. 5 for two different gear ratio pair cases corresponding CPSR = 4. As can be seen in Fig. 5a, b, the proposed objective function is expensive to evaluate and non-smooth. This is because the function is carried out by separated speed conditions and respecting dynamic constraints. In such conditions, the derivative-based methods fail to optimize the objective function or find a local solution and miss the global one. One more reason, although PSO shares similarities with Evolutionary Computation (EC) techniques, the PSO process does not require genetic operators such as crossover and mutation. It is mathe­ matically less complex and converges to the solution faster. However, it is noticed that PSO is chosen to use for the gear ratio optimization method, but it is not the only method that can be employed. The pro­ posed methodology does not limit any optimization algorithm depend­ ing on the designer’s decision. PIMmax V (7) (8) Rwheel (9) (10) in which FIMmax and TIMmax are the maximum traction force and the maximum torque of IM; FPMmax and TPMmax are the maximum traction force and the maximum torque of PMSM; ωIMbase and ωIMmax are the based rotational speed and the maximum rotational speed of IM; ωPMbase and ωPMmax are the based rotational speed and the maximum rotational speed of PMSM; kgearIM and kgearPM are the gear ratios of IM and PMSM; ηtran is the transmission efficiency; Rwheel is the wheel radius. Fig. 3 describes the logic used in the optimization of the objective function under two typical cases of different gear ratios of IM and PMSM. The power envelope of the reference model is shown in the black curve according to the selected CPSR of 4. IM and PMSM characterize the red power envelope and the blue one, respectively. The combination of two electric motors creates the power envelope of the dual-motor model plotted in purple. The proposed methodology is to minimize the power envelope difference of the two models. In other words, minimizing the objective function (J) in (4) is to minimize the green area. It is noticed that the various gearbox ratios change the motor power envelope characteristics. Using (7)–(10), the higher gear ratio increases the maximum force but reduces the speed of the vehicle and vice versa. The effect of motor F − V characteristics combined with the different gear ratios shapes the purple curve. The green area which corresponds to the mismatch between the targeted vehicle and dualmotor model is also different under each configuration. The optimal solution is found out by minimizing the green area. Fig. 3a and b of Case 1 (kgearIM = 15, kgearPM = 10) and Case 2 (kgearIM = 10, kgearPM = 20) 2.3. Power envelope definition of the targeted vehicle The reference vehicle offers three main driving criteria involving the top speed, the acceleration time, and the climbing ability. A maximum power value determined by these criteria characterizes the F − V curve well known as the power envelope of the vehicle [20]. The computation of the maximum power value of the targeted vehicle is presented below. This process is also summarized in the flow chart in Fig. 6. Fig. 3. The effect of gear ratios on the power envelope of the dual-motor vehicle as well as the objective function corresponding to CPSR = 4. 5 C.T.P. Nguyen et al. Energy Conversion and Management xxx (xxxx) xxx Fig. 4. Hierarchical optimization framework. Fig. 5. The 3D plots of the objective function performance in two different gear ratio pairs with CPSR = 4. 2.3.1. Power envelope based on maximum speed performance A maximum power value is required to hit the maximum vehicle speed (Vmax ) calculated by: Ptargetmax (Vmax ) = Ftargetmax (Vmax ) .Vmax 1 2 + Mgsin(α) Ftargetmax (Vmax ) = Mgfroll cos(α) + ρair Cx AVmax 2 (12) where M is the vehicle total mass; g the acceleration of gravity; froll the rolling coefficient; ρair the air density; Cx A the aerodynamic standard; and α the climbing angle of the road. In this case, it is assumed that α = 0 . (11) where Ptargetmax (Vmax ) is the maximum power at Vmax ; and Ftargetmax (Vmax ) is the maximum force at Vmax of the targeted model. 6 C.T.P. Nguyen et al. Energy Conversion and Management xxx (xxxx) xxx Fig. 6. Flow chart of maximum power definition of the targeted vehicle. 2.3.2. Power envelope based on acceleration performance The acceleration time (tacc ) is how long it takes the vehicle to accelerate from zero to a given speed with a full load. This time is used to measure the vehicle’s acceleration ability. The drivetrain system generally needs to be developed to shorten the acceleration time. The consumed power during accelerating up to the speed (Vacc ) is: Ptargetmax (acc) = Ftargetmax (acc) .Vacc at Vacc ; and Ftargetmax (acc) is the maximum force at Vacc of the targeted EV, as presented in the following: 1 dV Ftargetmax (acc) = Mgfroll cos(α) + ρair Cx AV 2 + Mgsin(α) + M 2 dt (14) The acceleration time from zero to Vacc is deduced from (14) with α = 0 as follows: (13) where Vacc is the acceleration speed, Ptargetmax (acc) is the maximum power 7 C.T.P. Nguyen et al. Energy Conversion and Management xxx (xxxx) xxx ∫ Vacc tacc = Ptargetmax (acc) V 0 ∫ Vbase ∫ Vacc ( + Vbase Ptargetmax (acc) V − M ( = 0 ⎧ ) ( ⎨ vi (k + 1) = wvi (k) + c1 r1 Pbest,i (k) − xi (k) + c2 r2 (Gbest (k) − xi (k) ) ⎩ xi (k + 1) = xi (k) + vi (k + 1); i = 1 to n M ( ) dV − Mgfroll + 12ρair Cx AV 2 Ptargetmax (acc) Vbase − ) dV where c1 and c2 are the cognitive and social parameters, respectively, which impact directly on the position of Pbest,i (k) and Gbest (k) ; r1 and r2 the vectors random numbers generated at every iteration (0 ≤ r ≤ 1) ; w the inertia factor balancing global and local search; and n the swarm size. Once the particle’s velocity and position have been calculated, Pbest,i (k) and Gbest (k) are updated simultaneously. Mgfroll + 12ρair Cx AV 2 M ) dV (15) Mgfroll + 12ρair Cx AV 2 The acceleration time is calculated in two separate regions as the constant force region and the constant power region divided by the base speed (Vbase ). Nevertheless, the base speed is changed by the CPSR factor as expressed in (2). The previous work [20] shows that at the same maximum power and maximum speed, higher CPSR results in a base speed decrease and a larger maximum traction force. This can reduce the acceleration time. In the fact of automobile applications, the CPSR value is limited by the mechanical structure up to 10:1 [21]. To handle (15), firstly, the maximum power value (Ptargetmax (Vmax ) ) computed in (11) is recalled into (15) for an acceleration time evalua­ tion from 0 to Vacc . Compared to the desired acceleration time, if the calculated time by (15) is longer, the maximum power value will be increased in accordance with the result of (1). After that, the updated Ptargetmax is used to re-calculate the acceleration time according to (15). Because the tacc is related to the CPSR factor, at first this calculation is applied for the case of the minimal ratio of the CPSR. A low ratio of CPSR implies a long acceleration time period. Therefore, at this CPSR value, if the required acceleration time is satisfied, the updated Ptargetmax is suffi­ cient for the expected acceleration. Otherwise, this action is repeated for a new larger CPSR until the tacc is equal or less than the expected time. 2.4.1. Optimization control variables and dynamic constraints As presented, after defining the maximum power value, which is enough to cover all expected driving performances, the power envelope of the targeted model is generated. This curve is limited by the targeted maximum vehicle speed and maximum traction force based on CPSRs. And then, the design variables including the motor gear ratios standing for kgearIM , kgearPM1 , and kgearPM2 and CPSR are embedded in the opti­ mization problem to minimize the power envelope difference of the targeted vehicle and the studied dual-motor vehicle. The feasible ranges of these design variables are confirmed based on the EV dynamic con­ straints as the following. 2.4.1.1. Maximum force constraint. Under the departure condition or climbing, the cooperation of two electric motors is implemented to in­ crease the traction force. The gear ratio pair in this event would be kgearIM and higher kgearPM1 . The maximum traction force provided by two mo­ tors is necessary to exceed the total resistance force in (16) given by: TIMmax 2.3.3. Power envelope based on gradeability performance Off-road vehicles are often featured by the greater traction force at the low-speed range in uphill situations than on-road vehicles are. In the case of this targeted model, the gradeability in which the vehicle can ascend a maximum road slope (α) at the speed (Vgrade ) is supplied by the maximum force defined by: 1 2 Ftargetmax (grade) = Mgfroll cos(α) + ρair Cx AVgrade + Mgsin(α) 2 (16) Ptargetmax (grade) = Ftargetmax (grade) .Vgrade (17) (18) kgearIM kgearPM1 + TPMmax ≥ Ftargetmax (grade) Rwheel Rwheel (19) 2.4.1.2. Maximum speed constraint. The top vehicle speed is basically limited by the maximum motor speed. Meanwhile, kgearIM and kgearPM2 can make a decision on the value of the maximum speed of IM and PMSM (see (9),(10)). For this reason, the upper bounds of the motor gear ratios are constrained by the maximum vehicle speed (Vmax ) given by: ⎧ ωIMmax 3.6Rwheel ⎪ ⎪ ⎪ ⎨ kgearIM max = Vmax (20) ⎪ ωPMmax 3.6Rwheel ⎪ ⎪ ⎩ kgearPM2 max = Vmax where Vgrade is the given speed in gradeability, Ptargetmax (grade) is the maximum power at Vgrade , Ftargetmax (grade) is the maximum force at Vgrade of the targeted EV. In conclusion, in the comparison of three maximum power values in (11), (13), and (17), the highest value is selected as the best maximum power of the targeted vehicle, which satisfies all performance re­ quirements. The targeted power envelope is then extracted by this power value with the possible CPSRs. 2.4.2. Objective function According to the general function in (4), two separate objective functions are suitable for the studied drivetrain structure where IM uses the fixed-speed gear and PMSM takes the two-speed gear. They are purposed to optimal solutions over two different speed regions. In the low-speed region, the objective function (J1 ) looks for the design vari­ ables as kgearIM , kgearPM1 to reach the targeted maximum traction force. 2.4. Optimal gearbox design using PSO J1 = PSO algorithm was introduced in 1995 [22] as a meta-heuristic optimization method. PSO simulates the behavior of a social group, such as a bird, fish, or ant flock. Each particle represents a point in the search space. The current particle’s position is determined by the n -dimension vector xi (k) . In the kth iteration, each particle’s position is renewed by its own updated velocity vector vi (k) , where i is the indi­ vidual particle and k is the iteration number. The particles fly towards the best personal solution Pbest,i (k) and the best global solution denoted by Gbest (k) . The optimal solution can be represented by Gbest position. The updated velocity and position vectors of the particle swarm are formulated by: On the other hand, in the high-speed region, the best combination of kgearIM , kgearPM2 is selected by the objective function (J2 ) for a maximum speed satisfaction given by: Vmax ∑ ( Fvehtarget (CPSR, V) − Fvehpro1 (V) ) (21) V=0 J2 = Vmax ∑ ( Fvehtarget (CPSR, V) − Fvehpro2 (V) V=0 Subject to: 8 ) (22) C.T.P. Nguyen et al. Energy Conversion and Management xxx (xxxx) xxx Fig. 7. Flow chart of the gear ratio optimization. accordance with different CPSRs in the outer loop, the minimum value depicted to fopt is selected. The proper set of CPSR and gear ratios is found out. kgearIM min ≤ kgearIM ≤ kgearIM max kgearPM1 min ≤ kgearPM1 ≤ kgearPM1 max kgearPM2 min ≤ kgearPM2 ≤ kgearPM2 max (23) 3. Results and discussion CPSRmin ≤ CPSR ≤ CPSRmax TIMmax kgearIM kgearPM1 + TPMmax ≥ Ftargetmax (grade) Rwheel Rwheel 3.1. Studied EV system It is observed by (4), (6), the power envelope of the targeted model can be reshaped by the CPSR range. This means CPSR also contributes to the objective function evaluation. For this reason, the optimal gear ratio determination should take care of the best CPSR selection within (1, 10) interval as explained in Subsection 2.3.2. A two-loop optimization schedule is demonstrated in the flow chart in Fig. 7. Firstly, the minimum value of CPSR is selected in the outer loop. Corresponding to the initial CPSR, the inner loop shown detailed in the right-side box realizes the specified targeted power envelope for the gear ratio design using PSO. This algorithm evaluates the objective functions in (21) and (22) to find the best values of gear ratios. This process is iterated on gradually increasing CPSR to its maximum value. Eventually, in comparison of objective function results obtained by PSO in The target vehicle is based on the off-road e-Commander of e-TESC laboratory at the University of Sherbrooke. The original e-Commander 4WD off-road topology (T1) includes one IM driving two or four wheels selectable by a shaft driven with a lockable rear differential. The 48 V IM with the peak power of 22.5 kW and the continuous power of 8.2 kW connects to the wheels by a fixed gear ratio of 20.5. This motor mounted on the front axle fed by a three-phase inverter and supplied by a 6.9 kWh battery pack provides the power to the studied e-Commander. The vehicle model and the control scheme are organized and simulated using energetic macroscopic representation (EMR) in Matlab/Simulink™. The EMR model includes the battery model. The traction system uses the electric motor with its gearbox transmission to transfer torque from the 9 C.T.P. Nguyen et al. Energy Conversion and Management xxx (xxxx) xxx Table 2 Full-scale system parameters. Parameters Values Vehicle (e-Commander) Vehicle total mass Aerodynamic standard Rolling coefficient Air density (at 20 ◦ C) Wheel radius Transmission efficiency Fig. 8. Power envelopes of the T1 and T2. Acceleration time from 0 to 45 km/h Grade slope at 20 km/h t0− 45km/h 13.3 Induction motor Maximal power PIMmax 22.5 kW Maximal efficiency ηIMmax 86 % TIMmax 60 Nm nIMmax 8000 rpm Permanent magnet synchronous motor Maximal power PPMmax 16 kW Maximal efficiency ηPMmax 93 % TPMmax 34 Nm Maximal speed nPMmax 5000 rpm Battery Battery bank capacity Battery bank resistance (at 80% SoC) No. of cells in series No. of cells in parallel Cbat rbat Ns Np 110 Ah 6 mΩ 4 3 Scenarios Number of particles (n) Number of iterations (ite) Inertial weight (w) Cognitive coefficient (c1) Social coefficient (c2) S1 100 0.4–0.8 2 2 S2 {100, 200, 300} 100 0.4–0.8 2 2 S3 100 {100, 200, 300} 100 2 2 S4 S5 100 100 100 100 {0.4, 0.8, 1.0} 0.4–0.8 0.4–0.8 {0, 1, 2} 2 2 {0, 1, 2} ▪ Mean value of the best global solutions f(xG ) over the number of iterations for the best run expressed by: ∑itemax f (xG (ite)) Mean = ite=1 (24) itemax ▪ Standard deviation (SD) value given by: √̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ √ite∑ √ max √ (f (xG (ite) − Mean )2 √ SD = ite=1 itemax (25) The sensitivity analysis is implemented under five scenarios as included in Table 3. In each scenario, the tested parameter is tuned in its value range, while the others keep recommended values. The previous works in the literature notice that the bigger number of particles increases the chances of searching successfully but also wastes processing time. Danial et al. [26] suggest a selected population from 25 to 500 individuals. In some other cases, the swarm size is decided by the design variable number. The number of times that the particles move to a new place is called the iteration number. As an agreement of the convergence rate and the computational cost, a potential range of the iteration is between (100, 1000) by [26]. The effect of the previous Values Vacc 20.5 kgearT2 Table 3 PSO parameter testing scenarios. Table 1 EV requirements. Vmax kgearT1 Gearbox ratio in T2 Maximal torque In PSO, tuning control parameters could considerably affect computational behavior. Hence, a parameter-sensitive analysis should be investigated to determine the impact of these parameters on the overall PSO performance. The sensitivities of five control parameters including the number of iterations (ite), the number of particles (n), the inertial weight (w), the cognitive coefficient (c1), and the social coeffi­ cient (c2) are evaluated by the following criteria: Acceleration speed Gearbox ratio in T1 ρair Rwheel Maximal speed 3.2. Drivetrain design optimization results Maximum speed ηtranIM,PM 871.1 kg 1.3 m2 0.035 1.223 kg/m3 0.3175 m 91 % Maximal torque motor armature to the driven force. The interaction between the traction force and the environment force delivers the vehicle speed in the EV chassis system. The global EV model using EMR representation was provided in [23,24]. After that, an on-road driving test with the tra­ jectory at the University of Sherbrooke campus conducted on this EV prototype by the e-TESC Lab validates the vehicle’s mathematical model described in previous studies [25]. The dynamic performance of T1 is presented in the green solid line in Fig. 8. The gear ratio used in the T1 comes from the datasheet provided by the manufacturer for the original e-Commander version. The top speed is limited at 45 km/h, and the maximum force is 3500 N. The acceleration time to reach 45 km/h is 12 s. One 16 kW (peak power) PMSM is added on the rear axle of T1 to become the dual-motor topology (T2). The same gear ratio of 13.3 is used for two motors in T2. Since the gear ratios were not considered as a part of the design in the past, despite the larger traction power capacity, the dynamic performance of dualmotor T2 was very poor. The blue dashed line in Fig. 8 referring to the T2 performance shows that the top vehicle speed ends at 45 km/h, and the maximum traction force is around 3600 N. In comparison with T1, using two electric mo­ tors in T2 does not see any driving performance increase. In other words, the dual-motor and one fixed-speed gearbox do not take full advantage of the high-power system. Motivated from the T2 performance limita­ tion, this study is carried out to deliver the desired performance following the maximum power capacity of the vehicle. As a result, T3 is introduced as an improvement of T2. The drivetrain structure of T3 considering the gear ratios and CPSR is appropriately designed by applying the proposed methodology. The T3 drivability is expected to satisfy the high-performance requirement as given in Table 1. Full-scale parameters of the e-Commander 4WD off-road EV are summarized in Table 2. Parameters M CxA froll 70 km/h 45 km/h 5.5 s 40% 10 C.T.P. Nguyen et al. Energy Conversion and Management xxx (xxxx) xxx Fig. 9. PSO parameter selection procedure and the results. velocity on the current velocity is denoted by the inertial weight (w). The larger w stimulates global exploration but probably overlooks local exploitation. The w is introduced between (0.4, 1.2) for the best PSO actions in [27]. The acceleration coefficients (c1, c2) represent the par­ ticle moving toward the personal best and the global best, respectively. The sum of both values should not be higher than four as the conception of PSO in [22]. As a result of this analysis, the swarm size is examined from 100 to 300 particles; the number of iterations is within a range of (100, 300); the inertial weight is between (0.4, 1) and c1, c2 are varied from zero to two. As seen in Fig. 9, in S1, the SD values show that the more particles covering a search space allow the more efficient performance. Also, due to the higher number of iterations, the seeking actions in S2 are more successful with lower SD values. Although the computation time in S1 and S2 is longer when the population size (n) and the iteration number (ite) grow, the process just takes about 10 s or less. S3 results indicate that the particles are more likely to explore the new space by a longer step based on a larger w (higher SD). Conversely, a smaller w motivates the particles to exploit the expected local area (lower SD). The selected w value should therefore take care of both global and local search. Addi­ tionally, the coefficients c1, c2 are expected to remember the last best own position and also share the information to the others as reflected by the smallest SD values in S5 results. The best PSO behavior is evaluated by the minimal values of SD as well as balancing between the global and the local search. After the above sensitive analysis, the PSO control parameters are used including the number of particles of 300, the iter­ ation of 300, the inertia weight between (0.4, 0.8), and the acceleration coefficients of 1. The optimization results are found that the best value of CPSR is 4; the IM-connected gear ratio is 13.5; the first and the second gear ratio for PMSM are 24.5 and 7.75, respectively. The sum of two motor power envelopes is the power envelope of the dual-motor vehicle as plotted in the red dot line in Fig. 10. The gear shifting of PMSM occurs at the vehicle speed of 23.6 ± 0.1 km/h to avoid the eventual chattering around the switching point. 3.3. Performance validations of the optimized drivetrain design The studied dual-motor EV is modeled and controlled based on MATLAB/SimulinkTM. The topologies T2 and T3 employ the same rulebased torque distribution strategy which is presented in our previous study [23]. Below 800 rpm of the PMSM speed, the torque demand is completely produced by IM. After 800 rpm, both PMSM and IM share the same torque amount to the wheels with the torque split ratio of 0.5. To verify the effective performance of the optimal drivetrain design, the performance of three topologies T1, T2, and T3 is compared and analyzed in the following driving conditions: ▪ Vehicle operation during a modified New European Driving Cycle (NEDC) with the top speed limited at 70 km/h; ▪ Vehicle acceleration from 0 to 45 km/h on a flat road; ▪ Vehicle gradeability with 20 km/h; ▪ Vehicle operation under a typical off-road driving cycle (ORDC); Fig. 10. Power envelope of the dual-motor T3 with optimized gear ratios. 11 C.T.P. Nguyen et al. Energy Conversion and Management xxx (xxxx) xxx Fig. 11. Vehicle speed response under NEDC cycle. Fig. 14. Vehicle speed response under a 0–45 km/h acceleration. performance is executed by applying full throttle to the given speed and then analyzing the results. The acceleration is evaluated by the accel­ eration time from 0 to 45 km/h with the maximum motor torque. Thanks to the two-speed gear solution of PMSM, the higher gear ratio allows a significant acceleration improvement to around 23.6 km/h. T3 provides a maximum traction force up to 4300 N followed by 3250 N and 3070 N of T2 and T1, respectively (see Fig. 13). On the other hand, after 23.6 km/h known as the gear switching speed of PMSM, the lower gear ratio is then engaged. The maximum force is reduced leading to slower acceleration. In Fig. 14, T1 accelerates from rest to 45 km/h in 12 s, while it takes only 5.3 s to accelerate T2 and T3 up to this given speed. In short, the acceleration time from stop to 45 km/h of the proposed design (T3) is less than 5.5 s of the required time in Table 1 and is reduced by 6.7 s compared to T1. Fig. 12. EV operation points under NEDC cycle. 3.3.3. Gradeability performance In this case, it is assumed that throughout acceleration from 0 to 20 km/h, EV will be asked for further gradeability. This test validates the maximum road slope that the vehicle can ascend respected to its dy­ namic limitations. As discussed in the previous part, smaller than 23.6 km/h, T3 shows the incredible traction force involved in the higher gear ratio of PMSM (in red points in Fig. 15). For this reason, the dual-motor model can climb up to 42% uphill grade. This figure is much higher than the 30% and 16% of the maximum slope for T2 and T1 climbing, respectively. ▪ Vehicle overall efficiency under ORDC. 3.3.1. High-speed performance The results in Fig. 11 and Fig. 12 demonstrate that the maximum speed of T3 with optimal gear ratios is increased up to 70 km/h, whereas T1 and T2 can only reach 45 km/h as the top vehicle speed. This is because the motor speed range is limited by using the single-speed gear in T1 and T2. As consequence, to deliver the possible maximum force for acceleration and climbing, the top vehicle speed should be lower (as at 45 km/h). Furthermore, by means of sharing the same gear ratio for two motors, T2 does not give IM a chance to hit 8000 rpm of the maximum motor speed. The IM speed here is being restricted by the PMSM maximum speed at 5000 rpm. However, the T3 design enables two motors to take full their speed range. 3.3.4. Off-roading ability performance The efficient performance of the proposed T3 is validated by the real conditions of the off-road vehicles through a typical driving profile. The off-roaders are actually ready to adapt to the load mass change on any extreme trail or terrain. The dual-motor vehicle using optimized gear ratios is operated under ORDC in Fig. 16. Over the first 500 s, the model is driven on a flat road without load, and sometimes the downhill spots occur. Between 500 s and 600 s, 272 kg, the full cargo capacity, of the extra load mass is filled onto the vehicle. From 600 s to 1300 s, the eCommander is asked to climb different road slopes from 0% to about 30% with the full load. As presented in Fig. 16, the off-roading abilities are realized by the optimal drivetrain design of T3. On smooth roads and no-load conditions, the vehicle could run up to 70 km/h of the top speed. The large maximum force allows the model to climb on a hill of 32% grade at 10–20 km/h. In the higher-speed climbing conditions, at around 30–40 km/h, after 900 s, the EV can ascend a lower gradient under 10%. The red operation points of T3 in Fig. 17 almost cover the working space of the force-speed map. They are located up to 70 km/h of the top vehicle speed, and 4300 N of the peak force. These red points also follow well the boundary of the power envelope of T3. There is no doubt that, in some places, T3 can work outside the working space of T1 and T2. By these results, the driving performance improvement of T3 is totally validated. 3.3.2. Acceleration performance A common procedure to test the accelerating or climbing Fig. 13. EV operation points under a 0–45 km/h acceleration. 12 C.T.P. Nguyen et al. Energy Conversion and Management xxx (xxxx) xxx Fig. 15. EV operation points under a climbing at 20 km/h. 85%. However, the maximum speed of IM in T2 is limited by 5000 rpm, so the high efficiency at the high-speed region is omitted. Meanwhile, T3 enables the red points to have high efficiency in the full speed range. Seeing the blue points representing the T2 operation in the PMSM effi­ ciency map (Fig. 18b), at the motor speed lower than 2000 rpm, the motor must work in less than 70% of efficiency conditions. In addition, the T2 operation points are not observed in the highest efficiency area (more than 90%) at above 3500 rpm. Compared to T2, the designed gearbox of T3 can adjust the position of the red points into the higher efficiency area at the high-speed region. As we can see, PMSM efficiency can be more than 90% in the 3500–4500 rpm range. In conclusion, T3 operates in the areas that both motors are the most efficient and this can be observed by the position of red points. As a result, the overall efficiency is enhanced by the proper drive­ train design of T3 under the ORDC as presented in Fig. 19. The average efficiency of T2 is 70.1% whereas it figures 72.0% for T3. The maximum value of the T3 efficiency hits 81.1% followed by 78.8% at the highest efficiency value of T2. Due to the efficiency enhancement of the proposed powertrain design, the driving range of the vehicle is also extended. To examine the range autonomy of the topologies, we employ a repeated speed profile of Worldwide harmonized Light vehicles Test Cycles (WLTC). The cycle has been modified for suitable to the speed range of the studied topologies with the maximum speed of 45 km/h. The vehicle stops when the battery is discharged to 20% of its state-of-charge (SoC). As revealed in Fig. 20, with the T3 topology, the vehicle can run 1.4 and 3.7 km longer than the one with T2 and T1, respectively. Moreover, it is noteworthy that this comparison is even not fair for the proposed T3 configuration because it only runs up to 45 km/h (the limit of T1 and T2) whereas its capability is 70 km/h. Hence, T3 does not work at its optimal efficiency region in this comparison. It should be noted that the comparative study conducted in this section is not to validate the advantages of the optimization method which is driven by the well-known PSO algorithm. The optimization is a part of the powertrain design methodology proposed in this paper. The output of the design process is the configuration T3 which has been justified being superior to the conventional unoptimized topologies T1 and T2. Fig. 16. Off-road driving cycle (ORDC). 4. Conclusion The paper has proposed a novel comprehensive approach for the drivetrain structure optimization of the dual-motor off-road EV. The proposed drivetrain design for the dual-motor model produces the high driving performance referenced by the targeted vehicle. The proposed methodology is based on minimizing the power envelope difference between the targeted and the dual-motor models. The optimization process is systematically presented in two layers. The first step is to build the targeted power envelope according to the pre-determined driving criteria in the upper layer. It is noted that CPSR plays an important role in the power envelope as well as the dynamic performance of the tar­ geted vehicle. Hence, CPSR needs to be considered in the design process. Next, the minimization of the power envelope gap between the two Fig. 17. EV operation points under ORDC. 3.3.5. Overall efficiency improvement Fig. 18 shows the motor operation points in T2 (blue points) and T3 (red points) during ORDC. In the efficiency map of IM (Fig. 18a), T2 and T3 operating points cover the efficiency regions which are higher than 13 C.T.P. Nguyen et al. Energy Conversion and Management xxx (xxxx) xxx Fig. 18. Operation points under ORDC in (a) IM efficiency map, (b) PMSM efficiency map. Fig. 19. System efficiency under ORDC. models is carried out in the lower layer. This layer finds the gear ratio solutions for two electric motors, corresponding to the optimal CPSR value by the PSO algorithm. The proposed design approach is applied to the drivetrain system of the dual-motor 4WD e-Commander off-road EV to evaluate its effectiveness. Compared to T1 and T2, the gear ratio optimization implemented in T3 offers a significant enhancement in multiple driving criteria. The top speed of T3 is 55% higher than T1 and T2; acceleration time from stop to 45 km/h is reduced by 55.83% in the comparison with T1; and the gradeability is improved 162.5% and 40%, respectively, compared to the T1 and T2. In real-world conditions, T3 verifies the great off-roading ability, also the average efficiency improvement to 72.0% from 70.1% of T2 overall efficiency. This novel methodology provides the comprehensive drivetrain design solution for high performance EV configurations. The proposed multi-step procedure is promising for a variety of the EV applications. In a future study, the torque distribution between two motors will be optimized to maximize the global energy efficiency. Thereby, the dualmotor multi-speed gear vehicle is expected for the most efficient driving performance and energy efficiency. Fig. 20. 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