Optimal Driving Torque Control Strategy for Front and Rear Independently Driven Electric Vehicles Based on Online Real-Time Model Predictive Control

Abstract

This paper presents a novel driving torque control strategy for the front and rear independently driven electric vehicle (FRIDEV) to reduce energy consumption and enhance vehicle stability. The strategy is built on a comprehensive vehicle model that integrates vertical load transfer, tire slip dynamics, and an electric system model that accounts for losses in induction motors (IMs), permanent magnet synchronous motors (PMSMs), inverters, and batteries. The torque control problem is framed with a nonlinear model predictive control (MPC) method, utilizing state-space equations as representations of vehicle dynamics. The optimization targets adjust in real-time based on road traction conditions, with the slip rate of front and rear wheels determining the torque control strategy. Active slip control is applied when slip rates exceed critical thresholds, while under normal conditions, torque distribution is optimized to minimize energy losses. To enable online real-time implementation, an improved sparrow search algorithm (SSA) is designed. Simulations in MATLAB/Simulink confirm that the proposed online strategy reduces energy consumption by 2.3% under the China light-duty vehicle test cycle-passenger cars (CLTC-P) compared to a rule-based strategy. Under low-adhesion conditions, the proposed online strategy effectively manages slip ratios, ensuring stability and performance. Improved SSA also enhances computational efficiency by approximately 44%–52%, making the online strategy viable for real-time applications.

1. Introduction

According to statistics provided by the International Energy Agency (IEA), in 2022, private cars and vans were responsible for over 25% of global oil consumption and approximately 10% of global energy-related CO₂ emissions. To reduce greenhouse gas emissions, a growing array of countries are implementing carbon neutrality initiatives and other climate change actions [1]. In the context of the current global push for carbon neutrality, reducing energy consumption and emissions in the automotive sector is urgent. Replacing the internal combustion engine vehicles (ICEVs) with the electric vehicle (EV) is a practical way to decarbonize transportation [2,3].

1.1. Literature Review

The overall performance of EVs is directly influenced by their powertrain systems [4]. Generally, EV powertrains can be classified into single-motor and multi-motor systems [5]. A single-motor powertrain uses one motor as the power source, transmitting torque to the drive wheels through a transmission system. In contrast, a multi-motor powertrain uses multiple motors, either coupled or independently driven, transmitting power to the drive wheels. Initially, single-motor powertrains were widely used in EVs because of their simplicity, reliability, and low cost [6]. However, a single motor provides limited power, limiting the vehicle’s power output. And the system’s average efficiency tends to be low because motor operating points on the efficiency map cannot be adjusted according to driving conditions. As an alternative, the multi-motor powertrain has been introduced for EVs to achieve higher average efficiency and better dynamic performance [7].

Among the various multi-motor configurations, the front and rear independent drive electric vehicle (FRIDEV) offers a simple yet reliable structure, making it a promising alternative to single-motor systems. Compared to single-motor and dual-motor coupled drives, FRIDEV can independently control the motors on the front and rear axles and optimize torque distribution. This configuration enhances vehicle traction and grip by delivering power to front and rear wheels, improving both performance and off-road capability. Moreover, by efficiently distributing power between the front and rear motors, energy losses are minimized or energy recovery is maximized, thus enhancing vehicle efficiency. Results in [8] show that the FRIDEV outperforms the traditional single-motor EV in terms of overall efficiency. Furthermore, the FRIDEV provides enhanced robustness and reliability. If one motor fails, the other can drive the vehicle independently, offering greater system redundancy [9]. In recent years, induction motors (IMs) or permanent magnet synchronous motors (PMSMs) alone have struggled to meet the increasingly stringent and specific demands of modern vehicle applications. To address this problem, there has been growing interest in improving EV designs by using combinations of different types of motors [10]. This approach leverages the unique characteristics of each motor type to optimize overall vehicle performance and efficiency, ensuring that the system can better handle diverse driving conditions and requirements. Research in [11] shows that the dynamic performance of the FRIDEV has been enhanced using both IM and PMSM.

The dual-motor configuration of FRIDEV, as opposed to a single-motor powertrain, presents additional complexities in torque optimization control. The primary challenge is to achieve a balance between minimizing energy consumption and ensuring vehicle stability. Various torque control strategies have been developed for the multi-motor EVs, which can be broadly classified into two main categories: rule-based methods and optimization-based approaches [5]. Rule-based control strategies dominate the current automotive industry due to their simplicity and ease of implementation [12]. However, their effectiveness can vary significantly depending on driving conditions, leading to limited adaptability. In response to the limitations, optimization-based control strategies have emerged as a more flexible and effective alternative. These strategies transform torque optimization control into a complex nonlinear constrained optimization problem, making use of mathematical models and an optimal algorithm to determine the torque of each motor. Optimization techniques applied in related studies include convex optimization methods [13], swarm intelligence methods [14], and model predictive control (MPC) methods [15]. To implement these approaches effectively, precise vehicle models and accurate state estimations are essential for defining the objective functions, setting constraints, and determining optimal or near-optimal solutions [16]. The results of related studies have shown that optimization-based strategies outperform rule-based ones in terms of energy efficiency.

Several studies have improved torque control strategies through detailed loss analysis using refined powertrain models. These models account for various energy losses, such as motor losses, inverter losses, tire slip losses, and mechanical losses in the transmission gears, among other factors affecting vehicle efficiency. By incorporating these specific loss elements, researchers aim to develop more accurate models that not only reduce energy consumption but also enhance overall vehicle performance. For instance, separate loss models for both the motor and inverter are developed to minimize electrical system losses [17,18]. Similarly, a comprehensive power loss model for the dual-induction motor system is presented to optimize motor torque distribution [19]. Moreover, tire slip losses are integrated into torque control strategies, as seen in studies like [20,21], aiming to minimize a cost function that considers both motor energy losses and tire slip losses. However, many of these studies primarily focus on energy efficiency under the assumption of ideal road conditions with optimal traction. While this assumption simplifies the modeling process, it overlooks the critical importance of vehicle dynamics, especially in real-world driving scenarios where road surfaces may be slippery. In such conditions, traction and stability become just as important as energy efficiency, with slip ratio management being crucial for maintaining vehicle control and safety. To address these challenges, MPC has gained prominence due to its ability to handle nonlinear, constrained optimization problems, making it particularly well-suited for torque control in FRIDEV. MPC predicts the vehicle’s future behavior based on its current state and uses this information to continuously adjust control inputs in real time, ensuring that slip ratios remain within a safe limit and reduce total energy loss. The results of [22,23] show that the MPC-based torque control strategy can automatically allocate the torque requirement according to the dynamic load and reduce power consumption. Studies such as [24,25] demonstrate that MPC-based torque control strategies are highly effective in tracking target slip ratios, even under challenging road conditions, thus improving vehicle stability, safety, and dynamic performance. However, MPC is limited by its relatively slow computation speed, indicating a need for further optimization to enhance real-time performance.

1.2. Motivation and Contribution

Although existing torque control strategies have made some progress in optimizing energy efficiency and slip rate control, several key issues still need to be addressed. First, insufficient consideration of vehicle dynamics and precise electric system modeling can reduce torque control accuracy and overall system performance. Additionally, torque control strategies should account for both energy efficiency and vehicle stability simultaneously. Moreover, the lengthy calculation process of optimization requires algorithm improvements to enhance computational efficiency. To tackle these challenges, the main contributions of this paper are as follows:

• This study uses a FRIDEV architecture with an IM on the front axle and a PMSM on the rear axle, modeling vehicle dynamics and electric system losses, including motor, inverter, and battery losses, while applying a motor loss minimization algorithm.
• A torque control strategy based on nonlinear MPC is proposed, which comprehensively considers electric system losses and tire slip losses. This strategy not only enables power allocation between front and rear axles but also provides anti-slip functionality, thereby enhancing energy efficiency and vehicle stability.
• To effectively solve the optimization problem of nonlinear MPC, an improved SSA was designed, which uses chaos mapping for population initialization and improved producer position updates and individual perturbation mechanisms.

1.3. Organization of the Paper

The rest of this paper is organized as follows. Section 2 introduces the FRIDEV architecture and analyzes and models the dynamic vehicle and electrical systems. Section 3 presents an offline optimized driving torque control strategy and an optimal driving torque control strategy based on real-time MPC, aiming at reducing energy loss in a FRIDEV powertrain. In Section 4, the simulation experiment is established based on the driving cycles and driver behavior to confirm the effectiveness of the proposed strategy. Finally, the conclusion is given in Section 5.

2. FRIDEV Architecture and Modeling

This section begins with an introduction to the architecture of the FRIDEV towed by both an IM and a PMSM. Following this, the dynamic vehicle system and the electrical system are analyzed and mathematically modeled. Additionally, a motor loss minimization algorithm is implemented to reduce motor power losses for given torque and rotational speed. The symbol list for this section can be found in Appendix A in

Table A1. Symbols list.

Symbol	Parameter	Symbol	Parameter
Ft	Vehicle total longitudinal traction force (N)	γ1	IM constant factor mechanical loss
Fx1, Fx2	Longitudinal traction force of front/rear wheels (N)	Ploss_m1	IM loss (W)
M	Vehicle mass (kg)	PIM	IM power (W)
vx	Vehicle longitudinal velocity (m/s)	ηIM	IM efficiency
ρ	Air density (Kg/m3)	id, iq	PMSM armature current of d-axis/q-axis (A)
Af	Windward area (m2)	iod, ioq	PMSM air gap current of d-axis/q-axis (A)
g	Acceleration due to gravity (m/s2)	icd, icq	PMSM iron loss current of d-axis/q-axis (A)
δ	Road slope angle (°)	Ld, Lq	PMSM inductance of d-axis/q-axis (H)
Cd	Coefficient of aerodynamic drag	ws	PMSM stator electrical angular speed (rad)
Cf	Rolling resistance coefficient	ψa	PMSM flux of permanent magnet (Wb)
Fz1, Fz2	Vertical reaction force of front/rear wheels (N)	Rc	PMSM iron loss resistance (Ω)
La, Lb	CG distance from front/rear wheels (m)	ud, uq	PMSM terminal voltage of d-axis/q-axis (V)
hg	CG height above the ground (m)	Ra	PMSM armature winding resistance (Ω)
gk	Speed ratio of reducer	wm2	PMSM angular speed (rad)
z11, z12	Number of teeth of stage 1 pinion/gear	Tm2	PMSM rotor shaft torque (Nm)
z21, z22	Number of teeth of stage 2 pinion/gear	p2	PMSM number of pole pairs
ωm1, ωm2	Agular speed of IM/PMSM (rad/s)	PCu	PMSM copper loss (W)
ωw1, ωw2	Angular speed of front/rear wheels (rad/s)	PFe	PMSM iron loss (W)
Tw1, Tw2	Torque of front/rear wheels (Nm)	PMe2	PMSM mechanical loss (W)
Tm1, Tm2	Torque of IM/PMSM (Nm)	γ2	PMSM constant factor mechanical loss
ηr1, ηr2	Efficiency of the front/rear reducer	Ploss_m2	PMSM loss (W)
ηd1, ηd2	Efficiency of front/rear differential	PPM	PMSM power consumption (W)
Jw	Wheel-tire moment of inertia (Kg·m2)	ηPM	PMSM efficiency
R	Dynamic tire radius (m)	K1, K2, K3, K4	Inverter loss coefficients
μx	Tire force coefficient	Ip	Amplitude of the phase current (A)
B, C, D, E	Magic Formula parameters	m	Modulation index (Bit/Hz)
sx	Longitudinal slip rate	φ	Motor power factor angle
vth	Wheel velocity threshold (m/s)	Ploss_inv2	PMSM inverter loss (W)
isd, isq	IM stator current of d-axis/q-axis (A)	Km2_1, Km2_2	Inverter loss coefficient of the PMSM
imr	IM magnetizing current (A)	Ip_m1, Ip_m2	Phase current of the IM/PMSM(A)
ir	IM rotor current (A)	Uoc	Open-circuit voltage (V)
if	IM iron current (A)	Ub	Terminal voltage (V)
Rr, Rs	IM resistance of rotor/stator (Ω)	Rint	Internal resistance of the battery (Ω)
Rr′	IM referred rotor resistance (Ω)	Pout_batt	Output power of the battery (W)
Rf′	IM referred iron loss resistance (Ω)	Ibatt	Discharge current (A)
wf	IM angular speed of rotor flux (rad/s)	SOC0	Initial SOC of the battery
we	IM electrical rotor speed (rad/s)	Cb	Battery capacity (Ah)
Lm′	IM referred magnetizing inductance (H)	Tdes_ff	Desired torque of feedforward control (Nm)
Lm	IM magnetizing inductance (H)	ξk	Proportional gain in the feedforward control
Lr, Ls	IM self-inductance of rotor/stator (H)	ξp	Proportional factor
usd, usq	IM terminal voltage of d-axis/q-axis (V)	ξi	Integral factor
wm1	IM angular speed (rad/s)	Tdes_fb	Desired torque for feedback control (Nm)
Tm1	IM angular torque (Nm)	Tdes	Total desired torques (Nm)
p1	IM number of pole pairs	o	Position of the accelerator pedal
Pcus	IM stator copper loss (W)	ξo	Proportional gain in the open-loop control
Pcur	IM rotor copper loss (W)	τo	Respond time constants
Piron	IM iron loss (W)	Tmax	Maximum driving torque (Nm)
PMe1	IM mechanical loss (W)

.

2.1. FRIDEV Architecture

The studied FRIDEV is driven by an IM in the front axle and a PMSM in the rear axle. Because IM and PMSM have different efficiency distributions, combining the two motors can benefit from higher energy efficiency [10]. An IM drives the front axle, and a PMSM drives the rear axle. The power distribution between the two motors is adjusted in real-time by the vehicle controller to optimize driving performance and efficiency. The motor controller adjusts the current and voltage according to the instructions from the vehicle controller, thereby controlling the speed and torque output of the motors. The motor torque is transmitted to the wheels via a two-stage reducer and a Torsen differential, driving the vehicle. The inverter converts the direct current (DC) from the battery pack into alternating current (AC), supplying power to the motors. The FRIDEV architecture is schematically shown in Figure 1.

2.2. Dynamic Vehicle System Modeling

2.2.1. Vehicle Longitudinal Dynamics Model

The vehicle longitudinal dynamics model is shown in Figure 2. The fundamental equation that describes the longitudinal dynamics of a vehicle is expressed as follows:

$$\left\{\begin{array}{c}{F}_t=2(F_{x1}+F_{x2})\\ M{\dot{v}}_x=F_t-\left(0.5\rho C_d{A}_f{v}_x^2+Mg\sin (\delta )+Mg\cos (\delta )C_f\right)\end{array}\right.$$

(1)

The distribution of tire loads between the front and rear axles will vary proportionally with longitudinal acceleration. The vertical reaction forces Fz₁ and Fz₂ received by the front and rear wheels, respectively, are calculated using the following formulas:

$$\left\{\begin{array}{c}{F}_{z1}={\displaystyle \frac{M\left(L_b\cos (\delta )g-h_g{\dot{v}}_x\right)}{2\left(L_a+L_b\right)}}\\ F_{z2}={\displaystyle \frac{M\left(L_a\cos (\delta )g+h_g{\dot{v}}_x\right)}{2\left(L_a+L_b\right)}}\end{array}\right.$$

(2)

2.2.2. Mechanical Gearing Dynamics Model

As shown in Figure 3, the reducer and differential transmit power from the motor to the wheel axle via gearings. Both the front and rear axles use the same type of reducer and differential. The speed ratio g_k is as follows:

$$g_k=(z_{12}/z_{11})(z_{22}/z_{21})$$

(3)

To improve energy analysis and achieve more accurate predictions, the mechanical gearing dynamics model that accounts for constant losses is introduced. Since kinematic constraints fix a speed ratio, these losses lead to a reduction in the torque at the output shaft. The model for the reducer and differential can be represented as follows:

$$\left\{\begin{array}{c}{\omega }_{m1}={\omega }_{w1}{g}_k,T_{w1}=0.5{\eta }_{r1}{\eta }_{d1}{T}_{m1}{g}_k\\ {\omega }_{m2}={\omega }_{w2}{g}_k,T_{w2}=0.5{\eta }_{r2}{\eta }_{d2}{T}_{m2}{g}_k\end{array}\right.$$

(4)

2.2.3. Wheel Rotation Dynamics Model

The wheel connects between the powertrain and the external environment. Based on the torque balance equation, the mathematical formulation of the wheel rotation dynamic can be written as follows:

$$\left\{\begin{array}{c}{J}_w{\dot{\omega }}_1=T_{w1}-R{F}_{x1}\\ J_w{\dot{\omega }}_2=T_{w2}-R{F}_{x2}\end{array}\right.$$

(5)

To describe the slip phenomenon, the slip rate is defined, and the relationship between the longitudinal slip rate and the longitudinal force of the longitudinal tire can be described using the semi-empirical Magic Formula tire model [26]. The Magic Formula tire model can be written as follows:

$$\left\{\begin{array}{c}{s}_{xi}=(w_{wi}R-v_x)/(\sqrt{v_{th}^2+v_x^2})\\ \begin{array}{c}{\mu }_{xi}=D\sin (C{\tan }^{-1}(B{s}_{xi}-E(B{s}_{xi}-{\tan }^{-1}(B{s}_{xi}))))\\ F_{xi}=F_{zi}{\mu }_{xi}\end{array}\end{array}\right.$$

(6)

where i = 1, 2 represents front and rear wheels. When wheel velocity is below the threshold, the slip calculation is modified to avoid the divide-by-zero problem [27]. The plots of longitudinal tire forces on different road surfaces are shown in Figure 4.

2.3. Electric System Modeling

2.3.1. IM Model

The IM, a type of AC motor, is powered by three-phase electric currents. For the AC motor, the d-q axis model is commonly used to simplify analysis and control by transforming three-phase currents and voltages into two-axes, where d and q denote the direct and quadrature axes, respectively. Figure 5 shows the d- and q-axes equivalent circuits of the IM in the steady state. The current and voltage equations can be expressed as follows [28]:

$$\left\{\begin{array}{c}{i}_{mr}=i_{sd}\\ i_r=i_{sq}-i_f=i_{sq}-(i_r{R}_r^{\prime }/R_f^{\prime }+{\omega }_e{i}_{sd}{L}_m^{\prime }/R_f^{\prime })\end{array}\right.$$

(7)

$$\left\{\begin{array}{c}{R}_r^{\prime }={(L_m/L_r)}^2{R}_r\\ L_m^{\prime }=L_m^2/L_r\end{array}\right.$$

(8)

$$\left\{\begin{array}{c}{u}_{sd}=R_s{i}_{sd}-{\omega }_f{L}_s{i}_{sq}\\ u_{sq}=R_s{i}_{sq}+{\omega }_f{L}_s^{\prime }{i}_{sd}+{\omega }_f{L}_m^{\prime }{i}_{mr}\end{array}\right.$$

(9)

For the IM, the angular speed w_m1 and torque T_m1 at the rotor shaft have the following relationship with the number of pole pairs p₁:

$$w_e=p_1{w}_{m1}$$

(10)

$$T_{m1}=1.5p_1{L}_m^{\prime }{i}_{mr}{i}_r=1.5p_1{L}_m^{\prime }\left({\displaystyle \frac{R_f^{\prime }}{R_f^{\prime }+R_r^{\prime }}}\right)i_{sq}{i}_{mr}-1.5p_2{\displaystyle \frac{{(L_m^{\prime }{i}_{mr})}^2}{R_f^{\prime }+R_r^{\prime }}}{\omega }_e$$

(11)

Since R_f′ >> R_r′ and (R_f′ + R_r′) >> (L_mi_mr)², torque Equation (11) can be approximated as follows:

$$T_{m1}\simeq 1.5p_1{L}_m^{\prime }{i}_{sq}{i}_{mr}$$

(12)

The losses of IM can be categorized into several components, including stator copper, iron, rotor copper, and mechanical losses. The stator copper loss P_cus occurs due to the resistance in the stator windings, which causes power dissipation as heat when current flows through them. The rotor copper loss P_cur is like the stator copper loss but occurs in the rotor due to current induced in the rotor bars. The iron loss P_iron occurs in the stator core due to alternating magnetic fields; therefore, the equivalent iron loss resistance is introduced into the circuit to represent the iron loss. The mechanical loss P_Me1 is due to friction and windage within the motor, increasing with the motor speed, represented by a constant factor γ₁. Then, the IM loss P_{loss_m1} can be calculated as follows:

$$P_{loss\_m1}=P_{cus}+P_{iron}+P_{cur}+P_{Me1}=1.5R_s\left(i_{sd}^2+i_{sq}^2\right)+1.5R_f^{\prime }{(i_{sq}-i_r)}^2+1.5R_r^{\prime }{i}_r^2+{\gamma }_1{w}_{m1}$$

(13)

Substituting i_r from Equation (7) into Equation (13) yields the following equation:

$$P_{loss\_m1}=1.5R_d{i}_{sd}^2+1.5R_q{i}_{sq}^2+{\gamma }_1{w}_{m1}$$

(14)

where R_d = R_s + ${\omega }_e^2$$L_m^{\prime 2}$/($R_f^{\prime }$+$R_r^{\prime }$) and R_q = R_s + $R_f^{\prime }$$R_r^{\prime }$/($R_f^{\prime }$+$R_r^{\prime }$).

The power consumption P_IM and efficiency η_IM of the IM when driving can be written as follows:

$$\left\{\begin{array}{c}{P}_{IM}=P_{loss\_m1}+P_{out\_m1}=(P_{cus}+P_{iron}+P_{cur}+P_{me1})+T_{m1}{w}_{m1}\\ {\eta }_{IM}={\displaystyle \frac{P_{out\_m1}}{P_{IM}}}={\displaystyle \frac{T_{m1}{w}_{m1}}{(P_{cus}+P_{iron}+P_{cur}+P_{me1})+T_{m1}{w}_{m1}}}\end{array}\right.$$

(15)

To minimize the loss of the IM, the differential of Equation (14) with respect to i_sd must be zero for a given torque as follows:

$${\displaystyle \frac{d{P}_{loss\_m1}}{d{i}_{sd}}}=3R_d{i}_{sd}-3R_q{\displaystyle \frac{i_{sq}^2}{i_{sd}}}=0$$

(16)

An optimum magnetizing current for the minimum loss of the IM can be obtained as follows:

$$i_{mr\_opt}=i_{sq}\sqrt{R_q/R_d}$$

(17)

Based on the equivalent circuit of the IM and the above loss minimization algorithm, the loss of the IM is represented by the function with the torque T_m1 and angular speed ω_m1, as follows:

$$P_{loss\_m1}=P_{loss\_m1}\left(T_{m1},{\omega }_{m1}\right)$$

(18)

By mapping the angular speed to the rotational speed, the loss of the IM for a given rotational speed and torque can be shown in Figure 6a. Additionally, for the loss of the IM, Figure 6b compares the data of the laboratory with the data calculated by the model.

2.3.2. PMSM Model

The PMSMs have been the preferred choice in EVs because of their high power density and efficiency [29]. The PMSM can be analyzed by converting three-phase currents and voltages into d- and q-axes, like the IM. Specifically, the rotor of PMSM has permanent magnets, which provide a constant magnetic field. Thus, the d-q axis model of the PMSM does not need to account for rotor currents. Figure 7 shows the d- and q-axes equivalent circuits at the steady state. The current and voltage equations can be expressed as follows [30]:

$$\left\{\begin{array}{c}{i}_d=i_{od}+i_{cd}\\ i_q=i_{oq}+i_{cq}\end{array}\right.$$

(19)

$$\left\{\begin{array}{c}{i}_{cd}=-{\omega }_s{L}_q{i}_{oq}/R_c\\ i_{cq}={\omega }_s({\psi }_a+L_d{i}_{od})/R_c\end{array}\right.$$

(20)

$$\left\{\begin{array}{c}{u}_d=R_a{i}_d-{\omega }_s{L}_q{i}_{oq}\\ u_q=R_a{i}_q+{\omega }_s(L_d{i}_{od}+{\psi }_a)\end{array}\right.$$

(21)

For the PMSM, the angular speed w_m2 and torque T_m2 at the rotor shaft have the following relationship with the number of pole pairs p₂:

$$w_s=w_{m2}{p}_2$$

(22)

$$T_{m2}=1.5p_2\left[{\psi }_a{i}_{oq}+(L_d-L_q)i_{od}{i}_{oq}\right]$$

(23)

The losses of the PMSM can be categorized into copper, iron, mechanical, and stray losses. The copper loss P_Cu occurs due to the resistance in the stator windings, where the flow of current generates heat and thus results in energy loss. The iron loss P_Fe is measured by introducing an equivalent iron loss resistance into the circuit, akin to the method used for the IM. The mechanical loss P_Me2 increases linearly with speed, represented by a constant factor γ₂. In particular, the stray loss is generally negligible due to its relatively small magnitude compared to the other types of losses [31]. Then, the PMSM loss P_{loss_m2} can be calculated as follows:

$$P_{loss\_m2}=P_{Cu}+P_{Fe}+P_{Me2}$$

(24)

$$P_{Cu}=1.5R_a\left({\left(i_{od}-{\displaystyle \frac{{\omega }_s{L}_q{i}_{oq}}{R_c}}\right)}^2+{\left(i_{oq}+{\displaystyle \frac{{\omega }_s({\psi }_a+L_d{i}_{od})}{R_c}}\right)}^2\right)$$

(25)

$$P_{Fe}=1.5R_c(i_{cd}^2+i_{cq}^2)=1.5\left({\displaystyle \frac{{({\omega }_s{L}_q{i}_{oq})}^2}{R_c}}+{\displaystyle \frac{{\omega }_s^2{({\psi }_a+L_d{i}_{od})}^2}{R_c}}\right)$$

(26)

$$P_{Me2}={\gamma }_2{w}_{m2}.$$

(27)

The power consumption P_PM and efficiency η_PM of the PMSM can be written as follows:

$$\left\{\begin{array}{c}{P}_{PM}=P_{PM}=P_{loss\_m2}+P_{out\_m2}=(P_{Cu}+P_{Fe}+P_{Me2})+T_{m2}{w}_{m2}\\ {\eta }_{PM}={\displaystyle \frac{P_{out\_m2}}{P_{PM}}}={\displaystyle \frac{T_{m2}{w}_{m2}}{(P_{Cu}+P_{Fe}+P_{me2})+T_{m2}{w}_{m2}}}\end{array}\right..$$

(28)

To minimize the loss of the PMSM, the differential of Equation (24) with respect to i_od must be zero for a given torque as follows:

$${\displaystyle \frac{d{P}_{loss\_m2}}{d{i}_{od}}}=0.$$

(29)

Thus, an optimum current for the minimum loss of the PMSM can be obtained as follows:

$$i_{od}={\displaystyle \frac{{\omega }_s^2{L}_d(R_a+R_c){\psi }_a}{R_a{R}_c^2+{\omega }_s^2{L}_d^2(R_a+R_c)}}$$

(30)

Like the IM, the loss of the PMSM can be represented as a function of the output torque Tm₂ and angular speed ω_m2 [18]. This relationship is given as follows:

$$P_{loss\_m2}=P_{loss\_m2}\left(T_{m2},{\omega }_m^2\right).$$

(31)

Figure 8a shows the loss of the PMSM for a given speed and torque. In addition, Figure 8b compares the data from the laboratory with the data calculated by the model.

2.3.3. Inverter Model

Inverters play a crucial role in controlling the output torque of both IM and PMSM by converting DC to AC using insulate-gate bipolar transistors (IGBTs). The inverter losses mainly include conduction losses, caused by the current flow through the IGBTs and diodes, and switching losses, which occur during the transition of IGBTs between on and off states. For a three-phase full-bridge inverter, the inverter loss Ploss_inv is expressed as follows:

$$P_{loss\_inv}=6K_1{I}_p+6K_2{I}_p^2+6K_3{\displaystyle \frac{m}{4}}{I}_p\cos \phi +6K_4{\displaystyle \frac{m}{3\pi }}{I}_p^2\cos \phi$$

(32)

where K₁, K₂, K₃, and K₄ are the inverter loss coefficients; I_p is the amplitude of the phase current; m is the modulation index; and φ is the motor power factor angle [18]. As discussed in [32], the last two terms are negligible. Hence, the inverter loss can be written as follows:

$$P_{loss\_inv}=6K_1{I}_p+6K_2{I}_p^2.$$

(33)

In the context of the inverter for the IM, the inverter loss is characterized by the amplitude of current:

$$\left\{\begin{array}{c}{P}_{loss\_inv1}=6K_{m1\_1}{I}_{p\_m1}+6K_{m1\_2}{I}_{p\_m1}^2\\ I_{p\_m1}=\sqrt{(i_{sd}^2+i_{sq}^2)/1.5}\end{array}\right.$$

(34)

Similarly, the inverter loss of the PMSM can be derived as follows:

$$\left\{\begin{array}{c}{P}_{loss\_inv2}=6K_{m2\_1}{I}_{p\_m2}+6K_{m2\_2}{I}_{p\_m2}^2\\ I_{p\_m2}=\sqrt{(i_d^2+i_q^2)/1.5}\end{array}\right.$$

(35)

The equivalent circuit models for IM and PMSM explain that the currents directly influence the output torque. According to the inverter model above, the inverter loss depends on the motor phase current. Consequently, the loss in the inverter is written as a function of the output torque, as follows:

$$\left\{\begin{array}{c}{P}_{loss\_inv1}=P_{loss\_inv1}(T_{m1})\\ P_{loss\_inv2}=P_{loss\_inv2}(T_{m2})\end{array}\right.$$

(36)

2.3.4. Battery Model

The internal resistance model is a commonly used equivalent circuit model for representing the behavior of the battery pack, as shown in Figure 9a. Using the internal resistance model of the battery, the open-circuit voltage U_oc and the terminal voltage U_b can be expressed as follows:

$$\left\{\begin{array}{c}{U}_{oc}=U_b+I_{batt}{R}_{int}\\ U_b=P_{out\_batt}/I_{batt}\end{array}\right.$$

(37)

Solving Equation (37), I_batt can be derived, written as follows:

$$I_{batt}={\displaystyle \frac{U_{oc}-\sqrt{U_{oc}^2-4R_{int}{P}_{out\_batt}}}{2R_{int}}}$$

(38)

Then, the battery SOC is given as follows:

$$SOC(k)=SO{C}_0-{\displaystyle \frac{1}{C_b}}{\displaystyle {\int }_0^t{I}_{batt}}dt$$

(39)

Based on Equation (38), the loss of the battery is shown in Figure 9b, calculated as follows:

$$P_{loss\_batt}=I_{batt}^2{R}_{int}={\displaystyle \frac{{(U_{oc}-\sqrt{U_{oc}^2-4R_{int}{P}_{out\_batt}})}^2}{4R_{int}}}$$

(40)

2.4. Driver Model

2.4.1. Closed-Loop Driver Model

For tracking the target speed, a closed-loop driver model is designed, combining feedforward control and feedback control. This dual approach helps anticipate control actions and corrects deviations in real time. The feedforward control calculates the required control action based on the longitudinal dynamics of the FRIDEV. The desired torque T_{des_ff} for feedforward control can be obtained using the Laplace transform [23]:

$$T_{des\_ff}=\mathrm{s}{\xi }_{k}MRv$$

(41)

The feedback control uses a PI regulator to correct deviations from the target speed. In this study, a self-adjusting PI regulator is utilized, which can inhibit the overshoot in speed response resulting from the unconstrained integration of tracking errors. The desired torque T_{des_fb} for feedback control can be determined through the application of the Laplace transform as follows:

$$T_{des\_fb}={\displaystyle \frac{\mathrm{s}{\xi }_p+{\xi }_i(1-|{\xi }_p△v_{err}|)}{\mathrm{s}}}△v_{err}$$

(42)

Ultimately, the overall torque requirement T_des is determined by adding the desired torques from both the feedforward and feedback controls:

$$T_{des}=T_{des\_ff}+T_{des\_fb}$$

(43)

2.4.2. Open-Loop Driver Model

On the other hand, a model based on driver behavior is also designed for open-loop control. The control input is determined directly by the driver’s action, such as the position of the accelerator pedal. Besides, the model considers a lag response to simulate real driver behavior.

$$T_{des}={\displaystyle \frac{o{\xi }_o}{{\tau }_o\mathrm{s}+1}}{T}_{\max }$$

(44)

3. Optimal Driving Torque Control Strategy

In this section, the offline optimized driving torque control strategy and optimal driving torque control strategy based on real-time model prediction control are introduced. A rule-based driving torque control strategy is introduced as the comparison for the proposed control strategies.

3.1. Rule-Based and Offline Optimization-Based Driving Torque Control Strategies

3.1.1. Rule-Based Driving Torque Control Strategy

The rule-based torque control strategy is frequently employed in FRIDEV because of its effectiveness in managing driving torque in dual-motor systems. This approach offers significant advantages, including simplicity and ease of implementation, making it a practical choice for the dual-motor EV. The strategy operates based on a predefined set of rules to determine the allocation of driving torque between the two motors.

In this paper, a rule-based driving torque control strategy is introduced as the comparison for the proposed control strategies. When the desired torque is less than or equal to the sum of the maximum torques of both motors, the torque is evenly distributed between the two motors. When the desired torque exceeds the maximum torque capability of one motor, the other motor should compensate for the deficit. T_{m1_max} represents the maximum driving torque of the IM, while T_{m2_max} denotes the maximum torque of the PMSM. The expression for the rule-based driving torque strategy can be expressed as follows:

$$\left\{\begin{array}{cc}{T}_{m1}=T_{m2}={\displaystyle \frac{T_{des}}{2}}& 0\le {\displaystyle \frac{T_{des}}{2}}\le \min (T_{m1\_\max },T_{m2\_\max })\\ T_{m1}=\min \left(T_{m1\_\max },{\displaystyle \frac{T_{des}}{2}}+{\displaystyle \frac{T_{des}-T_{m2\_\max }}{2}}\right),T_{m2}=T_{des}-T_{m1}& \min (T_{m1\_\max },T_{m2\_\max })<{\displaystyle \frac{T_{des}}{2}}\le {\displaystyle \frac{T_{m1\_\max }+T_{m2\_\max }}{2}}\end{array}\right.$$

(45)

When the desired torque exceeds the combined maximum torque capabilities of both motors, the system reaches its limit and cannot meet the full demand. In such cases, both motors should operate at their maximum torque output to provide the highest possible combined torque. The expression for this situation is as follows:

$$\left\{\begin{array}{cc}\begin{array}{c}{T}_{m1}=T_{m1\_\max }\\ T_{m2}=T_{m2\_\max }\end{array}& T_{des}>T_{m1\_\max }+T_{m2\_\max }\end{array}\right.$$

(46)

3.1.2. Offline Optimization-Based Driving Torque Control Strategy

The offline optimization-based driving torque control strategy is developed according to the analysis of the FRIDEV electric system presented in Section 2.3, with the objective of minimizing energy losses within the electric system. This strategy is iterating through various combinations of SOC, vehicle speed, and desired torque to determine the optimal front and rear motor driving torque distribution ratio, denoted as λ. Figure 10 shows the flowchart of the offline computation of this torque control strategy. The calculation process consists of the following five parts:

• Part 1: Parameter definition and variable initialization: T_List and v_List are the discretization sequences of desired torque and vehicle speed. λ_List represents the discretization sequence of driving torque distribution ratio, ranging from 0 to 1 in increments of 0.01. Additionally, variables for the losses of each motor are initialized.
• Part 2: Calculation of motor torque and total losses: For a given desired torque, the driving torque distribution ratio λ_List determines the driving torque of the PMSM and IM. For each torque distribution ratio, the total losses of the electric system are calculated, encompassing the losses of both the PMSM and IM motors, respective inverters, and the battery.

• Part 3: Handling of constraints: To ensure proper motor operation, constraints are applied during the calculation, as detailed in Equation (47).
• Part 4: Selection of optimal driving torque distribution ratio: For each combination of speed and torque, the driving torque distribution ratio that results in the minimum total losses is selected and recorded. This ratio ensures optimal system performance under the current SOC and load conditions.
• Part 5: Parameter iterating: In the main loop, the SOC value increases incrementally from 0.2 to 0.8. For each SOC value, all possible combinations of vehicle speed and desired torque are further iterated.

$$\left\{\begin{array}{c}{T}_{m1}\le T_{m1\_\max },T_{m2}\le T_{m2\_\max }\\ {\omega }_{m1}\le {\omega }_{m1\_\max },{\omega }_{m2}\le {\omega }_{m2\_\max }\end{array}\right.$$

(47)

Figure 11a shows the optimal driving torque distribution ratio MAP calculated offline for combinations of vehicle speed and required torque when SOC is 0.6. Figure 11b shows the MAP of the optimal driving torque distribution ratio across different SOC levels, based on offline optimization calculations.

3.2. Optimal Driving Torque Control Strategy Based on Real-Time Model Prediction Control

To reduce energy consumption, minimize tire slip losses, and achieve anti-slip control, a dynamic driving torque control strategy is proposed in this section. Figure 12 illustrates the framework of the proposed strategy. Initially, a state-space equation is established to predict the future states of the vehicle based on the dynamic vehicle system model. The control objectives are automatically adjusted based on the slip conditions of the front and rear wheels, which allows for efficient driving torque distribution and provides anti-slip functionality. Additionally, constraints are incorporated into the optimization process to address practical application requirements. Finally, to expedite the solution of the nonlinear MPC problem and improve real-time computational efficiency, an online control law is developed using an improved sparrow search algorithm. In particular, the proposed online strategy in this paper specifically targets motor torque control for driving conditions. However, during braking, the system follows a fixed-ratio hydraulic braking mechanism, and the motors do not participate in regenerative braking.

3.2.1. Predictive Model of Dynamic Vehicle

Based on the dynamic vehicle system model, which includes vehicle longitudinal dynamics, mechanical gearing dynamics, and wheel rotation dynamics, the continuous state-space equation representing the movement of the vehicle is as follows:

$$\left\{\begin{array}{c}{\dot{\omega }}_{w1}=\left(0.5T_{m1}{\eta }_{r1}{\eta }_{d1}{g}_k-R{F}_{z1}\left({\dot{v}}_x\right){\mu }_{x1}\left(s_{x1}\right)\right)/J_w\\ {\dot{\omega }}_{w2}=\left(0.5T_{m2}{\eta }_{r2}{\eta }_{d2}{g}_k-R{F}_{z2}\left({\dot{v}}_x\right){\mu }_{x2}\left(s_{x2}\right)\right)/J_w\\ {\dot{s}}_{x1}={\displaystyle \frac{\sqrt{v_{th}^2+v_x^2}{\dot{\omega }}_{w1}R-{\omega }_{w1}R{\dot{v}}_x}{v_{th}^2+v_x^2}}\\ {\dot{s}}_{x2}={\displaystyle \frac{\sqrt{v_{th}^2+v_x^2}{\dot{\omega }}_{w2}R-{\omega }_{w2}R{\dot{v}}_x}{v_{th}^2+v_x^2}}\\ {\dot{v}}_x={\displaystyle \frac{1}{M}}\left(\begin{array}{l}2F_{z1}\left({\dot{v}}_x\right){\mu }_{x1}\left(s_{x1}\right)+2F_{z2}\left({\dot{v}}_x\right){\mu }_{x2}\left(s_{x2}\right)-\\ 0.5\rho C_d{A}_f{v}_x^2+Mg\sin (\delta )+Mg\cos (\delta )C_f\end{array}\right)\end{array}\right.$$

(48)

The Euler differential method is a simple and widely used approach for discretizing continuous state-space equations into iterative equations. In this method, the derivative of the state variables is approximated by the difference quotient between two consecutive time steps. Specifically, the value of a state variable at the next time step is approximated by adding the product of its derivative and the time step to its current value. The above continuous differential equations can be transformed into discrete iterative equations as follows:

$$\left\{\begin{array}{c}x(k+1)=f(x(k),u(k))\cdot T_s+x(k)\\ x={[x_1{x}_2{x}_3{x}_4{x}_5]}^T={[{\omega }_{w1}{\omega }_{w2}{s}_{x1}{s}_{x2}{v}_x]}^T\\ u={[u_1{u}_2]}^T={[T_{m1}{T}_{m2}]}^T\end{array}\right.$$

(49)

where x(k) is the state vector at discrete time step k, x(k + 1) is the state vector at the next time step k + 1, and T_s is the time step. The detailed expansion equation can be written as follows:

$$\left\{\begin{array}{c}{x}_1(k+1)=\left(0.5u_1(k){\eta }_{r1}{\eta }_{d1}{g}_k-R{F}_{z1}\left(f_5(k,T_s)\right){\mu }_{x1}\left(x_3(k)\right)\right)/J_w\cdot T_s+x_1(k)\\ x_2(k+1)=\left(0.5u_2(k){\eta }_{r2}{\eta }_{d2}{g}_k-R{F}_{z2}\left(f_5(k,T_s)\right){\mu }_{x2}\left(x_4(k)\right)\right)/J_w\cdot T_s+x_2(k)\\ x_3(k+1)={\displaystyle \frac{f_1(k,T_s)\sqrt{v_{th}^2+x_5{(k)}^2}R-f_5(k,T_s)x_1(k)R}{v_{th}^2+x_5{(k)}^2}}\cdot T_s+x_3(k)\\ x_4(k+1)={\displaystyle \frac{f_2(k,T_s)\sqrt{v_{th}^2+x_5{(k)}^2}R-f_5(k,T_s)x_2(k)R}{v_{th}^2+x_5{(k)}^2}}\cdot T_s+x_4(k)\\ x_5(k+1)={\displaystyle \frac{1}{M}}\left(\begin{array}{l}2F_{z1}\left(f_5(k,T_s)\right){\mu }_{x1}\left(x_3(k)\right)+2F_{z2}\left(f_5(k,T_s)\right){\mu }_{x2}\left(x_4(k)\right)\\ -0.5\rho C_d{A}_f{x}_5{(k)}^2+Mg\sin (\delta )+Mg\cos (\delta )C_f\end{array}\right)\cdot T_s+x_5(k)\end{array}\right.$$

(50)

where f₁(k, T_s) = (x₁(k + 1) − x₁(k))/T_s, f₂(k, T_s) = (x₂(k + 1) − x₂(k))/T_s, and f₅(k, T_s) = (x₅(k) − x₅(k − 1))/T_s.

This predictive model is developed based on the vehicle’s dynamic system, providing a framework for transferring the strategy to other similar architectures. By leveraging the underlying vehicle dynamics formula, the model can be adapted to different drivetrains.

3.2.2. Optimization Objectives and Constraints

For the FRIDEV, the losses in the electric system account for most of the total losses. These losses include the losses in the IM, PMSM, inverters, and battery. As analyzed in Section 2.3, the losses of the IM, PMSM, and inverters will be represented by a function with the torques T_m1, T_m2, and angular speeds ω_m1, ω_m2. The battery loss is influenced by the SOC and battery output power. Due to the indirect relationship between battery loss and control variables, the battery loss is not optimized directly, considering the complexity of the optimization process. The battery output power is the sum of the output power of the IM and PMSM, along with their respective losses and the inverter losses. Since the battery loss increases with output power, reducing motor and inverter losses indirectly decreases battery loss. In summary, the first objective is to minimize the losses of the electric system for the FRIDEV. At the time step k, the cost function J_I is defined as follows:

$$\begin{array}{l}{J}_{{\rm I}}(k)={\displaystyle {\sum }_{i=1,2}\left(P_{loss\_mi}(k)\right)+{\displaystyle {\sum }_{i=1,2}\left(P_{loss\_invi}(k)\right)}}\\ ={\displaystyle {\sum }_{i=1,2}\left(P_{loss\_mi}\left(T_{mi}\left(k\right),{\omega }_{mi}\left(k\right)\right)\right)}+{\displaystyle {\sum }_{i=1,2}\left(P_{loss\_invi}\left(T_{mi}\left(k\right)\right)\right)}\end{array}$$

(51)

Under driving conditions, tire longitudinal slip occurs when the applied torque from the motors exceeds the available traction between the tires and the road surface. For optimal performance of traction, a certain degree of longitudinal slip is necessary to generate the required traction force. Part of the energy is converted into heat rather than effective mechanical motion due to the tire longitudinal slip, especially on poor road conditions or during rapid acceleration. The excessive slip can result in inefficient driving energy utilization, increased energy losses, and potential instability. Therefore, the second optimal goal focuses on minimizing the longitudinal slip-induced energy losses of the tires. According to the definition of the tire slip losses [33,34], at time step k, the cost function J_II is given as follows:

$$J_{\mathrm{II}}(k)=2{\displaystyle {\sum }_{i=1,2}\left(P_{loss\_trs}(k)\right)}=2{\displaystyle {\sum }_{i=1,2}\left(F_{xi}\left(k\right)v_x(k)s_{xi}(k)\right)}$$

(52)

In this paper, the nonlinear activation functions defined by the sigmoid function are introduced to balance energy consumption optimization and longitudinal slip rate control. The activation functions control the penalty terms in the optimization problem. When the longitudinal slip rates of the front or rear wheels exceed predefined thresholds, the cost function increases sharply. This sharp increase forces the optimization process to adjust the torque distribution between the front and rear motors, giving priority to reducing torque in wheels with a higher slip rate. Under normal conditions, the slip rates are below the thresholds. The penalty remains inactive, allowing the algorithm to focus on minimizing losses in motors, inverters, and tires. In summary, at time step k, the cost function J_III can be expressed as follows:

$$J_{{\rm I}{\rm I}{\rm I}}(k)={\sigma }_1(k)\inf +{\sigma }_2(k)\inf +\left(1-{\sigma }_1(k){\sigma }_2(k)\right)\left(J_{{\rm I}}(k)+J_{{\rm I}{\rm I}}(k)\right)$$

(53)

where inf is the penalty term. σ₁(k) and σ₂(k) are nonlinear activation functions, defined as σ₁(k) = 1/(1 + exp(ρ∙s_x1(k) − s_{x_th})) and σ₂(k) = 1/(1 + exp(ρ∙s_x2(k) − s_{x_th})). ρ is the scaling factor that determines how steeply the sigmoid function, and s_{x_th} is a predefined threshold value for the slip rate.

Now, the optimization problem for driving torque distribution in the prediction horizon N_p1 can be constructed as follows:

$$\begin{array}{ll}\min J_e& ={\displaystyle {\sum }_{j=1}^{N_{p1}}\left(J_{\mathrm{III}}(k+j|k)\right)}\\ & ={\displaystyle {\sum }_{j=1}^{N_{p1}}\left({\sigma }_1\left(k+j|k\right)\inf +{\sigma }_2\left(k+j|k\right)\inf \right)}\\ & +{\displaystyle {\sum }_{j=1}^{N_{p1}}\left(\left(1-{\sigma }_1(k+j|k){\sigma }_2(k+j|k)\right)\left(J_{{\rm I}}(k+j|k)+J_{{\rm I}{\rm I}}(k+j|k)\right)\right)}\end{array}$$

(54)

Additionally, under driving conditions, the motor torque must meet the traction demand specified by the driver model. Considering the operational characteristics of the motors, the output torque should be constrained within specified bounds. Therefore, the torque distribution is subject to the following:

$$\left\{\begin{array}{c}{T}_{m1}\left(k\right)+T_{m2}\left(k\right)=T_{des}\left(k\right)\\ 0\le T_{m1}\left(k\right)\le T_{m1\_\max }\\ 0\le T_{m2}\left(k\right)\le T_{m2\_\max }\end{array}\right.$$

(55)

When road surfaces provide insufficient traction, excessive slip rates in the front and rear wheels can lead to vehicle instability and loss of control, especially on low-adhesion surfaces. In such conditions, optimization priorities must shift from energy efficiency to maintaining stability by ensuring that slip rates stay close to the desired value s_{x_des}. Accordingly, the driving torque should be reduced to minimize wheel slip, and the associated constraints need to be adjusted accordingly. The slip rate control problem within the prediction horizon N_p2 can then be formulated by minimizing the cost function J_s as follows:

$$\min J_s={\displaystyle {\sum }_{j=1}^{N_{p2}}\left({\left(s_{x1}(k+j|k)-s_{x\_des}\right)}^2+{\left(s_{x2}(k+j|k)-s_{x\_des}\right)}^2\right)},$$

(56)

subject to:

$$\left\{\begin{array}{c}{T}_{m1}\left(k\right)+T_{m2}\left(k\right)\le T_{des}\left(k\right)\\ 0\le T_{m1}\left(k\right)\le T_{m1\_\max }\\ 0\le T_{m2}\left(k\right)\le T_{m2\_\max }\end{array}\right..$$

(57)

In this study, the dynamic driving torque control problem for the FRIDEV is formulated as a constrained nonlinear model predictive optimization problem. Notably, this approach is designed with flexibility and generality, making it adaptable to similar architectures. By adjusting system parameters and modifying the control variables and constraints within the model, this method can be extended beyond the discussed FRIDEV in this study. For example, the method is adapted to dual-IM FRIDEV, dual-PMSM FRIDEV, and other multi-motor drive architectures, accommodating a variety of torque distribution needs. This flexibility means the system can be readily tailored to provide efficient torque allocation and enhance vehicle stability on other EV platforms, including configurations with different numbers or types of motors on separate axles. As a result, the strategy can support optimal torque distribution across a wide range of EV architectures, improving driving stability and energy efficiency under various operating conditions.

3.2.3. Online Control Law Based on Improved Sparrow Search Algorithm

Addressing the nonlinear control challenges in dynamic FRIDEV driving torque control presents significant difficulties. The SSA, a swarm intelligence-based technique, has emerged as a promising alternative [35]. This study proposes an improved SSA to address these challenges. The improved SSA introduced several innovations, including chaotic mapping for population initialization, improved producer location updates, and individual perturbation mechanisms. The details are described below.

During initialization, the sparrow population size is randomly generated, often causing early aggregation and low solution space coverage. To improve this, circle chaotic mapping is used. This method enhances population randomness, reduces premature convergence, and ensures better exploration of the solution space by providing higher coverage of chaotic values. Circle chaotic mapping is defined mathematically as follows:

$$X_{n+1}=\mathrm{mod}\left(X_n+a-b\sin (2\pi X_n)/(2\pi ),1\right)$$

(58)

In the SSA, producers guide the population’s foraging but may lead to local optima. To overcome this problem, an improved method of updating the producer position is introduced, inspired by the Salp Swarm Algorithm [36]. This method expands the search range, enhancing the algorithm’s global exploration capability and reducing the risk of premature convergence. The updated method is described by the following equation:

$$X_{i,j}^{d+1}=\left\{\begin{array}{ll}{X}_{i,j}^d{\displaystyle \frac{c_1\left((u{b}_j-l{b}_j)c_2+l{b}_j\right)}{(1+c_3)u{b}_j}}\hfill & R_2<ST\hfill \\ X_{i,j}^d+QL\hfill & R_2\ge ST\hfill \end{array}\right.$$

(59)

where d represents the current number of iterations, d_max is the maximum number of iterations, $X_{i,j}^d$ denotes the location of the jth dimension for the ith sparrow at the dth iteration, Q is a random number following a normal distribution, L is a matrix of 1 × j for which each element inside is 1, R₂ ∈ [0,1] represents an early warning value, and ST ∈ [0.5,1] denotes a safety threshold. ub_j and lb_j are the upper and lower bounds of the solution space for the jth dimension, respectively, while c₂ and c₃ are random numbers between [0,1]. The parameter c₁ is defined as follows:

$$c_1=2\exp \left(-{\left(4d/d_{\max }\right)}^2\right)$$

(60)

To enhance population diversity and avoid local optima, dynamic individual perturbation mechanisms are introduced. Entropy measures the uniformity of fitness distribution: a small entropy indicates a concentrated population, requiring more disturbance to maintain diversity, while a large entropy suggests reducing disturbance to facilitate convergence to the optimal solution. The formula for calculating entropy H is as follows:

$$\left\{\begin{array}{c}H=-{\displaystyle \sum _{i=1}^P{p}_i}\log (p_i)\\ p_i=f_i/\max \left({\displaystyle \sum _{i=1}^P{f}_i,\epsilon }\right)\end{array}\right.$$

(61)

where P is the population size, the number of individuals. p_i is the relative fitness of the ith sparrow, and f_i is the fitness of the ith sparrow. ε is an extremely small constant that prevents the denominator from being zero. On the other hand, the disturbance intensity is adjusted according to the difference in fitness. Individuals with poor fitness are subjected to stronger disturbance, and individuals with fitness close to the optimal solution are subjected to smaller disturbance to avoid excessive dispersion. Therefore, the fitness difference ∆f_i is introduced, which is defined as follows:

$$\Delta f_i={\displaystyle \frac{f_i-f_{\mathrm{best}}}{f_{\mathrm{worst}}-f_{\mathrm{best}}+\epsilon }}$$

(62)

The disturbance intensity starts strong to ensure exploration and gradually weakens as iterations increase, with a dynamic factor ∂_d controlling the perturbation degree as follows:

$${\partial }_d=\cos \left({\displaystyle \frac{\pi }{2}}{\displaystyle \frac{d}{d_{\max }}}\right)$$

(63)

In summary, the individual perturbation formula can be written as follows:

$${\tilde{X}}_{i,j}^d=X_{i,j}^d+\tilde{\partial }{\partial }_d\Delta f_{i}HQL$$

(64)

where ${X(~)}_{i,j}^d$ is the location after individual disturbance and HQL is the normalization factor.

SSA, being a stochastic optimization algorithm, does not guarantee global optimality in each iteration. To improve control performance, two criteria are introduced: an error threshold Err_th to detect convergence when fitness changes are minimal, and a maximum iteration count d_max to terminate the algorithm if the optimal solution is not found, using the last computed values as temporary solutions.

In the original SSA, in addition to updating the location of producers, location updates are also required for the scroungers and vigilantes. In the improved SSA, the location of producers is updated according to Equation (59), while the location updates for the scroungers and vigilantes follow the same approach as in the original SSA. The location update formula for the scroungers is given as follows:

$$X_{i,j}^{d+1}=\left\{\begin{array}{ll}Q\cdot \exp \left({\displaystyle \frac{X_{\mathrm{worst}}^d-X_{i,j}^d}{i^2}}\right)\hfill & i>P/2\hfill \\ X_p^{d+1}+|X_{i,j}^d-X_p^{d+1}|\cdot A^{+}\cdot L\hfill & i\le P/2\hfill \end{array}\right.$$

(65)

where $X_{worst}^d$ is the worst location in the population at the dth iteration. A is a matrix of the 1 × j dimensions where the elements are randomly 1 or −1, A⁺ = A^T(AA^T)⁻¹, and A^T is the transpose matrix of A. In general, the scroungers will be close to the producer and look for food around it. When A > P/2, it shows that there are too many sparrows around the foraging position of the current discoverer, there is not enough food, and the scroungers need to fly elsewhere for food.

During the foraging process, a subset of the sparrow population functions as vigilantes. These vigilantes continuously monitor the environment for potential threats. When a threat is detected, they signal the population to move away from danger. The location update for vigilantes is governed by the following formula:

$$X_{i,j}^{d+1}=\left\{\begin{array}{ll}{X}_{\mathrm{best}}^d+\Lambda \cdot \left|X_{i,j}^d-X_{\mathrm{best}}^d\right|\hfill & f_i>f_{\mathrm{best}}\hfill \\ X_{i,j}^d+K\cdot \left({\displaystyle \frac{\left|X_{i,j}^d-X_{\mathrm{worst}}^d\right|}{\left(f_i-f_{\mathrm{worst}}\right)+\epsilon }}\right)\hfill & f_i=f_{\mathrm{best}}\hfill \end{array}\right.$$

(66)

where Λ represents the step size control parameter, $X_{\text{best}}^d$ is the best location in the population at the dth iteration. When f_i > f_best, it indicates that the vigilante sparrow is at the edge of the population and needs to move closer to other sparrows to ensure its safety. Conversely, when f_i = f_best, it signifies that a sparrow located in the center of the population has detected danger and moves closer to other sparrows for protection.

Comparing with the original SSA, the overall architecture of the improved SSA is illustrated in Figure 13, and the key improvements are highlighted. The specific flow of algorithmic interactions and updates for improved SSA is shown in

Table 1. Framework of the improved SSA.

Input: dmax, P, SD, ST
1:	Initialize a population of sparrows using Equation (58)
2:	while (d < dmax)
3:	Rank the fitness values and find the current best individual and the current worst individual
4:	for i = 1: PD
5:	Using Equation (59), update the location of producers
6:	end for
7:	for i = (PD + 1): P
8:	Using Equation (65) update the location of scroungers
9:	end for
10:	for i = 1: SD
11:	Using Equation (66), update the location of vigilantes
12:	end for
13:	for i = 1: P
14:	Apply dynamic perturbation to the individual location based on Equation (64)
15:	end for
16:	Get the current new location
17:	If the fitness of the new location is better than before, update the best location
19:	If the fitness threshold is met for consecutive iterations, break it
20:	d = d + 1
21:	end while
22:	Return Xbest

[37]. The proposed algorithm not only is limited to addressing the MPC in the discussed FRIDEV but also applies to other vehicle architectures that adopt MPC. Furthermore, it can be applied to a wide range of nonlinear control problems. The improved SSA introduces chaos mapping, an enhanced producer position update, and a single perturbation mechanism. These improvements allow for a more efficient balance between exploration and exploitation, reducing computational complexity and accelerating convergence. Whether applied to EV with other architectures that require dynamic control, the improved SSA can be used as a solution to improve the computational efficiency of MPC solving.

4. Analysis and Discussion of Results

To verify the effectiveness of the proposed driving torque control strategy for the FRIDEV, a simulation is conducted using MATLAB/Simulink 2022b. The simulation model encompasses the dynamic vehicle system, electric drive system, and driver model. To highlight the advantages of the proposed strategy, it is compared against rule-based and offline optimization-based control strategies. Key performance metrics for comparison include energy consumption, longitudinal slip ratio, and real-time performance. The simulation is carried out under the China light-duty vehicle test cycle-passenger cars (CLTC-P), the world-harmonized light-duty vehicle test cycle (WLTC), the new European driving cycle (NEDC), and the US06 driving cycle, with varying road adhesion coefficients. Both closed-loop and open-loop driver models are employed to comprehensively evaluate the performance of the proposed real-time model predictive torque control strategy.

Table 2. Related parameters of discussed FRIDEV.

Part	Parameter	Notation	Value
Vehicle	Vehicle mass	M	2000 kg
	CG distance from front/rear wheels	La, Lb	1.35 m, 1.65 m
	Height of the vehicle center of gravity	hg	0.7 m
	Windward area	Af	2.3 m2
	Coefficients of aerodynamic drag	Cd	0.26
Gearing	Efficiency of front/rear reducer	ηr1, ηr2	0.96, 0.96
	Efficiency of front/rear differential	ηd1, ηd2	0.94, 0.94
	Speed ratio of reducer	gk	9.04
Wheel	Dynamic tire radius	R	0.35 m
	Wheel-tire moment of inertia	Jw	1.8 kg‧m2
IM	Maximum driving torque of IM	Tm1_max	240 Nm
	Maximum rational speed of IM	Nm1_max	12,000 rpm
	IM number of pole pairs	p1	2
	Resistance of the rotor/stator	Rr, Rs	0.022 Ω, 0.039 Ω
	Referred iron loss resistance	Rf′	370 Ω
	Magnetizing inductance	Lm	16.6 mH
	Self-inductance of the rotor/stator	Lr, Ls	0.389 mH, 0.389 mH
PMSM	Maximum driving torque of PMSM	Tm2_max	300 Nm
	Maximum rational speed of PMSM	Nm2_max	12,000 rpm
	PMSM number of pole pairs	p2	3
	Flux of permanent magnet	ψa	0.13 Wb
	Armature winding resistance	Ra	0.087 Ω
	Iron loss resistance	Rc	110 Ω
	Inductance of d-axis/q-axis	Ld, Lq	0.64 mH, 0.64 mH
Inverter	Inverter loss coefficients for IM	Km1_1, Km1_2	0.507, 0.000396
	Inverter loss coefficients for PMSM	Km2_1, Km2_2	0.479, 0.000383
Battery (SOC = 0.6)	Open-circuit voltage	Uoc	355 V
	Internal resistance of the battery	Rint	0.0389 Ω

lists the related parameters of the discussed FRIDEV.

4.1. Energy Consumption Performance

In this part, FRIDEV is managed to track target vehicle speed under CLTC-P using a closed-loop driver model and a road adhesion coefficient of 0.8. For objectively assessing energy consumption performance across different control strategies, several quantifiable metrics are applied, including overall energy consumption, energy losses in the electric system, and losses due to tire slip.

The CLTC-P is selected as a validation condition for its relevance and benefits in evaluating energy efficiency. This cycle includes frequent stops, low-to-medium-speed driving, and varying acceleration demands, accurately reflecting the typical urban driving environment that electric vehicles commonly experience. Figure 14a,b illustrate the vehicle speed and tracking error of the FRIDEV vehicle under the proposed online strategy. The velocity error is generally less than 0.2 km/h, and the maximum deviation is 0.24 km/h, which indicates that the FRIDEV can effectively track the target velocity under the proposed online strategy.

Figure 14c,d illustrate the output torques of the IM and PMSM under the rule-based strategy and the proposed online strategy. In the high-speed range, both strategies show similar results, with the FRIDEV primarily operating in four-wheel-drive mode. This allows for optimal utilization of the available driving force, particularly due to reduced tire slip losses. However, in the mid-speed range, the proposed online strategy starts to dynamically allocate torque based on real-time vehicle status and road conditions. In the low-speed region, the FRIDEV predominantly operates in rear-wheel-drive mode, where the PMSM exhibits higher efficiency, further reducing energy consumption. In contrast, the rule-based strategy allocates driving torque according to a predefined ratio without considering vehicle dynamics or real-time conditions. This approach lacks the flexibility to adjust torque distribution based on varying driving scenarios, limiting overall energy efficiency. On the other hand, the proposed online strategy is based on the dynamic vehicle model, and the output optimized torque can better respond to the dynamic behavior of the vehicle and significantly improve the energy-saving effect compared with the rule-based method.

In the proposed online control strategy, driving torque for the FRIDEV is dynamically distributed between the IM and PMSM in real time. This allocation ensures efficient performance across various driving conditions. The strategy uses a predictive model and optimization algorithm to adjust motor torque according to vehicle status and driving demands, placing operating points in higher efficiency regions. Figure 15a shows motor operating points under the rule-based strategy, while Figure 15b displays the distribution with the proposed online control strategy. In driving conditions, the proposed online strategy positions more operating points in the 0.95 efficiency region, effectively reducing electric energy losses. This results in significantly enhanced energy efficiency for the FRIDEV under CLTC-P, particularly in motor efficiency and overall energy consumption.

Under the good road conditions, the longitudinal slip ratio remains below the threshold, leaving the penalty term inactive in the cost function. The proposed online strategy aims to minimize power losses across the motors, inverter, and tire slip while maintaining efficient performance. As shown in Figure 16, the proposed online strategy reduces total energy consumption by 2.3% compared to the rule-based strategy under CLTC-P. This improvement results from the proposed online strategy, which considers both electric system losses and tire slip losses. While the offline strategy may lower electric system losses, they often increase tire slip losses, offsetting energy savings, as shown in Figure 17. In contrast, the proposed online strategy balances reductions in both electric and tire losses, achieving a net decrease in energy consumption. This approach enhances FRIDEV’s energy efficiency and supports greater driving sustainability by addressing both electric system and tire losses. The proposed online strategy provides a comprehensive solution for optimizing FRIDEV’s performance in real-world conditions, enhancing both energy efficiency and vehicle stability.

To comprehensively evaluate the effectiveness of the proposed strategy for energy management under various driving conditions, NEDC and WLTC were included as validation cases. Detailed results for the energy consumption performance are listed in

Table 3. Comparison of the energy consumption performance.

Item	Strategy	NEDC	WLTC	CLTC-P
Electric system loss energy (kJ)	Rule	853.5	2015.9	1142.4
	Offline	828.8	1970.0	1098.7
	Online	834.6	1983.9	1110.1
Tire slip loss energy (kJ)	Rule	55.6	162.6	106.7
	Offline	72.7	202.0	134.5
	Online	60.1	173.4	115.1
Total energy consumption (kJ)	Rule	4909.6 (−)	13,154.1 (−)	7446.9 (−)
	Offline	4859.2 (↓1.03%)	12,988.7 (↓1.26%)	7345.5 (↓1.36%)
	Online	4846.2 (↓1.29%)	12,925.2 (↓1.74%)	7275.3 (↓2.30%)

. Regarding electric system losses, the online strategy performs better than the rule-based strategy, though it is slightly higher than the offline strategy. This indicates that it maintains relatively low losses during real-time adjustments. In terms of tire slip losses, the online control strategy consistently shows lower values across all driving cycles compared to the offline approach. This suggests that the online strategy effectively balances tire slip losses and electric system losses rather than focusing solely on electric system efficiency. Finally, when comparing the proposed online control strategy to the rule-based approach, total energy consumption is reduced to 4846.2 kJ for NEDC, 12,925.2 kJ for WLTC, and 7275.3 kJ for CLTC-P, achieving reductions of 1.29%, 1.74%, and 2.30%, respectively. These results underscore the advantages of the online control strategy in enhancing overall energy efficiency.

4.2. Longitudinal Slip Ratio Performance

In low-adhesion road conditions, maintaining longitudinal stability is essential, as controlling the slip ratio directly impacts vehicle safety and handling. Excessive slip can result in traction loss, increasing the risk. This section assesses the effectiveness of torque control strategies under these conditions. The first scenario involves tracking a target speed using a portion of the US06 driving cycle, which includes smooth acceleration and braking, as shown in Figure 18a. The road adhesion coefficient is set at 0.4, and a closed-loop driver model is employed. Although the adhesion coefficient is low, this scenario can meet the required driving force for the FRIDEV. Figure 18a shows that the actual speed coincides with the target speed, indicating that FRIDEV effectively tracked the target speed under the proposed online strategy.

Figure 18b,c show the torque curves for the rule-based and proposed online strategies, respectively. Meanwhile, Figure 18d,e show the wheel slip ratios under each strategy. During acceleration, the load on the front wheel decreases, which reduces the front wheels’ ability to provide adequate traction. For the rule-based strategy, the FRIDEV allocates torque proportionally between the axles. At 10–11 s, the slip rate of the front wheels becomes excessively high, resulting in a traction loss on the front wheels. While the rear wheels still have the potential to offer better traction, the fixed proportional allocation limits their contribution. In contrast, the proposed online strategy dynamically adjusts torque distribution based on slip rate thresholds and vertical load transfer conditions. When the slip rate of both the front and rear wheels remains within the defined thresholds, the online strategy reallocates torque demands according to the dynamic vertical load transfer. At 10 s, the proposed online strategy triggers the penalty term in the cost function through the activation function when the front axle slip ratio exceeds a predefined threshold. The optimization algorithm adjusts the torque distribution to reduce the front wheels slip. The penalty term in the cost function emphasizes the punishment for high slip, guiding the algorithm to decrease the torque on the front IM and increase the torque on the rear PMSM. This dynamic torque redistribution optimizes the traction distribution between the front and rear wheels. Figure 18f highlights the superiority of the proposed online strategy from the perspective of tire slip energy. Compared to the rule-based strategy, the proposed online strategy reduces tire slip energy, enhancing driving efficiency.

In the second scenario shown in Figure 19a, the vehicle undergoes rapid acceleration with an open-loop driver model. The road adhesion coefficient is set to 0.2, and the initial vehicle speed is 3.6 km/h. The simulation results in Figure 19b show that during acceleration, the low traction available from the road surface causes excessively high slip ratios on both the front and rear wheels. This results in potential instability, making it difficult to maintain control. To address these challenges, the proposed online control strategy actively manages the slip ratios, ensuring that they remain close to the desired values. When the slip ratios exceed predefined thresholds, the strategy triggers a condition switch. The cost function and constraints are changed, prioritizing slip control over other objectives. The adjustment to the cost function directs the optimization process to reduce motor torque, effectively managing wheel slip and bringing the slip ratios closer to the desired values. Additionally, the constraints are updated to ensure that total torque stays within the system’s limits while still allowing the necessary reductions in torque to maintain stability. As shown in the simulation results in Figure 19c, the strategy adjusts motor torque in response to the slip ratios, maintaining traction and stability during acceleration. This dynamic adjustment ensures that the vehicle performs more effectively in low-adhesion conditions. The simulation results in Figure 19d further demonstrate that under the proposed strategy, the FRIDEV accelerates smoothly even on low-adhesion road surfaces. The proposed strategy enhances vehicle stability by dynamically adjusting torque distribution based on real-time slip ratio feedback, effectively maintaining traction, and managing wheel slip under challenging conditions.

4.3. Real-Time Performance

The final section discusses the real-time computational performance of the improved SSA. To demonstrate the advantages of the improved SSA in solving MPC optimization, a comparison with the original SSA is conducted. Figure 20 shows computation times for different methods, indicating a substantial reduction in computation time with the improved SSA compared to the original. This improvement is crucial for the real-time application of nonlinear MPC, enabling rapid response in the online torque control strategy for FRIDEV. Notably, the online control strategy is inactive during vehicle braking, resulting in blank data. Further details on computational performance are presented in

Table A2. Comparison of the real-time performance.

Item	Algorithm	Average (s)	Maximum (s)	Standard Deviation (s)
CLTC-P	Original SSA	5.8 × 10−4	1.6 × 10−3	1.6 × 10−4
	Improved SSA	2.8 × 10−4 (↓51.7%)	9.9 × 10−4 (↓38.1%)	8.6 × 10−5 (↓46.3%)
US06 part	Original SSA	5.2 × 10−4	1.4 × 10−3	1.9 × 10−4
	Improved SSA	2.9 × 10−4 (↓44.2%)	8.0 × 10−4 (↓42.9%)	9.4 × 10−5 (↓50.5%)

.

During the CLTC-P, the improved SSA reduces average computation time by 51.7%, from 5.8 × 10⁻⁴ s to 2.8 × 10⁻⁴ s. Maximum computation time also decreases, from 1.6 × 10⁻⁴ s to 9.9 × 10⁻⁵ s, with a 46.3% reduction in standard deviation. For both the CLTC-P on high adhesion surfaces and the US06 section on low-adhesion surfaces, the maximum computational time for each step remains below 1 ms, demonstrating that the improved SSA is suitable for real-time implementation. The reduction in standard deviation of computation time further highlights the robustness of the improved SSA. The effectiveness of the improved SSA arises from several factors: chaotic mapping for population initialization, a refined producer update mechanism, and targeted perturbation techniques. These enhancements make a better balance between exploration and exploitation of the optimal solution, effectively reducing computational complexity and accelerating convergence. Therefore, the solution speed of the MPC problem is significantly improved, and the real-time performance of the proposed strategy is guaranteed.

5. Conclusions

This study develops a dynamic driving torque control strategy for FRIDEV to reduce energy consumption, minimize tire slip, and enhance anti-slip performance. A comprehensive vehicle model is created, integrating vertical load transfer, tire slip, and losses from the IM, PMSM, inverter, and battery. The torque control problem is formulated in a nonlinear MPC framework, with optimization targets adjusting based on road traction conditions and slip rates. Active slip control is applied when slip rates exceed thresholds. An improved SSA accelerates convergence and reduces computational time. MATLAB/Simulink simulations show that, compared to rule-based strategies, the proposed online strategy reduces energy consumption by 2.3% in CLTC-P, balances electric system and tire slip losses, and improves performance in both NEDC and WLTC. On low-adhesion roads, they quickly adjust slip rates under rapid acceleration, ensuring stability. The improved SSA reduces computation time by 44–52%, with maximum computational times under 1 ms, demonstrating suitability for real-time applications in FRIDEV.