Energy-Efficient Optimal Torque Vectoring for a Four-Motor High-Performance Electric Vehicle

Abstract

The paper presents and compares an optimal control allocation (CA) and model predictive control (MPC)-based torque vectoring (TV) for improved energy efficiency of electric vehicle with four independent electric motors. Offline and online (instantaneous) optimisation-based CA are designed for front-rear torque distribution. For overall wheel torque allocation, a production-ready MPC-based TV is extended with energy consumption minimisation terms. CA and MPC rely on power loss curves of differently sized front and rear powertrains that are fitted with polynomial regression models. Performance of both strategies is evaluated in high-fidelity nonlinear simulation environment in terms of energy efficiency improvement on standard driving cycles and impact on the vehicle dynamics in lateral manoeuvres. Results demonstrate consistent reduction of the energy consumption and preservation of the vehicle handling behaviour.

1 Introduction

Battery electric vehicles with four independent motors offer improved vehicle dynamics performance and energy efficiency due to their all-wheel drive and torque vectoring capabilities, achieved through independent wheel torque control. In the literature, different approaches have been applied to solve this efficient torque allocation problem, typically with the assumption of equal powertrains. These approaches range from rule-based/analytical solutions [1], offline and online instantaneous optimisation [2, 3], and various model predictive control concepts for powertrains with [4] and without disconnect clutch actuation [5, 6]. For equal motor assumption, it is demonstrated in [7] that optimal longitudinal bias is either single axle for low torque demand or 50:50 for high torque demand. Similar conclusions have been drawn for lateral motion, where outer-track motors are first considered. In this paper, a 4MEV with differently sized front and rear powertrains is considered, and the optimal control allocation (CA) and model predictive control (MPC) based torque vectoring (TV) are designed with an emphasis on improving the overall vehicle energy efficiency and driving range, while accounting for induced effect on vehicle dynamics [8].

2 Energy-Efficient Torque Vectoring Design

2.1 Power Losses Modeling

The powertrain power losses models are developed based on experimental data obtained from bench tests of inverters, surface-mounted permanent-magnet electric motors, and gearboxes originating from an electric hypercar. The power loss maps are a function of respective torque, speed, and battery voltage. They are fitted with polynomial regression models for use in the online optimisation problems to capture the complex and nonlinear relationships between the variables.

Figure 1 shows the power loss characteristics of the front and rear powertrains as a function of the electric motor torque request for multiple motor speed operating points and constant battery voltage (normalized axes). Each rear axle powertrain can be used in a free-rolling configuration with inverter switches opened, i.e. in open circuit with mechanical losses predominant (represented by $*$ in Fig. 1b).

Fig. 1.

Individual power losses characteristics for specific EMs angular velocities

Tyres introduce controllable energy dissipation attributable to longitudinal slip, which is a function of wheel torque. Herein, the tyre losses are excluded from offline optimisation to reduce computational complexity. We observed in simulation that tyre power losses never exceed 10% of a Front-Wheel Drive (FWD) vehicle’s global power losses on a demanding driving cycle (EPA-US06).

2.2 Optimal Torque Control Allocation Problem

The CA strategy determines the longitudinal torque split between front and rear axles. The nonlinear multi-parametric optimisation problem, given by Eq. 1, minimises the overall powertrain power losses subject to driver-demanded force request and velocity-dependent actuator torque constraints. The optimisation yields a front-to-total bias for the overall range of achievable inputs.

$$\begin{aligned} J = \mathop {min}_{{\textbf {T}}} \sum _{i=1}^{2} P_{i}(T_{i},V_{bat},\omega _{i}) \end{aligned}$$

(1)

where $i = [Front, Rear]$, $T_{i}$ is the vector of front and rear motor torques [Nm], $V_{bat}$ is the battery voltage [V] and $\varOmega _{i}$ is the front and rear motor speed [rad/s].

The objective function value of optimal solution is compared to the one of a FWD strategy with rear powertrains disabled, i.e., free rolling (cf. Fig. 1b), and more efficient torque allocation is selected. Such switching between strategies can yield excessive bias transitions and, thus, instantaneous torque steps on both axles which compromise drivability. A rate limit is therefore applied to the bias after the optimisation to avoid selection of possible local minima as illustrated in Fig. 2 where points A, B, C, and D correspond respectively to previous sample time ($t-1$) solution, locally optimal solution with the rate constraint (red line) included in the optimisation, selected suboptimal solution, and globally optimal solution without rate constraints included in the optimisation. Rate limit can be scheduled with yaw rate to avoid unsafe behaviour and/or driver demand to protect drivetrain components from torque steps, while maximum value should be set for acceptable drivability (low jerk).

Fig. 2.

Power losses against force bias and force request at constant $v_{x}$ and $V_{bat}$

Fig. 3.

Optimal front-to-total force bias for constant $V_{bat}$, positive $F_{req}$ and low tyre slip

In the offline case using a fixed lateral distribution of 50:50, the results are stored in 3D map as shown in Fig. 3 to facilitate vehicle implementation and reduce computational effort. The optimisation tool is implemented in Matlab and uses the fmincon SQP solver.

The same optimisation problem is solved online using the fmincon SQP solver and implemented in real-time on a production ECU. The benefit of this approach is the adaptation to varying motor torque limits that can be changed by the driver (directly or through driving modes) or inverter due to, e.g., temperature derating. The offline optimisation in that case requires pre-computed maps.

2.3 Model Predictive Control Torque Vectoring Architecture

For the overall torque distribution control, a production-ready Linear Time-Varying (LTV) MPC TV [9] is expanded with energy efficiency terms. It is based on a 7-degree-of-freedom handling model as seen in Fig. 4, extended with energy loss state fed by the aforementioned, polynomial regression power loss functions for each powertrain where tyre slip losses can be considered independently. The cost function S that includes driver demand tracking terms (yaw rate and force request) and control input penalization terms is extended with energy state minimisation term to account for energy efficiency as shown in Eq. 2.

Fig. 4.

Simplified 7deg of freedom vehicle handling model including power losses calculation where $i = {Front, Rear}$ and $j = {Left, Right}$

A high cost set on yaw rate tracking $W_{\dot{\psi }}$ gives authority to the safety/performance character of the problem over the efficiency cost. The efficiency term weight $W_{E}$ is scheduled based on yaw rate and surface adhesion to maintain safety during high slip or high lateral dynamic manoeuvres.

$$\begin{aligned} \begin{aligned} S(X(k),T_{ij}(k),X_{dev}(k)) &=W_{\dot{\psi }}.(\dot{\psi }_{ref}(k)-\dot{\psi }(k))^2\\ &+ W_{Fx}.(F_{x,ref}-(\sum _{ij} T_{ij}(k).\frac{1}{R_{w,ij}}))^2\\ &+ W_{E}.({E_{loss}(k)})^2\\ &+ W_{Urate}.(T_{ij}(k-1)-T_{ij}(k))^2\\ \end{aligned} \end{aligned}$$

(2)

where X is the state vector, $X_{dev}$ is a vector of controller inputs, $\dot{\psi }$ is the yaw rate [rad/s], $F_{x,ref}$ is the longitudinal force request [N], $R_{w,ij}$ are the wheel radii [m], $E_{loss}$ is the energy losses estimation [J], k is the current discretization step, and W are the weights applied to each cost term of the objective function.

The MPC TV problem is solved using a quadratic problem solver with linear constrains and runs on a production ECU at a sampling rate of 10 ms.

3 Longitudinal Motion Simulations

A longitudinal vehicle model consisting of a closed-loop driver model, longitudinal vehicle dynamics with suspension effects included, powertrain model with compliance and inertia effects, and Pacejka tyre model is used to asses the vehicle efficiency on standard driving cycles. Table 1 compares the results of the different torque allocation strategies relative to a fixed 50:50 All-Wheel Drive (AWD) strategy in terms of energy losses $\varDelta E_{loss}$ and total energy consumed $\varDelta E_{cons}$. Results from online CA and energy efficient MPC TV are equal.

Table 1. Efficiency variation compared to fixed AWD for different drive cycles

It is evident that the FWD configuration presents efficiency results close to those of the offline and online CA and MPC efficient TV due to the unusually high capacity of the front powertrain for the low force demand of most driving cycles. The online CA exhibits minor improvement compared to the offline CA as it circumvents the inaccuracy of fixed step breakpoints in the offline map.

4 Lateral Motion Simulations

The weights associated to the cost function of the MPC TV are tuned to conserve the safety, performance and driving characteristic of the baseline MPC TV while improving efficiency. Steady-state skidpad tests in Fig. 5 demonstrate that both efficient CA and TV provide efficiency improvement at low values of lateral acceleration, with the efficient TV surpassing the CA for $a_{y} \in [2.5:5.5]$ m/s$^2$ due to a yaw moment request allowing for a favorable left-right torque distribution.

ISO Ramp steer test presented in Fig. 7 showcase the weight associated to energy efficiency at low values of lateral acceleration $a_y$ and the convergence of the energy efficient TV yielded torque commands towards the baseline TV ones. While the energy efficient MPC TV reduces the vehicle responsiveness (understeer gradient increased by approx. 9%), the maximum value of lateral acceleration is preserved. Note that the efficient TV torque allocation corresponds to the successive enabling of the outside and inside rear wheels as shown in Fig. 6.

Fig. 5.

Steady-state skidpad test ($R=42\,\textrm{m}/ \mu =1$) AWD relative power losses

Fig. 6.

Wheel torque during steady-state ($v_{x}=100\,\textrm{km}/\textrm{h}$) ramp steer manoeuvre

Fig. 7.

Steady-state ($v_{x}=100\,\textrm{km}/\textrm{h}$) ramp steer with tyre slip induced $Fx_{req}$ increase

The control allocation strategies are implemented on a production specification vehicle equipped with 4 electric powertrains, i.e. one for each wheel, to validate integration and real-conditions behaviour during dynamic manoeuvres.

Conclusion

This paper presents strategies to enhance vehicle efficiency and dynamic performance, including an optimal offline torque allocation strategy, an online torque allocation strategy, and a model predictive control-based energy-efficient torque vectoring system. Through simulation testing on an EV, these solutions proved effective for vehicles with multiple electric motors of varying sizes. The control allocation strategy efficiently distributes force requests between front and rear axles, enhancing overall vehicle efficiency by up to 32.05% compared to a baseline 50:50 AWD architecture and up to 5.68% compared to a fixed FWD architecture on typical driving cycles. However, the performance gains were tempered by the unusually high capacity of the demonstrator vehicle powertrains. Furthermore, the strategy preserved the dynamic behaviour of the baseline vehicle, ensuring stability, maneuverability, and safety across various driving conditions, including cornering manoeuvres at the handling limit. Vehicles equipped with efficient torque vectoring maintained lateral performance and improved overall efficiency by up to 17% compared to fixed lateral distribution configurations at specific vehicle operating points.