## 1 Introduction  
Concerning the urgent issues of climate change and environmental protection, promoting electric vehicles (EVs) has become a primary policy for many developed countries around the world [1, 2]. One major challenge for developing EVs lies in enhancing energy efficiency and driving range under the limited power capability and energy storage of electrochemical batteries [3]. A feasible way is to install a differential which can perform functionality of lateral torque vectoring, called torque vectoring differential (TVD) [4]. Many studies focus on the kinematics analysis for TVD. [PERSON] et al. [5] proposed a calculation method to evaluate the effect of the left and right TVD mounted in either the front or rear axle of front-wheel-, rear-wheel-, or four-wheel-drive systems. Thereby, the optimal longitudinal forces of wheels applied through the TVD of each drivtain are attained by analyzing the dynamic limits of the vehicle so that the cornering ability in terms of the maximum lateral acceleration can be found. [PERSON] et al. [6] carried out a kinematics analysis on an electric-powered lateral torque vectoring differential (E-TVD) with two degrees of freedom and five elements, consisting of a single electric motor, the first planetary gear, and the second planetary gear. The differential can be classified and characterized by analyzing the relationship between torque and speed of each component using speed diagram. As a result, analyzing the amplification factor G and the rotational speed NCmakes it possible to sort out a favorable type of E-TVD with simultaneous reduction in both the size of the motor machine and the acceleration resistance of the vehicle. Recently, developing an effective dynamic control method for TVD has become more popular due to the higher demand of safe handling and cornering performance. [PERSON] [7] developed a model-based integrated control system for a TVD, based on the transmission system structure referring to the MUTE project of the Technical University of Munich (TUM). A simulation software package called Modelica is utilized for constructing the TVD model so as to develop an effective control strategy for predicting the instant vehicle yaw rate and vehicle sideslip angle. The simulation results show that the predicted dynamics behaviors are well matched with the expected outcomes. The documentary work of [PERSON] et al. [8] presents a predictive design an integrated control system of active torque vector (ATV) and electronic stability control (ESC) based on continuous nonlinear vehicle models. The Karush-Kuhn-Tucker (KKT) optimization theorem is adopted to track the desired yaw rate response for better steering performance, and limit the sideslip angle within the allowable range for enhanced directional stability. The comparative results show that the proposed method possesses the advantages of fast controller speed, simplicity of calculation, enhancing the handling and stability performance, as well as energy efficiencies, without losing too much power. Some studies about TVD aim at not only optimizing dynamic control but also improving energy saving. [PERSON] et al. [9] implemented a so-called Markov decision process (MDP) based on deep reinforcement learning to formulate a torque distribution strategy for a four-wheel-drive electric vehicle. The goal of improving both active safety and energy saving has been fulfilled through the optimal control of torque distribution. This method uses the actor-critic network to train the agent to obtain better control performance. The CarSim-Simulink simulation results show that this torque distribution strategy significantly improves the average motor efficiency and reduces energy loss by 5.25-10.51%. [PERSON] et al. [10] specially incorporated a mathematical model of mechanical elastic electric wheels based on experimental data into a 7-DOF vehicle model. It optimizes the objective function based on the NMPC controller of the 4 IWMD EV stability control.