## 1 Introduction
in [31] investigated the channel constrained beamformer design for a more general SWIPT system in the presence of multiple Eves equipped with multiple antennas. The authors in [32] considered the resource allocation problem for the secure SWIPT multicasting scenario with channel uncertainties, where a low complexity optimization algorithm was proposed. Moreover, the authors of [33] also consider the SWIPT multicasting scenario and address the secure beamforming design problem. From the above discussion, it is worth noting that most of the existing works focused on the beamforming design supporting SWIPT under the linear EH model, where it is assumed that the collected energy power could be linearly increased by growing the input power of the received RF signals. However, the practical power conversion circuits usually perform a nonlinear feature rather than the linear one due to their nonlinear elements such as two-electrode valves or two-electrode valves connected to the transistor. Therefore, the authors of [34] proposed a nonlinear EH model by fitting over real sampled data based on a logistic function. Then, the resource allocation optimization algorithms for SWIPT and wireless powered communication networks have been attracting increasing interest (see [35; 36; 37; 38; 39; 40; 41]). In [35], authors addressed the max-min signal-interference-noise-ratio (SINR) beamforming design optimization problem in a V2X-SWIPT environment with imperfectly estimated channels at the base station, where a practical nonlinear EH model is considered. In [36; 37], the authors discuss the robust transmit beamformer design, joint user scheduling and power allocation for multiple user SWIPT scenario under nonlinear EH model. In [38], the authors designed the beamformer for energy efficiency maximization in a secure SWIPT system taking into account the influence of the nonlinear EH circuits. Then, the authors in [39] investigate the joint optimization of transmit precoding and power splitting ratios in MISO SWIPT multicasting networks with the objective of maximizing the minimum collected energy among all receivers under a generic (including nonlinear) EH model. The multiuser MIMO SWIPT system is also considered in [40; 41], where the authors study the transmit beamforming design to maximize the harvested power based on perfect CSI under a generic EH model. The results obtained in [35; 36; 37; 38; 39; 40; 41] demonstrated that a favorable performance gain could be achieved if the beamforming design takes into account the effect of the nonlinear EH circuit instead of a linear EH circuit owing to the mismatch between the linear EH model and the nonlinear EH model. To the best of our knowledge, most of the above works on SWIPT either focus on the optimization of secrecy capacity of information receiver or the transmit power of the base station. It is usually assumed that the CSIs are perfectly available at the base station or the receivers are equipped with one antenna. These motivate us to research uncommonly considered optimization problem of fair power harvesting among all energy vehicular receivers that are equipped with multiple antennas. In addition, the CSIs of base stations to energy vehicular receivers are imperfect. In this paper, we focus on the optimal secure beamformer design for V2X-SWIPT in a multiple vehicles MISO communication network with multiple multi-antenna energy vehicular receivers, where the nonlinear EH model proposed in [35] is utilized. The CSIs between the base station and the energy vehicular receivers are supposed to be imperfect. Moreover, the artificial noise-aided signal strategy and the power splitting receivers are taken into account. We aim for the design of a beamforming design algorithm maximizing the minimum harvested energy among multiple multi-antenna energy vehicular receivers for the case when the information vehicular receiver is able to collect energy from radio frequency signals.