## 2 System Model and Problem Formulation  
We consider the downlink of a V2X-SWIPT communication system that consists of a base station and two kinds of legitimate vehicular receivers, namely one information vehicular receiver and \(K_{E}\) idle energy vehicular receivers. The base station is equipped with a \(N_{T}\) transmit antenna while the information vehicular receiver is a single antenna device and is able to decode information and collect energy from radio signals, cf. Figure 1. In addition, the \(K_{E}\) idle energy vehicular receivers equipped with \(N_{R}\) antennae are supposed to collect energy from the radio frequency as they are inactive. The energy vehicular receivers can work in double functions of EH and information decoding. In this scenario, the idle energy vehicular receivers, which cause no interference to the communication channel, can potentially eavesdrop on the confidential information of the information vehicular receiver by switching its working pattern to information decoding since all the legitimate vehicular receivers are within service coverage. Therefore, the idle energy vehicular receivers are potential eavesdroppers that can take care of supplying physical layer security. The channel vector between the base station and the information vehicular receiver is denoted as \(\mathbf{h}\in\mathbb{C}^{N_{T}\times 1}\) and the channel matrix between the base station and the \(k\)-th energy vehicular receiver is denoted as \(\mathbf{G}_{k}\in\mathbb{C}^{N_{T}\times N_{R}}\), \(\forall k\in\{1,2,\cdots,K_{E}\}\). In our considered system model, we suppose that the vehicle speed is low. Several simulation and theoretical investigations have shown that V2X communication channels such as \(\mathbf{h}\) and \(\mathbf{G}_{k}\) can be modelled as Rayleigh channel fading [42; 43; 44; 45]. It is assumed that our considered system works in a Time Division Duplex (TDD) mode with slowly time-varying communication channels. Hence, the base-station-to-legitimate-vehicular-receiver fading gains, \(\mathbf{h}\) and \(\mathbf{G}_{k}\), could be reliably estimated at the base station at the start of each scheduling slot with negligible channel estimation errors. During the data transmission, the information vehicular receiver sends acknowledgement packets to tell the base station of successful reception of data packets. Hence, in this paper, the channel uncertainties for the base-station-to-energy-vehicular-receiver are considered while the CSI of the base-station-to-information-vehicular-receiver is perfectly known at the base station. We model the CSI of the link between the base station and the \(k\)-th energy vehicular receiver as:  
\[\mathbf{G}_{k}=\bar{\mathbf{G}}_{k}+\Delta\mathbf{G}_{k},\forall k\in\left\{1,2,\cdots,K_{ E}\right\}, \tag{1}\]  
\[\Xi_{k}\triangleq\left\{\Delta\mathbf{G}_{k}\in\mathcal{C}^{N_{T}\times N_{R}}: \left\|\Delta\mathbf{G}_{k}\right\|^{2}\leq v_{k}^{2}\right\},\forall k, \tag{2}\]  
where \(\bar{\mathbf{G}}_{k}\) is the channel estimate of the \(k\)-th energy vehicular receiver available at the base station while \(\Delta\mathbf{G}_{k}\) denotes the unknown channel uncertainty of the energy vehicular receiver \(k\). For notational simplicity, a set \(\Xi_{k}\) in Label (2), which includes all possible CSI uncertainties of energy vehicular receiver \(k\). Moreover, the radius \(v_{k}\geq 0\) denotes the size of the uncertainty region of the estimated CSI of the energy vehicular receiver \(k\). To provide vehicular secure communication and to facilitate EH at the desired information vehicular receiver, artificial noise signals and energy signals are generated at the base station. In particular, both signals could degrade the channels between the base station and the energy vehicular receivers and perform as an energy source for EH.