## 2 System Model and Problem Formulation
\begin{split}\Gamma_{I}&=\frac{\rho|\mathbf{h}^{H} \mathbf{w}|^{2}}{\rho|\mathbf{h}^{H}\mathbf{v}_{0}|^{2}+\rho\sum_{i=1}^{K_{E}}|\mathbf{h}^{H} \mathbf{v}_{i}|^{2}+\sigma_{I}^{2}}\\ &=\frac{\rho Tr(\mathbf{H}\mathbf{W})}{\rho Tr(\mathbf{H}\mathbf{W}_{0})+\rho Tr (\mathbf{H}\mathbf{V})+\sigma_{I}^{2}}.\end{split}  \tag{8}\]  
The input power at the information vehicular receiver can be written as  
\[ \begin{split}& P_{I}\bigl{(}\mathbf{W},\mathbf{V},\mathbf{V}_{0},\rho\bigr{)} =(1-\rho)|\mathbf{h}^{H}\mathbf{w}|^{2}+(1-\rho)\bigl{(}|\mathbf{h}^{H}\mathbf{v}_{0}|^{2}+ \sum_{i=1}^{K_{E}}|\mathbf{h}^{H}\mathbf{v}_{i}|^{2}\bigr{)}\\ &=(1-\rho)Tr(\mathbf{H}\mathbf{W})+(1-\rho)\bigl{[}Tr(\mathbf{H}\mathbf{V}_{0})+ Tr(\mathbf{H}\mathbf{V})\bigr{]}.\end{split}  \tag{9}\]Here, we also adopt the nonlinear EH model, and then the harvested energy at the information vehicular receiver is given by  
\[\Phi_{I}\big{(}\mathbf{W},\mathbf{V},\mathbf{V}_{0},\rho\big{)}=\frac{\Psi_{I}}{X_{I}}-Y_{I}, \tag{10}\]  
with  
\[\Psi_{I}=\frac{M_{I}}{1+\exp\big{(}-a_{I}\big{(}P_{I}\big{(}\mathbf{W},\mathbf{V},\mathbf{V} _{0},\rho\big{)}-b_{I}\big{)}\big{)}}, \tag{11}\]  
where \(X_{I}=\frac{\exp\big{(}a_{I}b_{I}\big{)}}{1+\exp\big{(}a_{I}b_{I}\big{)}}\) and \(Y_{I}=\frac{M_{I}}{\exp\big{(}a_{I}b_{I}\big{)}}\). \(M_{I}\) is a constant representing the maximum harvested power at the information vehicular receiver when the EH circuit is saturated. \(a_{I}\) and \(b_{I}\) are also constants associated with the detailed circuit specifications. Without loss of generality, the energy vehicular receivers can also perform as eavesdroppers to decode the confidential information for the information vehicular receiver. Due to the fact that the energy signal \(\sum_{k=1}^{K_{E}}\mathbf{v}_{k}s_{E,k}\) is the only Gaussian pseudo-random sequence that is known at legitimate receivers (including both information vehicular receiver and energy vehicular receivers), we can thus perform interference cancellation at each energy vehicular receiver. As a result, the SINR of the \(k\)-th energy vehicular receiver is given by  
\[\Gamma_{E,k}=\frac{\|\mathbf{G}_{k}^{H}\mathbf{w}\|^{2}}{\|\mathbf{G}_{k}^{2}\mathbf{v}_{0}\| ^{2}+\sigma_{E,k}^{2}}. \tag{12}\]  
Figure 1: (a) V2X-SWIPT system model for two energy vehicular receivers and one information vehicular receiver; (b) the block diagram of the information vehicular receiver model for wireless information and power transfer. The optimal beamforming design policy, \(\{\mathbf{W}^{*},\mathbf{V}^{*},\mathbf{V}_{0}^{*},\rho^{*}\}\), for maximizing the minimum harvest energy among all the \(K_{E}\) energy vehicular receivers, can be obtained by solving  
\[\max_{\mathbf{W},\mathbf{V},\mathbf{V}_{0},\rho}\min_{\mathbf{\Delta}\mathbf{\Theta}_{k }\in\Xi_{k}} \frac{M_{E,k}}{X_{E,k}\left[1+\exp\left[-\,a_{E,k}\left(\text{Tr} \left((\mathbf{\bar{G}}_{k}+\Delta\mathbf{G}_{k})^{H}(\mathbf{W}+\mathbf{V}+\mathbf{V}_{0})(\mathbf{ \bar{G}}_{k}+\Delta\mathbf{G}_{k})\right)-b_{E,k}\right)\right]\right]}-Y_{E,k}\] \[s.t. \text{C1}:\Gamma_{I}\geq r,\] \[\text{C2}:\max_{\Delta\mathbf{G}_{k}\in\Xi_{k}}\frac{\text{Tr} \left(\left(\mathbf{\bar{G}}_{k}+\Delta\mathbf{G}_{k}\right)^{H}\mathbf{W}\left(\mathbf{\bar{ G}}_{k}+\Delta\mathbf{G}_{k}\right)\right)}{\left(\left(\mathbf{\bar{G}}_{k}+\Delta\mathbf{G}_{k} \right)^{H}\mathbf{V}_{0}\left(\mathbf{\bar{G}}_{k}+\Delta\mathbf{G}_{k}\right)\right)} \leq r_{k},\forall k,\] \[\text{C3}:\Phi_{I}\big{(}\mathbf{W},\mathbf{V},\mathbf{V}_{0},\rho\big{)} \geq E_{\text{min}},\] \[\text{C4}:\text{Tr}\big{(}\mathbf{W}+\mathbf{V}+\mathbf{V}_{0}\big{)}\leq P.\]  
The cost function in (13) takes into account the CSI uncertainty set \(\Xi_{k}\) to provide robustness against CSI imperfection. In C1: \(r\) represents the minimum SINR of the information vehicular receiver for information decoding. Constraint C2 is imposed such that for a given CSI uncertainty set \(\Xi_{k}\), the maximum received SINR at the passive eavesdropper \(k\) is not larger than the maximum tolerable received SINR \(r_{k}\).