## 2 System Model and Problem Formulation
As a result, we express the transmit signal vector \(\mathbf{x}\) as  
\[\mathbf{x}=\mathbf{w}s_{I}+\sum_{k=1}^{K_{E}}\mathbf{v}_{k}s_{E,k}+\mathbf{v}_{0}, \tag{3}\]  
where \(s_{I}\in\mathbb{C}\) and \(\mathbf{w}\in\mathbb{C}^{N_{T}\times 1}\) are the information-bearing signal and the corresponding beamforming vector for the information vehicular receiver, respectively. Without loss of generality, we suppose that \(\mathcal{E}\left\{|s_{I}|^{2}\right\}=1\). \(\mathbf{v}_{k}\in\mathbb{C}^{N_{T}\times 1}\) and \(s_{E}\) are the energy beamforming vector and the energy-bearing signal with \(\mathcal{E}\left\{|s_{E}|^{2}\right\}=1\), respectively. \(\mathbf{v}_{0}\in\mathbb{C}^{N_{T}\times 1}\) represents the artificial noise vector generated by the base station to combat both passive and potential eavesdroppers, and we model it as \(\mathbf{v}_{0}\sim\mathcal{CN}(\mathbf{0},\mathbf{V}_{0})\)With the transmit signal \(\mathbf{x}\), the received signal at information vehicular receiver and \(k\)-th energy vehicular receiver can be expressed as  
\[ \begin{split} y_{I}&=\mathbf{h}^{H}\mathbf{x}+n_{I}\;and\\ y_{E,k}&=\mathbf{G}_{k}^{H}\mathbf{x}+\mathbf{n}_{E,k},\forall k \in\left\{1,\cdots,K_{E}\right\},\end{split}  \tag{4}\]  
respectively, where \(n_{I}\sim\mathcal{CN}(0,\sigma_{I}^{2})\) and \(\mathbf{\sigma}_{E,k}\sim\mathcal{CN}(\mathbf{0},\mathbf{n}_{E,k}^{2}\mathbf{I}_{N_{R}})\) represent the complex Gaussian noise at the information vehicular receiver and the \(k\)-th passive eavesdropper, respectively. \(\sigma_{I}^{2}\) and \(\sigma_{E,k}^{2}\) is the noise power of each other. In the considered networks, we exploit the energy and artificial noise signals transmitted by the base station to facilitate secure information transfer and charge both information vehicular receiver and energy vehicular receivers, respectively. In this paper, we adopt a practical nonlinear model. According to [35; 36], the harvested energy at the \(k\)-th energy vehicular receiver can be modeled as:  
\[E_{E,k}^{\text{Nonlinear}}=\frac{\Psi_{E,k}}{X_{E,k}}-Y_{E,k}, \tag{5}\]  
with  
\[\Psi_{E,k}=\frac{M_{E,k}}{1+\exp\bigl{(}-a_{E,k}\bigl{(}P_{E,k}\bigl{(}\mathbf{W},\mathbf{V},\mathbf{V}_{0}\bigr{)}-b_{E,k}\bigr{)}\bigr{)}}, \tag{6}\]  
\[ \begin{split} P_{E,k}\bigl{(}\mathbf{W},\mathbf{V},\mathbf{V}_{0}\bigr{)}& =\|\mathbf{G}_{k}^{H}\mathbf{w}\|^{2}+\sum_{k=1}^{K_{E}}\|\mathbf{G}_{k}^{H} \mathbf{v}_{k}\|^{2}+\|\mathbf{G}_{k}^{H}\mathbf{v}_{0}\|^{2}\\ &=Tr\bigl{[}\mathbf{G}_{k}^{H}\bigl{(}\mathbf{W}+\mathbf{V}+\mathbf{V}_{0}\bigr{)} \mathbf{G}_{k}\bigr{]},\forall k,\end{split}  \tag{7}\]  
where \(X_{E,k}=\frac{\exp\bigl{(}a_{E,k}b_{E,k}\bigr{)}}{1+\exp\bigl{(}a_{E,k}b_{E,k} \bigr{)}}\), \(Y_{E,k}=\frac{M_{E,k}}{\exp\bigl{(}a_{E,k}b_{E,k}\bigr{)}}\), \(\mathbf{W}=\mathbf{w}\mathbf{w}^{H}\) and \(\mathbf{V}=\sum_{j=1}^{K_{E}}\mathbf{v}_{j}\mathbf{v}_{j}^{H}\). \(P_{E,k}\bigl{(}\mathbf{W},\mathbf{V},\mathbf{V}_{0}\bigr{)}\) denote the received RF power at the \(k\)-th energy vehicular receiver. Three parameters, namely, \(M_{E,k}\), \(a_{E,k}\) and \(b_{E,k}\) in (6) are introduced to capture the joint effects of various nonlinear phenomena caused by hardware limitations in practical EH circuits. Furthermore, \(M_{E,k}\) denotes the maximum power that can be collected by the EH circuit while \(a_{E,k}\) and \(b_{E,k}\) rely on serval hardware phenomena, such as the capacitance and diode turn-on voltage and the resistance. As depicted in Figure 1, we adopt the power splitting architecture at the information vehicular receiver, the RF signals are divided into two streams by a power splitter in \(\rho\), i.e., the \(1-\rho\) part of the received signals is input into the EH receiver and the remaining \(\rho\) part is sent to the information receiver. Therefore, the received SINR can be expressed as  
\[