## 4 Simulation Results  
In this section, we provide simulation results to validate the performance of our proposed robust beamforming design algorithm. The considered vehicular secure downlink channel in Section 2 for Rayleigh flat-fading environments with zero-mean and unit variance is considered. Unless specified otherwise, it is assumed that there are \(K_{E}=3\) energy vehicular receivers. Indeed, we have some other configurations that can be employed with a different number of energy vehicular receivers, transmit antennas and receiver antennas; however, the simulation results will be similar, the only difference is the computation complexity. In our simulations, we set \(N_{T}=6,N_{R}=3,\ r=10\text{ dB},\ r_{k}=\bar{r}=0\text{ dB},\ P=25\text{ dBm},\ E_{\text{min}}=0\text{ mW},\ \sigma_{I}^{2}=\sigma_{E,k}^{2}=\sigma^{2}=-20\text{ dBm}\). For the nonlinear EH model, we set \(M_{I}=M_{E,k}=M=24\text{ mW},a_{I}=a_{E,k}=\bar{a}=1500,\ b_{I}=b_{E,k}=\bar{b}=0.014\). To the imperfect CSI between the base station and energy vehicular receivers, \(d_{E,k}^{2}=\frac{v_{k}^{2}}{\|\mathbf{G}_{k}\|^{2}}\) is defined as the channel uncertainty ratio to demonstrate the \(k\)th energy vehicular receiver's channel estimate error. Without loss of generality, it is assumed that \(d_{k}=\delta_{E}\). We solve the optimization problem (21) and compute the average minimum harvested energy power per energy vehicular receiver by averaging over 1000 channel realizations. Figure 2 depicts the average minimum harvested energy power per energy vehicular receiver versus the maximum power of the base station, \(P\), for \(K_{E}=3\) energy vehicular receivers, \(\delta_{E}^{2}=0.02\), and different beamforming design schemes. We learn that the average minimum harvested energy power per energy vehicular receiver of the proposed beamforming design scheme is a monotonically increasing function of \(P\). This is attributed to the fact that a higher harvested energy power is necessary for satisfying constraint C4 when the maximum transmit power requirement of \(P\) becomes more stringent. For a comprehensive comparison, Figure 2 also contains the average minimum harvested energy power per energy vehicular receiver of the proposed algorithm that the base station has perfect CSI, the non-robust scheme, the linear EH scheme and two baseline beamforming design schemes. For the non-robust scheme, we treat the estimated channel \(\bar{\mathbf{G}}_{k}\), \(\forall k\in\{1,2,\cdots,K_{E}\}\) as true CSI, and then optimize \(\{\mathbf{W},\mathbf{V},\mathbf{V}_{0},\rho\}\) jointly in (21) according to \(\bar{\mathbf{G}}_{k}\), \(\forall k\in\{1,2,\cdots,K_{E}\}\). For the linear EH scheme, we adopt the existing linear EH model for the beamforming design algorithm. In particular, \(\{\mathbf{W},\mathbf{V},\mathbf{V}_{0},\rho\}\) optimizes to the maximum the minimum harvested energy power per energy vehicular receiver subject to the constraints in (13). For the baseline scheme 1, we adopt the zero-forcing approach to null out the undesired interference to the information vehicular receiver. In particular, we choose \(\mathbf{V}\) and \(\mathbf{V}_{0}\) lying in the null space of the channel \(\mathbf{h}\). Furthermore, we employ the  
maximum ratio transmission for transmitting information to the information vehicular receiver. It can be observed that the proposed scheme closely approaches the performance of the perfect CSI scheme. On the other hand, it can be seen that the low computational complexity of the baseline scheme 1 comes at the expense of a significantly lower harvested energy power compared to the proposed scheme and the linear EH scheme. Indeed, the proposed scheme and the linear EH scheme fully use the CSI of all vehicular communication links and optimize the space spanned by the artificial noise and the energy signal for performing beamforming design.