## 4 Simulation Results and Discussion
### Comparative Studies  
A quite interesting point is to investigate what happens in the algorithm in terms of energy efficiency. The force vector \(u_{i}\) is used to calculate the energy of each vehicle and eventually the platoon's system in total. Specifically, \(Energy_{i}=\int_{0}^{800}u_{i}^{2}(\tau)\,d\tau,\;i=1,2,3,4\). Employing the same system as before, we compare the final energy values for the plain PPC (fixed inter-vehicular distances), the proposed PPC with ESCWAO, and the basic ESC with PPC, as designed in Figure 2. The setup is the same as before: in the plain PPC we set the desired distance \(\Delta_{i-1,i}=8,\;i=1,2,3,4\) and in the basic ESC with PPC we set \(a_{i}=1\), for \(i=1,2,3,4\). Figure 8 shows the comparative results. The total energy that the system requires in the plain PPC is almost double the energy required by the proposed ESCWAO one. Moreover, PPC with a basic ESC system energy requirement is also high and for the first two vehicles in the platoon is individually higher even than the plain ESC indicating that PPC with basic ESC is not worthy at all. It should be noted that the first vehicle behind the leading one requires in all cases more energy than the others due to the fact that its frontal area is set far bigger, demonstrating a larger truck, but still, in our case, the energy is less.  
Further investigation of the energy efficiency is provided in Table 1. We implemented the algorithm proposed in [19] for a system of two vehicles neglecting rolling and gradient resistance. For the leading vehicle, we adopted the same velocity profile as before, and for the following vehicle, the second spline-fitted profile for the drag resistance (\(C_{d_{2}}\)) is shown in Figure 4. For the following vehicle \(m=1200\) kg, \(A=2.5\) m\({}^{2}\), \(p=1.28\) kg m\({}^{-3}\), and the initial distance between the two vehicles was \(20\)m in both cases. Additionally, the sliding mode controller's (SMC) and high-gain observer parameter values were set the same as the authors suggested in their work. In our case, for the PPC scheme, \(k_{p}=1,k_{p}=149\), \(\Delta_{Col_{1}}=2.5,\Delta_{Con_{1}}=20,H_{1}=L_{1}=10\) use the same performance functions as in the previous section and for our observer \(\epsilon_{1}=10^{-3}\).  
Concerning the extremum seeking scheme, in the implementation of the ESCWAO with the sliding mode controller \(\omega_{h}=1.1,\omega_{l}=0.4,\omega=1,k=0.6,\lambda_{2}=0.2\), \(\gamma_{2}=6,g(a)=a,a(0)=4\), \(y=-100\mathcal{C}_{d}^{\,2}\), and \(\hat{x}_{d}=\hat{\theta}(0)=17\). In our case, the parameters are set \(\omega_{h_{1}}=3.27,\omega_{l_{1}}=0.5,\omega_{1}=1,k_{1}=0.7,\lambda_{2_{1}} =0.5,\gamma_{2_{1}}=6,g(a_{1})=a_{1},a_{1}(0)=4\), \(y_{1}=-700\mathcal{C}_{d}^{\,2}\), and \(\Delta_{0,1}(0)=\hat{\theta}_{1}(0)=17\). Following the same energy calculations for a  
Figure 7: The position errors \(e_{pi}=p_{i-1}-p_{i}-\Delta_{i}\) illustrated in blue solid lines along with the corresponding performance bounds depicted in red dashed lines. The zoomed plots clarify the transition from the transient to the steady state and the convergence to zero.  
simulation time \(t=800\) s, it is clear that energy demanding in both cases is close, although in SMC, implementation is slightly less. The important fact is that the computational time in our proposed scheme is almost half the time needed for the computational time of the SMC implementation. Hence, the algorithm proposed in this research provides a different aspect of minimizing the aerodynamic drag force while considering collision and connectivity issues without risking higher energy losses or higher hardware demands.