# Enhancing Freeway Traffic Capacity: The Impact of Autonomous Vehicle Platooning Intensity
## 3 Numerical Analysis
### Numerical Analysis of Mixed Flow  
Using Equation (17) and the identified parameters, the maximum theoretical capacity of mixed flow with different degrees of platooning intensity can be further calculated, as shown in Figure 2a-e. It can be seen from Figure 2a that the maximum capacity is positively correlated with MPR, speed, and platooning intensity. The capacity reaches its highest value when both MPR and speed are at their maximum. This trend becomes even more pronounced in Figure 2b-e. It should be pointed out that the platooning intensity \(P_{CC}\) is assumed to have a fixed value in each situation. The different colors of mesh surfaces represent the changes in the theoretical capacity. Under the same MPR, the higher the platooning intensity, the higher the achievable capacity. Mesh surfaces with different values of \(P_{CC}\) shrink with the reduction in this value. This is because of the inherent relationship between \(P_{CC}\) and \(P_{C}\). When the MPR is less than 50%, \(P_{CC}\) can take any value in the range [0, 1] because any CAV can always find a CAV or HV to follow when there are fewer CAVs than HVs in a fleet. However, when the MPR exceeds 50%, the value of \(P_{CC}\) should be larger than \(2-1/P_{C}\) because there are not enough HVs for all the CAVs to choose to follow in this condition, indicating a minimum threshold for \(P_{CC}\). In Figure 2e, it can be observed that the theoretical capacity of mixed flow can reach 3673 veh/h when the market penetration rate reaches 100% and the speed reaches 110 km/h. This condition is considered the most favorable. The increase in capacity between the maximum \(P_{CC}\) and minimum \(P_{CC}\) is shown in Figure 3a; the percentage increment is also shown in Figure 3b. It can be seen that \(P_{CC}\) has a significant impact on capacity, especially when the MPR is approximately 50%. When the speed reaches its maximum value and the MPR is around 50%, the increment percentage reaches its peak at 7.52%. Consequently, the capacity reaches its theoretical maximum value when MPR approaches 100%. However, the capacity increment reaches its maximum value when the MPR approaches 50%. It should be mentioned that the numerical variation is not linear in Figure 3a, which is also reflected in Figure 3b. This is because of the insufficient precision of the selection of values. However, this error can be corrected by applying finer precision. As proven above, for a given \(P_{C}\), the desired value of \(P_{CC}\) can be calculated using Equation (10), and \(P(N_{CC})\) is a function of \(P_{C}\) and \(n\). We try to determine the relationship between \(P_{C}\), \(P_{CC}\), and \(n\) through numerical analysis. Table 3 shows the value of the weighted average of platoon intensity \(\sum P_{CC}\times P(N_{CC})\) with different values of MPR and \(n\). It can be seen that as \(n\) increases, the weighted average of platoon intensity \(\sum P_{CC}\times P(N_{CC})\) gradually approaches \(P_{C}\) and the term \(\sum\left(1-P_{CC}\right)\times P(N_{CC})\) gradually approaches \(1-P_{C}\). Consequently, Equation (17) can be reformulated as Equation (18) when \(n\) is relatively large. Equation (18) can also be rewritten as Equation (19). It should be pointed out that \(n\), which represents the total number of vehicles in the fleet, is only analyzed here as a theoretical value. Due to the complexity of the calculation, \(n\) is only calculated up to a value of 150. The value of \(n\) cannot be too large in real scenarios due to technical and personnel reasons. \begin{table}
\begin{tabular}{c c} \hline
**Parameters** & **Value** \\ \hline \(h_{buffer}\) & 0.9 m \\ \(h_{teror}\) & 0.1 m \\ \(h_{lead}\) & 4.5 m \\ \(\tau_{CC}\) & 0.8 s \\ \(\tau_{CH}\) & 1.2 s \\ \(\tau_{HC}\) & 1.5 s \\ length & 4.5 m \\ \hline \end{tabular}
\end{table} 
Table 2: Parameters used in the study.