# Enhancing Freeway Traffic Capacity: The Impact of Autonomous Vehicle Platooning Intensity
## 4 Case Study
### Simulation Result  
The simulation result is shown in Figure 6. The capacity of mixed flow reaches the highest value, 3376 veh/h, when both MPR and speed reach their maximum. The impact of MPR on capacity intensifies with the increase in speed. When the speed limit is 110 km/h, the incremental percentage of capacity from 0%MPR to 100%MPR is 57.4%, while when the speed limit is 10, the capacity is slightly decreased. This is mostly because of the vehicle parameter settings of the simulation and the random uncertainties. For example, some vehicles are not always configured with minimum following distances and may undergo randomized slowdowns to improve the authenticity of the simulation. Another possible reason is the influence of the car-following model, which will be verified in the next section. Similarly, the impact of speed on capacity also intensifies with the increase in MPR. When the MPR is 100%, the incremental percentage of capacity from 10 km/h to 110 km/h is 245%, while when the MPR is 0, the incremental percentage of capacity reduces to 108%. Similar to the result of the numerical analysis, both the impacts are not linear. The slope of the speed-affected curve gradually decreases, whereas the slope of the MPR-affected curve gradually increases, revealing the different effects of speed and MPR on capacity. This can also be observed from the capacity in Equation (19). By deriving the derivatives of velocity and MPR separately, it can be seen that capacity is a concave function with respect to velocity and a convex function with respect to MPR. Consequently, MPR has the dominant impact on capacity compared to speed.  
The capacity simulation results, together with the formulated theoretical results, are shown in Figure 7. The red mesh represents the formulated theoretical capacity, whereas the green mesh represents the simulated results for capacity. The simulated result basically reflects the changes in the formulated theoretical capacity, although there are some discrepancies. The percentage of differences are also shown in Figure 7b. Except for very few spots, the majority of the grid has percentage differences below 10%. Samples with errors of less than 10% account for over 89.2% of the data, while samples with errors between 10% and 25% account for 8.3%. There are only three special values with larger errors, namely 24.4%, 35.56%, and 38.11%. The likely reason for the spots with larger errors is the influence of the selected car-following models. The simple car-following model used in this study was derived from a previous experiment conducted to represent the production ACC and the new CACC in microscopic simulations of their impacts on traffic flow dynamics. Its calculated value at low speed with a high penetration rate may have errors. Regarding small errors over large areas, the main reason for the error lies in the vehicle parameter settings of the simulation and the random uncertainties mentioned earlier. The theoretical model assumes perfect conditions, while the simulator considers the errors present in real situations, resulting in slightly lower values compared to the theoretical predictions. Considering the factors mentioned before, the simulated results are basically consistent with the theoretical values, which proves the rationality of the proposed theoretical model.  
Figure 6: (**a**) Change trends of capacity simulation results with different MPR and speed values. (**b**) Exact values of capacity simulation results with different MPR and speed values.  
Figure 7: (**a**) Comparison of the capacity between simulation results and theoretical results. (**b**) The percentage of capacity difference between simulation results and theoretical results.