# A Digital Twin for Intelligent Transportation Systems in Interurban Scenarios
## 6 Performance Evaluation Results
### Results of Congestion Use Case Evaluation
#### 6.1.2 Re-Routing  
Re-routing is a last resort strategy, as the primary interest of the highway operator is to keep as many vehicles as possible on the highway. It is important to note that this re-routing strategy focuses exclusively on evaluating the reduction of internal congestion on the highway. The impact of diverted vehicles on external or alternative routes is not modeled within the digital twin. The objective is to measure how reducing the number of vehicles within the simulated highway segment can improve traffic performance metrics during an incident, independently of any downstream effects on the surrounding road network. The best percentage of re-routing is calculated using the fundamental traffic diagram. The fundamental traffic diagram, also referred to as the fundamental diagram of traffic flow, is a graphical representation used to depict the relationship between traffic flow, density, and speed on a roadway [33]. There is a unique relationship among these three macroscopic traffic parameters. Under uninterrupted conditions, these traffic variables are related by Equation (2) as follows:  
\[Flow[\mathrm{veh}/\mathrm{s}]=speed[\mathrm{m}/\mathrm{s}]*density[\mathrm{ veh}/\mathrm{m}] \tag{2}\]  
In Figure 7, a basic model of a fundamental traffic diagram is presented, where three traffic states can be distinguished. The first is free flow, where the vehicles can move at or near their maximum speed (\(\
u\)). The second is optimal flow, which occurs when the traffic volume is at its highest (\(k_{m}\)) without causing congestion. Lastly, the congested flow is where the traffic volume exceeds the road capacity, so the speed and flow decrease until reaching the jam density (\(k_{j}\)). Figure 7 displays three graphs: flow-density, speed-density, and speed-flow. The flow-density graph demonstrates that as the density  
Figure 6: Simulation results with incident A for the variable speed limit strategy applied to the incident and previous sections. increases, the flow initially increases, reaches a maximum at \(q_{m}\), and then decreases. The speed-density diagram illustrates how the speed decreases as the density increases. Finally, the speed-flow graph shows that when the flow is low, there are two possibilities: either the speed is high due to a low number of vehicles on the road, or the speed is low due to a high number of vehicles and the road is congested. Between these two possibilities, the optimal flow is reached at \(q_{m}\) with speed \(v_{m}\). The fundamental traffic diagram has been replicated using the highway scenario from this paper. Figure 8 illustrates the graphs obtained using data from section S_6 (cf. Figure 4) in a scenario without incidents. To increase traffic density, the scaling factor was adjusted. Rather than displaying the density, the graphs in this paper depict the number of vehicles entering the section, i.e., the input count. This modification enables the calculation of the percentage of vehicles that need to exit the highway to achieve a specific average speed within the section. Upon examining the graphs, it is evident that the results exhibit only the free flow state, due to simulation issues. As the scaling factor increases, the density (input count), flow, and speed change until the maximum capacity is reached, at which point the parameters stay constant. This is attributed to the fact that the results presented in the graphs are only of section S_6, so once the section's maximum capacity is reached, the change is reflected in the previous sections, not in section S_6. The bottom-left graph, which is the most crucial for this particular use case, demonstrates that as the density (input count) increases, meaning that the number of vehicles on the highway increases, the speed decreases. To determine an accurate re-routing percentage, it is essential that the fundamental traffic diagram reflects the real scenario, including the corresponding incident.