# A Two-Level Rolling Optimization Model for Real-time Adaptive Signal Control
## 3 Case Study and Discussion  
As shown in Figure 4, there is an actual road network in Chengdu, China, which has a typical grid structure. Geometric data were collected in the field to reflect real conditions and were further modeled into the microscopic simulation software Vissim [28]. Of the five intersections in total, one of those intersections, which is marked 5 in Figure 4, was chosen as the testing intersection for the proposed control system. The area surrounding the intersection is necessary for traffic flow prediction model [26] and to ensure realistic traffic flows. Full-actuated control was applied to all the other intersections. As a reasonable simplification, no right-turn traffic was modeled in this study, and only straight and left-turn traffic flow were modeled. The simulation pre-warm time was set at 900 secs, and the effective simulation time was 3600 secs. Different traffic volume levels were modeled in Vissim [28] for testing the compliance of the proposed control system with real traffic conditions. In addition, to analyze the sensitivity of the rolling time step, we use three rolling time steps: 2 secs, 4 secs, and 6 secs. The average delays of the two control methods were collected from simulation data and are plotted in Figure 5.  
As shown in Figure 5, the average vehicle delay in the two methods increased with the increase of traffic volume. However, compared with the COP algorithm, the proposed control method always had a lower average vehicle delay. The benefits are mainly due to the proposed method being based on a rolling strategy, which can capture the real-time chematistics of traffic flow. Therefore, the proposed method had a smaller vehicle delay. In addition, as shown in Figure 5, the smaller the time step of rolling optimization, the better the effect of the proposed control method. This showed that smaller rolling time steps can bring better performance. Next, the average vehicle delay of each phase was obtained as shown in Tables 2-4. Figure 3: Diagram of rolling horizon optimization strategy. \begin{table}
\begin{tabular}{l c c c c c c c c c c} \hline \hline
**Methods** & **Phase 1** & **Phase 2** & **Phase 3** & **Phase 4** & **Phase 5** & **Phase 6** & **Phase 7** & **Phase 8** & **Average** & **Improvement** \\ \hline COP & 35.01 & 24.95 & 44.64 & 30.74 & 28.76 & 38.24 & 40.73 & 29.01 & 34.01 & NA \\ Proposed-6 secs & 35.22 & 24.29 & 43.83 & 27.98 & 28.15 & 37.67 & 39.38 & 26.92 & 32.59 & -4.17\% \\ Proposed-4 secs & 31.98 & 23.41 & 43.11 & 27.63 & 25.29 & 36.79 & 36.40 & 26.18 & 31.35 & -7.82\% \\ Proposed-2 secs & 29.46 & 23.02 & 42.10 & 25.38 & 24.73 & 34.56 & 34.56 & 24.74 & 29.82 & -12.32\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The average vehicle delay of each phase (volume level is 3500 veh/h). Figure 4: Diagram of the simulated road network. Figure 5: The control performance versus volume for Controlled Optimization of Phases (COP) and the proposed method. \begin{table}
\begin{tabular}{l c c c c c c c c c c} \hline \hline
**Methods** & **Phase 1** & **Phase 2** & **Phase 3** & **Phase 4** & **Phase 5** & **Phase 6** & **Phase 7** & **Phase 8** & **Average** & **Improvement** \\ \hline COP & 35.01 & 24.95 & 44.64 & 30.74 & 28.76 & 38.24 & 40.73 & 29.01 & 34.01 & NA \\ Proposed-6 secs & 35.22 & 24.29 & 43.83 & 27.98 & 28.15 & 37.67 & 39.38 & 26.92 & 32.59 & -4.17\% \\ Proposed-4 secs & 31.98 & 23.41 & 43.11 & 27.63 & 25.29 & 36.79 & 36.40 & 26.18 & 31.35 & -7.82\% \\ Proposed-2 secs & 29.46 & 23.02 & 42.10 & 25.38 & 24.73 & 34.56 & 34.56 & 24.74 & 29.82 & -12.32\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The average vehicle delay of each phase (volume level is 3500 veh/h). As shown in Tables 2-4, the reduction of average delay is observed for all phases. For the studied intersection, the proposed control method reduced the average vehicle delay in each phase, compared with the COP algorithm. The results show that the proposed method was able to reduce both total vehicle delay of the intersection and for each phase.