# Research on a Collaborative Control Strategy of an Urban Expressway Merging Bottleneck Area
## 4 An Illustrative Example
### Parameter Calibration  
According to the above, the parameters to be calibrated in the improved CTM are the free flow speed and capacity of the mainline and ramp cells, the congestion wave propagation speed, and the decline rate of the merging cell's capacity. Based on the detector data of each segment uncontrolled output by VISSIM (sampling interval 6 s, \(\Delta t\leq\min\left(l_{i}/v_{f}\right)\)), GA is used as the optimization algorithm, and the sum of squares of deviation between flow and speed is taken as the solution target to optimize the parameters of the improved CTM prediction model. The initial boundary values of each genetic variable parameter are shown in Table 2.  
After setting the initial range of genetic variables, python is used to realize the procedural expression of improved CTM and GA optimization solution. The parameter values of the genetic algorithm are shown in Table 3.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Variable Name** & **Meaning** & **Boundary** \\ \hline \(v_{f}\) & Mainline free flow speed & [65 km/h,75 km/h] \\ \(v_{f}^{\prime}\) & Ramp free flow speed & [45 km/h,55 km/h] \\ \(q_{m}\) & Mainline Capacity of single & [1800 pcu/h,2200 pcu/h] \\ \(q_{m}{}^{\prime}\) & Ramp capacity of single lane & [1600 pcu/h,2100 pcu/h] \\ \(w\) & Congested wave speed & [10 km/h,18 km/h] \\ \(\alpha\) & Capacity decline rate & [7\%,15\%] \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameter range of genetic variables.  
Figure 14: Simulation results. (**a**) Traffic flow output; (**b**) Traffic flow speed output.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Algorithm Parameters** & **Parameter Value** \\ \hline Population size \(P\) & 100 \\ Maximum number of iterations \(G\) & 200 \\ Crossover probability \(P_{x}\) & 0.7 \\ Mutation probability \(P_{var}\) & 0.1 \\ Fitness function Fitness 0 & \(\frac{1}{\min{I_{d}}}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Genetic algorithms’ population parameter value.  
After the parameter setting, the genetic algorithm optimization operation is performed, as shown in Figure 15 and Table 4, which is the iterative result of the improved CTM parameter calibration.  
Under the optimal parameters obtained by satisfying the maximum iterations, the temporal and spatial distribution of the average absolute percentage error between the speed-flow data output by the improved CTM and the detector data for the Cell1-Cell9 segments at each sampling interval is shown in Figure 16. From the results of error distribution, it can be seen that the error of the Cell1-Cell2 segment is at a good level (within 10%) in the whole prediction period. Due to the interference of the entrance ramp, the traffic conditions on the Cell3-Cell6 segment are more complicated, and the error is slightly higher than in other segments. The error increases greatly during the peak period, and the error value is up to 14%, but the error is at an acceptable level in most periods. The Cell7-Cell9 segment was at a good level during the flat peak period (within 10%), and its error values were within 10% for most of the peak period. In general, the mean absolute percentage error (MAPE) of the prediction results of the improved CTM in the whole prediction period is about 8.6%, and the prediction results of the traffic flow on each segment are within the acceptable range, and there is no obvious deviation, indicating that the improved CTM has high accuracy in simulating and predicting the expressway merging bottleneck.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & & & & & & \\ \hline \(v_{f}\) & \(v_{f}^{{}^{\prime}}\) & \(q_{m}\) & \(q_{m}^{{}^{\prime}}\) & \(w\) & \(w^{{}^{\prime}}\) & \(\rho_{m}\) & \(\alpha\) \\ (km/h) & (km/h) & (pcu/h) & (pcu/h) & (km/h) & (km/h) & (pcu/km.In) & \% \\ \hline
70.2 & 49.4 & 2095 & 1807 & 16.7 & 14.4 & 32.3 & 13.7 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Improved CTM parameter calibration results.  
Figure 15: Genetic algorithm iteration.