# Research on a Collaborative Control Strategy of an Urban Expressway Merging Bottleneck Area
## 2 Improved CTM and Parameter Calibration, Validity Verification
### Objective Function  
The urban expressway control target is generally composed of efficiency and safety. For the traffic efficiency target, we use the comprehensive index of total travel time and total turnover as the efficiency control target function. The expression is as follows:  
\[\begin{array}{c}I_{1}=\sum\limits_{k=1}^{T}\Delta t\bigg{[}\sum\limits_{i=1} ^{N}\left(\rho_{i,k}\lambda_{i}l_{i}-\delta_{N}\rho_{i,k}v_{ijk}\lambda_{i}l_{ i}\right)\\ \qquad\qquad+\sum\limits_{i=1}^{M}\left(\omega_{i,k}-\delta_{M}\omega_{i,k}v_{ ijk}\right)\bigg{]}\end{array} \tag{14}\]  
where \(\rho_{i,k}\) represents the average density of segment \(i\) at the \(k\)th sampling period. \(\lambda_{i}\) represents the number of lanes on segment \(i\). \(l_{i}\) represents the length of segment \(i\). \(v_{ijk}\) represents the average speed of segment \(i\) in the \(k\)th sampling period. \(T\) represents the predicted number of time intervals. \(N\) represents the number of expressway segments. \(M\) represents the number of entrance ramps. \(\omega_{i,k}\) represents the number of queuing vehicles at the entrance ramp \(i\) at the \(k\)th sampling period. \(\delta_{N},\delta_{M}\) represents the dimensional balance coefficient, which balances the influence of each index on the objective function, is taken as \(1/v_{f}\). \(\Delta t\) represents a single sampling interval.  
Figure 4: Basic structure of MPC.  
Figure 5: Structure of traffic control system based on MPC.  
For safety, considering the complexity and uncertainty of traffic conflicts and accidents, the prediction of serious conflicts is selected as the objective, and the objective function expression is as follows [19]:  
\[J_{2}=\sum_{k=1}^{T}\sum_{i=1}^{N}\begin{bmatrix}591.634-14.872\rho_{i,k}+1.625 \Delta v_{i,m}+\\ 1.638 SVT+0.877 SVD\end{bmatrix} \tag{15}\]  
\[SVT=\left|v_{i,k+1}-v_{i,k}\right| \tag{16}\]  
\[SVD=\left|v_{i,k}-v_{i-1,k}\right| \tag{17}\]  
where \(\Delta v_{i,m}\) represents speed difference between mainline and ramp. \(SVT\) represents the speed difference between adjacent periods of the same segment of mainline, that is the time speed difference. \(SVD\) represents the speed difference between adjacent segments of the mainline in the same period, that space speed difference.  
Before determining the comprehensive control objective function, due to the dimensional difference between the efficiency objective and the safety objective, the value of each objective function needs to be dimensionless processing. The processing method adopts the (0,1) interval method. The expression is as follows:  
\[{{I_{i}}^{\prime}}=\frac{({{I_{i}}-{I_{i,\min}}})}{({{I_{i,\max}}-{I_{i,\min} }})} \tag{18}\]  
where \({{J_{i}}}\) represents a dimension value. \({{J_{i}}^{\prime}}\) represents a dimensionless value. \({{J_{i,\max}}}\) represents the maximum dimension value. \({{J_{i,\min}}}\) represents the dimension minimum dimension value.  
An objective function based on safety and efficiency is obtained:  
\[J={{J_{1}}^{\prime}}+{{J_{2}}^{\prime}} \tag{19}\]