# Research on a Collaborative Control Strategy of an Urban Expressway Merging Bottleneck Area
## 2 Improved CTM and Parameter Calibration, Validity Verification  
### Cell Space Division Improvement  
In the basic CTM, the road is divided into several equidistant cells, and the research time is also divided into equal time steps. The cell length is fixed and not less than the single-time step vehicle driving distance in the free flow state. However, the basic CTM has defects in cell length division, basic diagram relation of merging cells, and cell description of the on-ramp. In this paper, the spatial division method in the basic CTM is improved, and the cell state representation variable is changed, so the density is used to describe the cell traffic state so that the cell length in the model can be changed according to the actual conditions of the expressway. The variable-length CTM is shown in Figure 1.  
In the variable-length CTM:  
\[\rho_{i,k+1}=\rho_{i,k}+\frac{\Delta t}{\lambda_{i}\cdot l_{i}}(q_{i,k}-q_{i+ 1,k}) \tag{1}\]  
where \(\rho_{i,k}\) represents average density of cell \(i\) at the beginning of \(k\)th time step (pcu/km/ln). \(q_{i,k}\) represents traffic flow into cell \(i\) in \(k\)th time step (pcu/h). \(\lambda_{i}\) represents number of lanes in cell \(i\). \(l_{i}\) represents length of cell \(i\). \(\Delta t\) represents time step, which should satisfy \(\Delta t\leq\min\left(l_{i}/v_{f}\right)\) in variable-length CTM. Formula (1) is a flow conservation equation that describes the cell state update with traffic flow density and traffic flow rate. At this time, the sending and receiving capabilities of the cell also use the sending rates \(s_{i,k}\)(pcu/h) and the receiving rate \(r_{i,k}\)(pcu/h) to replace the number of vehicles:  
\[s_{i,k}=\min\left\{\lambda_{i}v_{f}\rho_{i,k},\lambda_{i}q_{m}\right\} \tag{2}\]  
\[r_{i,k}=\min\left\{\lambda_{i}q_{m},\lambda_{i}w\big{(}\rho_{j}-\rho_{i,k} \big{)}\right\} \tag{3}\]  
where \(v_{f}\) represents the free-flow speed of cell (km/h). \(\rho_{j}\) represents traffic flow average congestion density in cell (pcu/km). \(q_{m}\) represents single lane capacity in cell (pcu/h). \(w\) represents the speed value of the blocking wave propagating backward along traffic flow under high density, \(w<v_{f}\). Formula (1) can be derived as:  
\[q_{i,k}=\min\left\{s_{i-1,k},r_{i,k}\right\} \tag{4}\]  
Figure 1: Variable-length CTM. where \(s_{i-1,k}\) represents sending ability of upstream cell \(i-1\). Formulas (1)-(4) are basic models described by traffic flow density and traffic flow rate instead of vehicles' numbers in the variable-length CTM. Transform the basic CTM:  
1. Merging cell \(i\) can receive the maximum number of upstream mainline vehicles and ramp cells can send. \[\left\{ \begin{aligned} q_{UL,k}&=s_{UL,k}\\ q_{M,k}&=s_{M,k}\\ q_{i,k}&=q_{UL,k}+q_{M,k}&=s_{UL,k}+s_{M,k }\end{aligned} \right.,\ r_{i,k}\geq s_{UL,k}+s_{M,k}\] (5)
2. Merging cell \(i\) cannot receive the maximum number of upstream mainline vehicles and ramp cells can send. \[\left\{ \begin{aligned} q_{UL,k}&=\text{ mid}\left\{s_{UL,k},r_{i,k}-s_{M,k},r_{i,k}\cdot(1-\theta)\right\}\\ q_{M,k}&=\text{mid}\left\{s_{M,k},r_{i,k}-s_{UL,k},r_{i,k} \cdot\theta\right\}\\ q_{i,k}&=q_{UL,k}+q_{M,k}=r_{i,k}\end{aligned}  \right.,\ r_{i,k}<s_{UL,k}+s_{M,k}\] (6) where \(q_{UL,k}\) represents the number of vehicles entering the upstream mainline. \(q_{M,k}\) represents the number of vehicles entering the upstream ramp. \(q_{i,k}\) represents the number of vehicles flowing into cell \(i\). \(s_{UL,k}\) represents sending ability of upstream mainline cell. \(s_{M,k}\) represents sending ability of upstream ramp cell. \(r_{i,k}\) represents receiving ability. Assuming that the number of vehicles flowing into the mainline cell L is \(q_{L,k}\), the number of vehicles flowing into the ramp cell is \(q_{M,k}\), and the diverging rate of the ramp cell is \(\beta\), the diverging model is transformed into:  
\[\left\{ \begin{aligned} q_{i+1,k}&=\min_{\left\{s_{i,k}, r_{L,k}/(1-\beta),r_{M,k}/\beta\right\}}\\ q_{L,k}&=q_{i+1,k}\cdot(1-\beta)\\ q_{M,k}&=q_{i+1,k}\cdot\beta\end{aligned} \right.