## 2 Truck Platooning Speed Control Based on Multi-Objective Optimization
### Model Construction Based on Model Predictive Control
#### 2.1.1 Improved Cell Transmission Model  
The CTM is a type of macroscopic traffic flow model that is used to describe changes in road traffic conditions. The model divides the road into several cells and predicts their densities to calculate the current input and output flow between cells, detect the flow of input, and update the subsequent state values, such as speed and density.  
A variable speed limit is typically used to optimize speed in the on- and off-ramp weaving sections of highways. Combined with the CTM, a variable speed limit was used to control the upstream bottleneck cell speed to relieve congestion. However, due to the general nature of variable speed limit control, it cannot be ensured that human drivers who enter its range will comply with speed limit requirements.  
Based on this idea, the speed control of truck platooning in the weaving section can affect the downstream traffic flow state where the traffic capacity decreases. In contrast to the variable speed limit strategy, truck platooning speed control yields a higher compliance rate. In addition, the control range of truck platooning varies depending on location.  
In this study, when the downstream cell density was predicted to exceed the critical density, truck platooning speed control was performed. Based on prior research [10], this study comprehensively considers the speed of lanes in the presence and absence of truck platooning.  
Assume that in the \(t\)-th time interval, the speed of truck platooning is controlled according to \(v_{p}(t)\), the speed of the corresponding cell is \(v_{p}^{\prime}(t)\), the corresponding density is \(\rho_{p}(t)\), and the critical density is \(\rho_{c}^{p}\). \(\omega\) indicates the propagation speed of congestion wave. As shown in the basic diagram, the critical density is calculated as follows:  
\[\rho_{c}^{p}=\frac{\omega*\rho_{jam}}{\omega+v_{p}^{\prime}(t)} \tag{1}\]  
The maximum capacity of a single lane is obtained by the following formula:  
\[Q_{m}^{p}={v_{p}}^{\prime}(t)*\rho_{c}^{p} \tag{2}\]  
The calculation formulas for demand flow \(D_{i}(t)\) and supply flow \(S_{i}(t)\) of the cell corresponding to truck platooning control are:  
\[D_{i}(t)=min\Big{\{}v_{i}(t)*\rho_{i}(t)*n_{i}*\Delta t,Q_{m}^{p}*n_{i}*\Delta t \Big{\}} \tag{3}\]  
\[S_{i+1}(t)=min\Big{\{}\big{(}\rho_{jam}-\rho_{i}(t)\big{)}*\omega*n_{i}*\Delta t,Q_{m}^{p}*n_{i}*\Delta t\Big{\}} \tag{4}\]  
where, \(v_{i}(t)\) is the density of cell \(i\) in the \(t\)-th interval. The respective adjusted formula for speed within the cell is, then:  
\[v_{i}(t)=\left\{\begin{array}{ll}min\Big{\{}v_{f},v_{p}^{\prime}(t)\Big{\}}& \rho_{i}(t)\leq\rho_{c}^{p}\\ \frac{\big{(}\rho_{jam}-\rho_{i}(t)\big{)}*\omega}{\rho_{i}(t)}&\rho_{i}(t)> \rho_{c}^{p}\end{array}\right. \tag{5}\]  
where, \(v_{f}\) represents free-flow velocity. This study predicts the control module, CTM state quantity (that is, whether the input quantity is the current cell density), \(\rho_{i}(t)\), cell speed \(v_{i}(t)\), inter-cell transmission flow \(q_{i}(t)\), input flow of the main lane and on-ramp, location of truck platooning, and truck platooning speed \(v_{p}(t)\). The predictive quantity--that is, the output quantity--comprises the density, speed of each cell, inter-cell flow, and location of truck platooning in the forecast period.