# Freeway Traffic Congestion Reduction and Environment Regulation via Model Predictive Control
## 3 Traffic Flow Models
### Integrating METANET Model with VT-micro Model  
In this paper, the VT-macro model proposed in [1], which integrated the METANET model and the VT-micro, is used to calculated total emissions and fuel consumptions.  
In order to balance the performances of computation time and accuracy, the VT-micro model proposed by the authors of [22] is used to estimate emissions and fuel consumption in this paper. Figure 2 describes the temporal and spatial traffic flow in the METANET model.  
The instantaneous speed of segment \(i\) at \(kT_{s}\) in the VT-micro model can be replaced by average speed \(v_{m,i}(k)\) in the METANET model. \(a^{t}_{m,i}(k)\) is defined as the temporal acceleration in the segment \(i\) of link \(m\) at time \(kT_{s}\), while \(a^{s}_{m,i,i+1}(k)\) is spatial acceleration of the vehicles leaving segment \(i\) to segment \(i+1\) of a link \(m\). The corresponding number of vehicles is \(n^{t}_{m,i}(k)\) and \(n^{s}_{m,i,i+1}(k)\). As for the on-ramp,  
Figure 2: Temporal and spatial traffic flow in METANET.  
its acceleration and number of vehicles are marked as \(a_{on,o}\) and \(n_{on,o}\). As shown in Figure 2. Following equations calculate the accelerations and the number of vehicles:  
\[a_{m,i}^{t}(k)=\frac{v_{m,i}(k+1)-v_{m,i}(k)}{T_{s}} \tag{10}\]  
\[a_{m,i,i+1}^{s}(k)=\frac{v_{m,i+1}(k+1)-v_{m,i}(k)}{T_{s}} \tag{11}\]  
\[n_{m,i}^{t}(k)=L_{m}\lambda_{m}\rho_{m,i}(k)-T_{s}q_{m,i}(k) \tag{12}\]  
\[n_{m,i,i+1}^{s}(k)=T_{s}q_{m,i}(k) \tag{13}\]  
\[a_{on,o}(k)=[v_{m,i}(k+1)-v_{on,o}(k)]/T_{s} \tag{14}\]  
\[n_{on,o}(k)=T_{s}q_{on,o}(k), \tag{15}\]  
where \(v_{on,o}\) denotes the average speed of the on-ramp \(o\), and \(q_{on,o}\) is the outflow of \(o\). It is noted that there is no \(v_{on,o}\) in the METANET model. Therefore, considering the regularity of traffic at peak time, this paper will estimate spatial acceleration of on-ramp based on history data [1]. Following equations calculate the emissions and fuel consumption with temporal acceleration:  
\[J_{y,m,i}^{t}(k)=T_{s}n_{m,i}^{t}(k)\exp(V_{m,i}(k)P_{y}A_{m,i}^{t}(k)) \tag{16}\]  
\[V_{m,i}(k)=[\begin{array}{cc}1&v_{m,i}(k)&(v_{m,i}(k))^{2}&(v_{m,i}(k))^{3} \end{array}] \tag{17}\]  
\[A_{m,i}^{t}(k)=[\begin{array}{cc}1&a_{m,i}^{t}(k)&(a_{m,i}^{t}(k))^{2}&(a_{ m,i}^{t}(k))^{3}\end{array}]^{\rm T}, \tag{18}\]  
where \(y\in\{CO,HC,NO_{x},Fuel consumption\}\). \(P_{y}\) is parameter matrix indexed from [1]. In this paper, the model proposed in [1] is used to consider the environmental effect, in which these four emission and fuel consumption factors are considered.  
VT-micro emission model does not mention the estimated value of \(CO_{2}\) emission rate. However, the authors of [1] show that there is an almost affine relationship between fuel consumption and \(CO_{2}\) emission. Then \(CO_{2}\) emission can be computed as follows:  
\[J_{\alpha,CO_{2}}(l)=\delta_{1}v_{\alpha}(l)+\delta_{2}J_{\alpha,fuel}(l), \tag{19}\]  
where \(J_{\alpha,CO_{2}}(l)\) and \(J_{\alpha,fuel}(l)\) denote the \(CO_{2}\) emission rate and fuel consumption rate of vehicle \(\alpha\) for time step \(l\) respectively, \(\delta_{1}\) and \(\delta_{2}\) are model parameters. Although \(CO_{2}\) is related to fuel consumption [23], the model considered in this paper is unfit for considering the environmental effect on the prospect of \(CO_{2}\). It may take many changes by considering it in the model, therefore, we will only address this in future study.  
It is the same to calculate emissions and fuel consumption with other two accelerations, named as \(J_{y,m,i}^{s}(k)\) and \(J_{on,o}(k)\). Thus, total emissions for various gases or fuel consumption at \(kT_{s}\) of the freeway \(m\) is described as follows:  
\[J_{y}=\sum_{k}\sum_{m}\sum_{i}\left(J_{y,m,i}^{t}(k)+J_{y,m,i}^{s}(k)\right)+ \sum_{k}\sum_{o\in O_{ramp}}J_{on,o}(k) \tag{20}\]