## 3 Methods
### System Architecture and Related Models
#### 3.1.2 Second-Order Vehicle Model  
Suppose that a larger number of traffic detectors on the traffic network are used to capture traffic information, including the vehicle position, speed, and acceleration of each sampling time. The state of vehicles will be stored in the roadside unit (RSU) and has a unique identification, \(j\), where \(j\) (\(j\subset\mathbb{M}\)) is the vehicle index. The state equation of each vehicle \(j\) can be expressed as follows:  
\[\begin{split} X_{j}(t)=[p_{j}(t),v_{j}(t),a_{j}(t)]^{T}\\ p_{j}(t)=v_{j}(t)\\ v_{j}(t)=a_{j}(t)\end{split} \tag{1}\]  
where \(p_{j}\in\mathcal{P}_{j}\), \(v_{j}\in\mathcal{V}_{j}\), \(a_{j}\in\mathcal{A}_{j}\), denoting the position, speed, and acceleration/deceleration of each vehicle in the traffic network, respectively; \(x_{j}(t)\) denotes the state of each vehicle at \(t\). Note that the parameters of each vehicle have boundary conditions, considering vehicles have limited driving force.  
\[\begin{split} p_{j}\in\mathcal{P}_{j}=\{p\in\mathbb{R}\}=\{x_{j},y_{j}\}\\ v_{j}\in\mathcal{V}_{j}=\{v\in\mathbb{R}\mid 0\leq v\leq 120\}\\ a_{j}\in\mathcal{A}_{j}=\{a\in\mathbb{R}\mid-4\leq a\leq 8\} \end{split} \tag{2}\]  
For different vehicles, the parameters in the second-order model are different. The vehicle is chosen as the implementation medium of the proposed strategy. And the vehicles on the different segments will be controlled at different speeds. Moreover, as part of the strategy implementation process, the segment speed is gradual, considering the acceleration/deceleration of the vehicle is limited.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Definition** & **Unit** \\ \hline \(i\) & The identification of the road segment & \(\backslash\) \\ \(v_{i}(t)\) & The speed of the \(i^{th}\) road segment at time \(t\) & km/h \\ \(k_{i}(t)\) & The density of the \(i^{th}\) road segment at time \(t\) & veh/km \\ \(q_{i}(t)\) & The traffic flow of the \(i^{th}\) road segment at time \(t\) & veh/h \\ \(L_{i}\) & The length of the \(i^{th}\) road segment & km \\ \(r_{i}(t)\) & The traffic flow of the on-ramp at time \(t\) & veh/h \\ \(s_{i}(t)\) & The traffic flow of the off-ramp at time \(t\) & veh/h \\ \(T\) & The length of the time horizon & s \\ \(\tau\) & Relaxation time & s \\ \(\xi\) & Numerical stability parameter & veh/km \\ \(\zeta\) & Numerical stability parameter & veh/km \\ \(l\) & Positive real parameter & \(\backslash\) \\ \(m\) & Positive real parameter & \(\backslash\) \\ \(v_{f}\) & Free flow speed of the road segment & km/h \\ \(k_{cr}\) & Critical density of the road segment & veh/km \\ \(k_{b}\) & Jam density of the road segment & veh/km \\ \(v_{m}\) & Critical speed of the road segment & km/h \\ \(v_{d}^{d}(t)\) & The desired speed of the \(i^{th}\) road segment at time \(t\) & km/h \\ \(k_{i}^{d}(t)\) & The desired density of the \(i^{th}\) road segment at time \(t\) & veh/km \\ \(q_{c}\) & The capacity of the road segment & veh/h \\ \(j\) & Identification of the detected vehicle & \(\backslash\) \\ \(X_{j}\) & The stored information of each vehicle in RSU & \(\backslash\) \\ \(p_{j}\) & The stored position information of each vehicle in RSU & \(\backslash\) \\ \(x_{j}\) & The stored lateral position of each vehicle & km \\ \(y_{j}\) & The stored longitudinal position of each vehicle & km \\ \(v_{j}\) & The stored speed information of each vehicle in RSU & km/h \\ \(a_{j}\) & The stored acceleration information of each vehicle in RSU & m/s\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Definitions of the modeling parameters.