## 3 Methods
### System Architecture and Related Models
#### 3.1.3 Macro-Traffic Flow Model  
The traffic flow model can generally be divided into three types: macro, micro, and mesoscopic. The macro model focuses on the collective behavior of vehicles. The speed and density of the road segment are the research objects. The micro model aims to describe the maneuver of the individual vehicle in detail. The description range of mesoscopic traffic modeling lies between microscopic and macroscopic scales, making it suitable for medium-sized traffic networks. However, given the control objective based on VSL, the macro-traffic flow model in this paper is selected. In this paper, the macro-traffic model named MetaNet [34; 35] is explored and improved based on the characteristics of long road segments without on-ramps or off-ramps. Studies have shown that this model can accurately describe various traffic conditions, including the elimination of frequent events, occasional traffic congestion on roads with few lanes, and significant traffic inflow from ramps. Model-based control methods offer the advantages of reduced computational effort and lower cost. Therefore, based on this model, an extended macroscopic traffic flow model is proposed and will be used to verify the model's reliability on the simulation platform. Generally, a freeway can be divided into \(N\) segments, each with one on-ramp and one off-ramp, as shown in Figure 2. \(i^{th}\) (\(i\subset\mathbb{N}\)) denotes the identification of the road segments. \(v_{i}\), \(k_{i}\), and \(q_{i}\) represent the speed, density, and traffic flow of the \(i^{th}\) road segment, respectively. \(r_{i}\) and \(s_{i}\) represent the traffic flow of the on-ramp and off-ramp, respectively. \(L_{i}\) represents the segment length of \(i^{th}\). \(t\) represents time. The macro-traffic flow model can be expressed as follows:  
\[q_{i}(t)=k_{i}(t)v_{i}(t) \tag{3}\]  
\[k_{i}(t+1)=k_{i}(t)+\frac{T}{L_{i}}[q_{i-1}(t)-q_{i}(t)+r_{i}(t)-s_{i}(t)] \tag{4}\]  
\[v_{i}(t+1)= v_{i}(t)+\frac{T}{\tau}\{V_{e}[k_{i}(t)]-v_{i}(t)\} \tag{5}\] \[+\frac{T}{L_{i}}\frac{k_{i-1}(t)v_{i-1}(t)}{k_{i}(t)+\xi}(\sqrt{v _{i-1}(t)v_{i}(t)}-v_{i}(t))-\frac{\mu(t)\cdot T}{\tau\cdot L_{i}}\omega_{i}(t)\] (6) \[\mu(t)=\left\{ \begin{array}{cc}\mu_{1}\frac{\eta}{k_{\tau}-k_{ i+1}(t)+\sigma}&k_{i+1}(t)>k_{i}(t)\\ \mu_{2}&k_{i+1}(t)\leq k_{i}(t)\end{array} \right. \tag{7}\]  
\[V_{e}[k_{i}(t)]=v_{f}\Bigg{\{}1-\left(\frac{k_{i}(t)}{k_{\sigma}}\right)^{l} \Bigg{\}}^{m} \tag{8}\]  
More details can be found in [34; 35]. The proposed strategy is designed for scenarios involving long segments without on-ramps or off-ramps. Therefore, the formula can be replaced as follows:  
\[k_{i}(t+1)=k_{i}(t)+\frac{T}{L_{i}}[q_{i-1}(t)-q_{i}(t)] \tag{9}\]  
Figure 2: A road is divided into \(N\) segments. The models in previous studies reveal the relationship between density and traffic flow. Van Aerde is a classic traffic flow model with four parameters used to describe the characteristics of multiple structures [36]. There is sufficient freedom in the process of modeling. It is suitable for various traffic conditions, is simple in structure, and is easy to calibrate. The specific formula of this model is as follows:  
\[Van(k)=\frac{1}{c_{1}+\frac{c_{2}}{v_{f}-v}+c_{3}v} \tag{10}\]  
\[c_{1}=v_{f}\frac{2v_{m}-v_{f}}{k_{b}v_{m}^{2}} \tag{11}\]  
\[c_{2}=v_{f}\frac{(v_{f}-v_{m})^{2}}{k_{b}v_{m}^{2}} \tag{12}\]  
\[c_{3}=\frac{1}{q_{c}}-\frac{v_{f}}{k_{j}v_{m}^{2}} \tag{13}\]  
where \(k\) represents the traffic density; \(v\) represents the spatial average speed; \(v_{f}\) represents the free flow speed; \(q_{c}\) represents the capacity of the road segment; \(v_{m}\) represents critical speed; \(k_{b}\) represents jam density; \(c_{1}\), \(c_{2}\), and \(c_{3}\) represent the intermediate variables. Considering the fitting characteristics of traffic flow and density, the Van Aerde model can more accurately represent the relationship between speed and density. Therefore, Formula (8) will be replaced by the Van Aerde model. The time-discrete speed formula can be expressed as follows:  
\[