## 2 The Extended IDM
### Car-Following Model for CAVs  
In an intelligent network environment, information can be accurately transmitted between CAVs in real time via vehicle-to-vehicle (V2V) communication technology. If the considered vehicle and the front vehicle happens to be CAVs, the former runs in the CASC mode. If the vehicle considered is a CAV while the front vehicle is an RV, that is, the V2V communication conditions are not met, then the former runs in the Adaptive Cruise Control (ACC) mode. ACC mainly relies on onboard sensors to acquire information about front vehicles, so that the vehicle considered can be controlled by the model algorithm. Based on the function of ACC, CASC also has a connected driver assistant system, which can not only identify the driving state of front vehicles but also acquire the state information about other CAVs in the traffic flow through information technology. The above two scenarios are further discussed below.  
(1) RV followed by CAV  
When a CAV is following behind an RV, the CAV is considered to run in the ACC mode. In other words, the CAV can acquire the speed and acceleration information about the front RV via the onboard sensors. From Formula (1), therefore, the acceleration of the CAV can be expressed as:  
\[a_{n}(t)=a_{0}[1-\bigg{(}\frac{v_{n}(t)}{v_{0}}\bigg{)}^{4}-\bigg{(}\frac{s_{ n}^{*}(t)}{s_{n}(t)}\bigg{)}^{2}]+\mu a_{n-1}(t) \tag{5}\]  
\[s_{n}^{*}(t)=s_{0}+Tv_{n}(t)+\frac{v_{n}(t)\Delta v_{n}(t)}{2\sqrt{a_{0}b}} \tag{6}\]  
Combined with the parameter settings of RV and CASC, the characteristic parameters of the car-following behavior in the ACC mode are set as follows: \(a_{0}=2\) m/s\({}^{2}\), b = 2 m/s\({}^{2}\), \(v_{0}=10\) m/s, \(s_{0}=2\) m, \(T=2\) s, and \(\mu=0.16\)[9].  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Driver Type** & \(\tau_{n}\) & \(v_{0}\) \\ \hline Type I & 1.1 & 11 m/s \\ Type II & 0.9 & 13 m/s \\ Type III & 1 & 12 m/s \\ Type IV & 1.2 & 10 m/s \\ \hline \hline \end{tabular}
\end{table}
Table 2: Parameter settings for different driver types.  
(2) CAV followed by CAV  
When a CAV is following behind another CAV, the follower is considered to run in the CACC mode, and the state information about the front CAV can be acquired via the follower's onboard sensors. In addition, the vehicle considered can get access to data of multiple front vehicles in the fleet via the network [20]. From Formula (1), therefore, the acceleration of the follower CAV can be expressed as:  
\[a_{n}(t)=a_{0}[1-\left(\frac{v_{n}(t)}{v_{0}}\right)^{4}-\left(\frac{\tau_{n}s _{n}^{*}(t)}{\sum_{q=1}^{Q}\varphi_{q}s_{n+q-1}(t)}\right)^{2}]+\mu\sum_{q=1}^ {Q}\beta_{q}a_{n+q-1}(t) \tag{7}\]  
\[s_{n}^{*}(t)=s_{0}+Tv_{n}(t)+\frac{v_{n}(t)\left(\sum_{q=1}^{Q}\alpha_{q} \Delta v_{n+q-1}(t)\right)}{2\sqrt{a_{0}b}} \tag{8}\]  
According to existing research [21], the parameters that can more accurately describe the car-following characteristics in the CACC mode are set as follows: \(a_{0}=2\) m/s\({}^{2}\), \(b=2\) m/s\({}^{2}\), \(v_{0}=10\) m/s\({}^{2}\), \(s_{0}=2\) m, \(T=2\) s, and \(\tau_{n}=1\). \(\varphi_{q}\), \(\alpha_{q}\), and \(\beta_{q}\) are the weight factors in the model, with \(\varphi_{q}\geq 0\), \(\alpha_{q}\geq 0\), \(\beta_{q}\geq 0\), and \(\sum\limits_{q=1}^{Q}\varphi_{q}=1\), \(\sum\limits_{q=1}^{Q}\alpha_{q}=1\), \(\sum\limits_{q=1}^{Q}\beta_{q}=1\). The weights \(\varphi_{q}\), \(\alpha_{q}\), and \(\beta_{q}\) are assigned by the following formula [22]:  
\[\varphi_{q},\alpha_{q},\beta_{q}=\left\{\begin{array}{ll}\frac{Q-1}{Q^{n}},q \
eq Q\\ \frac{1}{Q^{n-1}},q=Q\end{array}\right. \tag{9}\]