# A Microscopic Traffic Flow Model Characterization for Weather Conditions
## 4 Model Stability  
In this section, the stability of the ID and proposed models is evaluated for different values of the acceleration exponent and weather severity index, respectively, over a circular road of length 800 m for 180 s. The time headway for both models is 2 s [42], and the minimum and maximum acceleration are 0.73 m/s\({}^{2}\) and 1.67 m/s\({}^{2}\), respectively [20]. The length of a vehicle is 5 m [41], the distance headway is 25 m [40], and the jam spacing is 7 m [41]. For the ID model, the acceleration exponent ranges between 1 and \(\infty\), and is typically 4 [20]. Hence, this model is evaluated using \(\delta~{}=~{}1,~{}4\), and 20. The proposed model is evaluated using \(\alpha~{}=~{}0,~{}0.3,~{}0.55,~{}0.7,~{}0.8\), and 0.9, where 0 denotes normal weather and 0.9 denotes the most adverse weather. The initial equilibrium velocity is 15 m/s. A disturbance at the maximum acceleration 1.67 m/s\({}^{2}\) is introduced at 30 s. The parameters for the ID and proposed models are given in Table 1.  
The velocity of the ID and proposed models is shown in Figure 1. The red line indicates the trajectory of the 1 st vehicle, and the black lines indicate the trajectories of the following vehicles. For the ID model with \(\delta~{}=~{}1\), the disturbance at 30 s causes the velocity of the 1 st vehicle to decrease to 14.2 m/s. It then increases to 15.2 m/s at 47 s and fluctuates, as shown in Figure 1a. These fluctuations increase with time, and the velocity varies between 14.9 m/s and 15.2 m/s. With \(\delta~{}=~{}4\), at 30.5 s, the velocity of the 1 st vehicle decreases to 14.4 m/s. It then increases to 15.46 m/s at 40.5 s and fluctuates, as shown in Figure 1b. The fluctuations increase with time, and the velocity varies between 15.0 m/s and 15.6 m/s. With \(\delta~{}=~{}20\), the velocity of the 1 st vehicle decreases to 14.1 m/s at 30.5 s. It then increases to 15.2 m/s at 40.5 s and fluctuates, as shown in Figure 1c. The fluctuations increase with time, and the velocity varies between 14.43 m/s and 15.7 m/s.  
\begin{table}
\begin{tabular}{l l} \hline
**Parameter** & **Value** \\ \hline Initial equilibrium velocity, \(v\) & 15 m/s \\ Maximum severity index, \(\alpha_{m}\) & 1 \\ Distance headway, \(H\) & 25 m \\ Time headway, \(T\) & 2 s \\ Weather severity index, \(\alpha\) & 0, 0.3, 0.55, 0.7, 0.8, and 0.9 \\ Jam spacing, \(s_{j}\) & 7 m \\ Minimum acceleration, \(a\) & 0.73 m/s\({}^{2}\) \\ Maximum acceleration, \(b\) & 1.67 m/s\({}^{2}\) \\ Vehicle length, \(L\) & 5 m \\ Time step, \(\Delta t\) & 0.5 s \\ Acceleration exponent & 1, 4 and 20 \\ \hline \end{tabular}
\end{table}
Table 1: Stability evaluation parameters for the ID and proposed models.  
Figure 1: _Cont._  
14.9 m/s and 15.5 m/s. With \(\alpha~{}=~{}0.8\), the velocity of the 1 st vehicle decreases to 14.2 m/s at 30.5 s. It then increases to 15.2 m/s at 41.5 s and fluctuates, as shown in Figure 1h, varying between 14.9 m/s and 15.3 m/s. With \(\alpha~{}=~{}0.9\), the velocity of the 1 st vehicle decreases to 14.4 m/s at 30.5 s. It then increases to 15.4 m/s at 45 s and fluctuates, as shown in Figure 1i, varying between 15.1 m/s and 15.4 m/s.  
The results obtained indicate that for the ID model, an increase in \(\delta\) also increases the velocity fluctuations, as shown in Figure 1a-c. Thus, employing a constant \(\delta\) leads to instability. Conversely, for the proposed model, an increase in \(\alpha\) decreases the velocity fluctuations, as shown in Figure 1d-i. This indicates that the traffic behavior with the proposed model is stable, even during adverse weather.