# A Microscopic Traffic Flow Model Characterization for Weather Conditions
## 3 Traffic Flow Models  
The ID model is a microscopic model that characterizes traffic based on forward vehicles [39]. Acceleration is based on driver response, distance between vehicles, and time headway. Driver response is defined as the ratio of average velocity to maximum velocity. For a smooth flow, this ratio is 1 and acceleration is low. Thus, acceleration is given by [20]  
\[\frac{dv}{dt}\ =\ a\bigg{(}1-\bigg{(}\frac{v}{v_{max}}\bigg{)}^{\delta}-\bigg{(} \frac{D}{s}\bigg{)}^{2}\bigg{)}, \tag{1}\]  
where \(a\) is the maximum acceleration; \(v\) and \(v_{max}\) are the average and maximum velocities, respectively; \(\delta\) is the acceleration exponent; \(s\) is the bumper-to-bumper distance between the vehicles; and \(D\) is the distance headway during alignment, which can be expressed as [11]  
\[D\ =\ s_{j}+Tv+\frac{v\Delta v}{2\sqrt{ab}} \tag{2}\]  
where \(s_{j}\) is the distance between vehicles during congestion, known as the jam spacing; \(T\) is the time headway; \(\Delta v\) is the change in velocity during alignment to forward vehicles; and \(b\) is the minimum acceleration. The ID model uses (1) and (2) to characterize traffic flow based on driver response and distance headway for traffic alignment during transitions. The ID model acceleration exponent \(\delta\) is a constant determined considering vehicle alignment during transitions. Thus, it is not based on traffic physics, which leads to unrealistic and inadequate traffic characterization. In particular, traffic behavior does not change according to the environment, including road surface conditions due to weather. Therefore, a variable exponent is proposed in this paper to accurately characterize traffic behavior based on these conditions. The time required for velocity alignment during a transition is the transition time headway \(T\), and the distance covered during this time is the transition distance headway \(H\). These headways affect driver response during transitions [40] and vary based on weather conditions. During adverse weather, the headway ratio is smaller than during normal weather. Thus, the acceleration exponent based on weather conditions can be characterized as  
\[\delta\ =\ \frac{H}{T}\bigg{(}1-\frac{\alpha}{\alpha_{m}}\bigg{)} \tag{3}\]  
where \(\alpha\) is the weather severity index and \(\alpha_{m}\) is the maximum severity index. When \(\alpha\ =\ 0\), the weather is considered normal and the road surface is clear, thus there is maximum friction between the road and tires. When \(\alpha\ =\ \alpha_{m}\), the friction between the road and tires is minimum due to adverse weather conditions, such as snow, compacted snow, and ice, which make roads slick. The proposed model is obtained by substituting (3) in (1), so that  
\[\frac{dv}{dt}\ =\ a\bigg{(}1-\bigg{(}\frac{v}{v_{max}}\bigg{)}^{\frac{H}{t}(1- \frac{\alpha}{\alpha_{m}})}-\bigg{(}\frac{D}{s}\bigg{)}^{2}\bigg{)} \tag{4}\]  
This model can characterize traffic during transitions based on the road surface conditions; hence, it is more accurate and realistic than with a constant exponent \(\delta\). The acceleration given by (4) varies according to the weather conditions, unlike the ID model. The density is given by \(\rho~{}=~{}\frac{1}{s_{e}}\)[41], where \(s_{e}\) is obtained by substituting (2) in (1). At equilibrium, \(\Delta v~{}=~{}0\), thus the distance headway for the ID model at equilibrium is  
\[s_{e}~{}=~{}(s_{j}+Tv)\Bigg{(}1-\bigg{(}\frac{v}{v_{max}}\bigg{)}^{\delta} \Bigg{)}^{-1/2} \tag{5}\]  
while for the proposed model it is  
\[s_{e}~{}=~{}(s_{j}+Tv)\Bigg{(}1-\bigg{(}\frac{v}{v_{max}}\bigg{)}^{\frac{H}{H}( 1-\frac{s_{e}}{2m})}\Bigg{)}^{-1/2} \tag{6}\]  
According to (5), the equilibrium distance headway between vehicles with the ID model is the same, regardless of the conditions. Conversely, (6) indicates that the distance headway with the proposed model varies according to the weather conditions.