# A New Driver Model Based on Driver Response
## 2 Traffic Flow Models  
The ID model provides a microscopic characterization of traffic based on forward vehicles [39]. In this model, acceleration is a function of driver response, the distance between vehicles, and the time required for vehicles to align to forward conditions. Driver response is based on the ratio of average velocity to maximum velocity. This ratio is nominally 1 for a smooth flow with low acceleration and deceleration. In this case, traffic velocity is near the maximum. Acceleration in the ID model is given by [15]  
\[\frac{dv}{dt}=a\Bigg{(}1-\bigg{(}\frac{v}{v_{max}}\bigg{)}^{\delta}-\bigg{(} \frac{D}{s}\bigg{)}^{2}\Bigg{)}, \tag{1}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Model** & **Mathematical Description** & **Interpretation** \\ \hline Intelligent Driver (ID) [15] & where & \begin{tabular}{c} A model that employs acceleration to characterize driver response. Driver behaviour is related to the velocity and distance headway of leading vehicles. It characterizes changes in traffic in an accident-free environment. However, a constant acceleration exponent is used for all traffic conditions. \\ \end{tabular} & \begin{tabular}{c} A model that maintains large distances between vehicles to characterize freeway traffic. It has been used for connected and autonomous vehicles. \\ \end{tabular} \\ \hline Revised ID [33] & \begin{tabular}{c} \(a=a_{max}\bigg{(}1-\bigg{(}\frac{v}{v_{max}}\bigg{)}^{\delta}-\bigg{(}\frac{D}{s }\bigg{)}^{2}\bigg{)}\) \\ where \\ \(D=s_{j}+s_{1}\sqrt{\frac{v}{v_{max}}}+c\frac{v^{2}}{b}+\max\Big{[}0,\,vT+\frac {v\Delta v}{2\sqrt{a_{max}b}}\Big{]}\) \\ \(c\) = safety impact factor, \\ \(s_{1}\) = non-linear jam spacing, and \\ \(a_{max}\) = maximum acceleration \\ \end{tabular} & \begin{tabular}{c} A model that maintains large distances between vehicles to characterize freeway traffic. It has been used for connected and autonomous vehicles. \\ \end{tabular} \\ \hline Proposed & \
\begin{tabular}{c} \(\delta=\bigg{(}\big{(}1-\frac{h_{i}+T_{T}}{h_{N}}\big{)}T\bigg{)}_{r}+\bigg{(} \big{(}\frac{h_{i}+T_{T}}{h_{N}}\big{)}T\bigg{)}_{f}\) \\ \(\frac{dv}{dt}=a\bigg{(}1-\bigg{(}\frac{v}{v_{max}}\bigg{)}^{((1-\frac{h_{i}+T_{T}} {h_{N}})T)_{r}+(\big{(}\frac{h_{i}+T_{T}}{h_{N}}\big{)}T)_{f}}-\bigg{(}\frac{D }{s}\bigg{)}^{2}\bigg{)}\) \\ \(D=(s_{j}+Tv)\bigg{(}1-\bigg{(}\frac{v}{v_{max}}\bigg{)}^{((1-\frac{h_{i}+T_{T}} {h_{N}})T)_{r}+(\big{(}\frac{h_{i}+T_{T}}{h_{N}}\big{)}T)_{f}}\bigg{)}^{-1/2}\) \\ \end{tabular} &
\begin{tabular}{c} A model that improves on the ID model by incorporating both forward and rearward vehicle conditions. The acceleration exponent is a variable based on traffic conditions. \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._where \(a\) is the maximum acceleration, \(v_{max}\) is the maximum velocity, \(v\) is the average velocity, and \(\delta\) is the acceleration exponent. The term \(D\) is the distance headway during traffic alignment. During alignment, vehicles adjust their velocity according to forward conditions in order to achieve the equilibrium velocity. The distance headway can be expressed as [15]  
\[D=s_{j}+Tv+\frac{v\Delta v}{2\sqrt{ab}}, \tag{2}\]  
where \(s_{j}\) is the bumper-to-bumper distance between vehicles during congestion (zero velocity), \(T\) is the time headway, \(b\) is the minimum acceleration, \(\Delta v\) is the change in velocity required to align to a forward vehicle, and \(s\) is the bumper-to-bumper distance between vehicles as shown in Figure 1. The ID model employs (1) and (2) to characterize traffic flow based on the driver response and distance headway during transitions. In the ID model, driver response to a change in traffic conditions is characterized with the acceleration exponent \(\delta\). This exponent cannot characterize traffic behaviour in different conditions because it is a constant. It indicates that driver behaviour is the same for all conditions, which is unrealistic. Further, it is not based on traffic physics.