# A Macroscopic Traffic Model based on Driver Reaction and Traffic Stimuli
## 2 Traffic Flow Modeling  
[PERSON] and [PERSON] independently studied macroscopic traffic behavior and developed a similar model [18], which is known as the Payne-Whitham (PW) model. The first equation of this model is the same as the LWR model [9; 10], while the second equation characterizes vehicle acceleration. The PW model [29; 30; 31] for traffic is  
\[\frac{\partial\rho}{\partial t}+\frac{\partial(\rho v)}{\partial x}=0, \tag{2}\]  
\[\frac{\partial v}{\partial t}+v\frac{\partial v}{\partial x}+\frac{C_{0}^{2}} {\rho}\frac{\partial\rho}{\partial x}=\frac{v_{e}(\rho)-v}{\tau}. \tag{3}\]  
Driver spatial adjustment to forward conditions is characterized by the anticipation term \(\frac{C_{0}^{2}}{\rho}\frac{\partial\rho}{\partial x}\). Traffic alignment occurs during the relaxation time \(\tau\). During alignment, traffic achieves the equilibrium velocity \(v_{e}(\rho)\) based on the density distribution and is characterized by the relaxation term \(\frac{v_{e}(\rho)-v}{\tau}\). The constant \(C_{0}\) is the driver spatial density adjustment parameter. It is a nonnegative constant, which, in the literature, varies between 2.4 and 57 m/s [29; 32]. However, it cannot characterize variations in driver behavior and so can produce unrealistic results. The PW model anticipation term can create large changes in acceleration and deceleration at abrupt changes in density [12]. To solve this problem, a variable anticipation term can be employed, which is based on traffic parameters. In this paper, a new anticipation term is proposed for the PW model. Acceleration is given by  
\[a=\frac{v_{m}-v}{\tau}, \tag{4}\]  
where \(v_{m}\) is the maximum velocity. There are large vehicle interactions with a small \(\tau\) and quick alignment in traffic occurs. This term represents the reaction of a driver to the forward conditions. The Greenshields equilibrium velocity distribution [33] is considered here, which is given by  
\[v_{e}(\rho)=v_{m}\left(1-\frac{\rho}{\rho_{m}}\right), \tag{5}\]  
where \(\rho_{m}\) is the maximum density. This model is widely employed [34] and has been verified using data recorded in Yokohama, Japan [35], and San Francisco, CA [36]. This model is suitable for both free flow and congested traffic. The change in the equilibrium velocity is the stimulus for driver reaction and is given by  
\[v_{e}^{{}^{\prime}}(\rho)=\frac{d}{d\rho}\left(v_{m}\left(1-\frac{\rho}{\rho_ {m}}\right)\right)=-\frac{v_{m}}{\rho_{m}}. \tag{6}\]  
A driver is more sensitive in congested traffic as the distance headway \(h\) is small. During free flow traffic, the distance headway is large, which makes drivers less sensitive to traffic conditions. A driver covers the distance headway during the relaxation time \(\tau\) and the transition velocity is [29]  
\[v_{t}=-\frac{h}{\tau}. \tag{7}\]  
The negative sign shows that the velocity is a monotonically decreasing function of density [8]. As the density increases, the headway decreases so that  
\[h=\frac{1}{\rho}.\]  
A driver is more sensitive to a large transition velocity and vice versa. Substituting \(\tau\) from Equation (7) into Equation (4) gives  
\[a=-\frac{v_{t}(v_{m}-v)}{h}. \tag{8}\]The change in velocity is given by  
\[\Delta v=at, \tag{9}\]  
where \(t\) is the time during which acceleration or deceleration occurs.