# Evaluating the Effect of Road Surface Potholes Using a Microscopic Traffic Model
## 2 Model Formulation  
The Intelligent Driver (ID) model is a microscopic traffic model which uses a second-order differential equation to characterize vehicle dynamics in congested and free-flow conditions [23]. It considers forward vehicles and the driver response according to the relative velocity [25]. The driver response is determined by the ratio of average velocity to desired velocity. When this ratio is equal to 1, the flow is smooth, and acceleration and deceleration are small [37]. According to the ID model, acceleration is influenced by driver response, the ratio of desired distance headway to distance headway, and the velocity difference between vehicles. Acceleration with this model is given by [15]  
\[\dot{v}=a_{max}\bigg{(}1-\bigg{(}\frac{v}{v_{D}}\bigg{)}^{\delta}-\bigg{(}\frac{s ^{*}}{h}\bigg{)}^{2}\bigg{)} \tag{1}\]  
where \(a_{max}\) is the maximum acceleration; \(v\) and \(v_{D}\) are the average velocity and desired velocity, respectively; \(h\) is the distance headway; and \(\delta\) is the acceleration exponent. The desired distance headway, denoted as \(s^{*}\), is given by [15]  
\[s^{*}=J_{s}+Tv+\frac{v\Delta v}{2\sqrt{a_{max}b}} \tag{2}\]  
where \(J_{s}\) is the distance between vehicles during congestion, called the jam spacing, as shown in Figure 1; \(T\) is the time headway; \(\Delta v\) is the velocity difference between vehicles; and \(b\) is the deceleration (minimum acceleration). The ID model employs (1) and (2) to characterize traffic behavior. The acceleration exponent \(\delta\) in the ID model is a constant which can result in unrealistic traffic characterization as it is not based on traffic physics. Traffic behavior is affected by the environment, including road surface conditions. Therefore, a variable exponent is proposed to realistically characterize traffic flow based on potholes. The road surface irregularity can be expressed as [1; 38; 39]  
\[-\frac{D}{2}\bigg{(}1-\cos\frac{2\pi\theta}{W}\bigg{)} \tag{3}\]  
where \(D\) and \(W\) are the depth and width of the pothole, respectively; \(\theta\) represents a location within the pothole measured from the farthest point on the left; and \(D<0\) denotes a bulging road surface whereas \(D=0\) denotes a smooth road surface without any holes or cracks. It is presumed that potholes have a conical shape but with a smaller height. Thus, the size of a pothole can be approximated as [1]  
\[P_{s}=-\frac{1}{2}\pi W\sqrt{\frac{W^{2}}{4}+D^{2}} \tag{4}\]  
The impact of potholes on the traffic flow primarily depends on the distance headway. For a large distance headway, the distance between vehicles is large so drivers have more time to react to road surface irregularities. Thus, they can maneuver around potholes. Conversely, if the distance headway is small, the impact of these irregularities is more significant as drivers have less time to react. Thus, (4) can be rewritten as  
\[P_{s}=-\frac{1}{2}\pi W\bigg{(}\frac{\tau}{\tau_{N}}\bigg{)}\bigg{(}1-\frac{h }{h_{s}}\bigg{)}\sqrt{\frac{W^{2}}{4}+D^{2}} \tag{5}\]  
Figure 1: The traffic model parameters. where \(\tau\) is the reaction time of a driver; \(\tau_{N}\) is the typical reaction time; and \(h_{s}\) is the safe distance headway. Driver sensitivity is defined as \(\frac{\tau}{\tau_{N}}\). A driver is aggressive when \(\tau<\tau_{N}\), sluggish when \(\tau>\tau_{N}\), and typical when \(\tau=\tau_{N}\). Aggressive drivers have a short reaction time and often maintain their speed when potholes are present [1]. In contrast, sluggish drivers have a large reaction time and tend to slow when potholes are present regardless of their size. Substituting (5) for \(\delta\) in (1) gives the acceleration for the proposed model as  
\[\dot{v}=a_{max}\Bigg{(}1-\bigg{(}\frac{v}{v_{\text{D}}}\bigg{)}^{-\frac{1}{2} \pi W(\frac{\tau}{\tau_{N}})(1-\frac{h}{h_{s}})\sqrt{\frac{W^{2}}{4}+D^{2}}}- \bigg{(}\frac{s^{*}}{h}\bigg{)}^{2}\Bigg{)} \tag{6}\]  
This characterizes traffic based on pothole size and driver sensitivity and so is more accurate and realistic than with a fixed \(\delta\).