# Evaluating the Effect of Road Surface Potholes Using a Microscopic Traffic Model
## 3 Performance Evaluation
### Fundamental Diagrams  
Figure 2 presents the fundamental diagrams for the ID model with \(\delta=1.0,~{}4.0\), and 200. This shows that a larger \(\delta\) results in a greater traffic flow. The maximum flow, critical density at maximum flow, and critical velocity are given in Table 3. These results indicate that an increase in \(\delta\) decreases the critical density at maximum flow but the corresponding velocity increases. Figures 3-5 present the fundamental diagrams for the proposed model with small, medium, and large potholes, respectively. Table 4 gives the maximum flow, critical velocity, and critical density at maximum flow for aggressive, sluggish, and typical drivers and small, medium, and large potholes. This shows that with small potholes, the road capacity is greater with a typical driver as the flow is larger when the density is high as shown in Figure 3a. Furthermore, for all driver types the maximum flow and critical velocity increase as the pothole size increases as shown in Table 4. In addition, an increase in the pothole size results in a greater change in maximum flow with an aggressive driver compared to sluggish and typical drivers. Thus, with the proposed model the relationship between flow, density, and velocity is based on road surface conditions and so is more realistic than with the ID model.  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Value** \\ \hline Desired velocity, \(v_{D}\) & 33.3 m/s \\ Time headway for the ID and proposed model, \(T\) & 1.0 s \\ Proposed model distance headway, \(h\) & 21.0 m \\ Safe distance headway, \(h_{s}\) & 5.0 m \\ Reaction time of an aggressive driver, \(\tau\) & 0.5 s \\ Reaction time of a sluggish driver, \(\tau\) & 6.0 s \\ Reaction time of a typical driver, \(\tau\) & 3.0 s \\ Typical driver reaction time, \(\tau_{N}\) & 3.0 s \\ Jam spacing, \(J_{s}\) & 2.0 m \\ Maximum acceleration, \(a_{max}\) & 0.73 m/s\({}^{2}\) \\ Minimum acceleration or deceleration, \(b\) & 1.67 m/s\({}^{2}\) \\ Vehicle length, \(L\) & 5.0 m \\ Acceleration exponent, \(\delta\) & 1.0, 4.0, and 200 \\ Time step, \(\Delta t\) & 0.5 s \\ Maximum normalized density, \(d_{m}=\frac{1}{J_{s}}\) & 0.5 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Simulation parameters.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Pothole** & **Width, \(W\)** & **Depth, \(D\)** \\
**Size** & **(m)** & **(m)** \\ \hline Small & 0.7 & 0.1 \\ Medium & 1.7 & 0.2 \\ Large & 3.0 & 0.3 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Pothole sizes and the corresponding dimensions.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Acceleration Exponent (\(\delta\))** & **Maximum Flow** **(veh/s)** & **Critical Density** & **Critical Velocity** **(m/s)** \\ \hline
1.0 & 0.69 & 0.076 & 9.1 \\
4.0 & 0.86 & 0.050 & 17.1 \\
200 & 0.94 & 0.029 & 32.4 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Maximum flow, critical density, and velocity for the ID model.  
Figure 4: Fundamental diagrams for the proposed model with medium potholes (**a**) flow–density and (**b**) velocity–density.  
Figure 3: Fundamental diagrams for the proposed model with small potholes (**a**) flow–density and (**b**) velocity–density.  
Figure 2: Fundamental diagrams for the ID model (**a**) flow–density and (**b**) velocity–density.