# Modeling of Traffic Flows Sustainability on Highway Network Stretches
## 3 Results  
The study of the transport flow sustainability (robustness), RR, Equation (11), when driving along urban streets and highways, will be carried out by changing the differential Equation (1) right side. This parameter is shown as the density gradient product, Equation (6) and velocity gradient, Equation (8):  
\[gradp\cdot gradv=\frac{\sigma_{N}\cdot\sigma_{a}}{L^{3}\cdot p},\left[\frac{1 }{m^{3}\cdot s}\right] \tag{12}\]  
In addition, the input parameters on which the traffic flow robustness depends will be: vehicle length \(l_{a}\), m; vehicle weight \(M\), kg; vehicle engine power \(N_{e}\), kW; driver's reaction time to the traffic situation \(t_{1}\), s; vehicle maneuver time according to traffic situation changes \(t_{2}\), s; total delays time driving route \(t_{3}\), s; lanes quantity on the roadway \(n\), unit; pedestrian crossings \(k\) and traffic lights quantities \(s\) in the controlled stretch, unit. The next input parameter values (Table 1) were chosen during the simulation of traffic flow robustness. The sustainability of traffic flow depends on three indicator groups, which are displayed as parameters. During modeling, three levels of values were utilized for each parameter: the minimum, average, and maximum. \begin{table}
\begin{tabular}{l c c c c} \hline \hline \multirow{2}{*}{**Parameter for Modeling**} & \multirow{2}{*}{**Label**} & \multicolumn{3}{c}{**Value**} \\ \cline{3-5}  & & **Minimal** & **Average** & **Maximal** \\ \hline Vehicle Length, m & \(l_{a}\) & 4 & 12 & 20 \\ Vehicle weight, kg & \(M\) & 2000 & 10,000 & 18,000 \\ Vehicle engine power, kW & \(N_{e}\) & 100 & 150 & 200 \\ Driver’s reaction time to the traffic & \(t_{1}\) & 0.6 & 1.0 & 1.4 \\ situation, s & \(t_{2}\) & 4 & 7 & 10 \\ Vehicle maneuver time according to traffic situation changes, s & \(t_{3}\) & 50 & 150 & 250 \\ Lanes quantity on the roadway, unit & \(n\) & 1 & 2 & 4 \\ Pedestrian crossings and traffic lights & \(k\)/s & 2/2 & 5/5 & 8/8 \\ quantities in the controlled stretch, unit & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters Value for Transport Flow Robustness Simulation. For clarity displaying the modeling process conducted in the study, Figure 3 depicts the incoming and decision variables and obtained results as target functions. Figure 3 fully presents factors that influence the sustainability in the traffic flows model. Figure 4 presents the dependencies of traffic flow robustness across various density and speed gradients, with changes in vehicle lengths considered. If the robustness criterion is \(RR=1\), the traffic flow loses sustainability, i.e., traffic stops or a traffic jam is formed. According to a study [56], the presence of vehicle traffic in the flow can be identified by multiple values of the robustness criterion. When the value of the \(RR\) criterion is higher, it indicates that the margin for traffic robustness has increased. If the \(RR\) value is less than one, it indicates a traffic stop and the appearance of congestion. According to the analysis of the dependencies displayed in Figure 4, vehicle length is a significant factor that affects traffic flow robustness. In curve 3, when the vehicle length measures 20 m, sustainability loss occurs at low gradient values as indicated by the crossing with \(RR=1\) line. Conversely, with a vehicle length of \(l_{a}=4\) m, the sustainability margin is achieved at 40 units, as demonstrated in curve 1 for cars. Vehicle weight is the second most crucial factor that impacts traffic flow robustness. Figure 5 illustrates the variations in traffic flow sustainability under different density and velocity gradient scenarios while accounting for changes in vehicle weight. According to  
Figure 4: Dependencies of traffic flow robustness range for different values of density and speed gradients when changing vehicle lengths: 1—\(l_{a}=4\) m; 2—\(l_{a}=12\) m; 3—\(l_{a}=20\) m.  
Figure 3: Modeling Structure. displayed dependencies, increasing vehicle mass reduces the traffic flow robustness.