# Modeling of Traffic Flows Sustainability on Highway Network Stretches
## 2 Materials and Methods  
The paper [56] presents a third-order differential equation written in operator form, where \(p=d/dt\), representing the differentiation operator:  
\[(T_{1}T_{2}T_{3})p^{3}+(T_{1}T_{2}+T_{1}T_{3}+T_{2}T_{3})p^{2}+(T_{1}+T_{2}+T_ {3}+K_{2}K_{3}T_{1})p+K_{2}K_{3}+1=(K_{1}K_{2}T_{3})p+K_{1}K_{2} \tag{1}\]  
After analyzing the experimental data, the paper [56] clarified the time constants and gain factors of the mathematical model's adequacy. The value of the time constant \(T_{1}\) is represented by an equation. It characterizes the driver's inertia, depending on traffic flow density and intensity values:  
\[T_{1}=\frac{t_{1}^{2}\cdot N\cdot l_{a}\cdot\sigma_{a}}{70\cdot L\cdot v},\,s \tag{2}\]  
where \(t_{1}\)--the driver's reaction time to changes in traffic situations can range from 0.6 to 1.4 s; \(N\)--vehicle quantity on the controlled highway stretch; \(l_{a}\)--vehicle length, m; \(\sigma_{a}\)--standard deviation of vehicle acceleration in traffic flow, m/s\({}^{2}\), a formula for calculation is submitted in the paper [56]; \(L\)--the typical length of a controlled highway stretch is about 1000 m; \(v\)--vehicle speed in traffic, m/s. The value of the time constant \(T_{2}\), which characterizes the vehicle inertia and is expressed in maneuverability, is represented by the formula,  
\[T_{2}=\frac{M\cdot l_{a}^{2}}{1000\cdot N_{e}\cdot t_{2}^{2}},s \tag{3}\]  
where \(M\)--vehicle weight, kg; \(N_{e}\)--rated engine power of the vehicle, W; \(t_{2}\)--time for a maneuver that the vehicle can use when changing the traffic situation in the controlled area, s.  
The value of the time constant \(T_{3}\), which characterizes the inertia in the traffic situation changing, is represented by the formula:  
\[T_{3}=\frac{n\cdot t_{2}^{2}}{5\cdot t_{3}},s \tag{4}\]  
where \(n\)--lanes quantity on the roadway; \(t_{3}\)--total delay time in passing route, s.  
The formula representing the gain coefficient, \(K_{1}\), which indicates how traffic density affects a driver's reaction time rate, is as follows:  
\[K_{1}=\frac{N\cdot l_{a}}{L}. \tag{5}\]  
Traffic density gradient:  
\[gradp=\frac{\sigma_{N}}{L^{2}},\;1/m^{2} \tag{6}\]  
where \(\sigma_{N}\)--the standard deviation of vehicle quantity in the traffic flow, the formula for calculation is presented in the paper [56]. By comparing Equations (5) and (6), it can be concluded that the coefficient is \(K_{1}\) proportional to the density gradient of the transport flow. The gain coefficient \(K_{2}\), which characterizes the rate of dynamic traffic flow impact on the traffic delay time and sustainability loss, is represented by the formula:  
\[K_{2}=\frac{\sigma_{a}\cdot N^{2}\cdot l_{a}^{3}}{v^{2}\cdot L^{2}} \tag{7}\]  
Traffic velocity gradient:  
\[gradv=\frac{\sigma_{a}}{v\cdot L},\;1/ms \tag{8}\]  
By comparing Equations (7) and (8), the \(K_{2}\) factor is proportional to the traffic velocity gradient. The formula that represents the gain coefficient, \(K_{3}\), which determines how much the delay time and sustainability are affected by changes in traffic situations during driving in the flow, is as follows:  
\[K_{3}=\frac{N\cdot l_{a}\cdot k^{2}\cdot s^{2}}{10\cdot L} \tag{9}\]  
where \(k\)--pedestrian crossings' quantity in the controlled stretch, units; \(s\)--traffic lights quantity in the controlled stretch, units. In differential Equation (1), the right-hand side describes the input signal as a disturbance that impacts the transport flow as a dynamic system. These are coefficient values \(K_{1}\) and \(K_{2}\) and the time constant \(T_{3}\). In addition, on the right side of the Equation is the first derivative of the product of the listed parameters. The sustainability of traffic flow is affected by more than just the levels of \(K_{1}\), \(K_{2}\), and \(T_{3}\)--it also depends on the rate at which they change over time. Therefore, it is possible to formulate a factor on which to perform modeling and studies of transport flow for sustainability.