Module 4 : Macroscopic And Mesoscopic Traffic Flow Modeling

Lecture 19 : Traffic Progression Models

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	Robertson's Platoon Dispersion Model The basic Robertson's recursive platoon dispersion model takes the following mathematical form $$q_t^d=F_n*q_{t-T}+(1-F_n)*q_{t-n}^d$$ (1) where, $q_t^d$= arrival flow rate at the downstream signal at time t, $q_{t-T}$= departure flow rate at the upstream signal at time t-T, T = minimum travel time on the link (measured in terms of unit steps T =$\beta T_a$), $T_a$ = average link travel time, n = modeling time step duration, $F_n$ is the smoothing factor given by: $$F_n=\frac{1}{1+\alpha_n\beta_n T_a}$$ (2) $\alpha_n$ = platoon dispersion factor (unit less) $\beta_n$ = travel time factor (unit less) Equation shows that the arrival flows in each time period at each intersection are dependent on the departure flows from other intersections. Note that the Robertson's platoon dispersion equation means that the traffic flow $q_{t}^d$, which arrives during a given time step at the downstream end of a link, is a weighted combination of the arrival pattern at the downstream end of the link during the previous time step $q_{t-n}^d$ and the departure pattern from the upstream traffic signal T seconds ago $q_{t-T}$. Fig. 1 gives the graphical representation of the model. It clearly shows that predicated flow rate at any time step is a linear combination of the original platoon flow rate in the corresponding time step (with a lag time of t) and the flow rate of the predicted platoon in the step immediately preceding it. Since the dispersion model gives the downstream flow at a given time interval, the model needs to be applied recursively to predict the flow. Seddon developed a numerical procedure for platoon dispersion. He rewrote Robertson's equation as, $$q_t^d=\sum_{i=T}^\infty{F_n(1-F_n)^{i-T}*q_{t-i+T}}$$ (3) Figure 1: Graphical Representation of Robertson's Platoon Dispersion Model This equation demonstrates that the downstream traffic flow computed using the Robertson's platoon dispersion model follows a shifted geometric series, which estimates the contribution of an upstream flow in the $(t-i)^{th}$ interval to the downstream flow in the $t^{th}$ interval. A successful application of Robertson's platoon dispersion model relies on the appropriate calibration of the model parameters. Research has shown that the travel-time factor $(\beta_n)$ is dependent on the platoon dispersion factor $(\alpha_n)$. Using the basic properties of the geometric distribution of Equation 1, the following equations have been derived for calibrating the parameters of the Robertson platoon dispersion model. $$\beta_{n}=\frac{1}{1+\alpha_n}~~~OR~~~\alpha_n=\frac{1-\beta_{n}}{\beta_{n}}$$ (4) Equation 4 demonstrates that the value of the travel time factor $(\beta)$ is dependent on the value of the platoon dispersion factor $(\alpha)$ and thus a value of 0.8 as assumed by Robertson results in inconsistencies in the formulation. Further, the model requires calibration of only one of them and the other factors can be obtained subsequently. $$\beta_{n}=\frac{2T_a+n-\sqrt{n^2+4\sigma^2}}{2T_a}$$ (5) where, $\sigma$ is the standard deviation of link travel times and $T_a$ is the average travel time between upstream and downstream intersections. Equation  demonstrates that travel time factor can be obtained knowing the average travel time, time step for modeling and standard deviation of the travel time on the road stretch. $$F_{n}=n\frac{\sqrt{n^2+4\sigma^2}-n}{2\sigma^2}$$ (6) Equation 6 further permits the calculation of the smoothing factor directly from the standard deviation of the link travel time and time step of modeling. Thus, both $\beta_n$ and $F_n$ can be mathematically determined as long as the average link travel time, time step for modeling and its standard deviation are given.