Module 4 : Macroscopic And Mesoscopic Traffic Flow Modeling

Lecture 18 : Cell Transmission Models

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	Single source and sink CTM model Basic Premise The cell transmission model simulates traffic conditions by proposing to simulate the system with a time-scan strategy where current conditions are updated with every tick of a clock. The road section under consideration is divided into homogeneous sections called cells, numbered from $i$ = 1 to I. The lengths of the sections are set equal to the distances travelled in light traffic by a typical vehicle in one clock tick. Under light traffic condition, all the vehicles in a cell can be assumed to advance to the next with each clock tick. i.e, $$n_{i+1}(t+1) = n_{i}(t)$$ (1) where, $n_i(t)$ is the number of vehicles in cell $i$ at time $t$. However, equation 1 is not reasonable when flow exceeds the capacity. Hence a more robust set of flow advancement equations are presented in a later section. Flow Advancement Equations First, two constants associated with each cell are defined, they are: (i) $N_i(t)$ which is the maximum number of vehicles that can be present in cell $i$ at time $t$, it is the product of the cell's length and its jam density. (ii) $Q_i(t):$ is the maximum number of vehicles that can flow into cell $i$ when the clock advances from $t$ to $t+1$ (time interval $t$), it is the minimum of the capacity of cells from $i-1$ and $i$. It is called the capacity of cell $i$. It represents the maximum flow that can be transferred from $i-1$ to $i$. We allow these constants to vary with time to be able to model transient traffic incidents. Now the flow advancement equation can be written as, the cell occupancy at time $t+1$ equals its occupancy at time t, plus the inflow and minus the outflow; i.e., $$n_i(t+1) = n_i(t)+y_i(t) - y_{i+1}(t)$$ (2) where, $n_i(t+1)$ is the cell occupancy at time $t+1$, $n_i(t)$ the cell occupancy at time $t$, $y_i(t)$ is the inflow at time $t$, $y_{i+1}(t)$ is the outflow at time $t$. The flows are related to the current conditions at time $t$ as indicated below: $$y_i(t) = min~[n_{i-1}(t), Q_i(t), N_i(t) - n_i(t)]$$ (3) where, $n_{i-1}(t)$: is the number of vehicles in cell $i-1$ at time $t$, $Q_i(t)$: is the capacity flow into $i$ for time interval $t$, $N_i(t)$ - $n_i(t)$: is the amount of empty space in cell $i$ at time $t$. Figure 1: Flow advancement Boundary conditions Boundary conditions are specified by means of input and output cells. The output cell, a sink for all exiting traffic, should have infinite size ( $N_{I+1} = \infty$) and a suitable, possibly time-varying, capacity. Input flows can be modeled by a cell pair. A source cell numbered 00 with an infinite number of vehicles ( $n_{00}(O) = \infty$) that discharges into an empty gate cell 00 of infinite size, $N_{0}(t) = \infty$. The inflow capacity $Q_0(t)$ of the gate cell is set equal to the desired link input flow for time interval $t+1$.