Module 4 : Macroscopic And Mesoscopic Traffic Flow Modeling

Lecture 18 : Cell Transmission Models

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	Equivalence with Hydrodynamic theory Consider equations  & , they are discrete approximations to the hydrodynamic model with a density- flow (k-q) relationship in the shape of an isoscaled trapezoid, as in Fig.1. This relationship can be expressed as: $$q = min~[v_k, q_{max}, v(k_j - k)], for~0 \leq k \leq k_j,$$ (1) Flow conservation is given by, $$\frac{\partial q(x,t)}{\partial x} = \frac{\partial k(x,t)}{\partial t}$$ (2) To demonstrate the equivalence of the discrete and continuous approaches, the clock tick set to be equal to $\partial t$ and choose the unit of distance such that $v \partial t$ = 1. Then the cell length is 1, $v$ is also 1, and the following equivalences hold: $x \equiv i$, $k_j \equiv N$, $q_{max} \equiv Q$, and $k(x,t) \equiv n_i(t)$ with these conventions, it can be easily seen that the equations 1 &  are equivalent. Equation 3 can be equivalently written as: $$y_i(t) - y_{i+1}(t) = - n_i(t)+ n_{i+1}(t+1)$$ (3) This represents change in flow over space equal to change in occupancy over time. Rearranging terms of equation  we can arrive at equation , which is the same as the basic flow advancement equation of the cell transmission model. Figure 1: Flow-density relationship for the basic cell-transmission model