Module 4 : Macroscopic And Mesoscopic Traffic Flow Modeling

Lecture 17 : Traffic Flow Modeling Analogies

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	Analytical Solution Formulation The analytical solution, popularly called as LWR Model, is obtained by defining the relationship between the fundamental dependant traffic flow variable ( $k \,\, and \,\, q$) to the independent variable ( $x \,\, and \,\, t$). However, the solution to the continuity equation needs one more equation: by assuming $q = f(k)$ , ie., $q = k.v$. Therefore: $$\frac{\partial q}{\partial x} + \frac{\partial k}{\partial t} = 0, \,\, \mathrm{becomes}$$ $$\frac{\partial f(k)}{\partial k} + \frac{\partial k}{\partial t} =$$ 0 $$\frac{\partial k}{\partial t} + \frac{\partial (k.v)}{\partial x} =$$ 0 $$\frac{\partial k}{\partial t} + \frac{\partial [k.f(k)]}{\partial x} = 0, \,\,\,\, v = f(k)$$ Therefore, $$\frac{\partial [k.f(k)]}{\partial x} = \frac{\partial k}{\partial x} .f(k) + k. \frac{\partial f(k)}{\partial x}$$ $$= \frac{\partial k}{\partial x} f(k) + k. \frac{df}{dk} . \frac{\partial k}{\partial x}$$ $$= \frac{\partial k}{\partial x} \left[f(k) + k. \frac{df}{dk}\right]$$ Continuity equation can be written as $$\frac{\partial k}{\partial t} + \frac{\partial k}{\partial x}\left[f(k) +k. \frac{df}{dk}\right] = 0$$ where $f(k)$ could be any function relating density and speed. Eg: Assuming the Greenshield's linear model: $$v = v_f - \frac{v_f}{k_j}k$$ $$\mathrm{Therefore}, \,\, {}f(k) + k \frac{df(k)}{dk} = v_f - \frac{v_f}{k_j}k + k\frac{-v_f}{k_j}$$ $$= v_f - 2 \frac{v_f}{k_j}k$$ Therefore, $$\frac{\partial k}{\partial t}+ \frac{\partial k}{\partial x}\left[v_f - 2\frac{v_f}{k_j}k\right]= 0$$ (1) The equation 1 is first order quasi-linear, hyperbolic, partial differential equation (a special kind of wave equation).