Module 4 : Macroscopic And Mesoscopic Traffic Flow Modeling

Lecture 17 : Traffic Flow Modeling Analogies

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	Derivation of the Conservation equation Consider a unidirectional continuous road section with two counting station. Let $N_1$ : number of cars passing (1) in time $\Delta t$; $q_1$ : the flow; $N_2$ : number of cars passing (2) in time $\Delta t$; and $q_2$ : the flow; Assume $N_1 \, > \, N_2$, then queuing between (1) and (2) $$q_1 = \frac{N_1}{\Delta t}, ~ q_2 = \frac{N_2}{\Delta t}$$ $$\Delta q = q_2-q_1 = \frac{N_2 - N_1}{ \Delta t} = \frac{- \Delta N}{\Delta t}$$ Note that $q_2 < q_1$ and therefore $\Delta q$ is negative. Therefore, $$\Delta N = - \Delta q . \Delta t$$ (1) Similarly ($k_2 > k_1$), $$\Delta k=k_2 - k_1=\frac{N_1-N_2}{ \Delta x}=\frac{+ \Delta N}{ \Delta x},$$ Therefore $$\Delta N = \Delta k \Delta x$$ From the above two equations: $$\Delta k~\Delta x~+~\Delta q~\Delta t~=~0$$ Dividing by $\Delta t~\Delta x$ $$\frac{\Delta k}{\Delta t } + \frac{\Delta q}{\Delta x} = 0 \\$$ Assuming continuous medium (ie., taking limits) $\lim_{t \rightarrow 0}$ $$\frac{\partial q}{\partial x} + \frac{\partial k}{\partial t} = 0 \\$$ If sink or source is considered $$\frac{\partial q}{\partial x} + \frac{\partial k}{\partial t} = g(x,t)$$ where, $g(x,t)$ is the generation or dissipation term (Ramp on and off). Solution to the above was proposed by Lighthill and Whitham (1955) and Richard (1956) popularly knows as LWR Model.