Traffic Flow Modeling Analogies - Traffic Engineering and Management - Lecture Notes, Study notes of Software Project Management

Download Traffic Flow Modeling Analogies - Traffic Engineering and Management - Lecture Notes and more Study notes Software Project Management in PDF only on Docsity! Traffic Engineering And Management 17. Traffic Flow Modeling Analogies Chapter 17 Traffic Flow Modeling Analogies 17.1 Introduction If one looks into traffic flow from a very long distance, the flow of fairly heavy traffic appears like a stream of a fluid. Therefore, a macroscopic theory of traffic can be developed with the help of hydrodynamic theory of fluids by considering traffic as an effectively one-dimensional compressible fluid. The behaviour of individual vehicle is ignored and one is concerned only with the behaviour of sizable aggregate of vehicles. The earliest traffic flow models began by writing the balance equation to address vehicle number conservation on a road. Infact, all traffic flow models and theories must satisfy the law of conservation of the number of vehicles on the road. 17.2 Assumptions The traffic flow is similar to the flow of fluids and the traffic state is described based on speed, density and flow. However the traffic flow can be modelled as a one directional compressible fluid. The two important assumptions of this modelling approach are: • The traffic flow is conserved, or in other words vehicles are not created or destroyed. The continuity or conservation equation can be applied. • There is one to one relationship between speed and density as well as flow and density. The difficulty with this assumption is that although intuitively correct, in some cases this can lead to negative speed and density. Further, for a given density there exists many speed values are actually measured. These assumptions are valid only at equilibrium condition, that is, when the speed is a function of density. However, equilibrium can be rarely observed in practice and therefore hard to get Speed-density relationship. These are some of the limitations of continuous modelling. Dr. Tom V. Mathew, IIT Bombay 1 April 2, 2012 Traffic Engineering And Management 17. Traffic Flow Modeling Analogies The advantages of the continuous modelling are: • Better than input output models because flow and density are set as a function of time and distance. • Compressibility: ie., flow is assumed to be a function of density. • Solving the continuity equation (or flow conservation equation) and the state equation (speed-density and flow-density) are basic traffic flow equations (q = k.v). By using the equation that define q, k, and v at any location x and time t, we can evaluate the system using measures of effectiveness such as delays, travel time etc. 17.3 Formulation Assuming that the vehicles are flowing from left to right, the continuity equation can be written as ∂k(x, t) ∂t + ∂q(x, t) ∂x = 0 (17.1) where x denotes the spatial coordinate in the direction of traffic flow, t is the time, k is the density and q denotes the flow. However, one cannot get two unknowns, namely k(x, t) by and q(x, t) by solving one equation. One possible solution is to write two equations from two regimes of the flow, say before and after a bottleneck. In this system the flow rate before and after will be same, or k1v1 = k2v2 (17.2) From this the shockwave velocity can be derived as v(to)p = q2 − q1 k2 − k1 (17.3) This is normally referred to as Stock’s shockwave formula. An alternate possibility which Lighthill and Whitham adopted in their landmark study is to assume that the flow rate q is determined primarily by the local density k, so that flow q can be treated as a function of only density k. Therefore the number of unknown variables will be reduced to one. Essentially this assumption states that k(x,t) and q (x,t) are not independent of each other. Therefore the continuity equation takes the form ∂k(x, t) ∂t + ∂q(k(x, t)) ∂x = 0 (17.4) However, the functional relationship between flow q and density k cannot be calculated from fluid-dynamical theory. This has to be either taken as a phenomenological relation derived from Dr. Tom V. Mathew, IIT Bombay 2 April 2, 2012 Traffic Engineering And Management 17. Traffic Flow Modeling Analogies 17.5 Analytical Solution 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: ∂q ∂x + ∂k ∂t = 0, becomes ∂f(k) ∂k + ∂k ∂t = 0 ∂k ∂t + ∂(k.v) ∂x = 0 ∂k ∂t + ∂[k.f(k)] ∂x = 0, v = f(k) Therefore, ∂[k.f(k)] ∂x = ∂k ∂x .f(k) + k. ∂f(k) ∂x = ∂k ∂x f(k) + k. df dk . ∂k ∂x = ∂k ∂x [f(k) + k. df dk ] Continuity equation can be written as ∂k ∂t + ∂k ∂x [f(k) + k. df dk ] = 0 f(k) could be any function Eg: Assuming the Green shield equation: v = vf − vf kj k Therefore, [f(k) + k df(k) dk ] = vf − vf kj k + k( −vf kj ) = vf − 2 vf kj k Dr. Tom V. Mathew, IIT Bombay 5 April 2, 2012 Traffic Engineering And Management 17. Traffic Flow Modeling Analogies Therefore, [vf − 2 vf kj k] ∂k ∂x + ∂k ∂t = 0 (17.10) The equation 17.10 is first order quasi-linear, hyperbolic, partial differential equation (a special kind of wave equation). 17.6 Solution by method of Characteristics Consider k(x, t) at each point of x and t, and ∂k ∂t + ∂k ∂x [f(k) + df dk k] = 0 in the total derivative of k along a curve which has slope ∂x ∂t = f(k) + df dk k. ie., Along any curve in (x, t), consider x, k as function of t. x0 x (k) t Total derivative of k will be dk dt = ∂k ∂t + ∂k ∂x . dx dt = ∂k ∂t + [f(k) + df dk k]. ∂k ∂x At the solution, dk dt = 0 k is constant along the curve f(k) + k. df dk is constant along the curve. ie., x(t) = x0 + ( dx dt )t = x0 + [f(k) + k. dt dk ]t Note: Solution is to construct some curve e so that (a) kt + c(k).kx is the total derivative of k along the curve (ie., directional derivative) (b) slope of the curve dx dt = c(k) We know k(x, t). Therefore directional derivative k(x, t) along t Dr. Tom V. Mathew, IIT Bombay 6 April 2, 2012 Traffic Engineering And Management 17. Traffic Flow Modeling Analogies dk(x, t) dt = ∂k ∂t + dx dt . ∂k ∂x = ∂k ∂t + [f(k) + k. df dk ] ∂k ∂x = 0 ie., dk dt = 0 ie., k is constant along the curve e or dx dt = f(k) + k df dk is constant along curve e. Therefore e must be straight line x(t) = x0 + [f(k) + k. df dk ]t If k(x, 0) = k0 is initial condition x(t) = x0 + [f(k0) + k0. df dk ∣ ∣ ∣ ∣ k=k0 ]t This function is plotted below along with a fundamental q-k diagram. A wave velocity wave velocity B k = 0 kB kA kj k q = kv = k(vf − vf kj k) dq dk = vf − 2 vf kj k ∼= dxdt Dr. Tom V. Mathew, IIT Bombay 7 April 2, 2012