Chapter 3 A Path-base Assignment Model with
3.3 Estimation of Path Flow Considering Link Travel Time
3.3.2 Link Dynamics
Traffic simulation techniques appeared in the early 1950s in the field of transportation science. Computer-based traffic simulation tools, mostly developed in the past few decades, exist as a cost-effective assisting tool for researchers and practitioners to verify and evaluate traffic management strategies. Traffic simulation models can be characterized as microscopic, mesoscopic or macroscopic. The microscopic models simulate every vehicle in the network; mainly include three behaviors, accelerating, decelerating and lane changing. This kind of models, try to describe the actions and reactions of the vehicle that make up the traffic as accurately as possible, are the so-called car-following model. In order to achieve accuracy in modeling traffic, it leads to a simulation model with high degree of parameters (50 parameters is common). The simulation time heavily depends on the number of vehicle that exist simultaneously in the simulated network, that make it hard to meet the requirements of simulating large-scale congested traffic networks for ITS applications, especially at real-time level (Yang and Koutsopoulos, 1996).
The macroscopic approach, based on an analogy between traffic flow and a real fluid flow, is also called continuum traffic-flow model. These models mainly based on traffic density, volume and speed have been widely analyzed in the past (Lighthill and Witham, 1955; Payne, 1979; Leo and Pretty, 1990; Helbing, 1995). Macroscopic models usually involve partial differential equations defined on appropriate domains with boundary conditions describing traffic phenomena. The models present a higher level of abstraction than the microscopic model and lead to some computing advantages. The computing time required for a macroscopic model do not increase with the number of existing vehicle on the simulation network, and this advantage makes it easier to implement on a large-scale network.
Macroscopic traffic continuum models had been classified into first-order continuum models, such as Lighthill, Whitham and Richards’ well-known flow conservation model (LWR model), and high-order continuum models, such as Payne’s momentum conservation models (PW model). The models are composed of one or several partial differential equations (PDEs) defined on appropriate domains with initial and boundary conditions.. The LWR model consists of the fundamental conservation principle in the form of a PDE,
( ) ( )
g( )
x ttogether with standard definition of flux function
( ) ( ) ( )
k k x t u kq = , × (3.20)
It is assumed that the empirical u-k relationships follow the Greenshields traffic stream model
( )
⎟where uf denote the free flow speed and kb the density with vehicles bumper to bumper.
Although the LWR model is widely cited in researches, it is also known to have some deficiencies. The steady state velocity assumption, which means that velocity changes instantaneously as density change is certainly not valid in traffic flow. Payne used a motion equation to obtain a more complex equation to describe speed dynamics (Payne, 1979),
( ) (
P( )
k) (
u( )
k u)
38
where ue
( )
k is an equilibrium speed-density relation and τ is the relaxation time.Since it is difficult to find the analytical solution of the traffic continuum model, numerical methods, such as Lax or Upwind method, had been used by researchers to simulate the numerical solutions for traffic continuum model. Lax-F finite difference method is used in solution scheme.
Lax-F scheme transfer PDEs to finite difference equations by using centered difference skill. The Lax-F difference equation or PW continuum model can be written as,
(
nj)
After the computation of the Lax-F scheme, densities on each mesh can be obtained.
In this dissertation, a link travel time estimation method for the continuum traffic models proposed by Hwang and Cho is used (Hwang and Cho, 2006).
3.3.3 Solution Framework
A solution framework that combines the link dynamics to address travel time issue of the state space model is suggested. It first set the travel time on each link to be the free-flow travel time and generate a initial path-link incidence matrix, H . The t path-link incidence matrix is then used in the path flow estimation. After the estimation of path flows, the path flows are transformed into link flows and the link dynamics are incorporated to give a travel time estimation of certain link. The link
travel time is then used to generate the path-link incidence matrix for the next iteration. The algorithm terminates while the difference of path flows in consecutive iterations are less than ε . The solution framework is illustrated in figure 3.2.
Figure 3.2 The solution framework of time-invariant coefficient state space model considering link travel time
Initialize
Set travel time on each link to be the free-flow travel time
Generate H by t travel time on each link
Estimate path flow by state space model
Multiply path-flow with path-link incidence matrix
Setup adequate boundary condition for link dynamics
Solve link dynamics by finite difference scheme
Estimate travel time on each link Convergence
Output path flows H matrix t
Densities on each time-space mesh
YES
NO
Travel Time on Each Link
Boundary conditions of traffic model on each link Link flows
Path Flows
Link Dynamics
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Chapter 4
A Path-base Assignment Model with Time-varying Coefficient Dynamic System
Chapter 3 introduced the path-base assignment model with time-invariant coefficient dynamic systems. That model assumes a time-invariant relationship among path flows in different times. In reality, the characteristic of traffic is different from time to time, i.e., peak hours and non-peak hours, trips heading to work or going home. This chapter tries to relax such time-invariant assumption by introduce a time-varying coefficient state space model.