Saturday, 1 December 2012

Macroscopic Model of Traffic

I work in the Intelligent Transport Systems (ITS) industry, at Transmax, and completed a PhD in applied mathematics at QUT. So it should come as no surprise that I want to combine and explore the two areas. To combine my two interests, I have investigated macroscopic models of traffic flow.

This post presents a simple model of traffic flow. It will form the baseline for further work and modifications. I learned of these equations from this paper by Paul Ross, published in 1988.

Model Equations

The first equation comes from the definitions of flow, density and speed:
$$q = \rho v,$$ where
\(q\) = traffic flow rate (veh/s) past the point;
\(\rho\) = vehicular density (veh/m); and
\(v\) = vehicle speed (m/s).

The above equation states that the number of vehicles that will pass by a point is equal to the density of traffic at that point times the speed of vehicles at that point. That is, flow rate increases with density and speed.

The second (partial differential) equation was given by Lighthill and Whitham in 1955. It describes the conservation of vehicles:
$$\frac{\partial \rho}{\partial t} + \frac{\partial q}{\partial x} = S(x,t),$$ where
\(\partial\) indicates partial differentiation;
\(t\) = time (s);
\(x\) = distance (m) along road; and
\(S\) = vehicles entering (+ve) or leaving (-ve) the road (veh/m/hr).

The third equation describes vehicle speed. The simplest relationship would have to be linear relationship between speed and density, as reported by Greenshields in 1934. This speed equation gives a free flow speed at zero density, and zero speed at the jam density:
$$v=v_f ( 1 - \rho/\rho_j )$$ where
\(v_f\) = free flow speed (m/s); and
\(\rho_j\) = jam density (veh/m), the average vehicles per metre in stationary traffic.

Let's say that the free flow speed is 100km/h. That means \(v_f = 27.78\) m/s. Also, let's assume that the average vehicle length is 5 metres, and that in a jam, they are spaced 2 metres apart. That means \(\rho_j = 1/7\) veh/m.

The equation for speed has an important effect on the traffic in the simulation, and has received a lot of attention in the literature. Various forms have been presented in the literature, and we will look at some of them in the future.

Boundary Conditions

A model with a partial differential equation is not complete without boundary conditions. There are a few options for us, but we must specify two boundary conditions - one for a point in space, and one for a point in time. First, some notation to help us specify the boundary conditions. We will refer to flow, density and speed as functions of two variables by writing \(q(x,t)\), \(\rho(x,t)\) and \(v(x,t)\).

Before we actually specify the boundary conditions, note that we have one partial differential equation, and two algebraic equations. With the two algebraic equations we can translate density to/from speed, and calculate flow from density or speed (but not the reverse - we'll see why below).

The time condition is usually chosen to be for time zero. We might choose that initially:
$$\rho(x,0) = 0$$ for all \(x\). In this model, it is equivalent to specifying \(v(x,0) = v_f\).

For this model, the space condition can be specified in terms of density or speed, as we can use the equations to determine the unspecified measures. Let's choose:
$$\rho(0,t) = \rho_j/4.$$ I don't think it's very important, but let's say that this condition overrides the time condition at \(x=t=0\).

In the future we may change the space condition to be a function of time, for example:
$$\rho(0,t) = \frac{(1+sin(2\pi t/T))}{2}\frac{\rho_j}{4}.$$

What's Next?

In another post we will analyse the relationship between speed, density and flow. After that we'll discretise the model equations so we can solve them numerically. And after that we may investigate the effect of different speed equations on model behaviour.

P.S. Thanks to MathJax and My technical memo for the script to render equations.

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