Urban-Scale Macroscopic Fundamental Diagrams
Background
Recently, it has been shown experimentally the existence of urban-scale macroscopic fundamental diagrams, the MFD, (Daganzo and Geroliminis, 2008a), and that in simple cases it can be approximated analytically (Daganzo and Geroliminis, 2008b).
It is still not clear, however, how the MFD depends on the different parameters governing an urban area. This applet lets you study the MFD using simulation. In particular, you can see the effects of signal timing (cycle length, proportion of green, coordination), and network irregularities such as large areas restricted from traffic (large squares, malls, etc).
The network simulation model in the Applet
The network simulation model implements the exact solution of the kinematic wave model with triangular fundamental diagram using Cellular Automaton (Daganzo, 2006). For a single link it reads:
Where n is the vehicle number, j is dimensionless time measured in units of 1/(jam density * wave speed), z is dimensionless position measured in units of the jam spacing, d, and q is the free-flow to wave speed ratio.
The network is assumed to be of the Manhattan-type grid with all the links of equal length, but admitting irregularities (called “parks" hereafter); see figure 1. There is a traffic light at each intersection.
Assumptions:
- All streets are single-lane and traffic moves in the North-South and West-East directions.
- Turn probabilities: at each intersection a driver decides to make a turn with probability p, which is uniformly distributed between 0 and Pmax. The driver will keep going straight with probability 1-p.
- The offset of an intersection located at (row, column) = (i, j) is defined relative to the intersection in the upper-left corner (0,0) as:
O_{i,j} = (i + j)u,
Where u is the “unit offset" between two intersection and it can vary between 0 and the cycle time. This formulation for the offset generates “green bands” that propagate symmetrically in both directions.
Example experiments:
By default, the network is loaded with capacity inflow at its access links; the ones located in the left and top edges of the network. Average flows and density were measured every five minutes for the whole network and for both directions separately.
In the following examples we have varied one parameter at a time and the rest remaining constant as in table 1.
Figure 2: the MFD for the parameters in table 1. The free-flow branch was obtained by regulating the inflow to the network. The congested branch was obtained by regulating the outflow from the network (the right and bottom edges of the network). Notice how there is a considerable scatter and the congested branch, but not so in free-flow. Also, the scatter is even more significant when each direction is measured separately. That the entire network has less scatter than each individual direction means that for any given point in the congested branch in the east-west direction with low flow and density, the corresponding point in the north-south direction tends to have higher values of flow and density.
Figure 3: effect of the cycle length. In this experiment, the cycle time has been varied between 60 and 300 seconds. It can be seen that larger cycle times give lower network capacity. This is a consequence of having closely spaced intersections (L=200m). It was verified that for larger blocks capacity increases before decreasing in this experiment.
Figure 4: effect of the offset. The unit offset between two intersections was varied between 0 and 60 seconds. It can be seen that (i) there is a large range of offset values (0<u<15) that give the same maximum capacity; (ii) a closed loop is formed as the offset is continuously increase; and (iii) there are a few critical values for the offset (u = 21, 22s) which makes the network unstable and prone to gridlock.
Figure 5: effect of irregularities. This figure shows the MFD in figure 2 (with no parks) superimposed with the MFD for the network in figure 1 (6 parks, 3 oriented vertically and 3 oriented horizontally). It is evident how network capacity decreases as a result of the presence of irregularities. In this case, the capacity reduction is approximately 20%, and we have observed that it is almost linearly related to the number of parks in the network.
References
N. Geroliminis and C.F. Daganzo, C.F. (2008a) Existence of urban-scale macroscopic fundamental diagrams: some experimental findings. Transportation Research Part B 42(9), 759-770.
C.F. Daganzo and Geroliminis, N. (2008b) An analytical approximation for the macroscopic fundamental diagram of urban traffic. Transportation Research Part B 42(9), 771-781.
C.F. Daganzo, (2007). Urban gridlock: macroscopic modeling and mitigation approaches, Transportation Research Part B 41 (1) (2007), pp. 49–62.
C.F. Daganzo, (2006). In traffic flow, cellular automata = kinematic waves. Transportation Research Part B, 40 (5)396-403.