Traffic Simulation Techniques

In order to consider the interactions between different junctions in the network topol- ogy, we need to be able to transform the output vehicles flow from a junction into an input flow in the next junction, over time. This can be achieved using traffic simulation techniques.

From a traffic flow perspective, it is possible to classify traffic simulation techniques in three different distinct types: microscopic, mesoscopic and macroscopic. The differences between these models are related to the abstraction levels in which vehicle movement is modeled and if the set of vehicles is treated as an aggregated mass or not.

2.1. Theory of Urban Traffic Signal Control 38

(a) (b)

Figure 2.5: Detailed junction. Panel 2.5a, adapted from Maroto et al. (2006), shows

possible conversions, while Panel2.5b shows the corresponding graph.

sented individually. Each vehicle is considered as an agent and there is an equation describing each one of them.

• macroscopic approach: vehicle movement is traced implicitly and they are treated

as an aggregated mass. Similar to fluid dynamics (Richards, 1959), it is necessary

to determine the following variables: flux q(x, t), the number of vehicles flowing in distance x during time t; space mean speed v(x, t), corresponding to the instant average velocity in a distance x and time t and density k(x, t), corresponding to the number of vehicles is a space unit.

• mesoscopic approach: vehicle movement is traced explicitly and they are represented as an aggregated mass. A probability function f(t,x,v) is defined, which expresses the probability of one vehicle existing in time t, in position x and velocity v. One

way of obtaining this function is resolving Boltzmann equations (Kutz,2004).

Macroscopic and microscopic approaches are the most common ones used, with clear advantages and disadvantages, depending on the application.

The macroscopic approach has the following advantages:

2.1. Theory of Urban Traffic Signal Control 39 • Less requirements related to map details;

• Less computational time to calculate outputs; • Model calibration is easier.

The LWR macroscopic model described by Lighthill & Whitham (1955); Richards

(1959) was one of the first models to make it possible to simulate traffic in environments

with few map details (Kutz, 2004).

However, macroscopic approaches are incapable of modeling events such as conflicts between cars, traffic light effects, among others, which are sometimes necessary. Micro- scopic approaches are capable of modeling these microscopic events, requiring on the other hand more computational power, that might become prohibitive, depending on the size of the network, when all inputs are given.

In the domain of microscopic approaches, the use of cellular automata (Nagel &

Schreckenberg, 1992) makes it possible to reconstruct traffic situations by modeling the

interaction between running vehicles. Volume data and route distributions are sufficient to guarantee satisfactory results, even when there is just small real time traffic data (Wahle et al.,2002).

For instance, consider the following cellular automata, adapted from Knospe et al.

(2002), which represents a microscopic description of the vehicles movement using a set of rules for each step.

The microscopic model structures, represented in Figure 2.6, are the following:

• Node: represents an intersection between roads; • Arc: a road connecting to nodes.

• Lane: road segment where vehicles flow.

• Cell: discrete parts of one lane that are occupied by vehicles.

The urban network graph described in section 2.1.2 is transformed into an automata

2.1. Theory of Urban Traffic Signal Control 40

Figure 2.6: Microscopic model structures

Vehicles move advancing through the cells, changing lanes and arcs. In this model, cells have an extension of 7.5 m, which is the space occupied by an average commercial vehicle. Therefore, each cell can be occupied or free at any time. One cell may contain obstacles, such as traffic lights, which might block vehicle flows. A bigger vehicle, like a truck, might occupy more than one cell. Road maintenance can be represented by special vehicles occupying many cells with velocity equals to zero.

A sophisticated behavior can be achieved with one simple strategy combining three

simple rules, as represented in Figure 2.7 (Knospe et al., 2002):

1. Vehicles accelerate until maximum velocity, in the long run. Stopped vehicles have a smaller acceleration. The velocity of the first car in the arc is anticipated, permitting smaller gaps and greater velocities.

2. Vehicles anticipate the need to break by watching next vehicle’s brake lights; 3. Vehicles always adjust velocity in order to maintain a security distance.

Vehicle movement is realized through the following variables, parameters and rules, which are applied in parallel for each vehicle:

• Variables:

– x - position

– v - velocity

2.1. Theory of Urban Traffic Signal Control 41

Figure 2.7: Road segment in the microscopic model. Arcs are divided in 7.5 m wide cells. Each car has a discrete velocity between 0, . . . , V max. Vehicles represent ideal drivers, who follow security rules, never accelerating more than the gap to the next vehicle.

– d - gap distance

– def f - effective gap distance

– b - brake light status

– p - deceleration probability

• Parameters:

– gapsaf ety - effectiveness of anticipation control – vmax - maximum velocity

– pb - deceleration probability

– p0 - deceleration probability

– pd - deceleration probability

– th - gap time

– ts - security gap time

1. Acceleration rule

• bn (t + 1) = 0

• if ((bn+ 1 (t) = 0) and (bn (t) = 0)) or (th >= ts ) then: vn (t + 1) = min(vn

(t) + 1, vmax ).

2.1. Theory of Urban Traffic Signal Control 42

• vn (t + 1) = min(deffn , vn (t+1))

• if (vn (t + 1) < vn (t)) then: bn (t + 1) = 1. 3. Stochastic deceleration rule

• if (rand() < p) then: • vn (t + 1) = max(vn (t + 1) - 1, 0) • if (p = pb ) then: bn (t + 1) = 1. 4. Movement rule • xn (t + 1) = xn (t) + vn (t + 1). Where:

• deffpred = dpred + max(vanti - gapsafety , 0)

• th = dn /vn (t)

• ts = min(vn (t), h)

• p = p(vn (t), bn+1 (t), th , ts ) =

– pb, if bn+ 1 = 1 and th < ts

– po, if vn = 0 and not (bn+ 1 = 1 and th < ts )

– pd, elsewhere

According toKnospe et al. (2002), these parameters give a realistic result:

• vmax = 2 • pd = 0.1 • pb = 0.94 • po = 0.5 • h = 6