Computational Modelling of Road Traffic SS Computational Project by David Clarke Supervisor Mauro Ferreira - Merging Two Roads into One As economies grow around the world there is an ever-increasing level of traffic. Traffic jams are a drain on these economies and are damaging to the environment. To keep roads at maximum throughput we need a better understanding of traffic dynamics and how different factors affect the flux (flow) and capacity of a road. The corrected flux: Creating a New Analytical Model Considering a road of length L, with N cars and a minimum car length of d 0 and an average separation d. The density ρ is given by: The velocity v of each vehicle is given by the headway to the car in front. Choosing to use a constant temporal headway analogous to the “2-second-rule” in road safety, a constant temporal spacing t 0 is used. The maximum velocity v max is the speed limit. The flux Φ is given by: Fundamental Diagram, Flux vs Density, with d 0 =4m, vmax = 27.7ms -1 (100km/h), t 0 =2s. The above assumes all drivers are perfect, which they are not. The model needs a stochastic factor. Consider a car travelling a distance L, with a velocity v 0 in a time t. Now consider it stops for a time τ i during that distance. In sufficiently dense traffic any car ahead that stops will also cause the subject car to stop. The reduced velocity v’ is given by: Introducing γ, a dimensionless constant as a measure of imperfect driving on a given road, by a given set of drivers. Expanding the NaSch Cellular Automata Numerical Model The road is divided into discrete cell sites. Each cell is either empty or occupied by exactly one car. Position, velocity, acceleration and time are discrete integer variables. Each car’s velocity is updated by the following rules: Accelerate all cars by 1 up to the speed limit. Decelerate if too close to next car. Randomization by slowing by 1 with probability q. Move, update all positions in parallel. Configuration at time t (velocity of each car shown). Position-time graph for various densities, each dot represents a car. Dense areas represent jams. The jams propagate upstream. The effect of speed limit on flux The effect of randomness on flux (or driver behaviour) Black points were collected by counting loops on a Canadian highway. The red line is a fit of the above equation using d 0 =7.18m, t 0 =1s and γ=0.318 Fitting New Analytical Model to Real Data As a first step to extending the single road NaSch cellular automata model, I merged two roads into one. Like an on-ramp merging onto a motorway, there is a merging zone before road B comes to an end. Conclusions The Analytical model I created fits well to experimental data. The NaSch cellular automata Numerical model for which I wrote a programme, agrees with the analytical model and shows true traffic dynamics such as Phantom traffic jams forming from small perturbations. The NaSch model has been extended to simulate two roads merging into one. The Numerical Programme is very efficient, it can complete a 1 hour simulation of 1 million cars in 10min. computer time. The combination of the newly developed analytical model coupled with the expansion of the NaSch model is worthy of further development. Traffic jams can sometimes form for no apparent reason, known as “Phantom Traffic Jams”; they grow from small fluctuations in traffic behaviour rather than being triggered by large, exceptional events. These phantom jams propagate upstream, against the flow of the traffic. The Fundamental Diagram in traffic modelling is a plot of traffic flux versus local time-averaged density. Motivation and Introduction - Developing a Computer Programme Using just the NaSch rules from above, I wrote a programme in C. Cars are randomly positioned at points along the road and the simulation is run. Flux is measured by counting cars passing a point and Density is averaged over regular time intervals. This data is outputted to analyse the effects of changing parameters. To allow the cars to merge physically, I had to create a set of complex rules. I found the system to be very dependent upon these rules. Density School of Physics, Trinity College Dublin © School of Physics, Trinity College Dublin