MATH3001 Overview of Traffic Modelling Projects
I would like the introduction for all of the proposed projects to have a similar structure, described below.
• Motivation. You should start by explaining why using mathematics to study traffic flow is a useful thing to do. In order to do this, I encourage you to find relevant documents published by, for example, the government and Highways England that describe the current problems with the road network and how they plan to tackle them. You could include some examples of the kinds of projects they are undertaking and how they incorporate modern technology.
• Empirical data. The technology installed in our road networks provides a lot of data about traffic that can be used to inform modelling. You should describe the various ways in which traffic data is collected (inductance loops for example) and what it shows. In particular, you should describe the Fundamental Diagram and what stop- and-go waves are. You might also describe the various types of congestion that are observed on motorways.
• Overview of modelling. You should give a brief description of each of the three main types of mathematical models of traffic: continuum models, car-following models and cellular automata models. These descriptions should be concise and should not be overly detailed.
If you need to, you can use figures from other sources (e.g. by taking screen shots), but make sure that you reference the source in the caption. Also, any figures that you use should be referred to in the text and you should adequately describe what the figure shows. Make sure you label everything (axes etc.) clearly.
Below are brief descriptions of the main project topics and ideas for further research. For each I have indicated relevant modules, although they are not necessarily prerequisites for the various topics. More details of the projects are written up in the corresponding project documents.
1 Continuum models
This project concerns partial di↵erential equation (PDE) models of traffic that describe the average density and velocity of traffic in continuous space and time. It would help a little if you’ve done MATH2620 Fluid Dynamics 1, although it is not essential (but some experience with PDEs is essential) . You will start off by deriving an equation that describes the conservation of cars, and how the fundamental diagram can be used to solve this equation. You will then describe how solutions can be found by the method of characteristics, giving rise to expansion fans and shock waves. You should also discuss ‘higher order’ models that include an equation for acceleration. Suggested areas to continue your research include:
• Daganzo’s criticism of continuum models oftraffic — does information flow downstream faster than individual vehicles? The various attempts to fix the recognised issues.
• Characterisation of travelling waves.
• Numerical methods for hyperbolic PDEs
• Deriving macroscopic continuum models from microscopic car-following models.
2 Car-following models
This project concerns ordinary differential equation models of traffic that describe the indi- vidual positions and velocities of cars in continuous space and time. Ideally you will have taken MATH2391 Nonlinear Di↵erential Equations and it would also be helpful if you are taking MATH3397 Nonlinear Dynamics. You will start by describing the general form of car-following models, how steady-state solutions relate headway and velocity and give some examples of such models. You should then describe platoon and string stability. Suggested areas to continue your research include:
• Derivation of string stability conditions for the general car-following model.
• Travelling waves in car-following models.
• Absolute and convective instabilities.
3 Cellular automata models
This project concerns stochastic cellular automata models of traffic that describe the in- dividual positions and velocities of cars in discrete space and time. Ideally you will have taken MATH2750 Introduction to Markov Processes. You will start by describing the Nagel- Schreckenberg model and the kinds of Fundamental Diagrams it can produce. You will con- sider simple cases where the Fundamental Diagram can be deduced and the site-orientated ‘mean field’ analysis. Suggested areas to continue your research include:
• The e↵ect of different update rules.
• The e↵ect of boundary conditions.
• Car Orientated Mean Field theories.
1 Continuum models of traffic flow
This project concerns partial diferential equation models of traffic that describe the average density ρ(x, t) and velocity v(x, t) of traffic in continuous space x and time t.
You should start of by reading about the Lighthill, Whitham and Richards (LWR) model [10, 9] . A very good book on this is Whitham’s Linear and Nonlinear Waves [11] (which can be found in pdf form online) . On a single lane road with no entrances or exits, the number of cars is conserved. You should be able to describe how this fact gives rise to the conservation law
where the subscripts refer to partial diferentiation. Describe how the Fundamental Diagram makes it possible to solve this equation, and how solutions can be drawn using the method of characteristics.
The LWR model captures many of the basic phenomena in traffic. However, it does not account for the generation or propagation of stop-and-go waves. You should illustrate this by sketching the solutions for the following initial condition:
where 0 < ρ0 < ρ1 and L > 0 are constants. This motivates “second order” models that include an equation for acceleration. Some examples have been proposed by Payne-Whitham (see Chapter 3.1 of Whitham’s book [11] and Kerner and Kohnh¨auser [6] . How and why do these difer to momentum equations in fluid dynamics? You should describe what the individual terms in such examples are meant to represent in real traffic.
Stop-and-go waves appear in these models at parameter values that are linearly unstable.
You should describe how to analyse the stability of second order models by deriving a dispersion relation that indicates which wave numbers are unstable (e.g. see Chapter 3.1 of Whitham’s book [11] .
The following subtopics are possible areas for you to explore further, but you are welcome to suggest alternative topics to me directly.
1.1 Daganzo’s criticism
Read Daganzo’s criticism of continuum models of traffic [3] and describe the problems that he presents. What is your take on these issues? These criticisms led to an efort to develop continuum models, in particular hyperbolic PDE models, that do not have the deficiencies described by Daganzo. How do you determine if a PDE is hyperbolic? Describe the fixes proposed by Aw and Rascle [1] . You will need to demonstrate how to work out the charac- teristic speeds for second order models, including some examples. This topic received a lot of attention in the academic literature [4, 13, 5], try to convey the essence of this discussion, but make sure you understand and include the mathematical elements.
1.2 Characterisation of travelling waves
Travelling waves are disturbances that propagate at a constant velocity c and with a fixed profile. Solutions of this type can be found by a change of variables z = x-ct, resulting in a system of ordinary diferential equations. What would such waves correspond to in traffic? For second order traffic models, the conservation law can be integrated directly and used to eliminate one variable in the acceleration equation. Try this out yourself. You can find help in Chapter 3.1 of Whitham’s book [11] and in Wilson and Berg [12] .
The goal of this subtopic is to explain Fig. 2 in Wilson and Berg [12] . The process to produce this figure is described in the paper but you will need to figure out the details, many of which are left out of the paper. You may find it helpful to discuss the various aspects of this figure separately and relate each region to the corresponding phase portrait. Taking Nonlinear Dynamics MATH3397 would significantly help with this subtopic.
1.3 Numerical methods for hyperbolic PDEs
An important class of traffic models are hyperbolic PDEs. It is particularly difficult to write good numerical solvers for such equations because solutions that start of smooth can develop discontinuities or shocks. You should describe how to approximate these equations numerically using finite diference and finite volume methods. You should discuss diferent numerical schemes, for example upwinding and the Godunov method. How and why do errors occur? An excellent book on this topic with detailed examples of traffic models is by Leveque [8] . Ideally you would code up your own numerical solvers and use them to illustrate the theory.
1.4 Deriving macroscopic continuum models from microscopic car- following models
How are macroscopic continuum models related to microscopic car-following models? In this subtopic you should describe various methods of deriving coarse-grained PDE models from microscopic car-following models. What approximations need to be made? In steady state conditions, or equivalently at equilibrium, density is the reciprocal of headway. Why isn’t this true out of equilibrium (i.e. when stop-and-go waves are present)? Two diferent approaches to coarse-graining are described in Berg, Mason and Woods [2], and Lee, Lee and Kim [7] . Try to work through these derivations (but a word of warning: in both papers some of the key steps are not particularly well described and may not even be correct) . Feel free to discuss any other approaches that you may find.
[1] A. Aw and M. Rascle. Resurrection of ”second order” models of traffic flow. Siam Journal of Applied Mathematics, 60(3):916–938, 2000 .
[2] P. Berg, A. Mason, and A. Woods. Continuum approach to car-following models. Phys.
Rev. E, 61(2):1056–1066, 2000 .
[3] C. Daganzo. Requiem for second-order fluid approximations to traffic flow. Transpora- tion Research Part B, 29(4):277–286, 1994 .
[4] J.M. Greenberg. Extensions and amplifications of a traffic model of aw and rascle. SIAM Journal of Applied Mathematics, 62(3):729–745, 2002 .
[5] D Helbing and AF Johansson. On the controversy around daganzos requiem for and aw- rascles resurrection of second-order traffic flow models. The European Physical Journal B, 69(4):549–562, 2009 .
[6] B. S. Kerner and P. Konh¨auser. Cluster e↵ect in initially homogeneous traffic flow.
Phys. Rev. E, 48(4):R2335–R2338, 1993 .
[7] H. K. Lee, H. W. Lee, and D. Kim. Macroscopic traffic models from microscopic car- following models. Phys. Rev. E, 64(5):056126, 2001 .
[8] R.J. LeVeque. Finite volume methods for hyprebolic problems. Cambridge University Press, 2002 .
[9] M. Lighthill and G. B. Whitham. On kinematic waves II. a theory of traffic on long crowded roads. Proc. Roy. Soc. London, Ser. A, 229:317–345, 1955 .
[10] P. Richards. Shockwaves on the highway. Operations Research, 4(1):42–51, 1956 .
[11] G.B. Whitham. Linear and Nonlinear Waves. John Wiley and Sons, New York, 1974 .
[12] R. E. Wilson and P. Berg. Existence and classification of travelling wave solutions to second order highway traffic models. In M. Fukui, Y. Sugiyama, M. Schreckenberg, and D.E. Wolf, editors, Traffic and Granular Flow ’01, pages 85–90 . Springer-Verlag, 2003 .
[13] H.M. Zhang. A non-equilibrium traffic model devoid of gas-like behaviour. Transporation Research Part B, 36(3):275–290, 2002 .
2 Car-following models
This project concerns ordinary diferential equation models of traffic that describe the indi- vidual positions xn and velocities vn of cars in continuous space x and time t.
Start by describing the general set-up of such models — good references for this are [11, 6, 10] . You should describe how the order of vehicles is represent by the subscripts n, what the headway hn is and how the acceleration of each vehicle is described by the general function f(h, h˙, v ) . Explain why the partial derivatives of f must have certain signs if we are to get sensible driver behaviour. You should describe how to write down a closed system of equations with equilibria that correspond to vehicles travelling with uniform velocities and headways. This is the general car-following model. How are the boundary conditions for a ring-road and a long straight road incorporated into the model? Give some examples of car-following models (e.g. the Optimal Velocity model [1] and the Intelligent Driver Model [7]) and compute what the corresponding partial derivatives of f are.
In a reasonable car-following model, drivers should be able to adjust their headway and velocity in response to the behaviour of the car in front without driving erratically. This notion is captured by the idea of “platoon stability”, see [4, 10] . Show that a car-following model that satisfies the constraints on the partial derivatives specified by Wilson [11] is platoon stable, meaning that small amplitude disturbances decay.
The following subtopics are possible areas for you to explore further, but you are welcome to suggest alternative topics to me directly.
2.1 String stability
Stop-and-go waves arise because successive vehicles break slightly harder than the vehicle in front. This growth of small disturbances as they propagate through the column of vehicles is known as “string instability” and is diferent to platoon instability.
Using Wilson’s paper [11], you should describe in detail how to analyse the linear stability of the general car-following model and the resulting condition for stability based on the partial derivatives of the model. This process assumes that the road is infinitely long. You could also consider the case where there are only a finite number of vehicles on a single lane ring-road. The method of doing this is described by Mason and Woods [5] and Gasser et al [3], but you should focus on the case where all drivers have the same behaviour and there are no time delays, and rewrite it in terms of the general car-following model. You should show that both methods result in the same dispersion relation, but the wave numbers in the finite sized case are discrete.
When do these two methods give diferent criteria for instability? You could try to compute the eigenvalues for finite sized case and compare with the dispersion relation for the infinitely long road case. You could also compute the stability criteria explicitly for the case where there are only 2 and 3 vehicles (not realistic, but illuminating!) . Note that the Jacobian matrix of the full system is not hyperbolic because
where L is the length of the road. You must use these facts to remove two of the variables and analyse the resulting hyperbolic system.
2.2 Travelling waves
Read the papers by Berg and Woods [2] and Ward and Wilson [9] . Describe how the analysis of travelling waves in the Optimal-Velocity model results in a second order delay diferential equation. Linearise this equation near to an equilibrium v* = V (h* ) and derive its characteristic equation. What is the corresponding characteristic equation for the general car-following model? You should consider how you might go about finding solutions to this equation numerically. One way to visualise them is to separate real and imaginary parts and plot the zero contours in the complex plane. Another is via brute force using a numerical root finder starting from lots of initial conditions (you could even code your own Newton-Raphson root finder) . What do the solutions to the characteristic equation tell us about the travelling wave solutions? For fixed model parameters, investigate what kinds of travelling wave solutions arise at diferent values of the up- and downstream headways h − and h+ respectively. This information can be captured in a “phase diagram” in which the boundaries between diferent regimes are plotted in (h − , h+ ) space. The case for the Optimal-Velocity at α = 2 is plotted in [2] . It would excellent if you could confirm some of these regions using your own simulations.
2.3 Absolute and convective instabilities
Read Treiber and Kesting [8] and Ward and Wilson [10] . What is the diference between absolute and convective instabilities? How can these diferent types of instability manifest themselves in traffic? Describe the notion of group velocity and how it might be used to distinguish between absolute and convective instabilities. Why doesn’t this work? Describe what the signal velocity is and how this can be used to find the edges of a growing wave packet in the frame of the vehicles using the method of steepest descents. How do you determine which point to deform the contour of integration through?
[1] M. Bando, K. Hasebe, A. Nakayama, A. Shibata, and Y. Sugiyama. Dynamical model of traffic congestion and numerical simulation. Phys. Rev. E, 51(2):1035–1042, 1995 .
[2] Peter Berg and Andrew Woods. Traveling waves in an optimal velocity model of freeway traffic. Physical Review E, 63(3):036107, 2001 .
[3] I. Gasser, G. Sirito, and Werner B. Bifurcation analysis of a class of ’car following’ traffic models. Physica D, 197(3-4):277–296, 2004 .
[4] Arne Kesting and Martin Treiber. How reaction time, update time, and adaptation time influence the stability of traffic flow. Computer-Aided Civil and Infrastructure Engineering, 23(2):125–137, 2008 .
[5] A. D. Mason and A. W. Woods. Car-following model of multispecies systems of road traffic. Physical Review E, 55(3):2203–2214, 1997 .
[6] G´abor Orosz, R Eddie Wilson, and G´abor St´ep´an. Traffic jams: dynamics and control. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical
and Engineering Sciences, 368(1928):4455–4479, 2010 .
[7] M. Treiber, A. Hennecke, and D. Helbing. Congested traffic states in empirical obser- vations and microscopic simulations. Phys. Rev. E, 62(2):1805–1824, 2000 .
[8] Martin Treiber and Arne Kesting. Evidence of convective instability in congested traffic flow: A systematic empirical and theoretical investigation. Transportation Research Part B: Methodological, 45(9):1362–1377, 2011 .
[9] J. A. Ward, R. E. Wilson, and Berg P. Multiscale analysis of a spatially heterogeneous microscopic traffic model. Physica D, 236:1–12, 2007 .
[10] Jonathan A Ward and R Eddie Wilson. Criteria for convective versus absolute string instability in car-following models. In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, page rspa20100437 . The Royal Soci- ety, 2011 .
[11] R. E. Wilson. Mechanisms for spatiotemporal pattern formation in highway traffic models. Philos. Transact. A Math. Phys. Eng. Sci., 366:2017–2032, 2008 .
3 Cellular automata models
This project concerns stochastic cellular automata models of traffic that describe the indi- vidual positions and velocities of cars in discrete space and time.
Start by describing the update rules for the Nagel-Schreckenberg (NaSch) model [1] .
You should try to write a program that implements the NaSch model — see the separate document on how to program the NaSch model. How do the parameters in the model a↵ect the dynamics? How can you measure the density and flow of traffic? What does the fundamental diagram look like? How does this change with different parameters?
The NaSch model can be analysed using the Master equation (also called the Chapman- Kolmogorov equation) . Describe what this equation represents. For the case where vmax = 1, derive the steady state Fundamental Diagram. What happens in the deterministic limiting cases where the random deceleration probability p is 0 or 1?
The following subtopics are possible areas for you to explore further, but you are welcome to suggest alternative topics to me directly.
3.1 Mean field approximations
Read the paper by Schreckenberg et al [4] . Work through the derivation of the mean field approximations for the Fundamental Diagram in the case where vmax > 1. Does it give a good approximation to that computed from simulations? Ideally you would compare the theory with your own simulations. You should also research attempts to get more accurate theories, for example cluster approximations [4, 2] and car-orientated mean field theory [3, 2] . What are Garden of Eden states?
3.2 Update rules
Typically the positions of vehicles in the NaSch model are updated in parallel, meaning that all vehicles move at the same time given the current state of traffic. There are other possible ways to update the system: random sequential with replacement, in which a vehicle is chosen at random from the population to update; random sequential without replacement, in which the order of vehicle updates is permuted; vehicles can be updated according to their position, either upstream or downstream. Why is the vmax = 1 steady state mean field Fundamental Diagram exact for random sequential updating?
3.3 Boundary conditions
Consider how the following two types of boundary conditions affect the dynamics observed: (i) periodic (i.e. a ring-road) where the first car follows the last; and (ii) long-straight road where the up- and downstream flows, α and β respectively, are specified. In the later case, how does the flow depend on α and β . Sketch a diagram for the various different regions in parameter space.
[1] K. Nagel and M. Schreckenberg. A cellular automaton model for freeway traffic. Journal de Physique, 12(2):2221–2229, 1992 .
[2] Andreas Schadschneider. The nagel-schreckenberg model revisited. The European Phys- ical Journal B-Condensed Matter and Complex Systems, 10(3):573–582, 1999 .
[3] Andreas Schadschneider and Michael Schreckenberg. Car-oriented mean-field theory for traffic flow models. Journal of Physics A: Mathematical and General, 30(4):L69, 1997 .
[4] Michael Schreckenberg, Andreas Schadschneider, Kai Nagel, and Nobuyasu Ito. Discrete stochastic models for traffic flow. Physical Review E, 51(4):2939, 1995 .