DYNAMICAL MODEL AND NUMERICAL SIMULATION OF TRAFFIC CONGESTION

Transcription

1 DYNAMICAL MODEL AND NUMERICAL SIMULATION OF TRAFFIC CONGESTION A Thesis Presented to the Faculty of California State Polytechnic University, Pomona In Partial Fulfillment Of the Requirements for the Degree Master of Science In Mathematics By Tiffany Ho 2014

2 SIGNATURE PAGE THESIS: DYNAMICAL MODEL AND NUMERICAL SIMULATION OF TRAFFIC CONGESTION AUTHOR: Tiffany Ho DATE SUBMITTED: Spring 2014 Department of Mathematics and Statistics Dr. Ryan Szypowski Thesis Committee Chair Mathematics & Statistics Dr. Jennifer Switkes Mathematics & Statistics Dr. Hubertus von Bremen Mathematics & Statistics ii

3 ACKNOWLEDGMENTS I would like to thank my friends and family who have given me words of encouragement and steered me away from the path of procrastination which has plagued my life ever since I can remember. Thanks to my boyfriend, Jayden, who has helped bring out the best in me and tolerated me throughout this ordeal. Additionally, I would like to thank Dr. Szypowski for helping sort through mathematical problems that I have run into over the past few months and dealing with my drafts that contained numerous grammatical errors. I would also like to thank my thesis committee members, Dr. Switkes and Dr. von Bremen, for being excellent professors who taught me in various classes throughout my undergraduate and graduate education. Lastly, I d like to thank the inventor of coffee, as it has helped me tremendously in my efforts to write this thesis. Without the help of everyone, I would not have made it this far and I cannot come close to expressing my gratitude. iii

4 ABSTRACT We will look at the dynamics of traffic congestion that is formed without any specific origin. This type of event can be viewed in metropolitan areas daily as highway traffic jams. We assume that these traffic congestions are formed from small disturbances in the flow of traffic. This will be modelled using a dynamical system where each differential equation represents a vehicle s behaviour. We will utilize Matlab s built-in numerical integration function ode45 which uses Runge- Kutta integration methods. Through our simulations, we are able to determine that with the addition of small perturbations to an equation in the steady-state flow solution, we will induce traffic congestion. iv

5 Contents Signature Page ii Acknowledgements iii Abstract iv List of Figures viii 1 Introduction 1 2 Background on Dynamical Systems One-Dimensional System Graphical Approach: Fixed Points and Stability Linearized Theory: Fixed Point and Stability Two-Dimensional Linear System Stability Analysis Rewriting Second-Order Ordinary Differential Equations Nonlinear Dynamical System Fixed Points and Linearization Dynamical Model 18 v

6 3.1 Dynamical Equation Stability of the System Analyzing the Linearized System when N = Stability in Linearized Theory Stability and Fourier Analysis Models and Numerical Simulation Simple Model Realistic Model Conclusion Further Research Bibiliography 47 A Main Traffic Code 49 B Simple Model Function Code 60 C Realistic Model Function Code 61 D Jacobian Code for 3x3 Case 62 E Solution to Complex Polynomial Code 63 F Velocity Graphs Code 67 vi

7 List of Figures 2.1 Vector field of one-dimensional system (2.4). [8] Stability of some of the different types of fixed points. [2] Stability criteria in the (f, α) polar coordinate plane of (a)numerical solution using Matlab and (b)analytical solution to equation (3.13). The blue line represents when u = 0, red is the circle when f = a 2, lastly green is the circle where f > a Graph of velocity function for simple model Comparison of simple model for stable and unstable case of the 50th vehicle. (a)velocity vs Time (b)distance vs. Time Simple, unstable model velocity snapshot at t= Fourier stability analysis of simple stable (a) and unstable (b) model. [4] Graph of velocity function for realistic model Realistic model snapshot of velocity vs. vehicle number at (a) t=100, (b) t=300, (c) t= Plot of the positions of all vehicles on the circuit with time development (x n, t) vii

8 4.8 Histogram displaying the first 100 time steps (blue), third 100 time steps (green), and the last 100 time steps (red) Bando s plot of the positions of all vehicles on the circuit with time development (x n, t). [4] Fourier stability analysis of realistic for (a) first 20 time steps and (b) whole 1000 time steps. [4] Unstable realistic model snapshot of velocity vs. vehicle number at (a) t=100, (b) t=300, (c) t= Stability of different velocities for (a)simple and (b)realistic model.. 44 viii

9 Chapter 1 Introduction Traffic flow problems are an interesting dynamic to study due to the relation it has to people s everyday life. There are various methods to study different traffic flow problems in mathematics that deal with three main variables speed, density and flow. In these problems, each car must obey common equations of motion which are determined by the relation to other vehicles moving in the flow of traffic. Analysts approach these traffic problems in three different ways [7]: 1. In a microscopic scale, every vehicle is considered as an individual and an equation is written for each as an ordinary differential equation. 2. A macroscopic scale uses a system of partial differential equations that balance laws of some gross quantities of interest such as density of vehicles or their mean velocity [1]. 3. Lastly, a mesoscopic (kinetic) scale defines a function f(t, x, V ) which expresses the probability of having a vehicle at time t in position x which runs with velocity V. The function follows statistical mechanics and can be computed using an integro-differential equation such as Boltzmann equation. 1

10 Some examples of traffic problems are variable speed limit assignment, road junctions, wave propagation of automobile brake lights, etc. We will be investigating the problem of the dynamical evolution of traffic congestion in the microscopic scale. We will look at different types of dynamical systems before developing our dynamical system that will simulate traffic congestion induced by small perturbations without any origin (i.e., no accidents and no traffic signal). The equation of motion of each vehicle is based on the assumption that each driver responds to a stimulus from other vehicles in a specific way. The response is expressed as an acceleration because that is the only direct control the driver has on traffic flow. The stimulus and sensitivity may be a function of position of vehicles, their time derivatives, etc. This function is decided by supposing all drivers obey traffic regulations to avoid accidents. There are two types of theories for regulation: 1. Each vehicle must maintain safe distance from the preceding vehicle, which depends on the relative velocity of the two successive vehicles. This theory is called follow-the-leader theories. 2. Each vehicle has a legal velocity, which depends on the following distance from the preceding vehicle. We will be consider the second theory for our model with the stimulus as a function of a following distance and the sensitivity as a constant. After the development of our model, we will analyze the system s stability then use Matlab to perform numerical simulations on a simple and realistic model. 2

11 Chapter 2 Background on Dynamical Systems Before we take a look at the model presented in Dynamical Model of Traffic Congestion and Numerical Simulation [4], we must introduce methods and definitions that will be utilized in later sections [8]. There are two main types of dynamical systems. The first is differential equations which deal with continuous time systems, and the other is difference equations which will deal with discrete time. We will only be focusing on differential equations because the problem that we dealing with is one in a continuous time space. The types of differential equations that we will be looking at are ordinary differential equations (ODEs) and the general structure of these is provided by the following first-order system: x 1(t) = f 1 (x 1 (t),, x n (t), t),. (2.1) x n(t) = f n (x 1 (t),, x n (t), t), 3

12 where dx i (i) x i = dt, (ii) x 1 (t),, x n (t) are the dependent variables at time t, (iii) f 1,, f n are functions that are determined by the problem at hand. The above system explicitly depends on time and this is called non-autonomous. A system that is not time-dependent is called autonomous. We will only be looking at automonous systems because non-autonomous systems are more difficult to deal with and may be converted into an autonomous system by removing the time dependence by introducing a new variable (extra dimension). Consider the previous first-order system (2.1) and let x n+1 = t. This will give us ẋ n+1 = 1 and will modify the non-autonomous system into an autonomous system given as: x 1 = f 1 (x 1,, x n, x n+1 ),. ẋ n = f n (x 1,, x n, x n+1 ), (2.2) ẋ n+1 = 1. The system is considered to be linear if all functions f i are first ordered polynomials with no product of x i x j for i, j = 1,, n + 1. Otherwise the system is said to be nonlinear. Common nonlinear terms of x i are x 1 x 2, (x 1 ) 3, or cos x 2. The solutions x = {x 1 (t),, x n (t)} can be viewed as trajectories in n-dimensional space. Theorem (Existence and Uniqueness Theorem) Consider the initial value problem ẋ = f(x) 4

13 with x(0) = x 0. Suppose that f is continuous and that all its partial derivatives f i x j, i, j = 1,, n, are continuous for x in some open connected set D R n. Then for x 0 D, the initial value problem has a solution x(t) on some time interval ( τ, τ) about t = 0, and the solution is unique. This theorem states that if f is continuously differentiable then a local solution exists and is unique. This theorem results in an important corollary which states that different trajectories never intersect. This is true because if the trajectories intersected then that solution is found with two different initial values which violates the Existence and Uniqueness Theorem stated above. 2.1 One-Dimensional System Consider the differential equation of the form ẋ = f(x), (2.3) where f(x) is a smooth real-valued function and the solution, x(t), is real valued function which can be easily solved as an ordinary differential equation using implicit differentiation given an initial condition x(0) = x 0. This equation is called a one-dimensional first order system. The Existence and Uniqueness Theorem can be restated for the one-dimensional case: Theorem (Existence and Uniqueness Theorem) Consider the initial value problem ẋ = f(x), x(0) = x 0. 5

14 Suppose that f(x) and f ' (x) are continuous on an open interval R, and suppose that x 0 is a point in R. Then the initial value problem has a solution x(t) on some time interval ( τ, τ) about t = 0, and the solution is unique. This theorem states that if f(x) is smooth enough, then local solutions exist and are unique Graphical Approach: Fixed Points and Stability If we think about x as the position of a particle moving at time t with a velocity ẋ then equation (2.3) represents the vector field. We will need to graph f(x) on the axes ẋ versus x and sketch the vector field on the real number line. When f(x) > 0 then the flow is to the right and to the left when f(x) < 0. To find solutions of the system, we start at an arbitrary point x 0 and watch how it is carried along by the flow. The flow that x 0 travels as time goes on is a function x(t) known as the trajectory based on x 0 that represents the solution to the differential equation. A picture which shows all the qualitatively different trajectories is called a phase portrait. At point x, where f(x ) = 0, there is no flow and these are called fixed points. A stable fixed point is denoted as a black dot on the graph and the local flow is going toward the fixed point. On the other hand, if the local flow is moving away from the fixed point, this would be considered an unstable fixed point, indicated as an open circle on the plot. In terms of the original differential equation, fixed points on a graph represent equilibrium solutions to the equation. An equilibrium is stable if all sufficiently small disturbances away from it decrease over time and unstable if they grow in time. 6

15 Figure 2.1: Vector field of one-dimensional system (2.4). [8] Consider the one-dimensional system ẋ = x 2 1. (2.4) To find the fixed points of the system we set ẋ = f(x ) = 0 and solve for x. This will give us x = ±1. We will need to determine the stability of each fixed point, so we will need to sketch the vector field (Figure 2.1). The vector field allows us to determine the flow of f(x) = x 2 1: (i) (, 1): f(x) > 0 so flow is to the right, (ii) ( 1, 1): f(x) < 0 so flow is to the left, (iii) (1, ): f(x) > 0 so flow is to right. This illustrates that the flow is going towards x = 1 hence this is a stable fixed point and the flow is moving away from x = 1 which makes this an unstable fixed point. 7

16 2.1.2 Linearized Theory: Fixed Point and Stability Another way to determine the stability of a fixed point in a more quantitative measure is to linearize about a fixed point. Let x be a fixed point and η = x(t) x be a small perturbation away from x. We have to derive a differential equation for η to determine if the perturbation grows or decays. Here, we have Thus, d η = (x x ) = x. dt η = ẋ = f(x) = f(x + η). Recall Taylor s expansion for an arbitrary function f(x) about a point a is: '' ' f (a) 2 f (n) f(x) f(a) + f (a)(x a) + (x a) (x a) n. 2! n! So performing a Taylor s expansion about a = x and η = x x so x = x + η we get: η = f(x + η) = f(x ) + ηf ' (x ) + O(η 2 ), where O(η 2 ) denotes quadratically small terms in η and x is a fixed point so f(x ) = 0. This gives η = ηf ' (x ) + O(η 2 ). If f ' (x ) = 0 then O(η 2 ) terms are negligible and, η ηf ' (x ). This is a linearization about x and shows that the perturbation about η(t) = 0 grows exponentially if f ' (x ) > 0 and decays if f ' (x ) < 0. 8

17 2.2 Two-Dimensional Linear System This section will go slightly beyond the previous section and look at two-dimensional linear systems. Consider the following system: x 1 =ax 1 + bx 2, x 2 =cx 1 + dx 2, (2.5) which is a general two-dimensional linear system. We can write this into a more compact by putting it in matrix form: ẋ = Ax, (2.6) where a b A = and x = x1 c d x 2 and the boldface letters are to denote vectors. It can be seen that when x = 0 then ẋ = 0, hence x = 0 is a fixed point for the system. The solutions of ẋ = Ax can be visualized moving on the (x, y) plane, called the phase plane Stability Analysis We begin by looking at solutions to the system (2.6) of the form x(t) = e λt v, (2.7) where v is a nonzero vector and λ is a growth rate. To find λ and v we substitute (2.7) into (2.6) and obtain λe λt v = e λt Av. 9

18 Dividing by e λt = 0 gives Av = λv. This becomes an eigenvalue problem, where v is an eigenvector of A and λ is the corresponding eigenvalue. So equation (2.7) is also called the eigenvalue solution. In general, the eigenvalues of a matrix A are given by the characteristic equation det(a λi) = 0, where I is the identity matrix and A = a b. c d Then the characteristic equation is a λ b det(a λi) = det c d λ =λ 2 (a + d)λ + ad bc =λ 2 τλ + Δ = 0, where Δ = det(a) and τ = a + d = trace(a). Then the eigenvalues will be given by τ + τ 2 4Δ τ τ 2 4Δ λ 1 =, λ 2 =. 2 2 So a general solution for x(t) can be written as λ 1 t λ 2 t x(t) = c 1 e v 1 + c 2 e v 2, where v 1 and v 2 are the eigenvectors corresponding to λ 1 and λ 2 respectively. This is a linear combination of solutions to the system ẋ = Ax where the coefficients 10

19 c 1 and c 2 are determined by the initial condition x 0. For every x 0, there exists a solution that is unique due to the existence and uniqueness theorem. There are different classifications for the fixed point, x, of the systems that depends on Δ and τ: (i) If Δ < 0, the eigenvalues are real and have opposite signs. This tells us that there is a stable manifold where trajectories approach x and an unstable manifold where trajectories move away from x. This is called a saddle point. (ii) If Δ > 0, the eigenvalues are either both real with the same sign, or a complex conjugate pair. (a) When τ 2 4Δ > 0 and τ > 0 then both eigenvalues are real and positive, so we will have unstable nodes where trajectories move away from x. On the other hand, when τ 2 4Δ > 0 and τ < 0 we have negative real eigenvalues which gives us stable nodes; so trajectories are attracted to x. (b) When τ 2 4Δ < 0 and τ > 0, we have complex eigenvalues with positive real parts and trajectories move away from x in a spiral motion; hence we have an unstable spiral(focus). Furthermore, when τ 2 4Δ < 0 and τ < 0, again we have complex eigenvalues but this time with negative real parts so trajectories move towards x in a spiral motion; we have a stable spiral(focus). (c) When τ 2 4Δ = 0 we get borderline cases between nodes and spirals; stars and degenerate nodes. This means that both eigenvalues are equal to each other, λ 1 = λ 2 = λ, and we have two cases. First, if there are 11

20 two independent eigenvectors corresponding to λ and λ = 0 then x is a star node. For star nodes, trajectories move in straight lines towards x (τ < 0) or move in straight lines away from x (τ > 0). Alternatively, if there is only one eigenvector then x is a degenerate node which means that as t + and t all trajectories become parallel to the one available eigendirection. Again, if τ > 0 then the degenerate node is stable and if τ < 0 it becomes unstable. (iii) If Δ = 0 at least one of the eigenvalues is zero so the origin is a non-isolated fixed point. (iv) When τ = 0, eigenvalues are purely imaginary and we get a neutrally stable center. These classifications are depicted in Figure (2.2). Figure 2.2: Stability of some of the different types of fixed points. [2] 12

21 2.2.2 Rewriting Second-Order Ordinary Differential Equations Let us consider the ODE for a damped harmonic oscillator which is given by: mẍ + bẋ + kx = 0. (2.8) This is a second-order differential equation. It is possible for us to rewrite equation (2.8) with a few transformations to make this ODE into a two-dimensional system. We will let x 1 = x, dx x 2 = = x. dt We will now take the derivative with respect to time of the above transformations So we get the system x 1 = ẋ = x 2, k b k b x 2 = ẍ = x ẋ = x 1 x 2. m m m m x 1 = x 2, k b x 2 = x 1 x 2. m m The system can also be written in matrix form, ẋ = Ax, where ẋ = x 1, x = x 1, and A = 0 1. x 2 x k b 2 m m Hence we have now rewritten our second order differential equation into a twodimensional linear dynamical system. This is an important technique which will be revisited later. 13

22 2.3 Nonlinear Dynamical System We will now focus on autonomous nonlinear dynamical systems in two-dimensions with general form ẋ 1 =f 1 (x 1, x 2 ), ẋ 2 =f 2 (x 1, x 2 ), where f 1 and f 2 are functions. This system can be written in vector form ẋ = f(x), where x = (x 1, x 2 ) and f(x) = (f 1 (x), f 2 (x)). In this context, x is a point in the phase plane and ẋ is the velocity vector at the point. The solution x(t) is found by tracing along the vector field that corresponds to the trajectory through the phase plane. Hence the entire phase plane is filled with trajectories since each point can play the role of an initial condition. For nonlinear systems, there s usually no chance of finding trajectories analytically and when they are, it is often too complicated to give much insight on the solution. We will find the system s phase portrait directly from properties of f(x). We will be using numerical integration methods to solve ẋ = f(x). The method that will always be used is the Runge-Kutta method, which in vector form is 1 x n+1 = x n + (k 1 + 2k 2 + 2k 3 + k 4 ), 6 where k 1 =f(x n )Δt, k 2 =f 1 x n + k 1 2 Δt, k 3 =f 1 x n + k 2 2 Δt, k 4 =f(x n + k 3 )Δt, 14

23 and Δt is the step size. By plotting the direction field for the phase portrait, this indicates the direction of the flow. Luckily, Matlab has a built in function that utilizes the Runge-Kutta method; one of the mostly commonly used is ode45 function Fixed Points and Linearization To study nonlinear systems, we will linearlize the system so that we can approximate the phase portrait near a fixed point. So let s consider the system ẋ =f(x, y), ẏ =g(x, y), with fixed point (x, y ). That is, f(x, y ) = 0 and g(x, y ) = 0. We will let u and v be some small perturbation from the fixed point, u = x x, v = y y. To determine whether these perturbations grow or decay, we need to derive the differential equations for u and v. Let s look at the u equation first. We will be differentiating u then doing a Taylor s series expansion to obtain u: u =ẋ =f(x + u, y + v) f(x, y ) f(x, y ) 2 2 =f(x, y ) + u + v + O(u, v, uv) x y f(x, y ) f(x, y ) 2 2 =u + v + O(u, v, uv). x y The partial derivatives are evaluated at the fixed points so they are numbers, not functions. Note that O(u 2, v 2, uv) is quadratic in u and v and since they are 15

24 small, we may ignore these higher order terms. Similarly we may find g(x, y ) g(x, y ) 2 2 v = u + v + O(u, v, uv). x y Using the same argument from previously we can ignore these higher order terms as long as the fixed points for the linearized system is not a borderline case and write this system as u = v f(x,y ) f(x,y ) x y u. v g(x,y ) g(x,y ) x y This become the linearized system where A is called the Jacobian matrix evaluated at the fixed point (x, y ) and the system can be written as ẇ = Aw where f f u, w = u, and A = x y ẇ = g g v v x y (x,y ). This system can now be analyzed using methods discussed for the two-dimensional linear system. We are only interested in the stability of the fixed point and not in the geometry of the trajectories so we can classify them as: (i) Robust cases: Repellers (sources): both eigenvalues have positive real part. Attractors (sinks): both eigenvalues have negative real part. Saddles: one eigenvalue is positive and the other is negative. (ii) Marginal cases: Centers: both eigenvalues are pure imaginary. Higher-order and non-isolated fixed points: at least one eigenvalue is zero. 16

25 Therefore, we would only need the Re(λ i ) < 0 for i = 1,, n for our system to be stable at the fixed point. 17

26 Chapter 3 Dynamical Model 3.1 Dynamical Equation We will now consider the traffic congestion model presented to us in the article [4] and we will make a few assumptions that will help us model this traffic phenomenon. This model will ignore the length of the car for simplicity and assume that all drivers will have common sensitivities. In today s world, we know that everyone does not obey all traffic regulations and one that is commonly broken everyday by an average person would be the legal speed limit. However for our model, we will also assume that each vehicle obeys the legal maximum velocity V and the driver will respond accordingly to the vehicle directly ahead. So the driver will speed up or slow down depending on the distance between him and the preceding vehicle. The dynamical system of the model that is presented to us is where, ẍ n = a[v (Δx n ) ẋ n ], (3.1) 18

27 (i) x n is the coordinate of the nth vehicle, (ii) Δx n = x n+1 x n for each vehicle number n for n = 1, 2,...N, (iii) a is a constant that will represent the drivers sensitivity (assume independent of n), (iv) V (Δx n ) is the legal velocity of vehicle number n that depends on the following distance of vehicle number n + 1. The legal velocity of each vehicle is dependent upon the following distance Δx n such that if the distance between car n and n+1 is small then velocity of car n would reduce to maintain a safe distance (i.e., no collision). If the following distance is large then the car may increase velocity but does not exceed the maximum velocity (speed limit). Hence the velocity V is a function that has the following properties: 1. Velocity is a monotonically increasing function of Δx n, 2. V (ΔX) has an upper bound. That is V max V (Δx n ). Lastly, these vehicles will travel on a circuit of length L so that the first car is equivalent to the (N + 1) car. This will be our boundary condition making the distance travelled by each vehicle periodic. The model that we are looking at is a single lane circuit where vehicles may not pass each other and (n + 1)th vehicle is always in front of the nth vehicle for n = 1,, N. So one can imagine N cars driving in a circle with a circuit length L with no passing allowed. 19

28 3.2 Stability of the System We will first take a look at the steady state flow of the dynamical system presented to us. The equation below will satisfy equation (3.1) (0) x n = bn + ct, (3.2) where, b = N/L and represents the constant spacing between two successive vehicles, c = V (b) and represents the constant velocity of the steady state traffic flow. Equation (3.2) is our system without congestion and the vehicles are all distributed uniformly along the circuit L all driving at constant velocity c with constant spacing b between each vehicle. Now that we have defined the steady state flow for the system, we will look at the original nonlinear equation (3.1) and linearize about the steady state (3.2) to determine its stability. We will consider y n = small deviation from the steady state flow such that x n = x (0) + y n, y n << 1. (3.3) n We will use equation (3.3) and take its derivative with respect to time and substitute it into equation (3.1). If we take the first and second derivative of (3.3) we get the following: (0) n Now let s reconsider equation (3.1): x n = x + y n = V (b) + y n, (3.4) (0) ẍ n = x n + ÿ n = ÿ n. (3.5) ẍ n = a[v (Δx n ) ẋ n ], 20

29 and substituting equations (3.4) and (3.5) we get ÿ n = a[v (Δx (0) n + Δy n ) (V (b) + ẏ n )]. (3.6) (0) We will do a Taylor s expansion of V (Δx n +Δy n ) about b because it is a nonlinear term that we want to linearize: (0) (0) V '' (b) (0) V (Δx n +Δy n ) = V (b)+v ' (b)(δx n +Δy n b)+ (Δx n +Δy n b) Higher Order Terms. 2! Note that (0) (0) (0) Δx n = xn+1 xn = [b(n + 1) + ct] [bn + ct] = bn + b + ct bn ct = b. Also, we will only take the linear terms and neglect the rest because y n << 1 so the higher order terms << 1. This give us (0) V (Δx n + Δy n ) = V (b) + V ' (b)δy n. (3.7) Now we will combine equations (3.6) and (3.7) y = a[v (b) + V ' n (b)δy n V (b) y n ]. Simplifying the above equation, we will arrive at the linearized system which matches the result found by Bando ÿ n = a(fδy n ẏ n ), (3.8) where n = 1,, N, and f is the derivative of V at b, f = V ' (b). 21

30 3.2.1 Analyzing the Linearized System when N = 3 We decided to start by considering a small case where N = 3 and determine the Jacobian to the linearized system (3.8), ÿ 1 =a(fδy 1 ẏ 1 ), ÿ 2 =a(fδy 2 ẏ 2 ), (3.9) ÿ 3 =a(fδy 3 ẏ 3 ). We will rewrite this system of second-order differential equations into a first-order system. Let w i = y i and w i+3 = ẏ i for i = 1, 2, 3. Differentiating with respect to time gives us the following first-order linear system ẇ 1 =w 4, ẇ 2 =w 5, ẇ 3 =w 6, ẇ 4 =afw 2 afw 1 aw 4, (3.10) ẇ 5 =afw 3 afw 2 aw 5, ẇ 6 =afw 1 afw 3 aw 6. Lets write this first-order system in matrix form ẇ = Aw, where ẇ 1 w ẇ 2 w ẇ 3 w, w = ẇ =, and A =. ẇ 4 w 4 af af 0 a 0 0 ẇ 5 w 5 0 af af 0 a 0 ẇ 6 w 6 af 0 af 0 0 a 22

31 The matrix A can be written in a simplified block matrix form given by A = I 3x3, (3.11) C ai 3x3 where O 3x3 is a zero matrix of size 3 3, I is the identity matrix of size 3 3 and af af 0 C = 0 af af. af 0 af We need to find the eigenvalues of A so that we can determine the stability of the fixed point w = 0 (this can be found by setting ẇ = 0). We will use Matlab to compute the characteristic equation for matrix A as det(a λi) = λ 6 +3aλ 5 +(3a 2 +3af)λ 4 +(a 3 +6a 2 f)λ 3 +(3a 3 f+3a 2 f 2 )λ 2 +3a 3 f 2 λ = 0. Notice that for N = 3 the characteristic equation is polynomial of degree 6 (or degree 2N) and is difficult to solve. So we can see that this method would not be feasible for N being large and is the reason why the paper uses a different method which we will explore next Stability in Linearized Theory Now let s look at the system from a different approach. The solution to equation (3.8) that Bando obtain from expanding Fourier series with e iα kn as an orthonormal set, y k (n, t) = exp{iα k n + zt}, (3.12) where α k = 2 π k for k = 0, 1, 2,, N 1 and z = u + iv (u and v are real). N 23

32 Let s test the solution (3.12) with our equation (3.8). To do this we will differentiate equation (3.12) with respect to time t twice Consider equation (3.8), y k (n, t) = exp{iα k n + zt}, ẏ k (n, t) = z exp{iα k n + zt}, ÿ k (n, t) = z 2 exp{iα k n + zt}. ÿ n = a(fδy n ẏ n ), and we will substitute in the above equations we just obtained: z 2 exp{iα k n + zt} = a[f(exp{iα k (n + 1) + zt} exp{iα k n + zt}) z exp{iα k n + zt}], z 2 exp{iα k n + zt} = a[f(exp{iα k n + zt} exp{iα k } exp{iα k n + zt}) z exp{iα k n + zt}], z 2 = a[f(exp{iα k } 1) z], 2 iα z = af(e k 1) az, 2 iα k z + az af(e 1) = 0. (3.13). We see that our solution (3.12) to the linearized system must satisfy equation (3.13). Now let s reconsider equation (3.12): y k (n, t) = exp(iα k n + zt) = exp(iα k n + ut + ivt) = exp(ut) exp[i(α k n + vt)] ut ivt+iα k n y k (n, t) = e e (3.14) Notice that Re(z) = u determines whether the perturbations, y k grow or decay: (i) if u < 0, then y k decay and the system will be stable, (ii) if u > 0, then y k grow and the system will become unstable, 24

33 (iii) if u = 0, then this is a marginal case. We used Matlab to compute the numerical solution to equation (3.13) and compared to the analytical solution presented in Bando s paper [4]. Both graphs in Figure 3.1 look very similar but, we can see that the numerical solution has numerical errors happening around the origin. The graphs display the (f, α) polar coordinate plane that is separated into stable (u < 0) and unstable (u > 0) regions by the critcal curve u(f, α) = 0, a f =. (3.15) 2 cos 2 α 2 Equation (3.15) is given in the paper [4] and can be found when the solution to equation (3.13), z, is solved when Re(z) = 0. Figure 3.1: Stability criteria in the (f, α) polar coordinate plane of (a)numerical solution using Matlab and (b)analytical solution to equation (3.13). The blue line represents when u = 0, red is the circle when f = a, lastly green is the circle 2 where f > a. 2 In Figure 3.1(a) of the numerical solution, we found that any circle of radius f 25

34 will always touch the blue curve making the system marginally stable. So we chose to use Bando s result because our numerical solution has errors that would not allow any stable solutions. In order for the state to be stable every corresponding u to the complete set of α k (k = 1, 2,, N 1) must be negative. Even one (0) positive mode (u > 0) will make the steady state flow solution x n unstable. We tested different values of f that will still allow us to maintain stability using both solutions and found that: (i) when f < a 2 the state is stable because u < 0, for all α k, (ii) when f > a 2 the state becomes unstable, because at least one u > 0 mode solution exists, (iii) when f = a 2 the state is marginal. Applying the above criteria, then the stability of our model would depend on the derivative of our legal velocity V (Δx) evaluated at constant car spacing b of (0) steady state flow x n : a f = V ' (b) < (3.16) Stability and Fourier Analysis We have discussed the stability of the steady flow state with linearized theory. Since we want to study the stability based off of the perturbation, we have to investigate numerically. We will define the amplitude of the Fourier mode ξ, N C αk + is αk = y k (n, t) exp( iα k n), (3.17a) n=1 ξ αk = C α 2 k + is α 2 k, (3.17b) 2π α k = k, (k = 0, 1, 2,, N 1), (3.17c) N 26

35 where C αk and S αk are time dependent functions. We can see that the amplitude of the Fourier modes depends on the real solutions of equation (3.13). If all u s are negative, then all their amplitude will shrink and the system will be stable. Conversely in early time stages, if even one u is positive that will cause those amplitudes to grow and other amplitudes with negative u to decline. As time increases, the evolution of each amplitude is governed by nonlinear equations which can only be studied by numerical simulations. 27

36 Chapter 4 Models and Numerical Simulation This chapter will recreate two models presented by Bando that will help us model traffic congestion. Before we look at these two models let s first revisit our dynamical system (3.1) introduced in previous sections, ẍ n = a[v (Δx n ) ẋ n ], for n = 1,, N and Δx n = x n+1 x n. We must rewrite this system of second order differential equations into a first order dynamical system. In the background section, it was shown that it is possible to rewrite one second order differential equation into a two-dimensional first order system using a change of variable or transformation. This is the technique that will be utilized here to rewrite our system of N second order ODEs into a first order system. So we will let z i = x i, z N+i = ẋ i, for i = 1,, N. 28

37 Differentiating z i with respect to time for i = 1,, 2N, we get: ż 1 =z N+1, ż 2 =z N+2,. ż N 1 =z N+(N 1) = z 2N 1, ż N =z 2N, ż N+1 =a[v (Δz 1 ) z N+1 ], (4.1) ż N+2 =a[v (Δz 2 ) z N+2 ],. ż 2N 1 =a[v (Δz N 1 ) z 2N 1 ], ż 2N =a[v (Δz N ) z 2N ]. The system (4.1) is a 2N dimensional nonlinear first order system where V is a nonlinear function and N Z. We will use Matlab ode45 solver to help us model this system for a simple and realistic case. First we will be looking at a simple model and later modify this into a more realistic model. 4.1 Simple Model For the simple model we will use the velocity function V that satisfies our criteria where this function must be bounded and monotonically increasing: V (Δx) = tanh(δx). (4.2) We will let a = 1 and N = 100 for numerical simulation reasons. We will use Matlab to analyze the system if at time t = 0 all vehicles move according to the 29

38 Figure 4.1: Graph of velocity function for simple model. steady state solution with one exception, i.e., one vehicle will move with a slight 0.1 unit shift ahead of the other vehicles. This gives us the following initial conditions: (0) x 1 (0) =x , x n (0) n (0) =x, n = 1, x n(0) =0. We will be looking at two different cases for the simple model: a stable and unstable case. (i) Stable case: Let L = 200, N = 100, this gives us L b = = 2, N f =V ' (b) = 1 tanh(2) = < a = (ii) Unstable case: Let L = 50, N = 100, So L b = = 0.5, N f =V ' (b) = 1 tanh(0.5) = > a =

39 We will apply the following parameters and design a program on Matlab that will solve system (4.1) using ode45 function with velocity function (4.2). In Figure 4.2, we compared the two models distance and velocity with respect to time for the 50th vehicle. It can be seen in graph (a) that the velocity for the stable case starts off slowly and reaches its optimal velocity. The velocity for the unstable case continually changes as cars speed up and slow down as time t increases and even reaches negative values. Let s analyze Figure 4.2(b) which shows the distance travelled by the 50th car for both cases. It is seen that for the stable case we have the vehicle gaining distance as time evolves. Conversely, we see in various time intervals, such as 230th to 240th step, where the distance travelled decreases which indicated that the vehicle reverses the car. This would cause a traffic accident in a single lane circuit. Figure 4.2 demonstrated that the 50th car had a negative velocity around time t = 240 which cause the vehicle to reverse. This only gives us information about one vehile Figure 4.3 is a snapshot at that time, t = 240, and shows all vehicles velocity at that time; this clearly shows numerous vehicles with negative velocities. From our analysis of the 50th vehicle we know that this indicates vehicle collisions in several locations rather than mimicking traffic congestion. Hence we will need to modify our simple model into a more realistic one to prevent this from happening. Let s check the behaviour of the Fourier modes for the simple model that were defined in equations (3.17a-3.17b) to further investigate this system s stability. Bando illustrated in Figure 4.4(a) and 4.4(b) the time evolution of several typical modes for the stable and unstable cases. Figure 4.4(a) for the stable case, shows that all amplitudes shrink as time increases. This tells us that any perturbations away from the steady state declines as time grows. On the other hand, Figure 31

40 Figure 4.2: Comparison of simple model for stable and unstable case of the 50th vehicle. (a)velocity vs Time (b)distance vs. Time. Figure 4.3: Simple, unstable model velocity snapshot at t=

41 4.4(b) for the unstable case, shows the amplitude for the positive u mode increases while the others decreases. We can clearly see that this case is unstable because we see perturbations increasing so this drives us away from our steady flow state. Figure 4.4: Fourier stability analysis of simple stable (a) and unstable (b) model. [4] 4.2 Realistic Model Now for the realistic model we will shift the velocity function in the simple model to eradicate the possibility of negative velocities, as seen with the simple model. We will choose the legal velocity to be V (Δx) = tanh(δx 2) + tanh(2). (4.3) 33

42 Figure 4.5: Graph of velocity function for realistic model. This case allows a driver to control the vehicle by gradually accelerating or braking in a way that any vehicle will never pass the preceding vehicle. We will take the parameters N = 100 and L = 200, which is the same as the simple model for the stable case. We use a similar program using Matlab s built-in solver ode45 and run the numerical simulation for 1000 time steps. Figure 4.6 shows snapshots of the realistic model s velocity for each vehicle at t = 100, 300, and It can be seen that there are no longer any negative velocities and that vehicles tend to have almost zero velocity or at nearly maximum velocity at t = 300 and t = This could suggest that there is traffic congestion is forming. However, to get a more definitive picture of what this model is portraying we will plot all vehicles on the circuit with time development (x n, t). Figure 4.7 graphs the time evolution of every vehicle along the circuit length L which is illustrated on the plot as traffic points, x n. This states that if a vehicle is at the beginning of the circuit it will be at traffic point 34

43 Figure 4.6: Realistic model snapshot of velocity vs. vehicle number at (a) t=100, (b) t=300, (c) t= and vise versa if it were at the end, it would be located at x n = 200. From our initial assumptions since we are on a circular track, once a vehicle reaches the end (x n = 200) then the next traffic point is at the beginning (x n = 0). The clusters of points represent traffic congestions on the circuit (i.e. traffic point) at time t = 0, 1,, We see that congestion is not formed in the initial time steps. However, as time increases there appears to be various clusters of traffic that is formed as a result of the small perturbation to our initial condition. After congestion is formed, we can see that it moves backwards as time increases. Bando estimated that the velocity of congestion moves backwards with a constant 35

44 rate given by: v c = V (Δx max )Δx min V (Δx min )Δx max Δx max Δx min Figure 4.7: Plot of the positions of all vehicles on the circuit with time development (x n, t). Let s take a closer look at the headway and velocity distribution accumulated over first 100 time steps (blue), third 100 time steps(green), and the last 100 time steps. In Figure 4.8, the y-axis depicts all 100 vehicles looped for 100 time steps and after each time steps we record the corresponding velocity and headway. When we look at the first 100 time steps, we can see that most vehicles have a headway Δx 2. This tells us that for the first 100 time steps the model moves towards the steady state solution where the constant spacing b = 2. As we start to look at the third 100 time steps and the last 100 time steps, we can clearly see that the vehicles no longer move toward the steady state solution and instead tend toward either Δx min or Δx max. This phenomenon is explained in the conclusion where we will investigate the velocity function further. 36

45 We can determine that in a high concentration area Δx min = 0.32 with corresponding velocity V (Δx min ) = 0.03 using our Matlab program. Similarly, in a low concentration area we have Δx max = 3.68 with corresponding velocity V (Δx max ) = Let s define the congestion region to be where the headway of vehicles is shorter than 2 units, i.e., Δx < 2. We can see that by analyzing Figure 4.8, roughly half the vehicles are in the congestion region. Figure 4.8: Histogram displaying the first 100 time steps (blue), third 100 time steps (green), and the last 100 time steps (red). We will now investigate the results Bando observed with the Fourier modes for the realistic model. Figure 4.10(a) shows the first 20 times steps. It is clear that there is instability in the uniform flow solution (3.2) with constant headway b = L/N and constant velocity c = V (b) because amplitudes are increasing. Now let s look at the Fourier modes for the entire 1000 time steps in Figure 4.10(b). Bando stated that the time interval can be divided into three stages. In the first 37

46 stage (0th to 50th stop), the perturbations grow and shows instability. The second stage (50th to 400th steps) we can see the amplitude is chaotically oscillating due because of the nonlinearity. Furthermore, the last stage (after 400th step) shows the amplitude changes slowly and approaches a constant value. Bando described that this behaviour in the third stage is caused by the combination of two congestions clusters which can be seen in Bando s plot in Figure 4.9 at around the 900th time step. After this congestion has been formed, the amplitudes of the Fourier modes become stable. Bando s simulation was run for an additional 200 time steps and each amplitude continued to remain constant after 900th time step. This implies that congestion maintains its structure after that point in time. We can clear see that by comparing our plot in Figure 4.7 with his results, the two congestions clusters observed will merge around t = 1000 in our results. Although we did not obtain the exact same results as Bando, we can clearly see that our results generated a similar congestion pattern with some slight deviation. Figure 4.9: Bando s plot of the positions of all vehicles on the circuit with time development (x n, t). [4] 38

47 Figure 4.10: Fourier stability analysis of realistic for (a) first 20 time steps and (b) whole 1000 time steps. [4] After our analysis of the realistic stable case, we were curious to see why Bando did not consider unstable case presented in Section 4.1 for the realistic simulations where we let L = 50, N = 100, L b = = 0.5, N f =V ' (b) = 1 tanh(0.5) = > a = So we decided to run our simulations to determine what happens for this case. Recall that in the simple model, this case resulted in traffic collision instead of traffic congestions. Figure 4.11 shows the velocity at three different time steps and it is very clear that the velocity of every vehicle always remains near V (Δx min ) =

48 Figure 4.11: Unstable realistic model snapshot of velocity vs. vehicle number at (a) t=100, (b) t=300, (c) t=1000. that was found for the stable case. This implies that at every point in time, every vehicle is stuck in a congested area and is barely moving. This is not a very accurate representation of modern day traffic because this would strongly suggest that there is constant traffic on highways and would be near impossible to get to any location. 40

49 Chapter 5 Conclusion We have taken a look at both the simple and realistic model and noticed that the stable simple model does not induce congestion with small perturbations whereas the realistic one does. Recall in the steady state solution we fixed headway b so we can determine the corresponding velocity, V (b) and f = V ' (b) determines the stability of the system. Hence, the stability of the system depends on which region b resides in. If b is in a stable region, small perturbations will not affect the structure of the traffic flow. On the other hand, if b is in an unstable region then each vehicle will be driven to stable regions. To take a closer look at this issue, lets recall the result (3.16) a f = V ' (b) <. 2 So the derivative of the velocity determines whether or not we are in a stable region. We must evaluate V ' (Δx) < a 2 (i) For the simple model: for the simple and realistic case where a = 1. V ' (Δx) = 1 tanh(δx) <

50 Solving the inequality yields the result that we will maintain stability if Δx < or Δx > These results tell us that the simple case has two stable regions. One of these regions is when Δx > This signifies that when headway of the vehicles are greater than this value then the vehicles will want to stay in this region where the velocity will be at near the maximum legal velocity. Hence signifying that this will result in no congestion. Alternatively, the other region where Δx < , implies that if vehicles have a negative headway it would stable. This means that there is the (n + 1)th vehicle is backed up into the nth which again shows us that accidents occur with the simple model. (ii) For the realistic model: V ' (Δx) = 2x+4 4e < 1. (e 2x + e 4 ) 2 2 Solving this, we determine that our system will be stable if Δx < or Δx > These results demonstrates that the realistic model also has two stable regions. The first is Δx > which is significant because if the vehicle s headway is greater than that amount then it will want to keep that following distance with its corresponding velocity near maximum. However, the other stable region Δx < also tells us that if the headway if less than this amount and even close to zero, our vehicle will also want to stay here. This is primary reason why the realistic model generates spontaneous traffic congestions because 42

51 vehicles with a small headway has a slow velocity and will tend to remain in this congested stable region. Figure (5.1) displays the plots of velocity versus Δx where both graphs are split into stable and unstable regions. Figure 5.1: Stability of different velocities for (a)simple and (b)realistic model. We have already determined for the stable realistic model that Δx min = 0.32 and Δx max = If we look at these values and compare with Figure 5.1(b), we can see that Δx min and Δx max are both in stable regions. Note that for the realistic model starts with a headway of b = 2 which lies within an unstable region and it can be seen in Figure 4.8 that as time increases vehicles tend to Δx min or Δx max. Also, since Δx min = , this guarantees the formation of traffic congestion. So for the stable case, any vehicle on the circuit will tend to either stable regions where there is no congestion near Δx max or to a congested area near Δx min. Let s also look at the location of b = for the unstable realistic model. We can see that b for the unstable case is located in a stable congested region. This indicates 43

52 that since we are already in a stable area, we will stay in here and is why the unstable case results in constant traffic congestion. Now let s investigate the simple model in a similar fashion. Using our program, we can determine that for the simple model Δx min = 1.66 and Δx max = In this case, we can clearly see that Δx min is in a stable region and is not close to Δx = 0, so it will not induce traffic congestion. However, notice that when vehicles are in the stable region that Δx min resides in, we have negative velocity which will cause traffic accidents. This cannot happen due to our initial assumptions that all vehicles follow the traffic regulations stated previously. Let s assess the location of b for the simple stable and unstable case. In the unstable case, b = 0.5 is in an unstable region so the model is unstable, whereas in the stable case b = 2 which is a stable region so small perturbations will not affect the traffic flow. Therefore, the realistic model is a better representation of traffic flow without accidents. It also demonstrates that small disturbances will disrupt the flow of traffic and cause traffic jams that do not originate from a specific source. If we were to compare this result to traffic in a large city such as Los Angeles, we would see that this happens on a daily basis where these small disturbances can be seen as slow drivers, police cars, or nosy drivers (i.e., accident on opposite side) and this would be enough to cause these traffic jams on a daily basis. 5.1 Further Research This thesis explored the formation of traffic congestion on a one lane circuit where the sensitivities of drivers are identical and have no dependence on velocity, headway, or the relative-velocity of the preceding vehicle. There have been various 44