Traffic Flow. June 30, David Bosworth

Transcription

1 Traffic Flow June 30, 2009 By David Bosworth

2 Abstract: In the following, I will try to eplain the method of characteristics, which is involved in solving many aspects of traffic flow, but not for traffic circles. Then I will eplain where the equation of the conservation of cars comes from using partial differential equation. I will then eplain the relationship between velocity and density, how that relates to the elementary traffic model. After that I will eplain the traffic flow at a red light, and finish up with an initial condition eample. Introduction The following is an eample and eplanation of the method of characteristics. Once this is eplained, it is shown how to be used involving traffic flow like the conservation of cars. From there we eplain the elementary model of traffic flow which is used for assumptions and the set up of traffic at a red light. We finish with an eample and then the conclusions. Method of Characteristics The method of characteristics is a method that is used to solve initial value problems for general first order partial differential equations. The goal of the method of characteristics is to change coordinates, for eample, from ( to a new coordinate system ( o, s) in which the

3 partial differential equation becomes an ordinary differential equation along certain curves in the t - plane, along the line = 0. This can be done by constructing the characteristics for the equations over the region of known initial conditions and proceeding along these lines to determine the solutions for later times or for new regions. Here is an eample of how to use method of characteristics to solve the partial differential equation: First set up the coordinates which are being worked in. The solution z = w( is a surface in the t z coordinate system. Now solve the partial differential equation w( / t + c( t, w) * w( / = Q( t, w), suppressing dependencies, w/ t + c * w/ = Q. Since the solution is w = w( you get w/ t = w/ t + / t * w/. Now choose / t = c so that we have w/ t = Q. And now the partial differential equation has become a system of two ordinary differential equations. Eample: Solve the following partial differential equation using the method of characteristics:

4 u u 2 t + = u, < < t (1) with the initial condition u ( 0) = The characteristic equations are dr = 1, t = r, by integrating the characteristic equation, the first equality, above we get the implied t = r and d dr = 2t d = 2t log = t also by integrating the second characteristic equation, the first equality, we get the implied. You can go from d to because they are just a dummy inde and rearranging we obtain 2 t = Ae = e 0 t c, Thus, 0 = e 2 t. On using, du dr u ds u = +, (1) is now written as dr t dr

5 du dr = u u = r Ae where we will introduce the constant A as actually being the constant along the characteristic defined by the value of 0. Thus, A is really a function of 0 and we write since t = r and the function F is determined by the initial conditions. At = 0 = t, we have 0 u ( 0,0) = so that ( ) 0 0 and 0 F =. Replacing 0 by the above epression involving an, we can obtain the final solution u( = e 0 t = e t 2 e t = e t t 2 as in reference [3]. The method of characteristics is used in many traffic flow problems as showed in the following situations. Conservation of Cars Suppose we have a highway of infinite length where the velocity and density are known; can we predict the pattern of traffic? First we consider r( and u( to be the two fundamental traffic variables. We have r( = traffic density, which is the number of cars at time t at position and u( = car velocity at position and time t, traffic flow = q(, which is the number of cars per hour passing position at time t, and thus q( = ρ( u(.

6 Then the initial traffic density is r( 0), which is the traffic density at position and time 0, and the traffic velocity field for all time remains the same, u(. The motion of each car is determined by taking the derivative of position with respect to t, which satisfies the following first order differential equation: d = u( with ( 0) = 0 (1) Solving equation (1) determines the position of each car at future times. To be able to calculate the traffic density at future times we would need to know the traffic velocity and the initial density. We want to be able to calculate the density easily if we know the velocity. We choose an interval on any particular roadway between say = a and = b, as illustrated below: Figure 1: cars entering and leaving a segment of roadway [1]. On this interval [a, b], the number of cars, denoted N, is the traffic density integrated: b ( N ( = = ρ, d (2) a

7 Even with no eits or entrances on this roadway the number of cars on the interval between = a and = b could still change in time. As cars enter at = a, the number of cars increases and as cars leave at = b, the number of cars decreases, therefore the traffic flow q (a, and q (b, is not constant in time. The rate of change of the number of cars with respect to time, dn, is equal to the number of cars per unit time entering the interval [a, b] at = a minus the number of cars per unit time eiting the interval [a, b] at = b, where the cars are always moving to the right, as illustrated in the equation below since the rate of change of the number of cars per unit time is the traffic flow at position a minus position b both at time t: dn q( a, q( b, = (3) Taking the derivative of both sides of equation (2) with respect to time gives the following: dn = d = b a ( ρ d (4) By combining equation (3) and equation (4), you get the result: d b a ( d = q( a, q( b, = = ρ (5) And q(a, q(b, can be rewritten by taking the partial derivative of the right hand side of equation (5) with respect to and then taking the integral from = b to = a gives the following equation:

8 d b q( a ( d = d ρ (6) = a b To have the integral with the same interval, we need to use an integral property, which is to take the negative of the right hand side of equation (6): d b q( b ( d = d ρ (7) = a a Moving the negative sign inside of the integral gives: d b q( b ( d = d ρ (8) = a a We can now move the d inside of the integral to get the following equation; we can do this because derivatives and integrals are interchangeable. If you move the derivative inside the integral and it has a function of two variables, then the derivative becomes a partial derivative: Equation (9) implies: Equation (10) implies: b q( t t a b ( d = d b = a ρ (9) = a q( t ( + ) d t ρ = 0 (10)

9 And from equation (11) we get q( ( + = 0 t ρ (11) dρ( q( + = 0 (12) Suppressing equation (12), which is just not including the variables of the function, gives us: dρ q + This is the equation of conservation of cars. = 0 (13) We know from above that following: q = ρu, and so therefore we can rewrite the q as the q = q( ρ, u). (14) Which implies q = q ρ q + ρ u u (15) Now combining equation (13) with equation (15) we get the following: ρ q ρ q u + + t ρ u = 0 (16) Which is still the conservation of cars since equation (16) is the same as equation (13). Now, assume that u = u ( ρ ).Taking the derivative to the velocity with respect to the position, gives: u = 0 (17)

10 Combining equation (16) and equation (17) we get the following result: ρ + t q ρ = 0 ρ (18) A Velocity-Density Relationship There are many factors that have an affect on the speed at which a car can go since it is operated by an individual. The person operating one car may want to drive faster than another person in a different vehicle. Once the traffic becomes a lot heavier, however, lane changing and speed are at a minimum for every driver on the road since it is difficult to change lanes when there are more vehicles on the road and it is not always possible to go the speed you want when there are more vehicles on the road. A lot of times you get stuck going the same speed at which the flow of traffic is moving. With all of these types of observations, we can make a simplifying assumption that at any point along the road the velocity of a car only depends on the density of cars. This is illustrated in the equation below, which was mentioned above in the eplanation of the conservation of cars: u = u(ρ) (19) As mentioned above, cars velocity can be at a imum when there are very little to no cars at all on the road with them. So when there are no other cars at all on the road, this means that the

11 density is at zero, and therefore the velocity will be at a imum as illustrated with equation (17) with the density at zero below: u ( 0) = u (20) As more and more cars per mile that join the road way their presence will slow down the car, and as the density increases more, the velocity of the cars would continue to decrease. Thus the rate of change, which is the derivative of the velocity with respect to density, is defined as below: du u' ( ρ) 0 dρ (21) Once density is at a imum, then cars will move at zero velocity, or stand still: u ( ρ ) = 0 (22) Therefore the car velocity vs. the traffic density is steady decreasing. Elementary Traffic Model As shown above, in general the car velocity is a decreasing function of density. At zero density, cars move the fastest which was denoted ρ u and the imum density was denoted where the car velocity is zero. The simplest relationship to satisfy these properties is let:

12 = u ( ρ ) u 1 (23) ρ ρ in which from the fact q = ρu the flow is given by ρ ( ρ) = u = ρ 1 u ρ ρ ρ 2 ρ q (24) And the density wave velocity satisfies 2ρ ( ρ) = q' ( ρ) = u 1 ρ c (25) As in reference [4]. Red Light Turning Green Now we assume the elementary model of traffic flow so that the traffic density satisfies ρ 2ρ ρ + u 1 = 0 t ρ (26)

13 Behind a red light position is at zero, = 0, which is traffic density at its imum, while ahead of the light the traffic density is at zero. So at time t = 0, the initial conditions for when the light turns green are: ρma < 0 ρ ( 0) = 0, > 0 The characteristic velocity is d 2ρ 1 ρ = u (27) The density is constant along the characteristics so that they satisfy 2ρ = u 1 t + ρ 0 (28) The characteristic velocity is u for = 0 ρ, while the characteristic velocity is u for ρ = ρ. And thus ρma < u ρ ( 0) = 0, > ut t The information that the traffic light turns green spreads backward at density velocity. u This is why each car has to wait until the car in front of them moves before they themselves can move. The characteristics are illustrated below:

14 Figure 2 [1] Fanlike characteristics are where they generate out from the origin, = 0. To obtain the density at other points, we note that the family of fanlike characteristics that all start at = 0 0. Thus in this region 2ρ u = 1 t ρ (29) Given and t in this region, equation (29), we can solve for the density: ρ ρ for u t < u t ( ) = t 1 2 ut (30) < As in reference [4].

15 An Initial Value Problem: Suppose that traffic, which is moving uniformly along a single lane road, comes to the end of a line of traffic which is stopped at a red light as illustrated below: Figure 3, [2], A model of traffic approaching stopped traffic. The cars that are already stopped are lined up with imum density u 1 approaching cars come to the end of the line have a uniform density u 0 cars per mile; while the cars per mile. Since u is the imum possible traffic density, the value of u 1 0 will satisfy 0 < u 0 < u1. From before, we know that ( u, is the density, or cars per mile, of traffic at position along the road at time t. The fluctuation φ ( represents the rate, cars per hour, that at which traffic passes by position at time t. Letting v 1 denote imum traffic velocity, the linear model for traffic velocity v u v 1 1 u = 1 (31) results in the constitutive equation, the equation relating fluctuation and traffic density, is

16 u = uv = v 1 u u 2 1 φ (32) Assuming that the road has no entrances or eits, the basic conservation law u + φ f with t = fluctuation φ and source f = 0 becomes 2u u 1 1 t + v u = 0 (33) u1 Let = 0 represent the location of the end of the stopped traffic at t = 0. It is assumed that the stopped traffic etends indefinitely in one direction and the incoming traffic etends indefinitely in the other. In this case, the initial value problem 2u u t + v1 1 u = 0, < <, > 0 u1 t (34) Models the profile of traffic density ( As in reference [2] u, at later times t. Conclusion: In conclusion, differential equations and partial differential equations are used in many aspects of traffic flow. The method of characteristics is very important in solving these traffic flow

17 problems, and the characteristics eplain the traffic flow velocity vs. the traffic flow density visually. There are many ways at looking into traffic flow, and many other ways to solve these situations, I did not, however, have the time to research them more.

18 References: [1] Haberman, Richard. Mathematical Models, Mechanical Vibrations, Population Dynamics, and Traffic Flow. Philadelphia: Society of Industrial and Applied Mechanics, [2] Knobel, Roger. An Introduction to the Mathematical Theory of Waves. Vol. 3. American Mathematical Society, [3] Hood, Alan. "Method of Characteristics." 30 Oct Mar [4] Haberman, Richard. Applied Partial Differential Equations. Edition 4. Prentice Hall, [5] Zachmanoglou, E.C. Introduction to Partial Differential Equations with Applications. Dover, 1986.

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