A continuum model for the dispersion of traffic on two-lane roads

Pergamon Trunspn Rex-B. Vol. 31, No. 6. pp. 413 485, 1997 Q? 1997 Elsevier Science Ltd All rights reserved. Printed in Great Britain 0191-261397 $17.00+0.00 PII: SO19L2615(97)00009-X A CONTINUUM MODEL FOR THE DISPERSION OF TRAFFIC ON TWO-LANE ROADS EDWARD N. HOLLAND Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, UK, CB3 9EW and ANDREW W. WOODS* School of Mathematics, University of Bristol, Bristol, UK, BS8 ITW (Received 6 February 1997; in revisedform 18 April 1997) Abstract-A continuum model for two-lane traffic flow is developed using the theory of kinematic waves in which the wavespeeds in the two lanes are assumed constant but unequal. The transient behaviour is found exactly using Riemann’s method of characteristics and an asymptotic model of the long time flow is descri- bed. It is shown, that for large times, the traffic concentration moves with a weighted mean wavespeed of the two lanes and disperses about this mean speed as a result of interlane concentration differences generated by the relative wavespeeds. The dispersion can be described by a virtual coefficient of diffusion proportional to the square of the differences of the two wavespeeds and inversely proportional to the rate of lane changing. The technique is extended to describe three-lane traffic flow and to include the dependence of wavespeed upon concentration. 0 1997 Elsevier Science Ltd Keywords: two-lane flow, continuum models, dispersion. I. INTRODUCTION As building new roads becomes more expensive and less attractive, it has become increasingly important to assess the efficiency of existing road networks. With technological advances, far more traffic data is available, real time computation is possible, and there will soon be the opportunity to control traffic dynamically at a macroscopic level. The main focus of such work is to control high flow traffic, where transitions from free flow to stop-start regimes often occur quite suddenly. A major priority of these projects lies in developing a quantitative description of congested traffic. When a road becomes congested the speed of each car is, to a large extent, controlled by the surrounding flow rather than the driver’s individual wishes. This observation suggests that the behaviour of individuals can be approximately described by averaged quantities, such as concen- tration and flow. Using such continuum models, it is possible to postulate macroscopic strategies that should improve safety and efficiency. The first major theoretical contribution to continuum traffic modelling was published by Lighthill and Whitham (1955). In this paper the concepts of concentration and flow as continuous variables on a single lane highway were introduced. Lighthill and Whitham showed that a conti- nuity equation, imposing a conservation law on the number of cars, combined with the assump- tion that flow is solely dependent on the concentration of traffic, leads to a kinematic theory of traffic in which the speed of individual cars may differ from the rate of propagation of concentra- tion fluctuations. A kinematic wave is a region of congestion that moves down the road with a fixed velocity. Plotting the evolution of these waves leads to many of them crossing. At these points a discontinuity in concentration is formed and the resulting motion of the shock can be calculated. Lighthill and Whitham investigated shock formation in the context of traffic lights, bottlenecks and regions of high concentration. They concluded that the presence of a traffic hump, *Author for correspondence. 473 414 Edward N. Holland and Andrew W. Woods or region of high concentration, leads to the formation of a shock if the wavespeed increases with concentration. This is because the waves in light (low concentration) traffic will tend to catch those in heavier (high concentration) traffic leading to a discontinuity. Lighthill and Whitham proposed that this model could be extended to a two-lane highway simply by doubling the flow for a given concentration. Munjal and Pipes (1971) and Munjal et al. (1971) adopted a different approach to investigate the coupling of several lanes of traffic, by explicitly accounting for the flow in each lane, extending the theory of Lighthill and Whitham. They assumed that the rate of lane-changing between each pair of adjacent lanes is proportional to the difference in their concentrations and that all lanes are equivalent in the sense that the wavespeeds are equal. They investigated the propagation of small amounts of extra traffic on a uniform background. Since the extra traffic does not have a significant effect on the concentration, they assumed that the wavespeeds can be approximated as constant. Their model predicts that traffic entering from an on ramp moves down the road whilst equili- brating between the lanes. The relative position of concentration along the road remains constant owing to the constant wavespeeds. They obtained some experimental evidence from American highways, which seemed to support their model (Munjal and Pipes, 1971). This showed that con- tinuum theories can, even in their simplest form, be used as models for multilane traffic flow. Since these papers, a number of further, mainly numerical, studies have investigated various aspects of one lane traffic flow in more detail. In the present work, we build upon the work of Munjal and Pipes by allowing the wavespeeds to vary between lanes and the lane changing law to become more general. This adds considerably more complexity to the problem and allows a number of important processes to occur. The phi- losophy of this work is to expose some of the effects in a theoretical framework and thereby pro- vide insight into some of the fundamental aspects of traffic dynamics, although realising that a real flow system is more complex than this simplified model. It is shown that the relative motion of traffic in different lanes causes local fluctuations in- traffic concentration to spread out along stream, thereby smoothing out concentration gradients. This process is analogous to Taylor dif- fusion in fluid dynamics (Taylor, 1953). This dispersion also tends to suppress the formation of shocks, described by Lighthill and Whitham, because the dispersion is able to smooth out the concentration gradients which develop prior to shock formation. We introduce our model of two-lane traffic flow in Section 2 and describe the long time asymptotic behaviour of this model in Section 3. Section 4 describes a method of obtaining exact solutions of the two-lane flow model, and in Section 5 we use this method to examine the evolution of(i) a local region in which the traffic concentration is elevated above a uniform background level and (ii) a continuous stream of traffic entering a steady background flow from a fixed point on the highway. In Section 6 we extend the model to describe traffic flow on a 3-lane highway and we also consider the case in which the wave-speed depends upon the concentration in order to identify the effect of the dispersion upon the formation of shocks, as described by Lighthill and Whitham. 2. A MODEL OF TWO-LANE HIGHWAYS The continuum variables necessary to describe two lane traffic flow are shown in the table below Symbol Physical description &(x7 t) the concentration (cars per unit distance) in lane i qi(x, t, &) the flow (cars per unit time) in lane i ci(x, t, Ki) the wavespeed of kinematic waves in lane i VI(X, t)j the space averaged velocity in lane i, qi/Ki Q(x, t, Kl. K2, VI, ~2) the lane changing flux from lane 2 to lane 1 The conservation of cars on a two lane highway can be represented by an equation for the concentration in each lane (Lighthill and Whitham, 1955) (KI), + (qi)x = Q(x, t, K1, K2, VI 7 ~2) (1) (K2Kz), +(42)x = -Q+. t, KI, Eci, VI. ~2) A continuum model for traffic dispersion 415 where subscripts 1, 2 label lanes and subscripts x, t denote partial derivatives with respect to road position x and time t. For simplicity we restrict attention to homogeneous highways on which Q does not depend on either x or t. We also adopt the kinematic approach of Lighthill and Whi- tham, and assume that the flow depends only on the concentration Ki. In all but the last section, we examine small fluctuations in concentration about a constant background and therefore, to leading order, the wavespeeds ci = & in each lane are constant (see Fig. I). In this limit, eqns (1) and (2) then reduce to the form (KI), + CI(KI), = QWI, K29 “19 “2) (3) (K2), + czW2), = -QVG 7 K2, VI, “2) (4) Equations 3 and (4) generalise the Munjal and Pipes model in that the wavespeeds in each lane, c,. are now allowed to be different and the lane changing term is now more general. For small differences in concentration, any genera1 lane changing function of the velocities and concentrations will reduce to a linear combination of KI, K2, VI and ~2, for example ai (K2 - )ci KI) + U~(VI - &IQ). Hence since velocity will be linear with concentration for these small differences, the lane changing will reduce to a(K2 - AK]) + B, where a, B and J. are all con- stants. The parameter a represents the rate of lane changing and A is the typical time for a lane change. The parameter h represents the balance between Kl and K2 about equilibrium, thus determining into what proportions any extra concentration is split, in this case 1%. B represents the balance between KI and K2 at equilibrium. Subtracting the background concentration, Klo and K20 say, from the respective lanes, k, = K,- Ka, gives the evolution equations for small perturbations to the equilibrium distribution of traffic. Pi), +cI&), = 4kz - kkl) Oh), + cz(kz)x = 4A.kl - W (5) (6) These equations are applicable when ki < Kio. It is possible to express the problem more simply by transforming from the original frame of reference S to a frame ?? moving with a weighted average of the two wavespeeds, C. This gives (kl), + hAc(kl)x’ = u(k2 - hkl) (7) (k& - Ac(k2)x’ = u(lk, - k2) (8) where x’ = x - Tt, T = & (ci + Lc~), AC = & (cl - ~2). In the frame s, variations in traffic den- sity appear to move with speeds AC, forwards in the faster lane and AAc backwards in the slower lane. Adding and subtracting eqns (7) and (8) gives Fig. I. Data from lanes I and 3 of the M25 in the UK showing the gradient of flow with respect to concentration approximately constant with small changes in concentration (data courtesy of Peter Gray. Transport Resarch Laboratory, UK). 476 Edward N. Holland and Andrew W. Woods k, + AcAkY = 0 (9) Ak, + Ac[Ak - (1 - i)Ak], = -( 1 + )c)aAk (10) where k = kl + kz, Ak = Akl - k2. In the case that the two wavespeeds are equal, AC vanishes to leave the sum k constant. The difference in concentration between lanes, Ak, decays exponentially as a result of the cross lane transfer of cars. This is the solution that Munjal and Pipes investigated and shows that concentration differences decay over a timescale of order i. In the case AC # 0, the process is more complex, and it is helpful to introduce a dimensionless time r and position in the moving frame z, where t = at (11) This simplifies the equations to the form it, + Ak, = 0 (13) Ak, + Ai& - (1 - QAk, = -(l + QAk (14) 3. LONG TERM BEHAVIOUR OF A TWO-LANE HIGHWAY Equation (14) suggests that differences in concentration Ak will initially decay as found by Munjal and Pipes. After a time r of order O(l), Ak decays to a sufficiently small value that the relative motion between the lanes represented by kz becomes important. A balance is then estab- lished between the generation of cross-lane concentration differences, which are produced by the along-lane shearing of concentration gradients, and the damping of cross-lane concentration gra- dients by the flow of cars from one lane to the other. In this limit, A& N -( 1 + )L) Ak and eqns (13) and (14) reduce to the form (15) This is a diffusion equation for the mean concentration and reveals the approximate evolution of vehicle concentration for t B 1, i.e. t >> i. In dimensional terms the effective diffusion coefficient has the form A(Ac)~/( 1 + A)a. As the equations are linear, complicated perturbations to the flow can be analysed by combining the solutions of a series of pulses. It is therefore interesting to look at the progress of a single traffic pulse containing N cars and entering the highway at t = 0. The distribution of this new traffic, in the limit t >> i, is Gaussian centred a distance Tt downstream and spread out over a distance (16) The diffusion eqn (15) also suggests that when there is a discrete change in concentration, as might occur on a congested road, the front becomes spread out a distance AC 211 r m after a time t where t > i. In this case the long time asymptotic concentration then has the form (17) In the next section a method of solution of the full system (13) and (14) is developed. This enables us to describe the temporal evolution of the traffic before these long term asymptotic solutions apply. A continuum model for traffic dispersion 4. THE GENERAL SOLUTION Recall eqns (5) and (6) for the concentrations in each lane @I), + cl&), = 42 - kkl) (18) (k2)t + C2(k2).r = 4hkl - k2) By making the substitution to scaled characteristic co-ordinates (19) (20) (21) (5) and (6) reduce to 2Akls = k2 - Ak, (22) 2k2, = kk, - k2 (23) Differentiating eqn (22) with respect to a, k2 may be eliminated via eqn (23) to yield a second order partial differential equation for kl . Similarly, kl may be eliminated to yield a partial differ- ential equation for k2. In both cases the same evolution equation is obtained, and has the hyper- bolic form (24) Substituting u(CY, /!?) = f&“+@‘f(cr, /?) (25) to eliminate the first derivative yields the simple equation which can be solved using Riemann’s method of characteristics (Zwillinger, 1989). This method uses a general&d Green’s function called a Riemann function R(cr, /3, t, q) with four arguments. The Riemann function associated with eqn (26) is a modified Bessel function of order zero, The solution forf(ar, j?) can then be written in terms of this function by integration along the line in (a, @space on which the initial data is specified. Further details may be found in the book by Zwillinger. The final result is Q u(a, j3) = -!e-ba+a, 2 R(Z’2f(P) + R(QHQ) - /(Rfq -fR,)d,, + (fRp - Rf)dt 1 (28) P where the integral is taken along the line PQ on which the initial data is specified in Fig. 2. In order to solve u(a, /I), a numerical scheme was developed to evaluate the integrals in eqn (28). These solutions give the exact representation of the flow at all times and places downstream of the source. The process of obtaining the derivatives from the initial data is described in Appendix for the dispersal of a pulse traffic in one lane. 478 Edward N. Holland and Andrew W. Woods Fig. 2. The integration domain in (6, Q) space. 5. SOLUTIONS FOR PARTICULAR TRAFFIC SOURCES In this section the method described in Section 4 is used to examine the evolution of both a single pulse of traffic and a continuous source of traffic. These full solutions are compared with the asymptotic analysis of Section 3. First, a pulse of traffic entering the slow lane is investigated. Second, a continuous source of traffic entering the slow lane is examined. 5.1. Solution for a traffic hump A pulse of traffic, superposed on a constant background, entering the slow lane may be repre- sented mathematically by the initial conditions kt (0, t) = sin2(wt)@(t)@(t - to) (29) where w = t. k2(0, t) = 0 (30) Figure 3 shows the concentration of traffic in each lane at four distinct times after this pulse of cars has entered the highway. These solutions are calculated using the method of Section 4. The first graph, at time T = 1, shows that there is still more traffic in the slow lane than in the fast lane. This is because the initial pulse of traffic enters the slow lane, and there has not been sufficient time for concentration equilibration between the lanes. At time T = 2, however, there has been enough time for a significant amount of traffic to change lanes. Initially cars move into the slow lane, and then advance ahead of the initial pulse of cars in the slow lane. Some of this traffic in the fast lane is then able to move back into the slow lane. As a result, at t = 4 a double peak in the concen- tration of traffic has developed in the slow lane, and cars naturally tend to circulate within the traffic wave. By the time T = 8, sufficient time has elapsed that the transient fluctuations in traffic concentration have decayed, and the traffic has equilibrated between lanes. In this case, the asymptotic dispersion model of Section 3 becomes approximately correct. The distribution of the sum of the concentrations along the road is approximately Gaussian [eqn (16)], and the concentra- tions in the slow and fast lanes become nearly symmetrical (allowing for the 1 :h balance) about the mean position. The effective diffusion process, caused by the interlane shearing and the cross-lane transfer of cars, causes the traffic to become progressively spread out, resulting in a continual decrease in the amplitude of the concentration anomaly, as can be seen from Fig. 4 at T = 4, 8 and 16. The dispersion process may be understood from the above numerical solutions. There is slightly more traffic in the fast lane ahead of the main peak. Therefore, these cars change lane, adding to the front of the distribution of cars in the slow lane. Also, there are slightly more cars in the slow lane behind the main peak. Some of these move into the fast lane and add to the back of the A continuum model for traffic dispersion 479 Fig. 3. Concentration of traffic in the slow and fast lanes subsequent to a sinusoidal input into the slow lane for ,I = 0.8. The concentration on the y axis is the additional concentration above background with I. 0 corresponding to the maxi- mum of the initial sinusoidal pulse. The non-dimensional distance is s. Curves are shown at dimensionless times r = I, 2, 4 and 8. distribution of cars in the fast lane. As a result, the velocity of the overall pulse of traffic approaches the average of the two wavespeeds while the cars become increasingly spread out about this mean. Figure 5 shows the exact position of the centre of mass of the distribution in comparison to the prediction of the asymptotic dispersion theory of Section 3. The figure shows that the asymptotic approximation of Section 3 is very accurate. There is a constant offset which is a relic of the initial distribution. This means that x = ?t + x0, where x0 is an effective origin of the source. This con- stant was taken into account when calculating the Gaussian fit curves in Fig. 3. Figure 5 also shows the variance of the actual distribution of traffic in comparison with the asymptotic solution of Section 3. Again, the agreement is very good, with a constant offset corresponding to the var- iance associated with the initial distribution. We deduce that after times of order T N 4, the asymptotic dispersion theory provides an accurate description of the traffic flow on a two lane highway in which the speed on each lane may be different. S.2. A continuous release of cars We now consider the case in which an additional constant traffic source at x = 0 is added to a uniform background flow at time t = 0. This is a simple model for example of the effect of dis- persion upon the merging of flow from a controlled slip road. The initial data can be summarised by the equations k,(O, t) = o(t) (31) kz(0, t) = 0 (