Modelingself-consistent multi-class dynamic
tra$c %ow
Hsun-JungCho, Shih-ChingLo
∗Department of Transportation Technology and Management, National Chiao Tung University, Hsinchu 30050, Taiwan
Received 15 January 2002
Abstract
In this study, we present a systematic self-consistent multiclass multilane tra$c model derived from the vehicular Boltzmann equation and the tra$c dispersion model. The multilane domain is considered as a two-dimensional space and the interaction amongvehicles in the domain is described by a dispersion model. The reason we consider a multilane domain as a two-dimensional space is that the drivingbehavior of road users may not be restricted by lanes, especially motorcyclists. The dispersion model, which is a nonlinear Poisson equation, is derived from the car-followingtheory and the equilibrium assumption. Under the concept that all kinds of users share the 6nite section, the density is distributed on a road by the dispersion model. In ad-dition, the dynamic evolution of the tra$c %ow is determined by the systematic gas-kinetic model derived from the Boltzmann equation. MultiplyingBoltzmann equation by the zeroth, 6rst- and second-order moment functions, integratingboth side of the equation and usingchain rules, we can derive continuity, motion and variance equation, respectively. However, the second-order moment function, which is the square of the individual velocity, is employed by previous researches does not have physical meaningin tra$c %ow. Although the second-order expan-sion results in the velocity variance equation, additional terms may be generated. The velocity variance equation we propose is derived from multiplyingBoltzmann equation by the individual velocity variance. It modi6es the previous model and presents a new gas-kinetic tra$c %ow model. By couplingthe gas-kinetic model and the dispersion model, a self-consistent system is presented. c 2002 Elsevier Science B.V. All rights reserved.
PACS: 89.40.+k; 05.60; 47.90.+a; 41.20.cv
Keywords: Boltzmann equation; Poisson equation; Macroscopic tra$c equations; Multiclass tra$c %ow; Multilane tra$c %ow
∗Correspondingauthor. No. 8, Lane 282, Wenhua St., Pingjen City, 324, Taiwan. Tel.: +886-931-040227;
fax: +886-945-866383.
E-mail address: [email protected] (S.-C. Lo).
0378-4371/02/$ - see front matter c 2002 Elsevier Science B.V. All rights reserved.
1. Introduction
Duringthe recent 6ve decades by developingkinetic tra$c %ow model, it is pos-sible to model more realistic tra$c phenomena for tra$c scientists in laboratories. Kinetic tra$c %ow models describe and forecast the time variant tra$c variables, such as density, tra$c volume and velocity. In addition, the performance of the tra$c-control alternatives and the network design can be evaluated by tra$c simulation.
Since Lighthill and Whitham [1] and Richards [2] 6rst proposed their kinetic model, the related subjects are broadly researched and debated. The LWR model was extended to second-order model, which includes the continuity equation and a phenomenolog-ical velocity equation. The second-order model was named PW model [3–14]. How-ever, this kind of models has a lot of arguments, so families of gas-kinematic mod-els [15–40] are presented. The development of gas-kinetic modmod-els includes the dis-cussion of multilane, multiclass users and overtaking, lane-changing, relaxation and interaction maneuvers. As the review of Boltzmann equation, Boltzmann equation is a phase-plane distribution. The macroscopic quantities are derived as follows. The 6rst step is multiplyingBoltzmann equation by the moment functions. The second step is integratingboth sides of the equations and usingthe chain rules. At last, the macroscopic quantities are obtained. Therefore, the resultingmacroscopic quantity and the moment function must have physical meanings. From the previous researches, the second-order moment function multiplied to Boltzmann equation is the square of indi-vidual velocity [18,19,22,30–40]. Nevertheless, the square of indiindi-vidual velocity, which denotes the individual kinetic energy in physics, is meaningless in tra$c. Although the second-order expansion results in the velocity variance equation, additional terms may be generated. For this reason, we multiply Boltzmann equation by the individ-ual velocity variance in order to modify the derivation of velocity variance equation herein.
A complete dynamic system should include motion equations and state equations. The state equation considered in this study is the vehicular dispersion model [41]. The model is derived from the car-followingtheory and the equilibrium assumption. Under a speci6c macroscopic situation, the most possible microscopic combination is de6ned as the equilibrium state. And the system is assumed to tend toward the equilibrium state. Accordingto the dispersion model, density is distributed on the road. By coupling the dispersion model to the kinetic system, a self-consistent system is obtained. Further-more, we consider the multilane model in a two-dimensional space because the driving behavior of road users may not be restricted to drive one by one, especially motor-cyclists [41,42].
The rest of this paper is organized as follows. Section 2 presents the historical evolution from LWR model to gas-kinetic tra$c %ow models brie%y. Section 3 introduces the concept of tra$c 6eld and the derivation from Boltzmann equation to macroscopic systems. In Section 4, the model of the multiclass users is presented. After that, the paper concludes with some perspectives in Section 5.
2. Historical evolution of dynamic macroscopic trac ow models
LWR model and the extended models are the most popular macroscopic dynamic tra$c %ow models. LWR model describes that tra$c is like %uid continuum %owing on highways. Researches based on the model extend the discussion to shock wave analyses, higher order eJect, tra$c control applications, multilane tra$c, and so on. As the purpose of this study is developinga dynamic macroscopic tra$c %ow system, the historical evolution of dynamic macroscopic tra$c %ow models is mentioned 6rst in this section.
2.1. The LWR-like models
Lighthill and Whitham [1] and Richards [2] are the 6rst persons who presented the macroscopic kinetic tra$c %ow model. They used kinematical concepts to describe waves in tra$c. The basic premises of their model are that tra$c is conversed and that there exists a one-to-one relationship between velocity and density. The LWR model can be viewed as a good and basic approximation. Mathematically, LWR model states that the density k and %ow Q satisfy
@k(x; t)
@t + ∇ · Q(x; t) = 0 ; (1)
where t denotes time and x denotes position. Eq. (1) expresses the conservation of vehicles. In addition, Q; k and velocity u are assumed to satisfy Q = ku(k). From these assumption, Eq. (1) has the solution k = F(x − ct), where F is an arbitrary function (the initial condition), c is the wave speed and c = dQ=dk. Eq. (1) implies that inhomogeneities, such as, changes in density of cars, propagate along a stream of cars with constant wave speed c with respect to a stationary observer.
The LWR model is a simple but su$cient tra$c theory if the size and end location of a queue is the things that one only cares about; such as, the time-space trajectory of a shock. Unfortunately, tra$c %ow phenomena are very complex, and some important phenomena that we are also interested in elude the LWR model. One such phenomenon is the stop–start waves in longqueues often observed on congested freeways. There-fore, the development of continuum models extends the LWR model by replacingthe instantaneous %ux function with a dynamic one. This is referred to higher order models, which is also named the PW model.
2.2. The PW-like models
The assumption of u=u(k) is a steady state assumption of velocity, which means that velocity changes instantaneously as density changes. It is certainly not valid in some tra$c %ow situations. To overcome the steady state assumption of velocity, Payne [4] used a motion equation to obtain time variant speed.
@u @t + u(∇ · u) = − 1 k∇ · (Pe(k)) + 1 (ue(k) − u) ; (2)
where ue(k) is an equilibrium speed–density relation. The motion equation (or the
so-called momentum conservation) coupled with the continuity equation is referred to PW model since the concept of employinga higher order function in LWR model is proposed by Whitham 6rst. The term −∇ · (Pe(k))=k = −[Pe(k)(∇ · k)]=k is an
anticipation term, which takes it into account that drivers beware of the precedingtra$c condition, where Pe(k) is an equilibrium tra$c pressure. Payne used an anticipation
term determined by P
e(k) = (1=2 )|ue(k)|. In general, we have Pe(k) = ke(k), where
e(k) denotes the equilibrium velocity variance. Papageorgiou et al. [5] substituted an
Euler-like discrete form for the anticipation term and then Michalopoulos et al. [6] developed a semi-viscous model. Zhang[7] also proposed a new additional model to a longlist of existingmomentum equation. Also, diJerent models are determined by diJerent assumptions of ue(k) and e(k). For example, KNuhne [8], and Kerner and
KohnhNauser [9,10] suggested that e(k) be a constant value c20, whereas Phillips [11]
suggested that e(k) be a linear relation e(k) = m(1 − k=km). The explicit functions
of ue(k) and e(k) can be derived from the equilibrium distribution function of kinetic
theory, respectively.
Daganzo [12] mentioned that although the result of higher order model is a little bet-ter than 6rst-order model, it needs more computation. He also pointed out higher order models bringthe wrongresult; that is, in some cases, vehicle speed will be negative. Aw and Rascle [13] explained that the phenomena is caused by the wrongassumption of tra$c pressure. Recently, GNunther et al. [14] presented a modelingprocedure to ensure that the PW-like models describe all situations correctly.
2.3. The gas-kinetic models
Gas-kinetic theory is a further modelingmethodology. This kind of models is 6rst employed to describe tra$c %ow by Prigogine and his colleagues [15–17] and is referred to the Boltzmann-like model. Hoogendoorn and Bovy [18–20] classi6ed the gas-kinetic models mesoscopic models. Boltzmann equation is widely applied in applied science, such as, gas dynamics, population analysis, tra$c %ow, semiconduc-tor and so on [21]. Generally, Boltzmann equation is used to describe properties of a %uid in the large domain by examining the statistics of motion of constituent particles. Prigogine described a tra$c %uid with a probability density for the velocity (v) of an individual car, f(x; v; t), which may vary with a function of time t and the coordinate x alongthe highway. This density is assumed to satisfy the equation
@f @t + v @f @x = @f @t relaxation+ @f @t interaction : (3)
The 6rst term of the right-hand element of Eq. (3) stems from the fraction that f(x; v; t) diJers from some desired velocity distribution f0(v). The second term
describes that a fast car will slow down owingto the in%uence of a slow car. The interaction term has been criticized. It has been argued that the collision term is only valid in the situation, which describes the incomingvehicle passes each
single car in the queue independently. Therefore, Paveri-Fontana [22] proposed an improved model to overcome the shortcoming of Prigogine’s approach. He generalized the phase-plane distribution as ˜f(x; v; v0; t), where v0 is the individual desired velocity.
f(x; v; t)= ˜f(x;v;v0; t) dv0. Nevertheless, his model still considered that the maneuver
on the lanes of a multilane road is the same and his model does not take queueing eJects into account. Transitory and stationary solutions and numerical simulations are proposed by researches [24–29] under diJerent interaction, relaxation, and adjustment terms.
The nature of the equilibrium solutions of a Boltzmann-like equation is likely to be re%ected in the nature of the associated hydrodynamic models [18,19,22,30–40]. Helbing[30] presents a gas-kinetic model for tra$c operations. In contrast to the model of [22], the model of [30] considers additional terms. The 6rst is a veloc-ity diJusion term, which takes the individual %uctuations of the velocveloc-ity into account due to imperfect driving. The other one is the rate of vehicles entering and leav-ingthe roadway. The macroscopic systematic equation Helbing[30] obtained from the mesoscopic equation has one more equation than PW-like model. The equation is the velocity variance equation. The previous models handle the velocity variance as an equilibrium quantity. However, in nonequilibrium situations, the velocity vari-ance may be better treated as a dynamic variable with a further equation describ-ingits evolution. In particular, to predict tra$c jams, an increase of the variance appears to be a very important indicator. The velocity variance equation is shown as @ @t + u(∇ · ) = −2(∇ · u) + 2 k(∇ · u)2+ 2 (e(k) − ) + k∇ · (∇u) + k∇ · (∇) ; (4)
where is velocity variance, ; are coe$cients, is relaxation time and e is
equi-librium velocity variance. e is assumed to depend on k only as it is assumed in the
PW-like models. Helbing[30] employed a numerical simulation and obtained following results:
(a) The section of high density induces low speed and small speed variance. (b) The section of low density induces high speed and large speed variance. (c) The largest speed variance takes place at the highest speed behind a platoon.
Empirical studies were also employed by Helbing[34,36] to validate the model. Mul-tilane tra$c can also be extended by the model [37,40]. Besides, Hoogendoorn and Bovy [18–20] derived a multiple user-classes tra$c %ow model from Boltzmann equa-tion by a similar approach. From the works of Paveri-Fontana, Helbing, Hoogendoorn and Bovy, macroscopic models can be derived from Boltzmann-like model, which is a microscopic model. Boltzmann-like models can be developed with behavioral analysis. Therefore, the macroscopic models derived from them improve the lack of behavioral analyses of macroscopic models.
3. Derivation from Boltzmann equation to macroscopic models
Since macroscopic models derived from Boltzmann equation can aggregate the microscopic behavior to be group behavior, this study proposes a Boltzmann equa-tion and derives it to macroscopic models. There are three main diJerences between this study and previous studies. The 6rst one is that a multilane road is considered as a two-dimensional domain. The second one is that the acceleration eJect of Boltz-mann equation is considered as the in%uence of tra$c 6eld. The last one is that the second-order moment function considered herein is individual velocity variance. Tra$c 6eld is derived from car-followingtheory, which describes the interaction between ve-hicles. The detail of derivation of tra$c 6eld is illustrated in Section 3.2. The concept of tra$c 6eld not only describes the interaction between vehicles, but also makes the macroscopic system consistent. Before introducingtra$c 6eld, de6nitions of variables and the relations amongvariables should be mentioned 6rst.
3.1. De5nitions
By reason of some drivingbehavior of road users cannot be restricted in one lane or even they derive in one lane they still not be restricted to drive one after one, such as, driving behavior of motorcycles. Therefore, a multilane highway is regarded as a two-dimensional space in this study. We assume that there exists a phase-plane distribution function f(x; v; t) at a given time and at a given point, where x = (x; y) denotes position, v = (vx; vy) denotes individual velocity and t denotes time. Since, v
denotes individual velocity, it is impossible to restrict a speci6c velocity at a speci6c time and place. Thus, v is independent to position x and t, i.e., ∇xv = 0, ∇x· v =
0; @v=@t = 0.
In addition, dx=dt = v and dv=dt = eE, where E denotes tra$c 6eld and is going to derive in detail in Section 3.2. By Taylor’s expansion or total derivation of f, the changing of f is shown by df(x; v; t) dt = @f(x; v; t) @t + v · ∇xf(x; v; t) + eE · ∇vf(x; v; t) = @f(x; v; t) @t coll ; (5) f(x; v; t)|@ v= 0 ; (6)
where f is de6ned on and @ is the boundary of · @ v is the boundary of
individual velocity. Since f is a distribution function, it is reasonable to assume that f is equal to 0 at the extreme value (i.e., boundary @ v). Thus, density is given by
k(x; t) =
and %ow density, which is de6ned by Cho and Lo [14], is given by q(x; t) =
vvf(x; v; t) dv = k(x; t)u(x; t) ; (8)
where u(x; t) denotes average velocity (or the so-called group velocity), which is de6ned as u(x; t) = vf(x; v; t) dv vf(x; v; t) dv : (9)
Flow is denoted by Q(x; t) =yq(x; t) dy, where y is the width of the road.
Next, three kinds of velocity variance are de6ned. The 6rst one is total velocity vari-ance, which is velocity variance between individual velocity and equilibrium velocity. Total velocity variance is de6ned by
(x; t) =
vv − u e2f(x; v; t) dv vf(x; v; t)
: (10)
The second one is individual velocity variance, which is the velocity variance between individual velocity and group velocity. Individual velocity variance is given by a(x; t) = vv − u(x; t) 2f(x; v; t) dv vf(x; v; t) : (11)
The last one is group velocity variance, which is the velocity variance between group velocity and equilibrium velocity. Group velocity variance is given by
e(x; t) = vu(x; t) − u e2f(x; v; t) dv vf(x; v; t) =u(x; t) − ue2 vf(x; v; t) dv vf(x; v; t) : (12) If tra$c %ow is uniform, individual velocity is equal to average velocity a(x; t) =
0. If tra$c %ow is equilibrium, average velocity is equal to equilibrium velocity e(x; t) = 0. The relationship amongthree diJerent velocity variances is
(x; t) = vv − u(x; t) + u(x; t) − u e2f(x; v; t) dv vf(x; v; t) = a(x; t) + e(x; t) ; since v[2(v − u(x; t)) · (u(x; t) − ue(k))]f(x; v; t) dv = 0 : (13)
That implies: (a) total velocity variance is equal to the summation of individual and group velocity variance, (b) ¿ a¿ 0 and ¿ e¿ 0, (c) =0 implies a=e=0.
Furthermore, equilibrium average velocity and equilibrium average variance are denoted by ue(k; u; ) and e(k; u; ), respectively. An equilibrium state is de6ned
as the most possible microscopic state under a speci6c macroscopic state. Since ue
Another variable, which appears in the derivation, is skewness. It is de6ned by #(x; t) = v(v − u(k))v − u(k) 2f(x; v; t) dv vf(x; v; t) ; (14)
which is the bias of the distribution. From the empirical study by Helbing[34,36], #(x; t) ≈ 0. Since the in%uences of the third or higher order moment functions are negligible, we do not have to expand the higher order conservation laws. The basic variables k; Q; ; a and e are scalar and q; u; #(x; t) are vectors.
The basic idea of derivingmacroscopic models from Boltzmann equation is the same as 6ndingthe expectation of a random variable. Therefore, 6ndingthe individual variables that are meaningful and multiplying them to distribution f will obtain macro-scopic variables (average or group quantities). The individual variables are named as moment functions and denoted as (x; v; t). The moment functions chosen by related researches are 1, v, and v2. However, v2 does not have physical meaningin tra$c.
For this reason, the moment functions chosen herein are 1, v, and v − ue2, where
v − ue2 is individual velocity variance. Thus, multiplyingEq. (16) by (x; v; t), we
have @f(x; v; t) @t (x; v; t) + [v · ∇xf(x; v; t)] (x; v; t) + [eE · ∇vf(x; v; t)] (x; v; t) = @f(x; v; t) @t coll (x; v; t) : (15)
The integration form of Eq. (15) is illustrated as v df(x; v; t) dt (x; v; t) dv = v @f(x; v; t) @t (x; v; t) dv + v[v · ∇xf(x; v; t)] (x; v; t) dv + v [eE · ∇vf(x; v; t)] (x; v; t) dv = v @f(x; v; t) @t coll (x; v; t) dv : (16)
By substituting1, v, and v − ue2 for (x; v; t) in Eq. (16), the macroscopic system
will be obtained. The derivation is shown from Sections 3.3–3.5. Before derivingthe macroscopic model from Boltzmann equation, the concept of tra$c 6eld [41] should be mentioned in brief 6rst.
3.2. Tra6c 5eld
Tra$c 6eld is employed to describe the tra$c pressure and the accelerated eJect in this study. Since the tra$c 6eld distributes density on a road, the relation between the tra$c 6eld and the density is named as the dispersion model. The derivation of the
dispersion model starts with the discussion of the interaction between a single vehicle and other vehicles by car-followingtheory [43–46]. Two assumptions are made. The 6rst one is that the in%uence of cars in the same lane is M times larger than that in the adjacent lanes, where M is a scalar. The second one is that the tra$c 6eld is independent of velocity. For the sake of safety, one vehicle on a road adjusts its velocity and spacingaccordingto the relative position between other vehicles so as to avoid the accident. It is assumed that each vehicle has its own 6eld. Vehicles exclude each other by their own 6eld. Thus, the interaction (in terms of tra$c force or tra$c pressure, which is denoted by F), which is produced by the tra$c 6eld ( ˜E), amongvehicles is a resistance. To simplify the complication of the problem, ˜E is assumed to depend on spacingand to satisfy the inverse-square law (the gravity model), which means the in%uence of other vehicles is larger when the spacing is smaller. If we consider the interaction between two vehicles (vehicle 0 and 1), the tra$c 6eld produced by vehicle 1 (leader) will act on vehicle 0 (follower). The tra$c 6eld actingon vehicle 0 can be formulated as ˜E01= e 0 ˜x0− ˜x1 | ˜x0− ˜x1|3i + ˜y0− ˜y1 M2| ˜y 0− ˜y1|3j ; (17)
where e is the passenger car equivalent, 0 is the interactingparameter and ( ˜x0; ˜y0) and
( ˜x1; ˜y1) are the position of vehicle 0 and 1, respectively. The in%uence between two
vehicles is larger as the distance between them decreases. Therefore, the assumption of the inverse-square law is reasonable herein. For convenience, we transform the domain from ˜ to , that is, let x = ˜x; y = M ˜y and tra$c 6eld actingon vehicle 0 in is denoted by E = N i (eiXi= iXi3) ; (18)
where N is the number of vehicles on the road, Xi denotes the spacing. In the
contin-uous space, Eq. (19) can be represented as E =e
((k − ks)=X
2) d ; (19)
where e denotes the passenger car equivalent and denotes the interactingparameter, if vehicles and drivingbehavior on the road are the same. k is the actual density and ks
is the unrestrained density that is the density which vehicles do not interfere with each other. The transformed tra$c 6eld is a conservative 6eld. Then, a potential function # exists by the potential theory. The potential function # satis6es E = −∇x#. Thus, the
magnitude of tra$c 6eld is illustrated as
div E = −R# = e(k − ks)= + Ka; (20)
where div E denotes the magnitude of tra$c 6eld, Ka= Ka(x), which depends on the
position x, is the adjust term of the road condition if the road condition is ideal Ka= 0:
3.3. Continuity equation (conservation of vehicle numbers)
After introducingtra$c 6eld and Poisson equation, derivation from Boltzmann equa-tion to the macroscopic system is presented. Firstly, let = 1, so Eq. (16) becomes Eq. (21), which is expectation of density.
v @f(x; v; t) @t dv + v[v · ∇xf(x; v; t)] dv + v[eE · ∇vf(x; v; t)] dv = v @f(x; v; t) @t colldv : (21)
Each term in Eq. (21) is discussed separately; the 6rst term of left-hand side (LHS) can be obtained from the de6nition (Eq. (7)):
v @f(x; v; t) @t dv = @ @t vf(x; v; t) dv =@k(x; t)@t : (22)
Since individual velocity is independent of position and Eq. (8), the second term of LHS is given by
vv · ∇xf(x; v; t) dv =
v[∇x· (fv) − f · ∇xv] dv = ∇x· q(x; t) : (23)
From Eq. (6), the third term of LHS is represented as v[eE · ∇vf(x; v; t)] dv = eE · v∇vf(x; v; t) dv = eE · [f(x; v; t)|@ v] = 0 : (24)
At last, the collision term is assumed to be equal to zero. Thus right-hand side (RHS) of Eq. (21) is given by v @f(x; v; t) @t colldv = 0 : (25)
Therefore, Eqs. (22)–(25) give the 6rst conservation law; that is, conservation of vehicle numbers.
@k(x; t)
3.4. Motion equation (conservation of momentum)
The motion equation is derived by substitutingthe 6rst order moment function, v, for in Eq. (16). Then the expectation function of velocity is illustrated as
v @f(x; v; t) @t v dv + v[v · ∇xf(x; v; t)]v dv + v[eE · ∇vf(x; v; t)]v dv = v @f(x; v; t) @t collv dv : (27)
Each term in Eq. (27) is also discussed separately; the 6rst term of LHS is obtained v @f(x; v; t) @t v(v) dv = v @ @t[f(x; v; t)v(v)] dv − vf(x; v; t) @ @t[v(v)] dv =@(ku)@t : (28)
By vector analysis, Eqs. (29) and (30) are true,
[v · ∇f(x; v; t)] = ∇ · (fv) − (f · ∇v) = ∇ · (fv); (29) [v · ∇f(x; v; t)]v(v) = ∇ · (fv)v = ∇ · (fvv) − fv · ∇v = ∇ · (fvv): (30) Thus, v[v · ∇xf(x; v; t)]v(v) dv = v∇x· (fvv) dv = ∇x· vfvv dv ; (31) where vv = (v − u + u − ue+ ue)(v − u + u − ue+ ue)
= [(v − u)(u − ue) + (v − u)ue+ (u − ue)(v − u) + ue(v − u)]
+ [uu] + (v − u)(v − u) (32)
is a tensor. Withvf(v − u)(u − ue) dv = k(u − u)(u − ue) = 0 and vf(v − u)uedv =
k(u − u)ue= 0. Eq. (31) becomes
v[v · ∇xf(x; v; t)]v(v) dv = ∇x· vfvv dv = ∇x· vf(v − u)(v − u) dv + vfuu dv : (33)
Assume the in%uence of velocity is dominated by the component of the same direc-tion, i.e., (vx− ux)2ii(vx− ux)(vy− uy)ij and (vy− uy)2jj(vx− ux)(vy− uy)ji.
De6nevf(v − u)(v − u) dv ≈vf(vv − uu)I dv ≡ kXa where I is the 2 × 2 identity
matrix. The summation of diagonal of Xa equals to a, i.e., tr(Xa) = a.
Since
vfuu dv = uu
vf dv = kuu ;
the second term of LHS becomes
v[v · ∇xf(x; v; t)]v(v) dv = ∇x· (kXa+ kuu) : (34)
The third term of LHS is computed by v[eE · ∇vf(x; v; t)]v dv = eE · v∇v(fv) dv − eE · vf∇vv dv = eE · [(fv)|@ v] − eE · vf dv = − ekE : (35)
The RHS of Eq. (27) is assumed to satisfy the relaxation time approximation, i.e., v @f(x; v; t) @t coll v dv = @ @t(ku) coll = −ku − kue m ; (36)
where m is the velocity relaxation time. Eq. (36) means that if the average
veloc-ity does not equal the equilibrium velocveloc-ity, it will become the equilibrium velocveloc-ity gradually after a period of time m(k). Therefore, Eq. (27) becomes
@(ku)
@t + ∇x· (kXa+ kuu) = ekE −
ku − kue
m : (37)
From Eq. (26), Eq. (37) is represented as @u @t + ux· u = − 1 k[∇x· (kXa)] + eE − u − ue m ; (38)
if k = 0. Thus, the conservation of momentum is obtained, which describes the changing of group velocity. Therefore, Eq. (38) also is known as a motion equation.
From Eq. (37), the explicit form of %ow density can be derived as follows: q = ku = − mk@u@t − mkux· u − m[∇x· (kXa)] + meE + kue: (39)
Under steady state and homogeneous velocity assumption, it’s reasonable to assume that u ˙ ue. Let ku − kue= &ku and = m=&. Eq. (39) becomes
where average velocity variance matrix equals to the equilibrium velocity variance (Xa= (e), and v = (e. The 6rst term of Eq. (40) is called the drift term, which is
induced by velocity (or 6eld). The second term of Eq. (40) is called the diJusion term, which is induced by distribution of density. Eq. (40) is the same as the fundamental diagram with diJusion eJect q =ekudrift−'·∇xk, where udrift=E = mE=&. The result
can also be derived from Fick’s law. Eq. (40) can be employed when velocity and velocity variance are homogeneous and stationary. Generally, Eq. (38) is employed as the explicit function of %ow density.
3.5. Variance equation (conservation of energy)
The last equation considered herein is the variance equation, which is obtained by substitutingthe second moment function, v − ue2, for in Eq. (16). Then the
expectation function of velocity variance is illustrated as
v @f(x; v; t) @t v − ue2dv + v[v · ∇xf(x; v; t)]v − ue 2dv + v[eE · ∇vf(x; v; t)]v − ue 2dv = v @f(x; v; t) @t collv − ue 2dv : (41)
Also, the derivation is done by each term of Eq. (41). The 6rst term of LHS is
v @f(x; v; t) @t v − ue2dv = v @ @t(fv − ue2) dv − vf @ @t(v − ue2) dv =@(k )@t : (42)
The second term of LHS is
v[v · ∇xf]v − ue 2dv = v∇x· fv(v − u 2+ 2(v − u) · (u − u e) + u − ue2) dv − vfv · ∇x(v − ue 2) dv = ∇x· [(k#) + (kau) + 2k(u − ue)Xa+ (keu)] ≈ ∇x· [(k u) + 2kXa· (u − ue)] ; (43) since, v(fu(v − u) · (u − ue)) dv = v(f(v − u)) dv · (u − ue) u = 0; # ≈ 0 ;
and
vfv · ∇x(v − ue
2) dv = 0
as mentioned before. The third term of LHS is
v[eE · ∇vf(x; v; t)]v − ue 2dv = eE · v∇v(fv − ue 2) dv − eE · vf∇vv − ue 2dv = eE · [(fv − ue2)|@ v] − eE · vf(2v − 2ue) dv = −2eE · k(u − ue) : (44)
The last term is the RHS of Eq. (41), which is also assumed to satisfy the relaxation time approximation, i.e.,
v @f(x; v; t) @t coll v−ue2dv = @ @t(k ) coll − v(f(x; v; t))coll @ @tv−ue2dv = −k − k e e ; (45)
where e is the relaxation time of velocity variance. Therefore,
@(k )
@t + ∇x· [(k u) + 2k(u − ue)Xa] = −2ekE · (u − ue) −
k − k e
e (46)
is obtained. From Eq. (26), Eq. (46) is represented as @ @t + u · ∇x = −2eE · (u − ue) − − e e − 2 k∇x· [k(u − ue)Xa] ; (47) if k = 0. Thus, the conservation of energy is obtained, which describes the changing of group velocity variance.
Other conservation laws can be derived by multiplyinghigher order moment func-tions to Boltzmann equation as longas the moment funcfunc-tions are meaningful and the macroscopic quantities obtained are signi6cant. As mentioned by Helbing [34,36], Boltzmann equation multiplied by the third-order moment function produces the expectation function of skewness, which is near zero. Therefore, higher order moment functions are not discussed in this study.
The system equations developed above includes three conservation laws derived from Boltzmann equation and Poisson equation derived from tra$c 6eld. These four equa-tions are transient equaequa-tions, which describe the changing of variables. However, a complete dynamic system not only needs transient equations, but also needs state equa-tions, which describe the state of variables. A state equation is needed so as to make
the system self-consistent. In this study, the dynamic system is assumed to become the equilibrium state gradually. Therefore, the equilibrium distribution is employed to be the state equation in the system. The derivation of the equilibrium distribution is shown in the followingsection.
3.6. Derivation of the equilibrium distribution
Since the tra$c 6eld aJects the movement of vehicles, density should be distributed by the 6eld (or potential). The relation between density and potential is obtained from the assumption that density will tend to become its equilibrium state under a spe-ci6c tra$c situation. The equation is derived by solvingthe followingmathematical programming: Max W = N! i ni! ∀i ∈ ; (48) s:t: i ni= n1+ n2+ · · · + nm= N ∀i ∈ ; (49) i ni i= n1 1+ n2 2+ · · · + nm m= tol ∀i ∈ ; (50)
where i is the number of intervals on the road, ni is the vehicle number of interval i,
tol is the total velocity variance, and i is the velocity variance in interval i. The
velocity variance of individual car is de6ned as Tui− ue2, where Tui is the average
velocity of interval i and ue is the equilibrium velocity. Eq. (48) is the objective
function and Eqs. (49) and (50) are the given macroscopic phenomena. Eq. (48) 6nds out the most possible combination of ni if the total number of cars on the road is N.
Eq. (48) is a simpli6ed form, which neglects total number of all-possible combination at denominator, since the denominator is a constant. Eq. (49) is the conservation of vehicle numbers and Eq. (50) is the conservation of total variance. The mathematical programming can be solved by the KKT condition. The solution obtained is the most possible density distribution, which is denoted by k = k( ). However, variance is not considered in some models, such as, LWR model and PW model. Therefore, the function k=k( ) must be converted to k=k(#). The transformation is made as follows. Since the tra$c pressure F is proportional to the acceleration, i.e., F ˙ du=dt, we have F ˙ −∇ from the relation between acceleration and energy because variance can be referred to energy. From the relation among the tra$c pressure, tra$c 6eld and tra$c potential, F ˙ E = −∇#, we have = e# while passenger car equivalent e is a relative scalar. Through the transformation, the equilibrium distribution of density not only can be coupled with the gas-kinetic model, but also can be coupled with the LWR-like model and PW-like model. Transformingvelocity variance into potential, the equilibrium distribution is given as
k = K0exp ((e − e#)= e) ; (51)
where K0 is the essential density, e is the equilibrium velocity variance, is the
potential equivalent of the velocity variance threshold. is named as the potential barrier here. We can infer several points from Eq. (51). The 6rst one is that density
Fig. 1. The density curves of diJerent potential barrier and equilibrium velocity variance, where k1= K0exp((e 1− e#)= e1), k2= K0exp ((e 1− e#)= e2) and k3= K0exp ((e 2− e#)= e1) · e1¿ e2
and 1¿ 2.
decreases as tra$c potential increases. The second one is as the equilibrium velocity variance increases, the variation of density increases, which means the tra$c is sen-sitive. The third one is as the potential barrier is low, the density is small; that is, drivers are aggressive. The three points are also illustrated in Fig. 1.
By couplingEqs. (20) and (51), the nonlinear dispersion model is obtained. If a set of boundary conditions of the tra$c potential is applied, vehicles are forced to drive through the road according to the path, which has the least resistance. Therefore, vehicles on the two-dimensional research domain will not move forward and backward or circle round. They will try to pass through the road as soon as possible.
3.7. Closure relations
The system presented herein also needs closure relations so as to determine the equilibrium velocity ue(k; u; ), equilibrium variance e(k; u; ), and relaxation time
m and e in Eqs. (37) and (46). There are a variety of possible closure relations,
which could be adopted from previous studies. [3,8–11,13,14,20,40]. The ue and e
proposed in study are represented by
ue(k; u; ) = u0− mpb(k)k (52)
and
e(k; u; ) = epp(k) · k u ; (53)
respectively. u0 is the average desired velocity, pb(k) ∈ [0; 1] is the
brakingproba-bility vector, and pp(k) ∈ [0; 1] is the passingprobability vector. The explicit forms
are obtained by specifyingexpressions for pb(k) and pp(k). Eq. (52) means that the
variance increases as pp(k) increases. Since ue(k) and e(k; u) are equilibrium
equa-tions, the functions suggested in this study are ue(k) = vvfe(x; v; t) dv vfe(x; v; t) dv (54) and e(k) = vv − u e2fe(x; v; t) dv vfe(x; v; t) dv (55) respectively. fe denotes the steady state homogeneous solution of Boltzmann equation.
The relaxation time m and e are shown as
m(k) =g(k)Tm and e(k) = g(k)Te ; (56)
which are modi6ed from the suggestion of Helbing [38]. g(k) is the proportion of freely movingvehicles, Tm is the reaction time of velocity and Te is the reaction time
of variance. As fe, g(k) Tm and Te are determined, the closure relations are expressed
speci6cally. Then, the self-consistent system is complete. 4. Multiclass users model
Dynamic multilane tra$c %ow model proposed in the previous studies includes three conservation laws, which control the motion of vehicles, and a nonlinear Poisson equa-tion, which distribute the density on the road. The set of equations can only describe one drivingbehavior. Fortunately, the system can be extended to multi-class users model by employingthe concept of Hoogendoorn and Bovy [18–20]. They considered that:
(a) each class has diJerent behavior and is described by diJerent conservation laws; (b) space of a road section is limited and all class of users share the space.
The 6rst assumption is easy to achieve. Accordingto diJerent drivingbehavior, a speci6c Boltzmann equation is derived. With the same procedure mentioned in Section 3, the macroscopic kinetic system of each class of user is derived from the speci6c Boltzmann equation. The concept of Hoogendoorn and Bovy’s second assumption is the same as Poisson equation (Eq. (20)) of our system. If there are i classes of users or vehicles on the road, Eq. (19) is modi6ed as
E = i ei i ki(x; y) X2 d − ek s+ Ka; (57)
where the subscript i denotes the variables of user i. Thus, div E = −R# = i eiki i − ek s+ Ka: (58)
The other equations in the multi-class users’ system are illustrated as follows: qi(x; t) = ki(x; t)ui(x; t) ; (59) @ki(x; t) @t + ∇x· qi(x; t) = 0 ; (60) @ui @t + ui∇x· ui= − 1 kc[∇x· (kiXia)] − ei∇# − ui− uie im ; (61) @ i @t + ui· ∇x i= 2ei∇# · (ui− uie) − i− ie ie − 2 ki∇x· [ki(ui− uie)Xia] ; (62) ki= Ki0exp −ei# − ei i ie : (63)
The system equations above mean that each class of user is controlled by his own conservation laws. By couplingwith the Poisson equation (Eq. (57)), a self-consistent multi-class users dynamic tra$c %ow model is obtained. Total %ow density and total density are represented as
q(x; t) = i eiqi(x; t) = i eiki(x; t)ui(x; t) (64) and k(x; t) = i eiki(x; t) ; (65) respectively.
5. Conclusions and perspectives
In this paper we have derived a macroscopic multilane tra$c model for multiple classes users. The system is a self-consistent system; it can be solved with proper initial conditions and boundary conditions. Our consideration is based on the following assumptions:
(a) A multilane road is considered as a two-dimensional domain; (b) the whole system will tend toward equilibrium;
(c) each class of user has diJerent behavior and is described by diJerent conservation laws (the gas-kinetic model);
(d) the individual velocity variance is employed as the second moment function; (e) the space of a road section is limited and all classes of users share the space.
Consideringa multilane road as a two-dimensional domain allows us to handle the drivingbehaviors, which are not restricted to drive one by one in a single lane. Another advantage of this consideration is to avoid modeling complicated lane-changing behavior. Lane-changing behavior is controlled by the nonlinear Poisson equation. If the
Fig. 2. The relationship between models.
research area is only a single lane road, the system can be reduced to a one-dimensional model.
This study derives a dynamic macroscopic tra$c %ow system from Boltzmann equa-tion. Boltzmann equation employed herein includes accelerated eJect, which is con-trolled by Poisson equation. Helbing[30–40] and Hoogendoorn and Bovy [18,19] followed Paveri-Fontana [22] to multiply Boltzmann equation by 1, v and v2 so as to
derive macroscopic systems. However, the second-order moment v2 does not make
sense in tra$c %ow. Although multiplying Boltzmann equation by v2 can obtain
velocity variance equation, it may also generate some meaningless terms. This study modi6es the second moment function as v − ue2, which is individual velocity
vari-ance, and reformulates velocity variance equation to be more reasonable. This study exposes three moment functions. If there still exists the other meaningful moment, it should be considered as its in%uence is signi6cant.
In addition, the Poisson equation plays an important role in the system. Since Poisson equation distributes vehicles on a road, it becomes the key point to extend the system to multiclass user tra$c. Besides, the equilibrium distribution is employed to determine the state of tra$c %ow. Thus, the system is self-consistent.
The system equations and the relationship amongthem are illustrated in Fig. 2. From Fig. 2, the system equations can be simpli6ed to adapt diJerent tra$c condition because
not all variables are signi6cant in each tra$c condition. For example, in uniform and equilibrium tra$c %ow, the in%uence of velocity variance equation may be ignored. The simpli6cation is needed because computation of the whole system takes a lot of time. Accordingto diJerent tra$c conditions, simpli6cation should be discussed and validated further. Also, numerical methods should be developed to solve the system. References
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