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International Journal of Advances inEngineering Sciences and AppliedMathematics ISSN 0975-0770Volume 5Combined 2-3 Int J Adv Eng Sci Appl Math (2013)5:158-176DOI 10.1007/s12572-013-0087-1
State-of-the art of macroscopic traffic flowmodelling
Ranju Mohan & GitakrishnanRamadurai
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State-of-the art of macroscopic traffic flow modelling
Ranju Mohan • Gitakrishnan Ramadurai
Published online: 15 August 2013
� Indian Institute of Technology Madras 2013
Abstract Macroscopic traffic flow models are suited for
large scale, network wide applications where the macro-
characteristics of the flow are of prime interest. A clear
understanding of the existing macro-level traffic flow
models will help in modelling of varying traffic scenarios
more accurately. Existing state-of-the-art reports on traffic
flow models have not considered macro-level models
exclusively. This paper gives a review of macroscopic
modelling approaches used for traffic networks including
recent research in the past decade. The modelling of the
two main components of the network i.e. links and nodes
are reviewed separately in two sections and solution pro-
cedures are discussed followed by a synthesis on the
advantages and disadvantages of these models. This review
should encourage efficient research in this area towards
network level application of these models.
Keywords Macroscopic traffic flow models �Link modelling � Cell transmission model �Riemann problem � Porous flow approach � Node modelling
1 Introduction
Traffic phenomena are complex and nonlinear depending on
the interactions of a large number of vehicles. Network level
traffic flow model must include three features—capturing of
bottleneck effects, computing wave propagation, and
modelling of intersections. The transportation system
infrastructure, time-dependent traffic demand, level of traffic
control, and stochastic behavior of people make the traffic
flow modelling difficult. Traffic flow can be analyzed at
macroscopic, mesoscopic, and microscopic levels of aggre-
gation. Macroscopic models are the aggregation of individ-
ual vehicle dynamics and mainly focus on describing the
overall stream features such as congestion, delay, and queue
formation. These models are suited for large scale, network
wide applications where the macro-characteristics of traffic
(speed, density, and flow) are of prime interest. Microscopic
models focus on individual vehicles and their interaction
with neighboring vehicles and describes phenomena such as
vehicle following, overtaking, lane changing, and gap
acceptance. Though these models are often seen as more
‘realistic’ representation of traffic flow, computational
complexity limits use of these models for network wide
applications. Mesoscopic models model traffic as vehicular
packets or even individual vehicles, but still governed by
macroscopic flow laws. Thus, retaining the advantage of less
complexity, mesoscopic models can capture the level of
detail closer to that by microscopic models.
At network level macroscopic models will be the best
choice to balance the tradeoff between solution detail and
computational effort. A clear understanding of the existing
macro-level traffic flow models will help in modelling of
varying traffic scenarios more accurately. Existing state-of-
the-art reports on traffic flow models have not considered
macro-level models exclusively (the last comprehensive
review was by Hoogendoorn and Bovy [1]—over a decade
earlier). This paper gives a review of traffic flow models,
restricted to macroscopic modelling approaches, for links
and nodes.
The important modelling approaches for link are clas-
sified in this paper as follows.
R. Mohan (&) � G. Ramadurai
Department of Civil Engineering, Indian Institute of Technology,
Madras, Chennai 600032, India
e-mail: [email protected]
G. Ramadurai
e-mail: [email protected]
123
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DOI 10.1007/s12572-013-0087-1 IIT, Madras
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(1) Continuum models
(a) First-order model (LWR model)
(b) Higher order models
(2) Non-continuum model.
Node models are classified as follows.
(1) First generation models
(2) Models based on constraint flow maximization.
Section 2 discusses the important macroscopic link
models and in Sect. 3, several approaches for node mod-
elling are reviewed. Section 4 shows a synthesis on the
choice of the above models and Sect. 5 draws conclusions.
2 Link modelling
2.1 Continuum models
Continuum models assume traffic flows similar to a com-
pressible fluid flow. This family of models started with the
simple, well-known first-order (time and space derivatives
are the order of one) LWR model [2, 3] and later developed
into higher order models. The basic, common equations for
all these models are the laws of conservation, fundamental
equation of traffic flow, and the speed–density relationship.
These models are reviewed in the following subsections.
2.1.1 First-order model (LWR model)
This model gives the dynamic equations for density
through Eqs. (1)–(3) [4].
ok
otþ oq
ox¼ 0; ð1Þ
u ¼ ueðkÞ; ð2Þq ¼ ku: ð3Þ
Here, x, t, u, k and q represent the space, time, speed,
density, and flow, respectively, and ue is the equilibrium
speed from the fundamental diagram of traffic flow.
Equation (1) is the flow conservation equation which is a
nonlinear hyperbolic partial differential equation and can
be rewritten as Eq. (4).
oU
otþ A
oU
ox¼ 0; ð4Þ
where U = [k], A = qF/qU and F = [q]. An equation
(system of equations) is said to be hyperbolic if Eigen
value(s) of A is (are) real (real and distinct). The dynamic
density, k(x, t) can be obtained by using the property of
characteristic curves. In LWR model, characteristic curves
are straight lines emanating from the boundary t = 0 and
passing through the points of equal density. The slope of
the characteristic curve is the speed with which the small
disturbances (flow change due to small change in density)
propagate and is given by the Eigen value of the matrix A,
dq/dk. The speed of the large disturbances—shockwaves
formed by two different steady states—is given by the
Rankine-Hugoniot condition as below:
w12 ¼q1 � q2
k1 � k2
; ð5Þ
where w is the shockwave speed and the indices 1 and 2
represent the two different steady states.
Though the model can represent explicitly the formation
of shockwaves, it is not capable of explaining other phe-
nomena such as steady state speed–density relationship,
discontinuities in the density, regular start–stop waves,
traffic hysteresis, localized clusters and phantom jams.
2.1.1.1 Extension of LWR model LWR model assumes
single lane, homogeneous traffic flow. As an extension to
the model, several papers tried to describe traffic with two
approaches. The first approach was to make a distinction
between lanes. Munjal and Pipes [5], Munjal et al. [6],
Holland and Woods [7] and Greenberg et al. [8] used this
approach and a separate LWR model and fundamental
diagram is used for each lane. This leads to a parallel
coupling of several LWR model using exchange terms. A
general formulation of this simple extension of LWR
model to multilane traffic is given in [4] as below.
oki
otþ oqi
ox¼ gi þ Qi i ¼ 1; 2; . . .;N; ð6Þ
Qi ¼ ai;i�1 ki�1ðx; t � sÞ � kiðx; t � sÞð Þ � kði�1Þ0; �ki0
� �� �
þ ai;iþ1 kiþ1ðx; t � sÞ � kiðx; t � sÞð Þ½� kðiþ1Þ0; �ki0
� ��i ¼ 1; 2; . . .;N;
ð7Þ
where N is the number of lanes, Qi is the lane changing rate
for lane i, i ? 1 and i - 1 are the neighboring lanes to lane
i, ai,i-1 and ai,i?1 are sensitivity coefficient describing the
intensity of interaction, gi is the rate of generation or loss at
entrance or exit ramps (equal to zero for all internal lanes)
and ki0 is the equilibrium density of the ith lane. For the
first (last) lane, i - 1 (i ? 1) should be set as i. Here, the
exchange of vehicles between neighboring lanes is pro-
portional to the difference of deviation of their densities
from equilibrium values. In Munjal and Pipe’s [5] model,
lane changing flows are prioritized over through flows on
target lanes. Laval [9] overcame this limitation proposing a
multilane cell transmission rule.
A second approach was to divide the vehicle population
into different classes having different driving characteris-
tics. The interaction between these classes will give a better
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description of heterogeneous traffic flow. Zhang and Jin
[10], Logghe and Immers [11] and Chanut and Buisson
[12] used this approach with Eulerian coordinate system.
These models used flow conservation equation separately
for each vehicle class and the class wise flows are formu-
lated as a fraction of the total flow from fundamental dia-
gram. In general, these models can be represented as [13]:
oki
otþ oqi
ox¼ 0 i ¼ 1; 2; . . .;N;
qi ¼ aiQe
ki
ai
� �;
ð8Þ
where N is the number of vehicle classes, Qe (.) is the
common fundamental diagram assumed for all classes and
ai is a fraction assumed for vehicle class i and is formulated
differently in the above models. Logghe [11] used the same
concept, but divided the flow regime into free flow, semi
congested and congested regimes. Class interactions are
modelled according to user equilibrium which states that
no vehicle can increase its speed any further without
influencing the speed of slower vehicles. Wong and Wong
[14] showed that multiclass LWR model can reproduce the
traffic phenomena such as discontinuity in density,
hysteresis and platoon dispersion. The model reveals that
the discontinuity in the flow–density diagram might not
necessarily be caused by the two operational regimes
(congested and uncongested), but is a result of user’s
interactions. The model is as shown below.
oki
otþXN
i¼1
cij
oki
ox¼ 0 i ¼ 1; 2; . . .;N;
cij ¼ Vidij þ ki
oVi
okj
;
ð9Þ
where N is the number of vehicle lass, cij is the kinematic
wave speed of vehicle class i in response to the presence of
class j, Vi = Vi(k1, k2,…,kN) is the equilibrium speed of class
i expressed as a function of different class wise densities and
dij = 1 if i = j; and dij = 0 if i = j. Few other multiclass
extensions of LWR model can be seen in [15–19].
Because of the simple formulation of equations and the
analytical solution techniques, modification and extension
of the LWR model is still an active area of research. But,
for a network level model, that includes different road
types and specific traffic situations, accuracy of flow pre-
diction by modified LWR model should be tested. Litera-
ture suggests higher order models can capture traffic
realism better. Though higher order models have compar-
atively higher computational effort, they could provide
more accurate results when applied at network level.
2.1.1.2 Cell Transmission Model When simulating LWR
model, outflow is typically specified as a function of
occupancy of the section from which it is emitted and
not as a function of downstream occupancy [20]. Such
approach does not converge to a desired solution and
cannot produce reasonable results [21]. Ensuring cell
occupancies between zero and the maximum possible do
not guarantee convergence, for example, stopped traffic is
predicted not to flow into an empty freeway [20]. These
issues are resolved when Daganzo [20, 22] formulated a
Cell Transmission Model (CTM) as a discretization of first-
order LWR model. The model divides the freeway stretch
into cells (cellular automation model [23] also use the same
concept, but a micro–macro approach) and uses a piece-
wise linear relationship between traffic flow and traffic
density (triangular fundamental diagram). In the model
formulation, given a time step, the length of the cells is
chosen such that under free flow conditions, all vehicles in
a cell will flow into the immediate downstream cell. The
model formulation is as follows:
Let Ni(t) be the maximum number of vehicles that can
be present in cell i at time t, Qi(t) be the maximum number
of vehicles that can flow into cell i when the clock
advances from t to t ? 1, ni(t) be the number of vehicles in
cell i at time t and yi(t) be the inflow to cell i in the time
interval (t, t ? 1), w is the disturbance propagation speed
(backward wave speed), and v is the free flow speed, then,
the recursive relationship by the CTM is given by:
niðt þ 1Þ ¼ niðtÞ þ yiðtÞ � yiþ1ðtÞ; ð10ÞyiðtÞ ¼ min ni�1ðtÞ; QiðtÞ; d NiðtÞ � niðtÞ½ �f g &d ¼ w=v:
ð11Þ
Daganzo [24] introduced a new version of the model
called lagged CTM (L-CTM) which enable variable cell
lengths and adapts a non-concave flow–density. Szeto [25]
modified L-CTM into an enhanced L-CTM removing few
of its drawback (negative densities or densities greater than
jam density). Laval and Daganzo [26] introduced lane
changing algorithms that can be incorporated in CTM. This
model consists of a discrete time formulation of the
multilane KW module by Munjal and Pipes [5] and a
module for the lane changing particle. This model is able to
reproduce the condition of drop in the discharge rate of
freeway bottlenecks when congestion begins and the
relation between the speed of a moving bottleneck and its
capacity. Hu et al. [27] introduced a variable CTM that
includes two parameters, namely, cell length and cell
density and obeys the flow conservation law.
The use of discrete numbers and the reliance of the
model on simple rules reduce the processing power when
compared to continuum models. In CTM, the number of
vehicles that enter a cell depend on the number of vehicles
from the previous cells, maximum flow and the occupancy
of the cell itself. Therefore the unusual situation where
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there may be still inflow to a fully occupied cell, the
common problem in LWR model that needs complicate
constraints to rectify that, is avoided. In the location spe-
cific CTM by Chen et al. [28], a modified CTM is used to
take into account the model variability influenced by sensor
locations, geometry features and many other factors. Using
various shapes of fundamental diagrams, the phenomena
like capacity drops, lane-by-lane variations, non-homoge-
neous wave propagation velocities and temporal lags can
be reproduced. Long et al. [29] used the concept of CTM
for urban traffic jam simulation. The proposed model
includes two components: the cell inflow model and the
flow conservation model and used two measures, namely,
traffic jam size and congestion delay to explain the effect
of congestion.
Though the use of discrete cell in the model formulation
makes the model simple and reduces the processing power in
computation, the need for practically sized cells and clock
intervals inevitably generate errors in actual applications.
The model assumes uniform velocity for all the vehicles, and
hence cannot accurately predict the platoon dispersion phe-
nomena at light traffic conditions and the effect of moving
bottleneck. Also, in the model vehicles are assumed to have
instantaneous acceleration and deceleration. Despite the
above limitations, CTM is widely applied and studied.
2.1.1.3 Link Transmission Model (LTM) The LTM by
Yperman [30] is a model for dynamic network loading
(DNL) and computes time-dependent link travel times and
route travel times using cumulative arrival–departure
curves. In this model, the flow conservation equation holds
for the whole link and sending and receiving flows are
determined using Newell’s [21] simplified theory of kine-
matic waves. Instead of the triangular fundamental diagram
by Newell, the model uses a piecewise linear fundamental
diagram. Since the entire link is analyzed as a whole, the
computational effort for this model is less when compared
to CTM, but requires more memory for storing cumulative
arrival and departure volumes for each time step.
2.1.1.4 A porous flow approach for heterogeneous traf-
fic Most of the traffic flow model formulations are intended
to be used for homogeneous traffic with perfect lane disci-
pline. In this type of traffic, vehicles moves one after another
in each lane, and changes lanes or overtake vehicles only at
permitted zones. In the case of heterogeneous traffic lacking
in lane discipline, vehicles with widely varying sizes will
move to front according to the space availability. Even
though, the vehicle heterogeneity is considered in LWR
model (see in Sect. 2.1.1.1), the problem of lacking lane
discipline traffic is not addressed in the model. Inspired from
the work of Logghe [13], Nair et al. [31] used LWR frame-
work with a porous flow approach to model heterogeneous
traffic having no lane discipline. The vehicles on the network
define a network of spaces (pores) and each vehicle class will
have different network of pores since the smaller vehicle can
use some pores that larger vehicles cannot. The traffic state
variables for each type of vehicle stream are defined sepa-
rately and equilibrium speed–density relationship is deter-
mined using the available empty spaces; not the density. The
model formulation is as follows: assume a traffic stream with
n vehicle classes indexed by m (m = 1, 2,…,n). In addition to
the traffic state variables flow (q(m, x, t)), density (k(m, x, t))
and speed (u(m, x, t)), the model defines a pore–space dis-
tribution with probability density function fp(rp, x, t) where rp
is the pore size with
fp rp; x; t� �
� 0 and
Z1
0
fp rp; x; t� �
drp ¼ 1: ð12Þ
Here fp(rp, x, t) denotes the fraction of pores within size
rp and rp ? drp. The fundamental diagram applies to each
vehicle class independently and the LWR model is extended
to several vehicle classes as a set of partial differential
equations:
okðm; x; tÞot
þ oðkðm; x; tÞfeðKðx; tÞÞÞox
¼ 0
8m ¼ 1; 2; . . .; n;ð13Þ
where K(x, t) = [k(1, x, t), k(2, x, t),…,k(n, x, t)] is the
vector of densities for all vehicle classes and fe(K(x, t)) is
the speed from the equilibrium speed–density relationship.
To account for the significant lateral movement, the traffic
stream for each vehicle class is considered to have two sub-
streams of ‘free’ and ‘restrained’ vehicles.
uðm; x; tÞ ¼ urðm; x; tÞZrðmÞ
0
fp rp; x; t� �
drp
þ uf ðm; x; tÞZ1
rðmÞ
fp rp; x; t� �
drp
8m ¼ 1; 2; . . .; n;
ð14Þ
where
urðm; x; tÞ ¼ uf ðmÞ 1�ZrðmÞ
0
fp rp; x; t� �
drp
2
64
3
75
ar
;
uf ðm; x; tÞ ¼ uf ðmÞ 1�ZrðmÞ
0
fp rp; x; t� �
drp
2
64
3
75
af
8m ¼ 1; 2; . . .; n:
ð15Þ
Here r and f stands for restricted and free vehicles,
ar C af, and uf(m) is the free flow speed of the vehicle class
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m. To solve the above system of equations, the finite
difference scheme cannot be applied directly, since the
pore–space distribution is not known and also the set of
n nonlinear partial differential equations have to be solved
jointly because of the K(x, t) term. The model’s algorithm
simulates pore size distribution using updated class wise
densities by Godunov method [32]. To calculate the
boundary flows, supply–demand analysis for each vehicle
class is performed independently. One drawback of this
model is that the number of vehicle class is restricted to
two since supply–demand analysis is cumbersome for more
number of classes. The model has not been calibrated with
real data.
2.1.1.5 Second-order LWR model A drawback in LWR
model is its inability to capture hysteresis effect. One of the
main reasons for the hysteresis phenomena observed in real
traffic is the driver’s response to frontal stimuli and inertial
effect (in the equilibrium speed–density relationship, speed
changes instantaneously with the density). In order to
account for these, Lighthill and Witham suggested second
order terms in the simple first-order model as shown
below [4].
ok
otþ c
ok
oxþ T
o2k
ot2� D
o2k
ox2¼ 0; ð16Þ
where T is the inertial time constant for speed variation, c is
the speed of small disturbances (dq/dk), and D is a diffu-
sion coefficient representing how vehicles respond to non-
local changes in traffic conditions. However, the simple
extension of first-order LWR equation to second-order
equation did not resolve the limitations of LWR model.
Following this second-order model, Payne [33], Ross [34],
Kuhne [35, 36] and Michalopoulos et al. [37] also proposed
higher order models of which Payne model aroused con-
siderable interest.
2.1.2 Higher order models
As mentioned in the above section, higher order models are
formulated to incorporate the inertial effect and effect of
drivers’ anticipation on vehicle speeds. Thus, differed from
the LWR model, these models contain additional equa-
tion(s) on velocity dynamics. The following models: Payne-
type model, Aw–Rascle model, Zhang model, Helbing’s
model and speed gradient (SG) model are presented in the
following subsections.
2.1.2.1 Payne-type models Second-order models remained
under-explored for some time until Payne [33] extended
LWR model using a simple car-following rule. One of
the main critiques of the simple LWR model is that the
mean velocity adapts instantaneously to the traffic density
(steady-state speed–density relationship). Along with the
LWR Eqs. (1) and (3), he proposed a partial differential
equation describing the dynamics of the velocity u. The
simple car-following rule is shown below.
uðxðt þ TÞ; t þ TÞ ¼ ueðkðxþ DÞ; tÞ; ð17Þ
where x(t) is the location of driver at time t, u(x, t) is the
velocity at x and t, ue is the equilibrium velocity expressed
as a function of density k, T is the reaction time and D is the
gross distance headway with respect to the preceding
vehicle. Applying Taylor’s expansion to both sides of the
above equation and substituting k = 1/D, the equation
becomes [1]:
ou
otþ u
ou
ox¼ ueðkÞ � u
T� c2
0
k
ok
ox; ð18Þ
where the constant c0 [ 0 is defined as the traffic sound
speed. The second term in the left hand side denotes con-
vection which describes changes in the mean velocity due
to in- and out-flowing vehicles. The first term in the right
hand side denotes relaxation which describes the tendency
of traffic flow to relax to an equilibrium velocity. The last
term constitutes the anticipation term that describes dri-
ver’s anticipation on spatially changing traffic conditions
downstream.
Kotsialos and Papageorgiou [38] used discretized Payne
model in the network model METANET along with net-
work relevant extensions (links are modelled separately as
motorway, origin, destination and store-and-forward links
and it also includes a node model). Whitham [39] also
proposed a second-order Payne-type model, the so called
Payne–Whitham model and in general, the velocity
dynamics of Payne-type model is given as [1]:
ou
otþ u
ou
ox¼ ueðkÞ � u
T� 1
k
oP
oxþ g
k
o2u
ox2; ð19Þ
P is the traffic pressure equal to c20k and g is the kinematic
traffic viscosity. The last two terms in the right hand side
stand for anticipation where qP/qx describes the local
anticipation behavior of the driver and the diffusion term
o2u=ox2 describes the higher order tendencies (immediate
acceleration or deceleration, stop-and-go behavior, etc.) of
drivers. Considering non-viscous flow and ignoring the
relaxation term, Eqs. (1) and (18) together can be expressed
in a nonlinear hyperbolic conservative form (see Eq. (4)).
The conservative variables U and F are given by:
U ¼ k
ku
� FðUÞ ¼ ku
ku2 þ c20k
� :
In Payne family of models, the state of the system at (x, t) is
determined using two characteristics (curves called Mach
lines) both emanating from t = 0 [1]. The speed of the
characteristics is the Eigen values of the Jacobian of the flux
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matrix F and is given in Table 1. In a certain density range,
Payne model is meta-stable. In this region, small variations in
the traffic density yields regions of increasing traffic densities
leading to the occurrence of start–stop waves or localized
traffic jams. This is one of the important properties of these
models [40, 41].
Daganzo [42] critically reviewed Payne model pointing out
that the kinematic theory of fluid should not be applied to
model vehicular flow. First, the fluid flow is isotropic, but
vehicular flow (driver reacts only to the downstream vehicles)
is anisotropic. Second, in fluid flow slow particles get affected
by faster particles, but in vehicular flow, slow vehicles remain
unaffected by fast vehicles. Finally, the driver’s personality is
also an important factor in the vehicular flow which is irrel-
evant in the fluid flow modelling. Other criticisms for Payne
model are: speed of one of the characteristic is higher than the
vehicular speed (information travels faster than the vehicles’
speed), and the model cause negative speeds at tails of the
congested region. Later, Liu et al. [43] pointed out that the
upstream moving curve is an indication of speed difference
between vehicles and not negative speed. Next section dis-
cusses the Aw–Rascle [44] model that overcomes this nega-
tive vehicle speed phenomenon.
2.1.2.2 Aw–Rascle model Aw and Rascle [44] formu-
lated a new anisotropic continuum flow model, addressing
the drawbacks noted by Daganzo [42] for the Payne model,
using a convective derivative of pressure instead of spatial
derivative. Thus the drivers’ anticipation to the traffic
ahead depends not on the spatial change in density, but on
the spatio-temporal change in density. The model includes
Eqs. (1), (3) and the velocity dynamics equation given in
Eq. (20).
oðuþ pðkÞÞot
þ uoðuþ pðkÞÞ
ox¼ 1
TueðkÞ � uð Þ; ð20Þ
where p(k) is the traffic pressure expressed as an increasing
function of density. They proved that with a suitable choice
of the function p(k), the model addresses two criticisms of
the Payne-type models (anisotropy and negative speed of
vehicles). Moreover, the model nicely predicts the
instabilities near vacuum i.e. for very light traffic. The
functional form of p(k) chosen in [44] is as follows:
pðkÞ ¼ C2kc; ð21Þ
where C is a constant equal to 1. Multiplying Eq. (1) by dp/dk
and adding to Eq. (20), the velocity dynamics equation can
be rewritten as:
ou
otþ ðu� kp0ðkÞÞ ou
ox¼ 1
TueðkÞ � uð Þ: ð22Þ
For the nonlinear hyperbolic conservative system, the
conservative variables are given by:
Table 1 Macroscopic models for traffic flow
Models Disturbance propagation speed Nature of
model
Literature on
Extension to multilane/multiclass traffic Large/network model
LWR model dq/dk Anisotropic See [5–8, 10–15, 17–19] Lebacque and
Koshyaran [90]
Payne model u ? c0, u - c0 Isotropic Nil Kotsialos and
Papageorgiou [38]
Aw–Rascle model u, u - cp(k) Anisotropic Bagnerini and Rascle [78], Colombo [79] Garavello and Piccoli
[91]b
Zhang model u, u ? ku0e(k) with u
0e(k) \ 0 Anisotropic Nil Nil
Helbing
modelaHelbing
[50]
u
u ? f1(k, u, H) - f2(k, u, H)
u ? f1(k, u, H) ? f2(k, u, H)
Isotropic Hoogendoorn and Bovy [60], Tampere et al.
[80], Ngoduy et al. [72]
Helbing et al. [58]
Helbing
[77]
u ? (H/u2) ? uf3(k, u, H)
u ? (H/u2) - uf3(k, u, H)
Speed gradient
model
u, u - c0 Anisotropic Jiang and Wu [65], Tang et al. [66, 68, 69] Nil
Porous flow
approach
dq/dk, depends on pore size
distribution function
Anisotropic Nair et al. [31] Nil
CTM w = a constant (backward wave
speed)
Anisotropic Laval and Daganzo [26], Tuerprasert and
Aswakul [103]
Lo and Szeto [104]
Non-continuum
model
– Anisotropic Nil Nil
a f1, f2 and f3 are nonlinear functions of k, u and Hb Separate modelling of mid-blocks and intersection, not the network as a whole
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U ¼ k
m
� FðUÞ ¼ m� kpðkÞ
m2
k� mpðkÞ
� ;
where m = k(u ? p(k)). The characteristic speeds of the model
as shown in Table 1 satisfy the anisotropic behavior of vehicles.
The model uses a simple pressure function satisfying few
qualitatively important conditions [44]. However, since the
model’s predictive ability depends on the assumed pressure
function, alternate formulations should be explored. A different
formulation of pressure term in the hybrid (macroscopic ?
microscopic) model by Moutari and Rascle [45]:
pðqÞ ¼uref
cqqm
�cc[ 0
�uref ln qqm
�c ¼ 0
8<
:; ð23Þ
Here q denotes the fraction of space occupied by vehicles (a
dimensionless local density), uref is a given reference
velocity, and qm is the maximal density (as fraction) equals 1.
2.1.2.3 Non-equilibrium model of traffic flow Defining
the concept of non-equilibrium traffic flow and to overcome
the limitation of ‘negative speed’ in Payne’s model, Zhang
[46] proposed a different model with a velocity dynamics
equation as in Eq. (24) and the LWR Eqs. (1) and (3).
ou
otþ u
ou
ox¼ 1
TueðkÞ � uð Þ � k u
0
eðkÞ �2ok
ox: ð24Þ
The model formulates the disturbance propagation speed
of non-equilibrium traffic flow as a function of the
equilibrium flow dynamics. LWR model is a special case
of this model where drivers do not anticipate traffic
conditions ahead (u = ue(k)). He showed that the higher
order terms in the model can be neglected if the temporal–
spatial scales are properly treated. However, this model
could not fully remove the isotropic behavior of vehicles.
Zhang [47] modified the velocity dynamics as follows:
ou
otþ uþ ku
0
eðkÞ � ou
ox¼ ueðkÞ � u
T: ð25Þ
The conservative variables for Zhang [47] model are
given by:
U ¼k
m
" #
FðUÞ ¼mþ kueðkÞ
m2
kþ mueðkÞ
2
4
3
5;
where m = k(u - ue(k)). Aw–Rascle [44] and Zhang [47]
models, together known as ARZ model is extended by
Lebacque et al. [48, 49] using a variable fundamental dia-
gram (fundamental diagram varying as a function of traffic
state) with inverse equilibrium speed–density relationship.
2.1.2.4 Model based on kinetic theory of gases In addi-
tion to the criticism by Daganzo [42] on Payne model,
Helbing [50] introduced three more conditions which are to
be fulfilled by a macroscopic traffic flow model. These are
finite space requirement of vehicles, velocity variance, and
finite reaction time and breaking time of driver–vehicle
units. For incorporating velocity variance, finite reaction
time and breaking time, a new model derived from gas
kinetic equations was proposed. The concept of gas kinetic
theory was first introduced by Newell [21], but, models
based on this received greater attention when Prigogine and
Andrews [51] used this theory with a Boltzmann-like
approach [52]. Gas kinetic theory has been widely used in
mesoscopic traffic flow modelling [51–55]. The macro-
model based on this theory is proposed by Helbing [50] and
consists of Eqs. (1), (3) and (19) along with the following
equation of velocity variance.
oHotþ u
oHox¼ �2
P
k
ou
oxþ 2
He �HT
� 1
k
oJ
ox; ð26Þ
where H and He are the velocity variance and equilibrium
velocity variance, respectively, P is the traffic pressure
equal to k(x, t) H and J is the flux of velocity variance
which is defined as the product of density k(x, t) and
skewness of the velocity distribution C(x, t) as shown
below.
Jðx; tÞ ¼ kðx; tÞCðx; tÞ: ð27Þ
Also,
ueðk; u; HÞ ¼ u0 � Tð1� pÞP;Heðk; u; HÞ ¼ Z � Tð1� pÞJ;
ð28Þ
where u0 and ue are the expected desired velocity and
equilibrium velocity, respectively, T is the reaction time, Z
is the covariance between the velocity and the desired
velocity and p is the immediate overtaking probability. The
propagation of disturbance is analyzed by the model using
three characteristic curves (one straight line and two Mach
lines). By this model, the small disturbances are transported
along with the mean traffic flow as well as in the up- and
downstream directions with respect to this mean flow.
Conserved variables in the model when expressed as a
nonlinear hyperbolic system are given by:
U ¼ U1; U2; U3½ �T
FðUÞ ¼ U2;U2
2
U1þ U1U3�U2
2
U1ð1�U1�s0U2Þ ;U2U3
U1þ 2ðU1U2U3�U3
2Þ
U21ð1�U1�s0U2Þ
h iT
;
where
U1 ¼ k; U2 ¼ ku; U3 ¼ ku2 þ r2kH; r ¼ffiffiffiffiffiffiH0
p
uf
;
s0 ¼ kjuf T :
Here kj, uf and H0 are respectively the jam density, free
flow speed, and maximum velocity variance of traffic [56].
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Helbing [57] modified the model for finite space requirement
of vehicles by replacing the P and J by:
P0 ¼ P
1� ksðuÞ J0 ¼ J
1� ksðuÞ ; ð29Þ
where s(u) = l ? uT, l is the average vehicle length, and
uT is the safe distance each driver should keep from the
next vehicle ahead. Helbing et al. [58] also proposed a gas
kinetic, non-local traffic model, MASTER that avoids
diffusion or viscosity terms and allows only forwardly
directed interactions. This model, similar to the other
higher order models (and differed from Helbing’s [50, 57]
original model), has two equations—flow conservation and
velocity dynamics equations—and hence two characteristic
curves. The main difference of this model from other
higher order models is in the velocity dynamics equation,
where, the equilibrium velocity also depends on the density
and average velocity at an interaction point that is
advanced by about the safe distance. This model allows
simulating synchronized congested traffic [59] that mainly
occurs close to on-ramps. The model is also extended to
multilane, heterogeneous traffic and successful calibration
with real traffic is reported [60]. Compared to the other
higher order models listed in previous sections, number of
parameters to be calibrated in this model is more, requiring
more computational effort at a network level.
2.1.2.5 SG model The SG model proposed by Jiang et al.
[61] is based on an improved car-following model. Helbing
and Trieber [62] pointed out that there exists in the real
world a common driver behavior that none of the earlier
car-following models can explain; when the distance
between the leader and follower become shorter than the
safe distance, the follower may not decelerate if the pre-
ceding vehicle drives faster than the follower. By com-
bining the classical car-following model [63] and the
optimal velocity model [64], the formulated improved car-
following model incorporating the effect of both the dis-
tance and the relative speed of two successive vehicles is as
follows [65]:
dunþ1ðtÞdt
¼ j UðDxÞ � unþ1ðtÞ½ � þ kDu; ð30Þ
where j is the reaction coefficient, U(Dx) represents the
legal velocity of the follower, un?1(t) is the speed of the
follower, k is the sensitivity coefficient and Du is the
relative speed of the vehicles. But this improved equation
represents the traffic flow condition in a microscopic point
of view. Transforming the discrete variables of individual
vehicles into continuous variables, the macro-approach for
the above formulation becomes [61]:
ou
otþ u
ou
ox¼ ue � u
Tþ c0
ou
ox; ð31Þ
where c0 is the propagation speed of disturbance. Thus, the
new model consists of Eq. (30) along with Eqs. (1) and (3).
The variables for the system of nonlinear hyperbolic
equations in the SG model are given by:
U ¼ k
u
� FðUÞ ¼ ku
u2
2� c0u
� :
Compared to other higher order models, the SG replaces
the density gradient in the anticipation term, and guarantees
the property that the characteristic speed is always less than
the macroscopic flow speed. Using hypothetical data, the
model proved to obtain shockwaves, rarefaction waves, stop-
and-go waves, and local cluster effects and is consistent with
the diverse nonlinear dynamical phenomena observed in the
freeway traffic.
Jiang et al. [65] extended the SG model for mixed traffic
consisting of fast cars and slow vehicles. The model for-
mulation is as follows:
oki
otþ oðkiuiÞ
ox¼ 0
oui
otþ ui
oui
ox¼ ueiðk1; k2Þ � ui
Ti
þ c0i
oui
ox
8>><
>>:
9>>=
>>;8i; ð32Þ
where i = 1, 2 represents fast cars and slow vehicle,
respectively, c0i is the kinematic wave speed of vehicle i,
uei is the equilibrium speed of the vehicle type i and Ti is
the reaction time for the vehicle type i. In the above
equation, effect of slow vehicles acting on fast cars is
neglected. To account for this, and also by considering the
difference in length of the two types of vehicles, Tang et al.
[66] modified the above expression as follows:
oki
otþ oðkiuiÞ
ox¼ 0
oui
otþ ui
oui
ox¼ ueiðk1; k2Þ � ui
Ti
þ c0i
oui
ox
� 1� k
kj
� �u1 � u2
si
km6¼i
kj
� �2
8>>>>>>><
>>>>>>>:
9>>>>>>>=
>>>>>>>;
8i; m ¼ 1; 2;
ð33Þ
where k = k1 ? k2 and kj is the jam density for the mixed
traffic. The right most term indicates that the frictional
effects between fast cars and slow vehicles are proportional
to the speed difference of these vehicles with si as the
proportional coefficient. The double exponential form of
the speed–density relationship by Del Castillo et al. [67] is
adopted for ue and extended for uei. The jam density kj for
the mixed traffic is calculated as follows:
kj ¼kj1
a� ða� 1ÞR ; ð34Þ
where kj1 is the jam density under the homogeneous traffic
where there are only fast cars, a is the length ratio of the
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slow vehicle to fast cars and R is the ratio of fast car
density to the total density.
Another extension to SG model is by Tang et al. [68]
where it is used to describe traffic flow on two lane free-
ways. Here, terms related to lane change are added into the
continuity equation and the velocity dynamic equation.
Tang et al. [69] extended the SG model for heterogeneous
traffic consisting of n different types of vehicle classes. The
new model, developed from an improved car-following
model, says that the acceleration of a vehicle depends on
the speed, spacing and the reactive coefficients of different
vehicle classes. The developed model is shown below.
oui
otþ ui
oui
ox¼ uieðkÞ � ui
Ti
þXN
j¼1
cijpj
ouj
oxþ pj
sij
uj � ui
� �� �;
ð35Þ
where N is the number of vehicle classes, uie is the equi-
librium speed for the vehicle class i, Ti and sij are the
reactive coefficients for vehicle class i with respect to a
leading vehicle of type j, pj is the proportion of leading
vehicle of type j, and cij is the disturbance propagation
speed of vehicle class j as result of vehicle class i. The
value of cij depends on the speed of the vehicle type j and
also on the reaction coefficient sij. SG model has not been
sufficiently validated with real data.
2.2 Non-continuum model
Darbha and Rajagopal [70] proposed a non-continuum
approach for macroscopic traffic flow modelling. They
suggested that the usage of the macroscopic variable
‘density’ has no intuitive meaning. In mechanics a section
of flow consists of infinite number of particles, so for
relating fluid flow to vehicular flow one must ensure there
are ‘sufficiently large’ number of vehicles. To have a
comparable number of vehicles in a section as particles (or
molecules) in a representative volume, the section lengths
to be considered must be at least millions of miles long
which is not practical.
In the non-continuum model, traffic is treated as a col-
lection of dynamical systems, with each vehicle in the
traffic treated as a dynamical system. A ‘representative
vehicle’ can be thought of as a limit of a collection of
dynamical systems of finite state space dimension. The
vehicle-following behavior of the representative vehicles
reflects the aggregate vehicle-following behavior of traffic.
The variables used to describe the flow of traffic are: the
number of vehicles in the section at any given time (N), the
aggregate following distance (D*), the aggregated speed of
traffic (v*), the number of vehicles entering the section
from upstream (Nen), the number of vehicles exiting from
the section to downstream (Nex), net inflow to the section
from ramps (dn*/dt), the length of the section (L), and the
speed correction factor (b) for travelling to the downstream
section. For the ith section of a freeway, the model of
traffic flow by the non-continuum approach is given by the
following system of equations:
dNi
dt¼ dNen
i
dt� dNex
i
dtþ dn�i
dt; ð36Þ
dD�idt¼ �ðL
�car þ D�i Þ
2
Ls;i
dNi
dtþ bi;i�1 þ v�i�1 � v�i ; i [ 1;
ð37Þ
dD�1dt¼ �ðL
�car þ D�1Þ
2
Ls;1
dN1
dt; ð38Þ
dv�idt¼ f v�i ; D�i ;
dD�idt
� �; ð39Þ
dNexi
dt¼ v�i Ni
Ls;i; ð40Þ
dNeni
dt¼ dNex
iþ1
dt: ð41Þ
Equation (36) is for vehicle balancing, Eqs. (37) and
(38) is for evolution of aggregate following distance,
Eq. (39) is an approximation to vehicle speed dynamics
and Eqs. (40) and (41) are to ensure compatibility. One of
the advantages of this approach is that the aggregate
vehicle following behavior is integrated in the macroscopic
flow model that replicates the effects of microscopic level
control on the macroscopic dynamics. However, this model
has not been studied in detail probably because of the
number of equations involved and the strong biased
towards continuum theory for traffic flow modeling.
2.3 Solution to macroscopic link models
This section discusses mainly the solution procedures of
continuum models. Macroscopic continuum traffic flow
models can be solved either analytically or by using
numerical simulation. Analytical method for solving LWR
model uses a set of characteristics that are straight lines
while higher order models contains more than one set of
characteristics which are curves. Hence, except for LWR
model, the analytical approaches for other models are
complex and numerical schemes are the widely used
solution procedure. Three important questions for picking a
numerical simulation technique are: the type of the scheme
to be used, time–space discretization, and the initial and
boundary conditions of the simulating domain.
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Numerical schemes can be classified as explicit or
implicit. Explicit methods calculate the state of a system at
a later time from the state of a system at the current time,
while implicit methods find a solution by solving an
equation involving both the current and later states of the
system. In realistic traffic simulations, data is continuously
fed into simulations for varying boundary conditions and
hence explicit methods are more useful. These methods are
more flexible for the simulation of on- and off-ramp or
entire road networks [62]. There also exist finite difference
methods (FDMs) and finite volume methods (FVMs) in
numerical schemes. Macroscopic continuum traffic flow
models are expressed as system of nonlinear hyperbolic
partial differential equations. In practice, solution to these
equations will not be smooth, but contain discontinuities
such as shockwaves. Classical FDM, in which derivatives
are approximated by finite differences (point wise
approximation at grid points), can be expected to break
down near discontinuities in the solution where the dif-
ferential equation does not hold. Thus, to solve continuum
traffic flow models, FVMs are more preferred. In FVM, the
domain or space is divided into grid cells and approximate
the cell average of the state of the system (integral of state
of the system divided by the cell volume). These values are
modified in each time step by the boundary fluxes of the
grid cells. The information that can be used to compute the
boundary fluxes can be obtained by solving the ‘Riemann
problem’. Riemann problem is simply a hyperbolic equa-
tion with piecewise constant initial data having a single
discontinuity at some point. An example is the hyperbolic
system (4), along with the following initial data:
Uðx; 0Þ ¼ UL; x\0
UR; x [ 0
; ð42Þ
where x = 0 is the point of discontinuity that separate two
different traffic states UL and UR (respectively on the left
and right of x = 0). The solution to the Riemann problem
consists of a finite set of waves that propagate away from
the point of discontinuity with constant wave speeds. A
detailed explanation of the available FVM methods can be
seen in [32].
Two simple schemes commonly used are upwind
scheme and Godunov scheme [32]. Zhang et al. [71] used
weighted essentially non-oscillatory numerical scheme in
multiclass LWR model and simulated results for a signal
control problem that are in good agreement with the ana-
lytical counterparts. Higher order schemes (if Dx and Dt
are simultaneously decreased by a factor of e, upper bound
of the local error will be proportional to e2), such as
Lax–Wendroff and flux splitting scheme [32] are not
necessarily more accurate than simple schemes and
sometimes even cause numerical instabilities [62]. Thus, it
is always recommended to implement different numerical
methods and compare their simulation results. Ngoduy
et al. [72] compared three different numerical schemes for
Payne model, namely, Steger–Warming flux splitting
scheme, Mac Cormack scheme, and Harten–van Leer–
Lax–Einfeldt (HLLE) scheme and obtained minimum total
relative mean square error in flow and mean speed by
HLLE scheme. There also exist high resolution schemes
where first-order scheme (upper bound of the error is
proportional to e) can be used for simulating congested
regime and higher order scheme can be used for uncon-
gested regime. Selection of the numerical scheme also
depends on the type of the problem to be solved and the
traffic flow model used. In the SG model, the system of
hyperbolic equations are not expressed in conservative
form (differed from other higher order models, when
expressed as a system of hyperbolic equations, the velocity
dynamics equation has not rewritten in terms of conserved
variables). Thus the numerical schemes for conservation
laws should not be applied directly for this model.
Time–space discretization for any numerical schemes
should be chosen to ensure two different stabilities,
namely, convective stability and relaxational stability.
Convective stability is ensured by the Courant–Friedrichs–
Levy (CFL) condition that the discretized time interval
Dt should be less than or equal to the minimum time for the
fastest vehicle to cross the discretized space interval
Dx. Relaxational instability can occur if Dt is greater than
the relaxation time of the vehicles. Traffic flow models
involving second-order viscosity term (for example, in
Payne model), may cause also a diffusion instability which
can be removed by a diffusional CFL condition [62].
Cremer and Papageorgiou [73] verified through simulation
that lowering discretization intervals does not create
amplified accuracy in macroscopic traffic flow models, and
in fact, increasing the discretization interval leads to much
lower computational effort. The choice of time–space
discretization also depends on the type of the numerical
scheme to be used. Mohan and Ramadurai [74] showed that
as grid size becomes smaller, LWR, Payne and Aw–Rascle
models produce more accurate results. However, as stated
by Papageorgiou [75], there exist optimum discretization
levels beyond which further improvements in accuracy
cannot be achieved.
Initial conditions for simulation is specified as the state
of the system in each grid cell at time t = 0. For boundary
conditions, different existing options are: Dirichlet condi-
tions, homogeneous von Neumann conditions, free and
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periodic boundary conditions [32]. Dirichlet boundary
conditions are given by the empirically measured values at
the boundaries and are best for realistic traffic simulations
with measured values of speed and flow at the boundary.
Homogeneous von Neumann and free boundary conditions
assume that the state of the system remains unchanged at
the boundaries and can be used if traffic situation outside
the boundaries is not of interest. Periodic boundary con-
ditions assume that the density and flow at entry and exit of
the considered segment are equal for considered time
period. Helbing and Trieber [62] pointed out few disad-
vantages of using Dirichlet boundary conditions. They
showed that if the imposed upstream boundary flow is
higher than the equilibrium flow or imposed downstream
flow is higher than the flow arriving from the simulated
section, continuity equation yields a lower equilibrium flow
or negative densities, respectively. To solve the above
problem they suggested hybrid boundary conditions, where
at boundaries Dirichlet condition is used if the direction of
information wave propagation pointed towards the simu-
lation stretch, and homogeneous von Neumann condition is
used otherwise.
Macroscopic traffic flow models other than continuum
models use first-order FDM in flow conservation or a
system of simultaneous equations for problem solution.
The CTM by Daganzo [20, 22] uses first-order Godunov
scheme and determine appropriate traffic flow between
two cell boundaries using supply–demand restriction. The
porous flow approach by Nair et al. [31] uses a hybrid
solution procedure where the densities are simulated using
finite difference (first order) scheme on conservation
equation and the supply side of the conservation equation
is determined by the incremental transfer principle [76].
The simulated densities are used to determine the pore
size distribution which in turn yields equilibrium speeds
of each vehicle class. The flow variables are then updated
using the fundamental equation of traffic flow. The non-
continuum model by Darbha and Rajagopal [70] is solved
as a simultaneous system of equations (given in Sect.
2.4).
2.4 Summary of link modelling
Link modelling has a strong and vast base in continuum
theory. The simple LWR model initiated traffic flow
modelling using the kinematic wave theory with the
explicit representation of shockwave formulation under
equilibrium state. Major limitations of the LWR model
(instantaneous speed–density relationship, prediction of
instability, stop–start waves, platoon dispersion, etc.) have
led to development of higher order models. These higher
order models are derived from simple follow-the-leader
rule and a review can be seen in [77]. These models seem
to have complex system of equations because of the type
and number of variables involved. However, the intuitive
conceptual basis, good analytical formulation, and ease of
implementation lead researchers to continue with the
continuum theory and to develop (or improve) new
(existing) models. Also, to capture the complex wave
interactions observed in real traffic, higher order models
are required. The CTM which is the discretized version of
the original LWR model was one of the milestones in
traffic flow modelling giving the concept of division of
road into spaces or cells and finding out the traffic state
using simple equations. Even though it holds the same
limitations as the LWR model, the advantage of less
computational effort when using this model led to wide
spread applications even up to network level. These models
use a direct mapping of micro-variables to macro-level
which is only an approximation even for ‘ideal’ equilib-
rium traffic. Two differed attempts in link modelling are
the non-continuum model and the porous flow approach.
Among the higher order models, two well known models
used at the network level are METANET in which the
underlying model is Payne model and MASTER in which
the underlying model is the Helbing’s [57] model. Exten-
sion of higher order models to multiclass traffic is limited
[60, 78–80], and is not explored to lacking lane disciplined
traffic. The discussed models are summarized in Table 1.
Numerical simulation is the widely used solution pro-
cedure in continuum macroscopic models where explicit
FVMs with hybrid boundary conditions are well proved.
Though the porous flow approach gave a good platform
for heterogeneous traffic flow modelling, the difficulty to
identify an appropriate pore distribution function and the
supply function analysis that restricts the number of
vehicle classes are limitations of this method. The non-
continuum model questioned the basic premise of stating
vehicular density analogous to that of fluid density and
assumes vehicles as a collection of dynamical systems.
Apart from the number of parameters involved, another
limitation of this model is the involvement of aggregate
following distance that restricts its extension to multive-
hicle traffic.
3 Node modelling
A node is a junction or intersection of two or more links in
the transport network, or can also be a point of abrupt
change in road characteristics (for example, a change in
number of lanes). Flow modelling for a node is difficult
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mainly because of the stochastic directional distribution of
traffic and also due to the existence of the various levels of
control for the node. Probably because of this complexity,
existing literature for node modelling in the macroscopic
level, is less when compared to the link modelling. Liter-
ature on DNL models discuss about two types of modelling
approach for intersection—point-like modelling and spatial
modelling. Many of the node models are point-like, i.e.
without physical dimensions, combing all the external and
internal constraints into a strongly coupled set of equations.
In spatial modelling, intersections and conflict zones are
represented through dummy links and nodes. This paper,
however, uses a different classification for node models,
i.e. first generation models that include simple merge or
diverge models and models based on constraint flow
maximization.
3.1 First generation models
First generation of the node models started with the merge
and diverge models [22, 81, 82]. The merge model of
Daganzo [22] consist of two incoming links (flows: q1 and
q2) and one outgoing link (total flow q = q1 ? q2) with an
objective function to maximize q subject to the following
demand and supply constraints:
qi� Si;X
i
qi�R 8i ¼ 1; 2; ð43Þ
where Si (demand) denotes the maximum flow that the
incoming link i could transfer if the node and the outgoing
link impose no constraints on the outflow of link i and R
(supply) is the maximum inflow that the outgoing link can
receive if the node and incoming link impose no
constraints. The model also uses ‘distribution factors (di)’
for the flow assignment to each link. Jin and Zhang [81]
modified the distribution factors in the model as:
di ¼SiPi Si
: ð44Þ
Later [83], they introduced a ‘level of reduction (a)’ to
represent the flows as a function of upstream variables.
a ¼ RP
i Si
; qi ¼ minð1; aÞSi: ð45Þ
Ni and Leonard [84] introduced the merge model in the
same way as that by Daganzo [22], but represented the
distribution factors in terms of capacities (Ci) of incoming
links, i.e.
di ¼CiPi Ci
: ð46Þ
The simple diverge model of Daganzo [22] contains one
incoming link (total flow: q) and two outgoing links (flows:
q1 and q2). The objective function is to maximize the total
flow, q subjected to the constraints as follows:
qi ¼ fiq�Ri; q� S; ð47Þ
where R and S are as defined for the merging model and fiis the turning fraction. The level of reduction, as in merge
model is given as:
ai ¼Ri
fiS; qi ¼ min 1; a1; a2ð ÞfiS: ð48Þ
Lebacque [85] proposed an exchange zone for
intersection flow modelling, where this zone is like a cell
with several entry and exit points. The traffic inside the cell
is disaggregated according to the entry and exit points. Let
N be the total number of vehicles, NIi be the number of
vehicles through the entry point i, NOj be the number of
vehicles through exit point j, Nij be the number of vehicles
entering through i and exiting through j, then the following
conservation holds:
NIi ¼X
j
Nij; NOj ¼X
i
Nij; N ¼X
i
NIi
¼X
j
NOj ¼X
ij
Nij:ð49Þ
A global fundamental diagram Qe(N) is defined for each
zone yielding total demand and total supply at the zone as
SN and RN, respectively. Let bi be the fraction of lanes at
entry i, then the split of SN and RN among upstream and
downstream links are:
Si ¼ biSN ; Rj ¼NOj
NRN : ð50Þ
If the downstream link has a supply of Rdj, the zone
outflow through exit point j is given by:
Qj ¼ min Sj; Rdj
� �ð51Þ
Similarly, if the demand of the upstream link i is Sui, the
zone inflow through entry point i is given by:
Qi ¼ min Sui; Rið Þ: ð52Þ
3.2 Models based on constraint flow maximization
Holden and Riserbo [86] were the first who developed a
mathematical model of traffic flow on a network with
unidirectional flow and solved a Riemann problem of a
highway intersection with m upstream links and n
downstream links. All links in the network have the same
fundamental diagram and traffic dynamics are described
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by the LWR model. The Riemann problem is solved by
introducing an entropy condition that maximizes a con-
cave function of node inflow (Qi; i = 1, 2,…,m) or out-
flow (Qj; j = 1, 2,…,n) subject to the following
constraints:
0�Qi� di; 8i; 0�Qj� rj; 8j;X
i
Qi ¼X
j
Qj;
ð53Þ
where di is the demand of the upstream link i, and rj is the
supply of the downstream link j. Few drawbacks of this
model are the lack of physical interpretation to the entropy
condition, oversimplification of the model by the assump-
tion that the drivers will choose their destination link based
only on the principle of least resistance, and the uniform
speed–density assumption for all the links. Herty and Klar
[87] followed the same assumptions and formulated a
multilane model. Coclite et al. [88] solved the Riemann
problem to maximize the total flux with an additional
constraint on turning proportions. They too assume a uni-
form fundamental diagram for all the links. Further, the
Riemann problem cannot be uniquely solved for a junction
with fewer downstream links, hence unable to apply to a
merging junction [85].
Durlin and Henn [89] proposed a delayed flow inter-
section model that captures three important effects of sig-
nal alteration at intersections. The effects, namely, cyclic
delay, bottleneck effect, and the shockwave propagation
are illustrated using a wave tracking resolution scheme.
The calculation of wave propagation (the waves separate
zones of different densities) is made possible by the
approximation of the fundamental relationship by a
piecewise linear functions. The flow through the intersec-
tion can be replaced by the use of a fictive arc (an artificial
arc between upstream and downstream arc), which has its
own fundamental diagram and the length of which can be
varied with respect to the inflow rate to the intersection.
The cyclic delay at intersection (s) can be computed using
the equation for uniform delay given by the Highway
Capacity Manual. The two bottleneck restrictions that are
used at the entry and exit of the fictive arc are as follows
[89]:
Qin ¼ min S; k � R; k � Rfictive
� �and
Qout ¼ min Sfictive; k � R� �
;ð54Þ
where Qin and Qout are respectively the inflow and outflow
for the fictive arc, k is the green ratio at the intersection, S
is the demand at the entry of the intersection, R is the
supply of the downstream arc, and Sfictive and Rfictive are the
demand and supply at the exit and entry of the fictive arc,
respectively. The above formulation of Qout holds only for
the constant inflow to the intersection where the length of
the fictive arc is constant. If inflow varies, length of the
fictive arc also changes; the speed with which the
downstream frontier of the fictive arc moves (Vfrontier)
and corresponding Qout are given by:
Vfrontier ¼ Vf 1� Qout
Qf
� �and Qout ¼ Qf 1� Vfrontier
Vf
� �;
ð55Þ
where Qf is the flow leaving the fictive arc and Vf is the
speed of vehicles on the fictive arc.
Lebacque and Koshyaran [90] introduced the ‘Invari-
ance Principle’ that any dynamic network model should
satisfy. It states that: ‘‘If the flow on any link i is supply
constrained, then link i enters a congested regime. After an
infinitesimal increment in time, because of the traffic flow
dynamics, Si (demand in link i) increases to the capacity of
the link, Ci. Then the flow in link i (qi) should not be
increased because of change in Si to Ci. Similarly, if qj is
demand constraint, then the solution should be invariant to
an increase in Rj (supply of link j) to Cj’’. Instead of linking
demand and supply directly, the new model incorporates
node demand and node supply constraints derived from a
global zone fundamental diagram for each node in the
network.
Among macroscopic continuum models, LWR model
(discretized or continuous) is the widely used one for
intersection [82, 85, 90]. However, there exists literature
[91–93] on intersection modelling using Aw–Rascle model.
They [91] used Aw–Rascle model without relaxation term
and solved the hyperbolic nonlinear equation system for
each node n as follows:
oUi
otþ oFðUiÞ
oUi
oUi
ox¼ 0; ð56Þ
where
U ¼ki
mi
" #
FðUÞ ¼mi � kipi kið Þ
m2i
ki
� mipi kið Þ
2
64
3
75
mi ¼ ki ui þ pi kið Þð Þ:
Here i represents the incoming and outgoing links of
node n.
Gentile et al. [94] introduced a node model capable of
the explicit representation of vehicle column formation
and dispersion, but sets no limits to the length of the
queue, therefore may well exceed the length of the arc.
Gentile et al. [95] extended this model to a network per-
formance model for modelling the spill back congestion.
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The new extended model eliminates the drawback in [94]
by translating the interaction among flows on adjacent
arcs by time varying arc entry and exit capacities. Spill-
back phenomenon is reproduced here as a hyper-critical
flow state either propagating backwards from the final
section of the arc or originating from the initial section of
the arc that reduces the capacities of the arc behind the
current arc.
Bliemer [96] formulated a node model that maximizes
the total flow subject to the constraints of turning fraction
[81] and flow conservation [87]. The model incorporates a
level of reduction aj equal to the ratio of Rj and the total
demandP
Sij towards j. The flow qij (flow from upstream
link i to downstream link j) is given by:
qij ¼ min Sij; a�j ðiÞSij
�a�j ðiÞ ¼ min
8j;Sij [ 0aj
� �; 8i; ð57Þ
when several links j impose a reduction on i.
Several alternatives have also been used to model flow
at intersections. Yuan et al. [97] developed a Hybrid Petri
Net model for unsignalized T-intersections, in which one
part represents the levels of traffic priority (Discrete Petri
Net) and the other part for the computation of the corre-
sponding time varying quantities (Timed Petri Net). The
model is capable of representing the relationship between
traffic flows, passing capacity and queuing vehicles for
dynamic traffic flows. Petri Net modelling of traffic flow
is well explained in Tolba et al. [98]. Liu and Dai [99]
came up with an interrupted traffic flow model which
consists of two parts: intersection buffer having inter-
rupted traffic flow and links having continuous traffic
flow. The link flow dynamics can be calculated using
LWR model and for the interrupted flow at junctions, two
additional equations are used. One is conservation of
vehicle at the intersections and the other is an entropy
condition that minimizes the uniform delay at junctions.
For the delay calculation, accumulation diagram with the
effect of physical queue is used. Jin [100] developed a
systematic approach for the kinematic wave solutions to
the Riemann problem of merging traffic flow in supply
demand space.
Tampere et al. [83] illustrated the violation of invariance
principle in the Bliemer [96] model and pointed out a set of
requirements from literatures, that need to be satisfied by
any first-order macroscopic node model. They are listed as
follows: (a) general applicability irrespective of the number
of incoming and outgoing links, (b) flow maximization,
(c) non-negative flow, (d) vehicle flow conservation,
(e) satisfying demand and supply constraints, (f) conser-
vation of turning fractions, and (g) satisfaction of invari-
ance principle. Along with this they also proposed some
node supply constraints and a supply constraint interaction
rule (SCIR—mandatory only for specific node model
instance) separately for unsignalized and signalized inter-
sections. The formulation of the model is as given below:
MaxX
i
X
j
qij
s:t: qij� 0 8i; j;
qi ¼X
j
qij� Si 8i
qj ¼X
i
qij�Rj 8j;
fij ¼Sij
Si
¼ qij
qi
8i; j;
if 9ijqi\Si; qi is invariant to Si ! Ci;
if 9jjqj\Rj; qj is invariant to Rj ! Cj;
SCIR constraints
optional ðnode supply constraintsÞ;
ð58Þ
where fij is the turning fraction. Node supply constraints Np
(defined through an internal node supply formulation) set
an upper bound to a selection Ap of all flows that make use
of the internal infrastructure p (e.g. green phase, conflict
point, arc on a roundabout).X
Up
f qij
� ��Np: ð59Þ
The SCIR is for answering the following important
questions: (a) each flow qij is limited by which of the
constraints (Si, Rj, Np), (b) for each supply constraint (Rj,
Np) how is the supply distributed to each of the competing
flows towards j or through p, and (c) if exactly one
constraint can be identified as the most restrictive one for
each link, define the composition of the sets Ui, Uj and Up,
each of which denotes the set of links that are demand
constrained, supply constrained and internal node supply
constrained, respectively.
Flotterod and Rohde [101] modified the generic node
model (GNM) of Tampere et al. [83] as an incremental
node model (INM) using Daganzo et al.’s [76] incremental
transfer (IT) principle. In this model flow transmissions are
computed in a multistage process with stage index
k = 0,…,K. The system of equations in the INM is given
by:
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Here ‘ represent one particular node, D(k) is the vector
of upstream and downstream nodes at stage k, u(k) is the
constant flow transfer rate at stage k, h(k) is the length of the
stage k, and bij is the non-negative turning fractions from
i to j.
The main limitation of Daganzo’s IT principle [76] and
the INM [101] is that it fails to explain the situation where
the increase in one flow leads to the decrease of another
flow. This situation occurs mainly when one traffic stream
yields to another and an increase in the high priority stream
decreases the throughput of the low priority stream. To
overcome this limitation, Flotterod and Rohde [101] further
extended the INM into the INM with node supply con-
straints. Corthout et al. [102] proposed additional supply
constraints in node models due to conflicts within the
intersection and analyzed the solution non-uniqueness for
point-like and spatial modelling of nodes.
3.3 Solution to node models
First generation models, as explained in the previous sec-
tion, are solved using simple supply–demand boundary
conditions (demand at upstream and supply at downstream
of the node) but can be applied only for merge or diverge
junctions. Node models formulated as directional flow
maximization problem is increasingly capturing attention
for further research, implementing additional constraints or
modifying the existing ones. Aw–Rascle model for inter-
section is solved in literature analytically as Riemann
problem with piecewise constant initial data at upstream
and downstream of the node. The model also incorporates
turning fractions which are obtained through constraint
downstream flow maximization problem [83]. Numerical
schemes as for link models can be used to solve spatial
node models where the concepts of grid points or cells
exist. However, representation of intersection as cells to
account for turning maneuvers is a difficult task following
restrictive assumptions, and thus analytical methods are
commonly used as a solution procedure.
3.4 Summary of node modelling
First generation node models (merge and diverge models)
provide a good base for node modelling and relies on
simple constraints of demand and supply. However, a
major limitation is that it cannot be applied for a general
case where there exist m upstream and n downstream links.
The problem remained unresolved until Holden and Ris-
erbo [86] formulated a constraint flow maximization
problem for intersection modelling. Following this model,
several other mathematical models [87, 88] are also
introduced giving better physical interpretation of the
entropy condition to the Riemann problem. The general
requirements for a node model are first introduced by
Lebacque and Koshyaran [90] and introduction of the
‘invariance principle’ became an inevitable condition for
the convergence of the solution by numerical schemes.
Incorporating all the reviewed requirements of a node
model, Tampere et al. [83] created a good algorithmic
platform for a GNM along with the supply constraint
interaction rule for modelling specific node instances.
Flotterod and Rohde [101] used GNM with the IT principle
and formulated an INM with node supply constraints.
Analytical methods are commonly used in literature for
qðkÞ ¼ qinðkÞ
qoutðkÞ
!
and qð0Þ ¼ 0;
qðkþ1Þ ¼ qðkÞ þ hðkÞu qðkÞ �
;
For any stage;
DðkÞ ¼ ‘
1� ‘� I ðupstream nodesÞ : qðinÞ‘ \D‘ and 8j; b‘j [ 0; q
ðoutÞ‘ \
X
j
I\‘� I þ J ðdownstream nodesÞ : qðoutÞ‘ \
X
‘
and 9i 2 DðkÞ; bi‘ [ 0
���������
8>>><
>>>:
9>>>=
>>>;
hðkÞ ¼ min‘2DðkÞ
D‘ � qinðkÞ‘
�=uin
‘ qðkÞ �
for ‘ upstream
X
‘
�qoutðkÞ‘
!
=uout‘ qðkÞ �
for ‘ downstream
8>>><
>>>:
9>>>=
>>>;
ð60Þ
172 Int J Adv Eng Sci Appl Math (April–September 2013) 5(2–3):158–176
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solving node models rather than solving numerically
(constrained optimization problem or a Riemann problem
with constrained optimization).
4 Synthesis
The models discussed above are synthesized in this section
based on four aspects as follows.
(1) Ability to map the real world traffic,
(2) Applicability to different flow regimes,
(3) Solution techniques and computational effort, and
(4) Scale of application.
Among link models, LWR model as well as its dis-
cretized model (CTM) limitedly captures traffic realism
when compared to higher order models. Anticipation and
inertial effect by drivers are two basic real traffic phe-
nomena which are lacking in the LWR model. When using
higher order models, formulation of velocity dynamics
leads results closer to real traffic. When applied to a
hypothetical mid-block section, LWR and higher order
models produce similar results, but for specific traffic sit-
uations such as lane reduction, higher order models dom-
inates to give better results [74]. In higher order models, as
the number of equations increase, the accuracy also
increases. Most of the continuum models discussed are
address vehicle heterogeneity and multilane effect in their
extensions, however, the lack of lane discipline behavior,
which is common in developing countries, should also be
studied to ensure transferability of these models. Porous
flow approach considered vehicle heterogeneity but
restricted the number of vehicle classes for tractability.
LWR model cannot model congestion dissipation well
and is more suited only for uncongested regime. Except
Payne model, all higher order models model congested as
well as uncongested regime equally well. In fact, Payne
model when applied to congested regime redraws the
cluster formation accurately; only at the tails of the con-
gestion, the model causes negative speed for vehicles.
Porous flow approach is also suited to model both the two
regimes, but not proven with actual data.
Easier analytical solution and less computational effort
are the two major highlights of LWR model. Though
analytical solution to higher order models is tiresome, the
computational effort when using simulation is comparable
with that of LWR model. Because of varying traffic con-
ditions on links and at boundaries, a hybrid approach of the
numerical schemes as well as of the boundary condition
will be the best option for accurate simulation results. The
high computational effort needed for porous flow approach
(because of the number of supply–demand conditions) and
for the non-continuum model (because of the number and
type of the variables involved) is a restriction for their
application in heterogeneous traffic flow.
Because of its simplicity, LWR model can be used for
large scale networks, but the errors may be large in con-
gested flow conditions when compared to other macro-
scopic models. For dynamic traffic assignment, when
applied to a single origin destination pair, higher order
models showed better convergence when compared to
LWR model [74]. There is a reasonable chance this effect
may hold at network level too. Not all higher order models
are used at a network level and is still an open research
area. Accuracy versus computational effort when applied to
a large scale network should be compared among higher
order models and also with LWR model.
Among node models, first generation models can only
be used for cases without any specific traffic instances or
constraints. The computational effort for these models are
less when compared to constraint flow maximization
approach, but the latter method will be the best choice for
modelling real flow behavior at intersections to avoid sig-
nificant compromise in the level of accuracy that can be
achieved. The GNM and its extensions should be explored
to large scale network and results should be compared with
that by existing network level node models both qualita-
tively and quantitatively. This promises to be an important
area of further research.
5 Conclusion
Links and nodes are the two major elements in a transport
network. Hence modelling of the network traffic flow
comprises these two elements either through their explicit
modelling and combination, or through any specific
approach that can simultaneously redraw the flow charac-
teristics on these elements. An efficient network model
should capture bottleneck effect and wave propagation
correctly and should also model the intersection flow
behavior well. Among the two major levels of models’
aggregation (micro- and macro-levels), macroscopic mod-
els are best suited for network wide applications where the
overall stream features (congestion, delay, etc.) are of
prime interest with comparatively less computational cost.
A clear understanding of the existing macro-level traffic
flow models will help in modelling of varying traffic sce-
narios with accurate qualitative and quantitative results.
The paper gives a broad over view of important, mac-
roscopic link and node models till date, and are described
and summarized separately in two main parts. Macroscopic
link models have a strong and vast base in continuum
theory. Earlier macroscopic link models were originally
formulated for homogeneous traffic conditions with perfect
lane discipline and hence the flow dynamics at stream level
Int J Adv Eng Sci Appl Math (April–September 2013) 5(2–3):158–176 173
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captures real conditions accurately. Research in the area of
modelling heterogeneous traffic lacking lane discipline is
insufficient. Non-continuum and porous flow approaches
for link modelling are complex because of the number and
type of variables involved. Literature in node modelling is
less when compared to link models because of the com-
plexity in modelling turning behavior. Compared to the
first generation node models, the constrained optimization
problem formulation for node modelling is gaining
increasing attention by researchers in this area. Numerical
schemes are the most used solution techniques in link
models where as analytical procedures are commonly used
for node models.
There exists numerous models of traffic flow in the
macroscopic level; yet not all the models are shown to be
used (or can be used) for network wide applications. Fol-
lowing an extensive review of the macroscopic traffic flow
models, the paper also presents a synthesis on the advan-
tages and disadvantages of the discussed models. The paper
provides the reader an understanding of the current status
of the macro-level modelling of traffic flow, highlighting
the positive features and hopefully will encourage efficient
research in this area to overcome drawbacks and witness
more applications particularly at network level.
Acknowledgements The authors thank the Ministry of Urban
Development, Government of India, for sponsoring the Center of
Excellence in Urban Transport at Indian Institute of Technology (IIT),
Madras that enabled this research work. The second author also
thanks the New Faculty Grant provided by IIT Madras that partially
funded this research work. All findings and opinions in the paper are
the authors and do not necessarily reflect the views of the funding
agencies.
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