Guido Gentile, Lorenzo Meschini, Natale Papola
Macroscopic Arc Performance Models For Within-Day Dynamic Traffic Assignment Guido Gentile Dipartimento di Idraulica, Trasporti e Strade University of Rome “La Sapienza” Rome, Italy
[email protected] +39 06 44585737 Lorenzo Meschini Dipartimento di Idraulica, Trasporti e Strade University of Rome “La Sapienza” Rome, Italy
[email protected] +39 06 44585737 Natale Papola Dipartimento di Idraulica, Trasporti e Strade University of Rome “La Sapienza” Rome, Italy
[email protected] +39 06 44585129
Submission date: 15/11/2002 Word count: 5200
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Guido Gentile, Lorenzo Meschini, Natale Papola
2
ABSTRACT The network travel time pattern plays a double role in within-day dynamic traffic assignment: on one side, it constitutes the main attribute in the context of users’ path choice; on the other side, it determines the arc flow pattern for given path choices. Then, it is important that in the supply model realistic arc performances are yielded for given arc inflows. On the other hand, the arc performance model has to be efficient in order to be employed in within-day dynamic traffic assignment, which is a complex problem of its own. In this paper, we present a new link-based macroscopic arc performance model with capacity constraints capable of taking implicitly into account the variability of the flow state along the arc accordingly to any fundamental diagram. Then we compare it, in terms of realism and efficiency, with three existing models, which have been to this end suitably modified and enhanced.
Guido Gentile, Lorenzo Meschini, Natale Papola
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3
INTRODUCTION
Within-Day Dynamic Traffic Assignment (WDDTA), regarded as a dynamic user equilibrium, can be formalized and solved as a fixed point problem in terms of arc inflow temporal profiles, accordingly with the model presented in (1) and depicted synthetically in FIGURE 1. [FIGURE 1 here] The network travel time pattern plays a double role in WDDTA: on one side, it constitutes the main attribute in the context of users’ path choice; on the other side, it determines the arc flow pattern for given path choices (dashed arrow in FIGURE 1). Then, it is important that in the supply model realistic arc performances are yielded for given arc inflows. On the other hand, the arc performance model has to be efficient in order to be employed in within-day dynamic traffic assignment, which is a complex problem of its own. This work focuses on those non-stationary macroscopic arc performance models which are based on the mono-dimensional fluid approximation (2). These models, usually expressed through differential equations and solved through finite difference methods, can be classified into two major groups. The models belonging to the first group, referred to as space continuous (e.g. METANET, (3); Cell Transmission Model, (4) and (5)), rely on a thick discretization in time and space; that is, they are point-based. Such models yield accurate results and allow any fundamental diagram to be used, but require considerable computing resources. The models belonging to the second group, referred to as space discrete, rely on time discretization only; that is, they are link-based. Such models can be, in turn, subdivided in whole link models and wave models. Whole link models (e.g. (6) and (7)), do not take into account the propagation of flow states along the arc, since performances are assumed dependent on a space-average state variable, such as density (8). This yields a poor representation of travel times, which gets worse as the arc length increases (9). Despite this major deficiency, these models allow any fundamental diagram to be adopted, and are widely used in WDDTA because of their simplicity (e.g. (10) and (11)). Wave models, based on the kinematic wave theory (see, for example, (12)), implicitly take into account the propagation of flow states along the arc, yielding arc performances as a function of the traffic conditions encountered while travelling throughout the link. So far, however, these models have been developed only for bottlenecks; that is when the fundamental diagram has a triangular shape and a capacity constraint is defined at the final section of the arc. In this case, only two speeds are admitted: the free-flow speed, when the inflow is lower than the capacity, and the queue speed, otherwise. Among them are the simplified kinematic wave model presented in (13), the deterministic queuing models ((14) and (15)), and the link-node model presented in (1). These models require minimal computing resources, but yield realistic results only in urban contexts. In this paper, we present a new wave model, named Average Kinematic Wave (AKW) model, as a generalization of the link-node model presented in (1) allowing any fundamental diagram to be used. We then analyse it together with three existing macroscopic arc performance models, namely one Space Continuous (SC), one Whole Link (WL) and the Simplified Kinematic Wave (SKW), in order to compare their computational efficiency and the realism of their output. To meet this objective, these three models have been suitably modified and enhanced in order to deal with general fundamental diagrams and with capacity constraints, thus obtaining four homogeneous models. 2
THE MATHEMATICAL FRAMEWORK
In this section we recall some significant results of traffic flow theory and introduce the mathematical framework underlying the four models discussed in this work. The following notation will be used throughout the paper: L arc length x∈[0, L] generic section of the arc δ infinitesimal distance τ0 initial instant of the period of analysis Θ final instant of the period of analysis τ∈[τ 0, Θ] generic instant of the period of analysis flow on arc section x at time τ q(x,τ) density on arc section x at time τ k(x,τ) speed on arc section x at time τ v(x,τ) Φ(x,τ) = [q(x,τ), k(x,τ), v(x,τ)] flow state on the arc section x at time τ speed of the kinematic wave on arc section x at time τ w(x,τ)
Guido Gentile, Lorenzo Meschini, Natale Papola t0(τ) t(τ) tL(τ) tW(τ) = tL(t(t0(τ))) f0(τ) = q(0,τ) f(τ) = q(δ,τ) e(τ) = q(L-δ, τ) eL(τ) = q(L, τ)
4
running link entrance time of a vehicle entering the arc at time τ exit time from the running link of a vehicle entering it at time τ arc exit time of a vehicle exiting the running link at time τ exit time from the arc of a vehicle entering it at time τ arc inflow at time τ running link inflow at time τ running link outflow at time τ arc outflow at time τ
τ
∫
Q ( x,τ ) = q ( x,σ ) ⋅ dσ
cumulative flow on arc section x at time τ
0
τ
∫f
F0 (τ ) =
0
(σ ) ⋅ dσ
= Q(0,τ)
cumulative arc inflow at time τ
0
F (τ ) =
τ
∫ f (σ ) ⋅ dσ
= Q(δ,τ)
cumulative running link inflow at time τ
0
τ
∫
E (τ ) = e (σ ) ⋅ dσ = Q(L-δ,τ)
cumulative running link outflow at time τ
0
τ
∫
E L (τ ) = eL (σ ) ⋅ dσ = Q(L,τ)
cumulative arc outflow at time τ
0
Based on the mono-dimensional fluid approximation, flow conservation and FIFO rule are satisfied; then, the following relations hold true (2): q ( x,τ ) = k ( x,τ ) ⋅ v ( x,τ )
∂k ( x,τ ) ∂τ
+
∂q ( x,τ ) ∂x
=0
F (τ ) = E ( t (τ ) ) f (τ ) = e ( t (τ ) ) ⋅
(1.1)
(1.2) (2.1)
dt (τ ) dτ
(2.2)
As usual, we assume the existence of a direct relation between speed and density. Then, under the hypothesis that the running link is a homogenous channel, we have: v ( x,τ ) = v ( k ( x,τ ) )
(3.1)
or equivalently:
k ( x,τ ) = k ( v ( x,τ ) )
(3.2)
Based on (1.1), equations (3) define also a relation between flow and density, called fundamental diagram: (4.1) q ( x,τ ) = q ( k ( x,τ ) ) k ( x,τ ) = k ( q ( x,τ ) )
(4.2)
and a relation between flow and speed: q ( x,τ ) = q ( v ( x,τ ) )
(5.1)
v ( x,τ ) = v ( q ( x,τ ) )
(5.2)
It is known from the kinematic wave theory ((12), (16)) that the solution in terms of flows to the system defined by (1.2) and (4.2) is such that the generic hypocritical flow state:
Guido Gentile, Lorenzo Meschini, Natale Papola
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Φ (x,τ ) = Φ (q(x,τ )) = [q(x,τ ) ,k(q(x,τ )) ,v(q(x,τ ))]
(6)
propagates along the running link at a constant speed: w ( x,τ ) = w ( q ( x,τ ) ) =
1 d k ( q ) dq
(7)
The instant u(τ) when a given state Φ( f(τ)) reaches the end of the running link and the corresponding cumulative outflow are given, respectively, by: u (τ ) = τ + L w ( f (τ ) )
(8)
G ( u (τ ) ) = F (τ ) + f (τ ) w ( f (τ ) ) − k ( f (τ ) ) ⋅ L (9) where the term [f (τ )/w( f (τ )) − k( f (τ ))] ⋅ L is the number of vehicles that would pass an observer travelling at speed w( f(τ)), entering in the running link at time τ and exiting it at time u(τ). To be notice that the temporal profile G, implicitly expressed by the system of equations (7), (8) and (9), is a multi-valued function of τ. Actually, not all the flow states generated at the beginning of the running link by a given inflow temporal profile reach the end of the running link, as some of them may be overwhelmed by other flow states. In (8) and (13), it is stated that the actual outflow temporal profile E is given by the lower envelope of G, that is:
E (τ ) = inf G (τ )
(10)
In the following, without loss of generality, we adopt the Greenshields linear model (17), where relations (3.1) and (3.2) have, respectively, the following form: k ( x,τ ) v ( x,τ ) = v0 ⋅ 1 − k j
(11.1)
v ( x,τ ) k ( x,τ ) = k j ⋅ 1 − v0
(11.2)
The linear model (11) is plausible and, combined with equation (1.1), allows expressing in a closed form the relations (4), (5) and (7). To be notice that the system of (1.1) and (11) in terms of flows has two solution, one for the hypocritical state and one for the hypercritical state. However, as it will be cleared later on, we need to model explicitly only hypocritical states; then, relations (4), (5) and (7) become, respectively: k ( x,τ ) q ( x,τ ) = k ( x,τ ) ⋅ v0 1 − k j k ( x,τ ) =
kj 4 ⋅ q ( x,τ ) ⋅ 1 − 1 − v0 ⋅ k j 2
v ( x,τ ) q ( x,τ ) = v ( x,τ ) ⋅ k j 1 − v0 v ( x,τ ) =
v0 2
4 ⋅ q ( x,τ ) ⋅ 1 + 1 − v0 ⋅ k j
w ( x,τ ) = v0 ⋅ 1 −
4 ⋅ q ( x,τ ) v0 ⋅ k j
(12.1)
(12.2)
(13.1)
(13.2)
(14)
In order to device a numerical method implementing the arc performance models analysed, the period of analysis is divided into I intervals identified by the sequence of instants (τ 0, … , τ i, … , τ I ) and, when needed, the running link is divided into Z sections identified by the sequence of progressives (x 0, … , x z, … , x Z ), where x 0 = δ and x Z = L-δ. In the following we assume to approximate flow temporal profiles through piece-wise
Guido Gentile, Lorenzo Meschini, Natale Papola
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constant functions, and cumulative flow and exit time temporal profiles through piece-wise linear functions; specifically: t0(τ00) = τ 0, t0 (τ ) = τ i −1 + (τ − τ 0i −1 ) ⋅ t(τ 0) = t 0, t (τ ) = t i −1 + (τ − τ i −1 ) ⋅
τ i − τ i −1 , τ∈(τ0 i-1, τ0 i], i = 1, … , I τ 0 i − τ 0 i −1
t i − t i −1 , τ∈(τ i-1, τ i], i = 1, … , I τ i − τ i −1
τ = (τ 0, … , τ I )
(15.1)
t = ( t 0, … , t I )
(15.2)
u(τ 0) = u 0, u (τ ) = u i −1 + (τ − τ i −1 ) ⋅
u i − u i −1 , τ∈(τ i-1, τ i], i = 1, … , I τ i − τ i −1
u = (u 0, … , u I )
(15.3)
tL(t 0) = tL0, tL (τ ) = tL i −1 + (τ − t i −1 ) ⋅
t L i − t L i −1 , τ∈(t i-1, t i], i = 1, … , I t i − t i −1
tL = (tL 0, … , tL I )
(15.4)
tW = (tW 0, … , tW I )
(15.5)
f0(τ00) = f00, f0(τ) = f0 i, τ∈(τ0 i-1, τ 0 i], i = 1, … , I
f0 = ( f00, … , f0I )
(15.6)
f(τ 0) = f 0, f(τ) = f i, τ∈(τ i-1, τ i], i = 1, … , I
f = ( f 0, … , f I )
(15.7)
tW(τ00) = tW0, tW (τ ) = tW i −1 + (τ − τ 0i −1 ) ⋅
tW i − tW i −1 , τ∈(τ0 i-1, τ0 i], i = 1, … , I τ 0i − τ 0i −1
e(t ) = e , e(τ) = e , τ∈(t , t ], i = 1, … , I 0
0
eL(tL0)
=
eL0,
i
eL(τ) =
i-1
eLi,
τ
i
∈(tLi-1, tLi],
i = 1, … , I
F0(τ00) = F00, F0 (τ ) = F0i −1 + (τ − τ 0i −1 ) ⋅ F(τ 0) = F 0, F (τ ) = F i −1 + (τ − τ i −1 ) ⋅
F i − F i −1 , τ∈(τ i-1, τ i], i = 1, … , I τ i − τ i −1
E (τ 0 ) = E 0 , E (τ ) = E i −1 + (τ − τ i −1 ) ⋅
E(t 0) = E 0, E (τ ) = E i −1 + (τ − t i −1 ) ⋅
F0i − F0i −1 , τ∈(τ0 i-1, τ0 i], i = 1, … , I τ 0 i − τ 0 i −1
E i − E i −1 , τ∈(τ i-1, τ i], i = 1, … , I τ i − τ i −1
E i − E i −1 , τ∈(t i-1, t i], i = 1, … , I t i − t i −1
G(u 0) = G 0, G (τ ) = G i −1 + (τ − u i −1 ) ⋅
G i − G i −1 , τ∈(u i-1, u i], i = 1, … , I u i − u i −1
EL(tL0) = EL0, EL (τ ) = EL i −1 + (τ − tLi −1 ) ⋅
E L i − E L i −1 , τ∈(tLi-1, tLi], i = 1, … , I t L i − t L i −1
0
I
(15.8)
eLI )
(15.9)
e = (e , … , e ) eL =
(eL0,
…,
F0 = (F00, … , F0I )
(15.10)
F = (F 0, … , F I )
(15.11)
E = ( E 0 , … , E I ) (15.12)
E = (E 0, … , E I )
(15.13)
G = (G 0, … , G I )
(15.14)
EL = (EL0, … , ELI )
(15.15)
In order to express E as the lower envelope of G, it is useful to introduce a subset (t 0̃ , … , t J̃ ) of the sequence of instants (u 0, … , u I ) such that its points are not dominated (see section 4.3): t j − t j −1 t (τ 0 ) = t 0 , t (τ ) = t j −1 + (τ − τ j −1 ) ⋅ j , τ∈(τ j-1, τ j], j = 1, … , J τ − τ j −1
t = ( t 0 , … , t J )
E j − E j −1 E (t 0 ) = E 0 , E (τ ) = E j −1 + (τ − t j −1 ) ⋅ j , τ∈(t j-1 ̃ , t ̃ j], j = 1, … , J t − t j −1
E = ( E 0 , … , E J ) (15.17)
(15.16)
The temporal profile E is then calculated in three different sequences of instants in (15.12), (15.13), (15.17). 3
THE ARC PERFORMANCE MODEL
In the following, the arc is modelled in three parts: a first bottleneck located at the arc initial section; a second bottleneck located at the arc final section; a running link between the arc initial and final sections, consisting of a homogeneous channel. The flow model introduced in section 2 associated to the running link may have any specification of (3); in this paper the linear model (11) is adopted as reference.
Guido Gentile, Lorenzo Meschini, Natale Papola
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The initial bottleneck, with a constant capacity CTL = qmax, maintains the inflow on the running link below the capacity implicitly defined by the fundamental diagram associated to the running link (for the Greenshields linear model it is: qmax = 0.25 ⋅v0 ⋅kj) and then guarantees the consistency of the traffic flow model in the context of WDDTA, where the arc inflows may assume any non-negative value. In order to avoid spillback modelling and to obtain spatially separable arc performance models, we assume that, when f0(τ) > CTL, a vertical queue is generated. The final bottleneck, with a constant capacity CHD < qmax, models the single hypercritic flow state (the queue) generated by a capacity reduction at the end of the arc (e.g. an intersection, a tollgate). To be noticed that the details of the actual temporal profile of the delay due to a variable capacity constraint (as a traffic light) are useless in the context of traffic assignment. At the instant when over saturation takes place, meaning that the capacity constraint is activated, a backward-moving shockwave generates from the second bottleneck and propagates the queue flow state along the running link, thus overwhelming any flow state propagating downstream on it. It will be shown that when a vehicle encounters the queue generated by the second bottleneck, the whole travel time tW(τ) -τ spent by the user on the three parts of the arc depends only on the capacity CHD and on the cumulative arc inflow-outflow difference F0(τ)-EL(τ), while it is independent of the vehicle speed before reaching the front of the queue; consequently, there is no need to represent explicitly the resulting arc flow pattern in order to evaluate tW(τ). The running link models only the congestion due to vehicles’ interaction along the arc due to hypocritical arc flow patterns. In fact, thanks to the first bottleneck, the incoming flow is lower than qmax and is always in the hypocritical state; then, since the running link is a homogeneous channel, either forward-moving shockwaves are generated by changes in the incoming flow states, or a backward-moving shockwave is generated by the final bottleneck. In the next section, the travel times on the running link will be evaluated through the SC model, discussed in section 4.1, the WL model, discussed in section 4.2, the SKW model, discussed in section 4.3, and through the new AKW model, discussed in section 4.4. We now examine, instead, the effect of the initial and final bottlenecks. To this end we will refer to a generic section where the capacity constraint is enforced. Let fB and FB be respectively the inflow and cumulative inflow temporal profiles, and CB be the bottleneck capacity; then, with reference to the initial bottleneck it is: fB (τ) = f0(τ), FB(τ) = F0(τ), CB = CTL; with reference to the final bottleneck it is: fB(τ) = e(τ), FB(τ) = E(τ), CB = CHD. As shown in FIGURE 6, the cumulative outflow temporal profile EB can be obtained as the lower envelope of profile FB and of the following set of lines: Qσ (τ ) = FB (σ ) + (τ − σ ) ⋅ C B , τ ≥ σ , σ : f B (σ ) = CB , ∂f B (σ ) ∂σ ≥ 0
(16)
[FIGURE 6 here] Then, based on FIFO and conservation rules, the outflow temporal profile eB and the exit time temporal profile tB can be determined by means of the following procedure: sub bottleneck(CB, τ, fB ; tB , eB) tB0 = τ 0 for i = 1 to I
{
(
)
tB i = max τ i , tB i −1 + τ i − τ i −1 ⋅ f B i CB
(
eB i = f B i ⋅ τ i − τ i −1
) (t
B
i
− tB i −1
)
}
(17)
next i end sub
where it is assumed that at time τ 0 there is no queue at the bottleneck. The arc exit time is given by:
(
tW (τ ) = tL t ( t0 (τ ) )
)
and can be evaluated, together with the outflow temporal profile eL, through the following procedure:
(18)
Guido Gentile, Lorenzo Meschini, Natale Papola
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sub global_exit_time(τ0, f0; tW, eL) call bottleneck(CTL, τ0, f0; τ, f ) call running_link(τ, f; t, e) call bottleneck(CHD, t, e ; tW, eL) end sub
The procedure running_link will be specified in the next section for each one of the four models considered. To be notice that, based on (15.1), (15.2), (15.4) and (18), it is tL i = tL(t (t0(τ i))) = tW(τ i) = tW i , i = 1, … , I . When the final bottleneck is active, equation (17) becomes:
(
)
tW i = tW i −1 + t i − t i −1 ⋅
ei ; CHD
(19)
Moreover based on the FIFO and conservation rules and equations (15.6), (15.7) and (15.8), the following holds true:
(
) (t − t ) ⋅ (τ − τ ) (τ − τ )
ei = f i ⋅ τ i − τ i −1 f i = f 0i
0
i
0
i −1
i
i −1
(20)
i −1
i
(21)
which, substituted in (19), show that when the queue is present at the final bottleneck the whole travel time depends only on CHD and temporal profile f0, and thus depends neither on the arc length, nor on the flow model adopted for the running link. 4
RUNNING LINK MODELS
In this section we specify the running link model in four different ways, where a flow state Φ(q0) is considered as initial condition. Recall that, if a queue is present at the final bottleneck, the output of the running link model, namely the profiles e and t, loose meaning. 4.1
Space Continuous Model
The SC model considered here is a slight modification of METANET (3), where the first order relation (11) is employed instead of the second order relation utilized by the authors. The model is implemented by the following procedure: sub running_link_SC(τ, f; t, e) F 0 = 0, E 0 = 0 for z = 0 to Z q z, 0 = q0 k
z ,0
(22.1)
(
= 0.5 ⋅ k j ⋅ 1 − 1 − 4 ⋅ q
next z for i = 1 to I q 0, i = f i for z = 1 to Z
z ,0
( v0 ⋅ k j ) )
q z ,i = k z ,i ⋅ v0
next z
( )( ⋅ (1 − k k )
( ⋅ (τ
z ,i
F i = F i −1 + q0 ,i ⋅ τ i − τ i −1 E =E
i −1
+q
(22.3)
(22.4)
k z ,i = k z ,i −1 − τ i − τ i −1 ⋅ q z ,i −1 − q z −1,i −1
i
(22.2)
Z ,i
i
−τ
i −1
j
) (x
z
− x z −1
)
(23) (24)
) )
next i call exit_time_and_outflow(τ; F, τ, Ē; t, e) end sub
Equations (22) are boundary conditions, while (23) and (24) result, respectively, from the discretization of (1.2) and (12.1) over the grid of points {( x z ,τ i ) ∈ [δ , L − δ ] × [0, Θ] : z = 0,1,..., Z , i = 0,1,..., I } .
Guido Gentile, Lorenzo Meschini, Natale Papola
9
Note that, in order to achieve a correct propagation of flow states through equation (23), the discretization grid must satisfy the following condition: x z − x z −1 1 ≥ max {w ( q ) :q ∈ [0,qmax ]} = = v0 i i −1 dk ( q = 0 ) dq τ −τ
(25)
If, for example, we have v0 = 25 m/sec and ( x z − x z −1 ) = 250 m, then ( τ i − τ i −1 ) must be smaller than 10 sec. Clearly, such a thick time discretization is unsuitable for WDDTA, where usually the period of analysis covers several hours. Based on equation (2.1), the exit time and outflow temporal profiles (15.2) and (15.8) are determined on the basis of the cumulative inflow and outflow temporal profiles, by means of the following procedure: sub exit_time_and_outflow(τ, F, σ; E; t, e) E J+1 = F I, σ J+1 = σ J j=0 for i = 0 to I do until E j > F i j=j+1 loop if f i = 0 then t i = max{t i-1, τ i + L / v0} ei = 0 else ti = τ
j −1
(
+ F i − E j −1
(
ei = f i ⋅ τ i − τ i −1
) (E
j
) (t − t ) i
)(
− E j −1 ⋅ σ j − σ
i −1
j −1
)
(26) (27)
end if next i end sub
Equations (26) and (27) are obtained, respectively, from (2.1) and (2.2), based on (15.7) and (15.8). Note that when f i > 0, it is always E j > E j-1 and t i > t i-1. This model gives very accurate results, so that in this paper it will be used as a term of reference to evaluate the efficacy of the other models. 4.2
Whole Link Model
The WL model considered here is based on the arc performance model proposed in (6), where the travel time of a vehicle entering the running link at time τ is determined as a function of the average density along the arc at the same instant. The linear time-density function utilized by the author is here replaced with the hyperbolic function that results from equation (11) assuming that the speed corresponding to the average density is maintained throughout the arc. The model is implemented by the following procedure: sub running_link_WL(τ, f; t, e) E -1 = 0, t -1 = 0, τ I+1 = τ I F 0 = 0, E 0 = 0 0
(
K = 0.5 ⋅ k j ⋅ 1 − 1 − 4 ⋅ q0
(
V 0 = v0 ⋅ 1 − K 0 k j t0 = τ 0 + L V 0 j=0 for i = 1 to I do until t j ≥ τ i j = j +1 loop
)
( v0 ⋅ k j ) )
(28.1) (28.2) (28.3) (28.4)
(
F i = F i −1 + f i ⋅ τ i − τ i −1
)
Guido Gentile, Lorenzo Meschini, Natale Papola
Ei = F i
(
)(
E i = E j −1 + τ i − t j −1 ⋅ E j − E j −1
(
Ki = F i − Ei
(
)
) (t
j
− t j −1
10
)
(29)
L
V i = v0 ⋅ 1 − K i k j
)
ti = τ i + L V i
(30)
i
if f = 0 then ei = 0
else
(
ei = f i ⋅ τ i − τ i −1
) (t − t ) i
i −1
(31)
end if next i end sub
where (28) are initial conditions, K i is the average density along the running link at time τ i and V i is the corresponding speed. In (29), the outflow Ēi is obtained by means of a linear interpolation on the points of its temporal profile given by (30) on the basis of (2.1). To this end, j | t j-1 < τ i ≤ t j must be not greater than i-1, otherwise t j would be still unknown when is needed. This implies:
{
t i −1 ≥ τ i , i = 1,..., I
}
{
⇒ min t i − τ i : i = 0,..., I = L v0 ≥ max τ i − τ i −1 : i = 1,..., I
}
(32)
Condition (32) yields an upper bound for the time intervals, which is analogous to that characterizing the SC model. 4.3
Simplified Kinematic Wave Model
We here present a solution method for the kinematic wave theory, based on cumulated flow temporal profiles, capable of handling any fundamental diagram. A similar approach can be found in (13) where, however, the solution method is provided only for the triangular-shaped fundamental diagram. The approach consists in evaluating the cumulative flow at a given point (x,τ) directly from boundary or initial conditions, without evaluating any state variable at intermediate times and positions. In this paper, specifically, profile G is evaluated directly through profile f, and the fundamental diagram expressed by relation (12.1) is employed. The model is implemented by the following procedure: sub running_link_SKW(τ, f; t, e) F 0 = 0, G 0 = 0 0
w = v0 ⋅ 1 − 4 ⋅ q0
u =τ +L/w for i = 1 to I 0
0
( v0 ⋅ k j )
(33.2)
0
(33.3)
(
F i = F i −1 + f i ⋅ τ i − τ i −1 wi = v0 ⋅ 1 − 4 ⋅ f i
)
( v0 ⋅ k j )
u i = τ i + L wi ki =
kj
2
(
(34) (35)
⋅ 1− 1− 4⋅ f i
(
(33.1)
)
( v0 ⋅ k j ) )
G i = F i + f i wi − k i ⋅ L
(36) (37)
next i call lower(u, G; t̃, Ẽ ) call exit_time_and_outflow(τ, F, t̃, Ẽ; t, e) end sub
where (33) are initial conditions, and (34), (35), (36) and (37) are respectively a discretization of equations (8), (9), (12.2) and (14).
Guido Gentile, Lorenzo Meschini, Natale Papola
11
Because of the more general fundamental diagram adopted in this paper, the determination of profile E from the multiple-valued profile G has required to develop the procedure lower described below, which selects, from the set of points defining G, only the points that belong to its lower envelope. Note that the resulting profiles E and e are defined over a subset (t̃ 0, … , t̃ j, … , t̃ J ) of points (u 0, … , u i, … , u I ). Procedure exit_time_and_outflow is the same described in section (3.1). sub lower(u, G; t̃, Ẽ ) i=1 1) for k = i + 1 to I if G k < G i then i=k end if next k j=0 Ẽ 0 = G i t 0̃ = u i do until i > I i=i+1 α = (G i-Ẽ j) / (u i-t ̃j) for k = i + 1 to I 2) if u k > t ̃j and G k < G i and (G k-Ẽ j) / (u k-t ̃j) ≤ α then i=k α = (G i-Ẽ j) / (u i-t ̃j) end if next k j=j+1 Ẽ j = G i t j̃ = u i loop end sub
Cycle 1) determines point (Ẽ 0, t̃ 0) finding the first point (Ẽ i, t̃ i) which is not dominated, that is the first point such that, for any k | i < k ≤ I, it is: Ẽ k ≥ Ẽ i. Successively, when a valid point (Ẽ j, t̃ j) is found, cycle 2) determines point (E j, t̃ j) finding the first point after (Ẽ j, t̃ j) that is not dominated, that is the point (Ẽ i, t̃ i) such that, for any k such that i+1 ≤ k ≤ I, point (Ẽ k, t̃ k) does not belong to the region shaded in FIGURE 2. To be notice that, in order to ensure the goodness of each point i, all the points k from i to I must be examined, yielding in the worst case I 2 operations to determine the entire profile E. [FIGURE 2 here] 4.4
The New Average Kinematic Wave Model
The new AKW model presented in this section is a generalization of the model presented in (1), as it allows also representing the congestion due to vehicle interaction along the arc, besides that caused by the bottleneck at the end of it. The main idea underlying the model is to determine, at a finite number of instants, a fictitious flow pattern, which synthesizes previous flow patterns along the arc and is employed, in its turn, in order to determine the successive flow patterns. The model is specified as follows. [FIGURE 3 here] Since the inflow temporal profile is given by (15.6), as depicted in FIGURE 3 the vehicle trajectories are piecewise linear. Then, the space (x,τ) comes out to be subdivided into flow regions Φ dependent on arc entering flow temporal profile and, in case, on over saturation of arc exiting section. Two neighbouring flow regions Φi and Φj are delimited by linear shock waves with slope: W ij =
q j − qi k j − ki
Based on (11.2) and (13.1), (38) becomes:
(38)
Guido Gentile, Lorenzo Meschini, Natale Papola
W ij = v i + v j − v0
12
(39)
In theory, knowing inflow temporal profile, with (13.2) and (39) is possible to determine the trajectory of a vehicle entered at the generic instant τ, and thus its exit time profile t. However, FIGURE 3 shows that these trajectories are extremely cumbersome to be determined and dealt with. For this reason, in order to obtain a workable model, we assume that the vehicle platoon entered during the interval (τ i-1, τ i] modifies its speed according to the average speed vmi-1 = L /(t i-1 – τ i-1) of the vehicle entered at instant τ i-1, as depicted in FIGURE 4. Thus, the trajectory of a vehicle entering in the arc at instant τ∈(τ i-1, τ i] is directly influenced only by the average trajectory of vehicle entered at previous interval’s end; this average trajectory is the synthesis of flow history trough the arc. Then, the slope W i-1 of the generic shock wave becomes: W i −1 = vmi −1 + v i − v0
(40)
[FIGURE 4 here] As depicted FIGURE 4, it’s worth notice that, although the modified model may allow local FIFO rule violations along the arc, it still ensures FIFO between initial and final sections of the arc. The running link travel time t is then determined through the following procedure, also depicted in FIGURE 5: sub running_link_AKW(τ, f; t, e)
(
( v0 ⋅ k j ) )
vm 0 = 0.5 ⋅ v0 ⋅ 1 + 1 − 4 ⋅ q0
t 0 = τ 0 + L / vm0 for i = 1 to I
(
v i = 0.5 ⋅ v0 ⋅ 1 + 1 − 4 ⋅ f i
( v0 ⋅ k j ) )
W i-1 = vmi −1 + v i − v0 i
(
i
ω = τ −τ
i −1
) ⋅W ( v − W ) i −1
if ω i ⋅ v i ≥ L then t i = τ i + L vi else t i = τ i + ω i + ( L − ω i ⋅ v i ) vm i −1 end if
(
ei = f i ⋅ τ i − τ i −1 vm i = L ( t i − τ i )
(41)
i −1
i
(42)
) (t − t ) i
i −1
next i end sub
[FIGURE 5 here] The generalization of this model to any fundamental diagram can be achieved as follows: a) since (38) holds in general, it can be evaluated (numerically, if needed) with any fundamental diagram knowing q i, q j, and thus k(qi ) k(q j ) , instead of using the analytical form (39); b) since for each τ i the model approximate the vehicle trajectory to a linear trajectory with constant speed vmi, the corresponding fictitious flow state Φmi = Φ( q(vmi ) , vmi, k(vmi)), may be determined; c) Φmi may be employed in (38), instead of using (40), in order to evaluate the resulting shock wave. 5
COMPARING THE DIFFERENT MODELS
In this section we compare, with respect to their efficiency and effectiveness, the different models presented in this paper. Regarding the effectiveness, the point based model presented in subsection 4.1 can be reasonably assumed as a term of reference, as it yields results close enough to the real ones. Regarding efficiency, it will result from the analysis of the algorithms.
Guido Gentile, Lorenzo Meschini, Natale Papola
13
Each model has been used to simulate traffic flow over an arc 10,000 meters long, adopting the Greenshields fundamental diagram with a capacity of 2025 veh/h, a free-flow speed of 90 km/h, and a jam density of 0,09 veh/m. With reference to the point based model, the arc was divided in Z = 40 sections 250 meters long; this required (see subsection (4.1)) to divide the period of analysis, 30 minutes long, into ISC = 180 intervals of 10 seconds. With reference to the link based models (where no spatial discretization occurs), in order to test the effect of time discretization size on the solution quality, the same period of analysis was divided into I = ISC /6 = 30 intervals of 60 seconds. First, in order to focus on the behaviour of the different models employed on the running link, we tested the models with three inflow temporal profiles (flow gradually increasing, gradually decreasing, and varying around an average value), lower than the arc incoming and outgoing capacities. [FIGURE 7 here] [FIGURE 8 here] [FIGURE 9 here] With respect to effectiveness, results show that the SKW model and AKW model behave much closer to the SC model than the WL model, especially if we look at the arc travel times (that, as said, are the relevant variables when performing WDDTA); in particular, the WL model shows a sort of “inertia”, with respect to the SC model, when representing travel times due to a rapidly varying inflow profile. Moreover, FIGURE 10 shows that the numerical approximation of the SKW model generates, in the presence of inflow discontinuities, a sequence of points t̃ i not uniformly distributed over time, yielding intervals where profile Ẽ, and consequently profiles E, e and t, are not determined. [FIGURE 10 here] Then, in order to test the complete arc model together with the new AKW model, a capacity constraint CTL = 2025 veh/h was set at the arc initial section, while a capacity constraint CHD = 0.6 ⋅ CTL = 1215 veh/h was set at the arc final section. A flow profile activating both bottlenecks was given to the model; results are shown in FIGURE 11. During interval (τB, τC), corresponding to interval (tB, tC) when the final bottleneck is active, the arc outflow is constant and equal to CTL, while the travel time profile is piece-wise linear with slopes equal to f i /CTL. To be noticed also that between τA and τB the inflow is greater than CTL, but the final bottleneck is not active; this is because the corresponding flow state, which raises in τA, reaches the arc final section only in τB: the running link thus acts as a sort of flywheel with respect to flow state propagations. [FIGURE 11 here] With respect to efficiency, the SC model has a complexity of O(ISC ⋅Z); thus the AKW model, having a complexity O(I), is much more efficient. Notice that the complexity of the SKW is not O(I), as it would appear, because of the procedure to evaluate profile E from profile Ẽ, which, as showed in subsection 4.3, has a complexity O(I 2). Then the AKW model is more efficient of the SKW model too. We recall that, with reference to the SC and WL models, (τ i – τ i-1) must be equal or greater than ∆τmin defined by (25) or (32), respectively; instead, with reference to the SKW and AKW models, numerical results show that (τ i – τ i-1) can be reduced up to ∆τmin /10 without a significant variation of the solution quality; this is important when applying the arc performance models to real-world systems, where run time is a critical issue. 6
CONCLUSIONS
The numeric results shows that the AKW model behave very closer to the point based model, requiring a computational effort much lower than all the other macroscopic models, both point and link based, yielding a valid tool to represent dynamically and macroscopically real-size systems where the short arc hypothesis doesn’t hold. REFERENCES
1. 2.
Bellei G., Gentile G., Papola N. (2002). A within-day dynamic traffic assignment model for urban road networks. Transportation Research, submitted Cascetta E. Transportation systems engineering: theory and methods. Kluwer Academic Publisher, Boston, 2001
Guido Gentile, Lorenzo Meschini, Natale Papola 3. 4. 5. 6. 7. 8.
9. 10. 11. 12. 13.
14. 15. 16. 17.
14
Messmer A., Papageorgiou M. METANET: a macroscopic simulation program for motorway networks. Traffic Engineering & Control Vol. 31, No. 8/9, 1990, pp. 466 - 470; No. 10, p. 549 Daganzo C.F. The cell transmission model: a dynamic representation of highway traffic consistent with hydrodynamic theory. Transportation Research part B Vol. 28, 1994, pp. 269-287 Daganzo C.F. The cell transmission model, part II: Network traffic. Transportation Research part B Vol. 29, 1995, pp. 79-93 Astarita V. A continuous time link model for dynamic network loading based on travel time function. In Proceedings of the 13th International Symposium on the Theory of Traffic Flow, Lyon, 1996, pp. 87-102 Ran B., Rouphail N.M., Tarko A., Boyce D.E. Toward a class of link travel time functions for dynamic assignment models on signalised networks. Transportation Research part B Vol. 31, 1997, pp. 277-290 Heydecker B.G., Addison J.D. Analysis of traffic models for dynamic equilibrium traffic assignment. In Transportation networks: recent methodological advances (ed MGH Bell). Pergamon, Oxford, 1998, pp. 35-49 Daganzo C.F. Properties of link travel time functions under dynamic loads. Transportation Research part B Vol. 29, 1995, pp. 95-98 Friesz T.L., Bernstein D., Smith T.E., Tobin R.L., Wie B.W. A variational inequality formulation of the dynamic network user equilibrium problem. Operations Research Vol. 41, 1993, pp.179-191 Tong C.O., Wong S.C. A predictive dynamic traffic assignment model in congested capacity-constrained road networks. Transportation Research part B Vol. 34, 2000, pp. 625-644 Daganzo C.F. Fundamentals of Transportation and Traffic Operations, Pergamon, 1997, pp. 97-112 Newell G.F. A simplified theory of kinematic waves in highway traffic, part I: general theory. Part II: queuing at freeway bottlenecks. Part III: multi-destination flows. Transportation Research part B Vol. 27, 1993, pp. 281-313 Arnott R., De Palma A., Lindsey R. Departure time and route choice for the morning commute. Transportation Research part B Vol. 24, 1990, pp. 209-228 Ghali M.O., Smith M.J. Traffic assignment, traffic control and road pricing. In Transportation and Traffic Theory, ed. C.F. Daganzo, Elsevier Science, Amsterdam, 1993, pp. 147-169 Newell G.F. Comments on traffic dynamics. Transportation Research part B Vol. 23, 1989, pp. 386-389. Huber, M.J. Traffic flow theory. Chapter 15 of Transportation and Traffic Engineering Handbook, Prentice-Hall, 1976
Guido Gentile, Lorenzo Meschini, Natale Papola
LIST OF TABLES AND FIGURES
TABLE 1 Taxonomy of the non stationary macroscopic arc performance models analyzed in this paper FIGURE 1 Dynamic User Equilibrium. FIGURE 2 Domination test for point (G i , u i ) when determining profile E FIGURE 3 Kinematic wave theory. FIGURE 4 Averaged kinematic wave theory. FIGURE 5 Travel time determination for under-saturated conditions. FIGURE 6 Construction of profile EB. FIGURE 7 Results for an increasing inflow. FIGURE 8 Results for a decreasing inflow. FIGURE 9 Results for a fluctuating inflow. FIGURE 10 Profile G generated by an inflow gradually increasing. FIGURE 11 Results of AKW model for an inflow.
15
Guido Gentile, Lorenzo Meschini, Natale Papola
16
TABLE 1 Taxonomy of the non stationary macroscopic arc performance models analyzed in this paper Type
Model
Discretization
Propagation of flow states along the arc
Fundamental diagram
point based
SC
time and space
Yes
any
link based
WL SKW AKW
time time time
No Yes Yes
any any any
Guido Gentile, Lorenzo Meschini, Natale Papola
17
path flows
path performances
network flow propagation model
path performance model
arc flows
arc performances arc performance model
FIGURE 1 Dynamic User Equilibrium.
arc perf. function
path choice model
network loading map
demand OD flows
Guido Gentile, Lorenzo Meschini, Natale Papola
18
E Gi
α Ẽj
point i is not dominated when no point k > i is in this area t̃ j
ui
t̃
FIGURE 2 Domination test for point (G i , u i ) when determining profile E
Guido Gentile, Lorenzo Meschini, Natale Papola
19
Trajectories of vehicles entered on the running link at instant τ i, i = 0, … , I Shock waves x
Outflow profile e
L W1,4 v0 v4
v1 v W0,1
W1,2
f1
τ0
FIGURE 3 Kinematic wave theory.
W4,5
W3,4 f4
f3
τ2
v5
v3
W2,3
f2
τ1
2
τ3
f5
τ4
τ τ5
Guido Gentile, Lorenzo Meschini, Natale Papola
20
Average trajectories of vehicles entered on the running link at instant τ i, i = 0, … , I Shock waves
x
Outflow profile e
L
vm0
vm0
v
vm1 vm2
1
vm
vm1 W0
τ0
v
τ1
W3 f4
f3
f2
τ2
FIGURE 4 Averaged kinematic wave theory.
vm5
4
2
W2
W1
f1
vm3
τ 3
W4 f5
τ4
τ τ5
Guido Gentile, Lorenzo Meschini, Natale Papola
21
ti
space
t i-1
vm
ωi
i-1
W i-1
τ i-1
τi
vm i-1 L
vi time
FIGURE 5 Travel time determination for under-saturated conditions.
Guido Gentile, Lorenzo Meschini, Natale Papola
Qσ 1
22
Qσ 2
FB
CB
σ1
σ 2 τ tB(τ)
FIGURE 6 Construction of profile EB.
τ
Guido Gentile, Lorenzo Meschini, Natale Papola
Inflow
outflow 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 0
300
SC
600
23
WL
900
SKW
1200
AKW
1500
1800
travel time 600 550 500 450 400 350 0
180
360
540
720 900 1080 time [sec]
FIGURE 7 Results for an increasing inflow.
1260
1440
1620
1800
Guido Gentile, Lorenzo Meschini, Natale Papola
Inflow
outflow 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 0
300
SC
600
24
WL
900
SKW
1200
AKW
1500
1800
travel time 600 550 500 450 400 350 0
180
360
540
720 900 1080 time [sec]
FIGURE 8 Results for a decreasing inflow.
1260
1440
1620
1800
Guido Gentile, Lorenzo Meschini, Natale Papola
outflow 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 0 travel time 550 530 510 490 470 450 430 410 390 370 350 0 180
Inflow
300
360
SC
600
540
25
WL
900
720 900 1080 time [sec]
FIGURE 9 Results for a fluctuating inflow.
SKW
1200
1260
AKW
1500
1440
1800
1620
1800
Guido Gentile, Lorenzo Meschini, Natale Papola
26
700 600 500 400 300 200 100 0 0
500
1000
Section x = 0 S i 3
1500
2000
Section x = L
FIGURE 10 Profile G generated by an inflow gradually increasing.
Guido Gentile, Lorenzo Meschini, Natale Papola
veh/sec 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 1000
27
arc inflow
first bottleneck capacity
first bottleneck outflow
last bottleneck capacity
arc outflow
200
400
600
800
1000
1200
1400
1600
1800
sec
800 600 400
τA τB
200
τC
tB
tC
0 0
200
400
600
800 1000 sec
first bottleneck delay = = tBtail t 0 (τ )i −- tτi
1200
1400
1600
arc delay = = tt iL-(tτ i)
1800
− τ
FIGURE 11 Results of AKW model for an inflow activating arc capacities.
