DYNAMIC TRAFFIC ASSIGNMENT: A CONTINUOUS MACROSCOPIC MODEL Guido Gentile Dipartimento di Idraulica Trasporti e Strade - Università degli Studi di Roma “La Sapienza” Via Eudossiana, 18 – 00184 Roma, [email protected]

ABSTRACT In this paper the Dynamic User Equilibrium is formulated and solved as a fixed point problem in terms of time-continuous real valued temporal profiles of arc inflows and arc performances, where travel demand is specified both through a deterministic and a stochastic implicit path choice model. By extending to the dynamic case the concept of Network Loading Map, yielding arc flows for given demand flows consistently with certain arc performances, the equilibrium is formulated without introducing the Dynamic Network Loading as a subproblem. The proposed assignment algorithm reflects this approach as the temporal consistency of the flow pattern through the arc performance model is attained only jointly with the equilibrium. In addiction, a new dynamic shortest path algorithm is presented to evaluate the Network Loading Map in the deterministic case, which can also be utilized within a Montecarlo simulation to evaluate it in the stochastic case. The main feature of this algorithm is its capability to handle a continuous travel time pattern by approximating the temporal profiles with piece-wise linear functions of time defined on long time intervals. Keywords: within-day dynamic traffic assignment, dynamic user equilibrium, fixed point problem, time-continuous temporal profiles, deterministic network loading map, implicit path choice model. 1

1

INTRODUCTION

This paper focuses on within-day dynamic traffic assignment regarded as a Dynamic User Equilibrium (DUE), or predictive assignment. Extending to the dynamic case Wardrop’s first principle, the deterministic DUE is defined as the state where no user can reduce his travel cost by unilaterally changing path (Smith, 1993; Heydecker and Addison, 1996). The stochastic DUE is defined analogously referring to the case where the costs perceived by users are modelled as random variables. It is assumed that users associate to each path its actual cost, given by the sum of the relative arc costs each one evaluated at the actual arc entering instant (that is the cost that would be actually experienced while travelling along the path), and then choose a minimum cost path, based on the hypothesis that they are rational decision makers. Note that DUE is different from the Dynamic User Optimum (DUO), or reactive assignment, where users are assumed to choose on the basis of the instantaneous path costs, given by the sum of the relative arc costs all evaluated at the departure instant from the origin. Due to the kind of temporal consistency that is to be ensured, DUE is more difficult to formulate and solve than DUO. In the literature, the deterministic DUE is mainly formalized as a finite-dimensional variational inequality problem in terms of, either path inflows and costs (Smith, 1993; Wie et al., 1995; Huang and Lam, 2002), or arc inflows and costs (Chen and Hsueh, 1998), based on the classical non-linear complementarity formulation of Wardrop’s first principle. However, other approaches exists: Janson and Robles (1993) formulate DUE as a bi-level nonlinear program in terms of arc inflows; Papageorgiu (1991) and Wisten and Smith (1997) propose an optimal control problem in terms of arc conditional probabilities; Bellei and Bielli (1996) and Kaufman, Smith and Wunderlich (1997) present fixed point formulations in terms, respectively, of arc inflows and arc conditional probabilities. With regard to the stochastic 2

DUE, a fixed point formulation in terms of arc inflows can be found in Cascetta (2001). In all these fixed point formulations, the Dynamic Network Loading (DNL), which consists in the seeking for given path flows of an arc flow pattern consistent with the travel time pattern through the arc performance model (Wu, Chan and Florian 1998; Xu et al., 1999), is introduced as a sub-problem of DUE. The algorithms proposed in literature to solve DUE are usually based on diagonalization methods implementing a bi-level approach, where iteratively an Upper Problem (UP) is solved for a given solution of a Lower Problem (LP) which is then consistently updated. In most cases, the UP consists in the non-linear problem of determining arc flows and costs resulting from a supply-demand equilibrium for a given travel time pattern, whose objective function is the classical integral sum employed in the formulation of traffic assignment in the static case, while the LP consists in the linear problem of updating the travel time pattern for given arc flows; the UP is solved with the Frank-Wolfe algorithm (Janson and Robles 1993; Chen and Hsueh, 1998), or with heuristic methods (Algorithm D, Wisten and Smith, 1997; swapping process, Wie et al., 1995). Alternatively, the UP is the path choice model, while the LP coincides with DNL problem; the latter is solved in Bellei and Bielli (1996) through the Method of Successive Averages (MSA), and in Tong and Wong (2000) using a traffic simulator. In a recent paper (Bellei, Gentile and Papola, 2002), a new fixed point formulation of the stochastic DUE in terms of time-continuous real valued temporal profiles of arc flows and arc performances (travel times and generalized costs) is presented, where the concept of Network Loading Map (NLM), yielding arc flows for given demand flows consistently with certain arc performances (Cantarella, 1997), is extended to the dynamic case, thus avoiding to introduce the DNL as a sub-problem. An implicit path algorithm is also proposed for the Logit case, 3

where the temporal consistency of the flow pattern through the arc performance model is attained only jointly with the equilibrium. The aim of this work is to extend this approach to the case where travel demand is specified through the deterministic or the Probit path choice model. Furthermore, a new dynamic shortest path (minimum cost) algorithm is presented to evaluate the Network Loading Map in the deterministic case, which can also be utilized within a Montecarlo simulation to evaluate it in the Probit case. The main feature of this algorithm is its capability to handle a continuous travel time pattern by approximating the temporal profiles with piece-wise linear functions of time. Our approach permits to avoid the thick time discretization (time intervals of few seconds), which is necessary to exploit the acyclic property resulting from the assumption of time-discrete temporal profiles (Pallottino and Scutellà, 1998). The reference instants on which the piecewise temporal profiles are based can define time intervals of several minutes. This, on one side, allows to capture the features of road traffic dynamic that are relevant for most planning purposes (some real-time application require by its nature a thick time discretization), and on the other side, results in very short computational times. In the numerical section of this paper a comparison has been carried out between our time-continuous approach with piece-wise linear approximated temporal profiles (with time intervals of 5 minutes) and the time-discrete approach described in Chabini (1998) (with a time discretization of 10 second), in terms of the equilibrium temporal profiles of arc flows and performances on one side, and computational times on the other. Considering one destination at the time, the algorithm computes in reverse chronological order the minimum cost from each node to the destination by means of a label correcting procedure, and then propagates in chronological order and in reverse topological order the 4

flows on the network toward the destination, regardless of their origin and departure time. This differs substantially from the approach of taking explicitly into account the trajectories on the network made by the demand flows, which has been adopted in Bellei, Gentile and Meschini (2003). Moreover the latter cannot handle generalized arc costs; on the contrary, another feature of the proposed dynamic shortest path algorithm is its capability to handle the case where the path costs are not proportional to travel times, thus allowing for effective modelling of toll pricing strategies in a dynamic framework. To focus our attention on the innovations here introduced and to simplify notation, a single user class and a fixed origin-destination demand will be considered, regarding only path choice as elastic. The case of elastic demand is addressed in Bellei et al. (2002), where a departure time choice model particularly suitable for DUE is also presented. On the supply side, the arc performances are evaluated by means of a simple link-based macroscopic model, which permits to represent vehicle queues explicitly by introducing a bottleneck (i.e. a capacity constraint) at the final section. This model is non-separable with respect to time, because the performances at a given instant depend on the previous history of inflows, but is separable with respect to space (e.g. the spillback phenomenon is not represented).

2

THE MATHEMATICAL MODEL

User trips on the road network are modelled through an oriented graph G = (N, A), where N is the set of the nodes and A is the set of the arcs. Each arc a is identified by its initial node TL(a), referred to as tail, and by its ending node HD(a), referred to as head, i.e. a = (TL(a), HD(a)). The origins and the destinations of user trips constitute a subset C of nodes, referred to as centroids. In the following it is assumed that, when travelling from node

5

x to destination d, users consider the finite set Kxd of paths from x to d with an opportune maximum number of cycles, ensuring that Kxd includes the shortest paths. Graph G is assumed to be strongly connected, i.e. Kxd ≠ ∅. The sets of the arcs exiting and entering, respectively, a given node x are referred to as forward star and backward star, and denoted FS(x) = {a∈A: TL(a) = x}, BS(x) = {a∈A: HD(a) = x}. Path topology is described through the following set notation: KaTL(a)d

sub-set of the paths belonging to KTL(a)d that begin with arc a ;

Akxd

set of the arcs constituting path k∈Kxd ;

Akxda

set of the arcs constituting the sub-path of path k∈Kxd between x and the tail of a∈Akxd .

When a cyclic path comprises a given arc more than once, in the above notation the arc a must be regarded with its order of appearance in the path. As the analysis is carried out within a dynamic context, the model variables are temporal profiles, here represented as piecewise C1 real valued functions of the continuous time variable τ. With reference to the flow pattern we use the following notation: dod(τ)

demand flow of users that travel from origin o to destination d ≠ o departing at time τ ;

d

temporal profile of the (D × 1) demand flow vector, where D = |C|⋅(|C|-1) ;

fa(τ)

flow of vehicles entering arc a at time τ ;

f

temporal profile of the (|A| × 1) arc inflow vector ;

Pkxd(τ)

probability to choose path k∈Kxd for users that travel from node x to destination d departing at time τ ;

pad(τ)

probability of using arc a, conditional on being at node TL(a) at time τ, when travelling to destination d ;

p

temporal profile of the (|A|⋅|C| × 1) arc conditional probability vector. 6

In the following the temporal profiles of the demand flows, referred to the departure time from the origin, are assumed to be fixed, bounded from above, and null outside the interval [0, Ω]. With reference to the performance pattern we use the following notation: ca(τ)

cost of arc a for users entering it at time τ ;

c

temporal profile of the (|A| × 1) arc cost vector ;

ta(τ)

exit time from arc a for users entering it at time τ ;

ta-1(τ)

entrance time in arc a for users exiting it at time τ ;

t

temporal profile of the (|A| × 1) arc exit time vector ;

ma(τ)

monetary cost of arc a for users entering it at time τ ;

Ckxd(τ)

actual cost of path k∈Kxd for users leaving node x at time τ ;

Tkxda(τ)

entrance time in arc a∈Akxd for users that follow path k∈Kxd leaving node x at time τ ;

wxd(τ)

minimum actual cost to reach destination d leaving node x at time τ ;

w

temporal profile of the (|N|⋅|C| × 1) minimum cost vector.

The supply model is founded on a link-based macroscopic flow model, where vehicles are represented as particles of a mono-dimensional partly compressible fluid, and capacity constraints are introduced in order to represent vehicle queues explicitly. In this context the First In First Out (FIFO) rule holds true, and the travel time temporal profiles are continuous (Cascetta, 2001). Then, if the inflow is positive between two given instants τ΄ < τ˝ , we have: ta(τ΄) < ta(τ˝) .

(1)

The monotonicity expressed by (1) ensures that the temporal profiles of the exit times are invertible. On the contrary, the flow temporal profiles are inherently discontinuous because, whenever the queue vanishes on an arc, the outflow reduces instantaneously from the capacity to a value related to the inflow. 7

Despite the FIFO rule, due to the presence of monetary costs (e.g. tolls), whose temporal profiles may be highly discontinuous, it may be convenient to wait at nodes in order to enter a given arc later (Ahuja et al. ,2002). In the following we assume that vehicles are not allowed to wait at nodes, so that cyclic paths may be convenient. We assume also that the arc costs are not smaller than a strictly positive constant (ca(τ) ≥ c > 0), so that the shortest paths include at the most a finite number of cycles. Clearly, if the departure time choice is included in the demand model, the motivation to consider cyclic paths becomes less relevant. The main variables of the mathematical model have been introduced; in the following subsections we focus on the relations among them.

2.1 Arc performance model The congestion phenomena are formally represented assuming that the temporal profiles of the arc exit times and of the arc costs depend on the temporal profiles of the arc inflows: t = t( f ) ,

(2)

c = c( f ) .

(3)

The performance functionals (2) and (3) are here specified with reference to a simple linknode model, where the generic arc a terminates with a bottleneck having a constant vehicle capacity Qa > 0 and is characterized by a constant under-saturated travel time ua , which takes into account the uncongested link travel time including the delay due to the final intersection. When the inflow exceeds the bottleneck capacity, an over-saturated queue occurs, causing a congestion delay. The resulting exit times are implicitly expressed by the following equation, which is based on vehicle conservation and FIFO rules: 8

Fa(τ) = Ea(ta(τ)) ,

(4)

where Fa(τ) and Ea(τ) are, respectively, the cumulative inflow and outflow at time τ. The former, by definition, is given by: τ

Fa(τ) = 0∫ fa(σ)⋅dσ ;

(5)

while the latter, based on the Newell-Luke minimum principle (Daganzo, 1997), is given by: Ea(τ) = min{Fa(τ -ua), Fa(σ -ua) +(τ -σ)⋅Qa : 0 ≤ σ ≤ τ} ,

(6)

which expresses the fact that the cumulative outflow cannot increase faster than the capacity as depicted in Figure 1. A numerical solution of the system of equations (5)÷(4) will be provided in section 4.1 .

ua Fa

Ea Qa

σ

1

τ

σ

2

time ta(τ)

Figure 1 - Newell-Luke minimum principle applied to the link-node model.

Alternatively, the more sophisticated models proposed in Gentile, Meschini and Papola (2003) can be adopted without difficulties. The arc cost for users entering at time τ is simply assumed to be: ca(τ) = η⋅(ta(τ)-τ) + ma(τ) ,

(7)

where η is the value of time.

9

2.2 Deterministic path choice model Assuming that the costs perceived by users are additive, the actual cost of path k∈Kxd at time

τ is the sum of the costs of its arcs a∈Akxd, each of them referred to the time Tkxda(τ) when the users leaving node x at τ reach node TL(a), i.e. Ckxd(τ) = ∑ a∈Akxd ca(Tkxda(τ)) .

(8)

Under the assumption that users are perfectly informed rational decision-makers, the resulting behaviour is such that only minimum cost path are utilized. The deterministic path choice model can then be formulated based on the following extension to the dynamic case of Wardrop’s first principle: - if the generic path k∈Kxd is used at time τ, that is its choice probability Pkxd(τ) is positive, then its actual cost Ckxd(τ) is equal to the minimum actual cost wxd(τ) to travel between x and d at the same time; - vice versa, if path k is unused, that is its choice probability is null, then its actual cost may not be smaller than the minimum actual cost. Formally, we have: wxd(τ) = min{Ckxd(τ): k∈Kxd} ,

(9)

Pkxd(τ) ⋅ (Ckxd(τ) - wxd(τ)) = 0 .

(10)

Clearly, the choice probabilities must satisfy the following conditions: ∑ k∈Kxd Pkxd(τ) = 1 ,

(11)

Pkxd(τ) ≥ 0 .

(12)

To be notice that, when there is more than one minimum cost path between any o/d pair, the choice probability pattern solving the system (10)÷(12) is not unique.

10

2.3 Implicit path formulation in the deterministic case We now extend to the deterministic case the implicit path formulation presented in (Bellei, Gentile and Papola, 2002), which is based on the concepts of node satisfaction and arc conditional probability. The main idea underlying the implicit path formulation is to decompose the path choice in a sequence of arc choices taken at each node among the arcs of its forward star, while travelling towards the destination, as depicted in Figure 2. In general, this decomposition is possible only if the arc conditional probabilities at each node are equal for all users directed to the same destination regardless of the sub-path so far utilized.

τ

tb(τ)

time d∈C

x ≡ TL(a)∈N

a∈A

b∈FS(x) cb(τ) -wHD(b)d(tb(τ)) Figure 2 - The arc choice.

Because the paths in Kxd can be partitioned on the basis of their first arc, equation (9) can be written as:

wxd (τ ) = min {Ckxd (τ ) : a ∈ FS ( x ) , k ∈ K axd } .

(13)

As the paths in KaTL(a)d coincide with the paths in KHD(a)d but for a, on the basis of (8) the right 11

hand side of (13) becomes:

{

}

( a )d min ca (τ ) + C HD ( ta (τ ) ) : a ∈ FS ( x ) , j ∈ K HD(a )d , j

which, because minimization is associative, can be written as:

{ {

}

}

( a )d min min ca (τ ) + C HD ( ta (τ ) ) : j ∈ K HD(a )d : a ∈ FS ( x ) . j

Because the term ca(τ) is common to all the terms of the inner minimization, based on (9) equation (13) becomes:

{

}

d wxd (τ ) = min ca (τ ) + wHD ( a ) ( ta (τ ) ) : a ∈ FS ( x ) .

(14)

Note that (14) coincides with the opposite of the deterministic specification of the satisfaction associated to the arc choice at the generic node x. For this reason the minimum actual costs can be referred to as node satisfactions. This recursive equation can be formally expressed by the functional: w = w(c, t) .

(15)

By definition, the choice probability Pkxd(τ) of the generic path k∈Kxd at time τ is equal to the product of the conditional probabilities of its arcs a∈Akxd, each of them referred to the time Tkxda(τ) when users departing from x at τ reach TL(a), i.e. Pkxd (τ ) = ∏ a∈Axd pad ( Tkxd a (τ ) ) .

(16)

k

With reference to the deterministic case, it is clear that, if a sub-path connecting a given node x to destination d has minimum cost, then it may be used by some traveller, and vice versa it may not, regardless of the sub-path utilized to reach x. However, when there is more than one minimum cost path from x to d, the conditional probabilities of the arcs exiting node x depend, in general, on the sub-path so far utilized, because different demand components may 12

split differently among minimum sub-paths. Yet, for a given cost pattern, there is always some solution point of the system (10)÷(12) consistent with the assumption that the arc conditional probabilities at each node are equal for all users directed to the same destination regardless of the sub-path so far utilized. Then, referring to these points, based on (16) it is: Pkxd (τ ) = pad (τ ) ⋅ PjHD( a )d ( ta (τ ) ) ,

(17)

where a is the first arc of path k∈Kxd and path j coincides with k but for a. Moreover, in order to determine the arc conditional probability of the generic arc a , we can refer to the users that depart from its tail, thus obtaining: pad (τ ) =

∑

TL a d k∈K a ( )

TL( a )d

Pk

(τ )

.

(18)

Multiplying both sides of (18) by wTL(a)d(τ), on the basis of (10) we have: d pad (τ ) ⋅ wTL ( a ) (τ ) =

∑

TL( a )d

TL a d k∈K a ( )

Pk

(τ ) ⋅ CkTL(a )d (τ )

.

(19)

As the paths in KaTL(a)d coincide with the paths in K HD(a)d but for a, on the basis of (17) and (8) the right hand side of (19) becomes:

∑

j∈K

HD ( a ) d

pad (τ ) ⋅ Pj

HD ( a )d

( t (τ ) ) ⋅ ( c (τ ) + C a

a

HL( a )d j

( t (τ ) )) , a

which can be written as:

⎛ HD a d pad (τ ) ⋅ ⎜ ca (τ ) ⋅ ∑ Pj ( ) ( ta (τ ) ) + ⎜ HD a d j∈K ( ) ⎝

∑

j∈K

HD ( a ) d

HD( a )d

Pj

( t (τ ) ) ⋅ C a

HL( a )d j

⎞

( t (τ ) ) ⎟⎟ . a

⎠

Based on (11) and (10), from (19) we then have:

(

)

d d pad (τ ) ⋅ ca (τ ) + wHD ( a ) ( ta (τ ) ) − wTL( a ) (τ ) = 0 .

13

(20)

By definition, the arc conditional probabilities must also satisfy the following conditions:

∑

a∈FS ( x )

pad (τ ) = 1 ,

(21)

pad (τ ) ≥ 0 .

(22)

Note that the system (20)÷(22) coincides with the deterministic specification of the probabilities associated to the arc choice at the generic node x, given the satisfactions of the nodes HD(a), a∈FS(x). Note that, when there is more than one arc choice alternative with minimum cost from any node, the arc conditional probability pattern solving the system (20)÷(22) is not unique. It is then formally expressed through the following point to set map: p ∈ p(w, c, t) .

(23)

Clearly, for a given performance pattern, the set of path choice probabilities obtained by composing the maps (23) through equation (16) is only a subset of the solution set of the system (10)÷(12). However, because when solving the assignment problem we just need to determine any point of the choice map, this circumstance does not constitute a relevant limitation of the implicit path formulation. The flow fad(τ) directed to destination d entering arc a at time τ is given by the arc conditional probability pad(τ) multiplied by the flow entering node TL(a) at time τ ; the latter is given, in turn, by the sum of the outflow fbd(tb-1(τ))⋅dtb-1(τ)/dτ of each arc b entering TL(a), and of the demand flow dTL(a)d(τ) from TL(a) to d, which is null if TL(a)∉C; i.e. ⎡ d TL( a )d (τ ) ⎛ d −1 dtb −1 (τ ) ⎞ ⎤ f (τ ) = p (τ ) ⋅ ⎢ + ∑ b∈BS (TL( a ) ) ⎜ f b ( tb (τ ) ) ⋅ ⎟⎥ , μ dτ ⎠ ⎦⎥ ⎝ ⎣⎢ d a

d a

(24)

where μ is the vehicle occupancy coefficient and the terms dtb-1(τ)/dτ stem from applying the FIFO and the vehicle conservation rule (Cascetta, 2001, p. 376). The total flow entering arc a 14

at time τ is then: f a (τ ) = ∑ d∈C f ad (τ ) .

(25)

Based on (24), (25) is expressed formally by the following functional: f = ω(p, t ; d) .

(26)

Note that, because the graph is strongly connected, the link capacities are strictly positive and the demand is bounded, then there exists a time Θ, with Ω < Θ < ∞, within which all the trips have been concluded, so that the arc inflow temporal profiles are null outside the period of analysis [0, Θ].

3

FORMULATION OF THE DYNAMIC USER EQUILIBRIUM

The implicit path formulation of the DUE model presented in the previous section is synthetically depicted in Figure 3, where the dotted arrow denotes any point of the corresponding point to set map.

network loading map

d

p(w, t, c) p

w

ω( p, t ; d)

w(c, t)

f

t

c

t( f ) c( f ) arc performance model

Figure 3 - Implicit path formulation of DUE. 15

In analogy with the static case, the NLM is a functional relation yielding an arc inflows f for given demand flows d consistently with certain arc performances t and c through the path choice model, which has been here formulated implicitly by introducing the node satisfactions w and the arc conditional probabilities p ; while the arc performance model yields the arc

costs c and the arc exit times t corresponding to certain arc inflows f . The system of the NLM and of the arc performance model formulates DUE as a fixed point problem.

3.1 The deterministic case The combination of (15) and (23) with (26) yields the implicit path formulation of the NLM: f ∈ ω(p(w(c, t), c, t), t ; d) .

(27)

Then, the deterministic NLM may be formally expressed through the following point-to-set map: f ∈ fD(c, t ; d) .

(28)

By combining (2) and (3) with (28), the deterministic DUE can be formalized as a fixed-point problem in terms of arc inflow temporal profiles: f ∈ fD(c( f ), t( f ); d) .

(29)

3.2 The Probit case The Probit path choice model, who’s formulation can be found in Cascetta (2001), is based on the random utility theory, where the travel costs are not known with certainty and thus are regarded as random variables. We assume that the perceived arc cost temporal profile ĉa of the generic arc a is equal to the sum of the arc cost temporal profile ca yielded by the arc performance model and of a random error temporal profile whose value at time τ is distributed as a normal variable, with null mean and variance proportional, through a

16

coefficient ξ, to a given cost term χa(τ). In order to improve numerical stability, we assume that the temporal profile χa is independent of congestion; however, χa is not necessarily constant in time, so that the level of uncertainty may vary during the period of analysis. The arc flow pattern resulting from the evaluation of the Probit NLM for given arc performances is accomplished through the well-known Montecarlo method, as follows. a) Get a sample of H perceived arc cost patterns: ĉah(τ) = ca(τ) + ψah⋅(ξ⋅χa(τ))0.5, ĉ h = ĉ(c; χ), h = 1, … , H ,

(30)

where each ψah is extracted from a standard normal variable N[0,1]. b) For each perceived arc cost pattern of the sample, determine through the deterministic NLM (27) a consistent arc inflow pattern; formally: y h ∈ ω(p(w(ĉ h, t), ĉ h, t), t ; d) , h = 1, … , H .

(31)

c) Calculate the average of the deterministic arc inflow patterns y h, thus obtaining an undistorted estimation of the Probit arc inflow pattern; formally: f = 1/H ⋅∑ h = 1, … , H y h .

(32)

Note that, based on (30), the same outcome ψah of the standard normal variable is used to perturb the whole temporal profile ĉah, so that the latter remains a continuous function of time. The Probit NLM, evaluated through (30)÷(32), can be formally expressed as: f = fP(c, t ; d) ,

(33)

and combined with (2) and (3) yields a formulation of the stochastic DUE: f = fP(c( f ), t( f ); d) .

(34)

Note that, because the Probit NLM is a point-to-point map, the resulting fixed point problem is based on a one-valued function. 17

4

SOLUTION ALGORITHM

In order to implement the proposed DUE model, the period of analysis [0, Θ] is divided into I time intervals identified by the sequence of instants (τ 0, … , τ i, … , τ I ), with τ 0 = 0 and τ I = Θ. In the following we assume to approximate the generic temporal profile x through either a piece-wise constant or a piece-wise linear function defined by the values taken at such instants, so that for the two cases we have, respectively: x(τ) = x i , τ∈(τ i-1, τ i] , i = 1, … , I ,

(35.1)

x(τ 0) = x0 , x(τ) = x i-1 + (τ -τ i-1) ⋅(x i -x i-1)/(τ i -τ i-1) , τ∈(τ i-1, τ i] , i = 1, … , I .

(35.2)

Specifically, the flow temporal profiles are assumed piece-wise constant, while the performance temporal profiles are assumed piece-wise linear. Based on this hypothesis, the generic temporal profile x can be numerically represented through the (1 × I+1) row vector x = (x 0, … , x i, … , x I ).

The initial and final state of the network is assumed to be known; here, without loss of generality the network is considered to be unloaded at times τ 0 and τ I. In the algorithm described below, the reference instants can define time intervals of several minutes, thus allowing to capture the features of road traffic dynamic that are relevant for most planning purposes, while keeping computational times to a minimum.

4.1 Arc performances Based on the assumption that the inflow temporal profiles are piece-wise constant as in (35.1), the temporal profiles yielded by the arc performance model presented in subsection 2.1 are actually piece-wise linear and their values at the reference instants can be obtained through the following procedure (Bellei, Gentile and Papola, 2002): 18

function [t, c] = γ(f) for each a∈A ta0 = ua for i = 1 to I tai = max{tai-1 +(τ i -τ i-1) ⋅fai/Qa , τ i+ua} cai = η⋅(tai-τ i) +mai next i next a end function

The only approximation affecting the corresponding performance temporal profiles (35.2) concerns the instants when the queues vanish, which do not coincide in general with any reference instant.

4.2 Network loading In this section we present a new dynamic shortest path algorithm to evaluate the deterministic NLM. The main feature of this algorithm is its capability to handle the continuous piece-wise linear temporal profiles of the travel time defined in (35.2) and obtained through the above procedure. In the following we aim at determining a specific value of the NLM, such that all users travelling toward a same destination d and traversing a given node x during the generic time interval (τ i-1, τ i] are propagated forward on one arc only, called the successive arc and denoted by σxd i, consistently with the assumption (35.1) referred to the arc conditional probabilities. Under this assumption, the flow traversing node x directed toward d during the time interval (τ i-1, τ i] utilizes arc a = σxd i and traverses node HD(a) during the interval (tai-1, tai]. Consistently with the deterministic path choice model, the successive arc σxd i shall belong to a minimum actual cost path connecting x to d at time τ i and can be determined coherently with equation (14) based on the performances of the arcs belonging to the forward star of x

19

and their head satisfactions. Note that by hypothesis the successor of each node is one arc only, the graph is strongly connected and the arc costs are non-negative; then, referring to any given instant τ i, the arcs σxd i, x∈N, constitute a tree. Beside the arc performances which are known at this stage, the node satisfaction (minimum actual cost) at a given reference instant depends on the satisfactions of other nodes at later reference instants and possibly at that same instant, but not on the node satisfactions at previous reference instants. In fact, based on (35.2), when a certain node satisfaction wyd i is improved, the Bellman triangular relation wTL(a)d i ≤ cai + wyd(tai) shall be verified for each arc a∈BS(y) such that tai < τ i+1, and, if the latter is not satisfied, the satisfaction wTL(a)d i shall be

consistently decreased and the successive arc σTL(a)d i possibly updated (see figure 4 below).

wyd i+1

TL(a) a

wyd(tai)

y

wydi

τi

ta i

τ i+1

d

Figure 4 - Dynamic shortest paths.

For this reason, it is convenient to perform the calculation of the node satisfactions and of the successor arcs in reverse chronological order, while it is necessary to employ either a label setting or a label correcting strategy (see for example: Bertsekas, 1993) for processing the nodes, where only the node satisfactions and the successor arcs of the current reference instant 20

are modified (in the case where the shortest arc travel time is greater than the longest time interval this is patently useless). Here we have adopted a label correcting approach, where the nodes whose satisfaction is updated are inserted at the top of a list of nodes to be still examined, while at each iteration the first node of the list is extracted and the Bellman relation is verified for its backward star. In order to enhanced the algorithm performance the list of nodes to be still examined is initialized with all the nodes in the order that they exit the list for the last time when determining the successive arcs for the next reference instant, so as to exploit the commonly experienced continuity over time of the path choice. Note that the latter is a node topological order, that is an ordering of nodes consistent with the tree topology of the successor arcs (for each successor arc, the order of its tail is greater than the order of its head).

vyd i vyd i+1 vyd i+2 tai-1

y

x

fa i

ta i

vxdi

τ i-1

τi

τ i+1

τ i+2

Figure 5 - Network flow propagation.

Note that the vehicles traversing a certain node at a given time instant have necessarily entered at a previous instant an arc of its backward star whose tail has a higher topological order. On this basis, to perform the network loading, the algorithm propagates in 21

chronological order the vehicle flows on the network toward the destination, regardless of their origin and departure time. Specifically, proceeding in reverse node topological order, the number of vehicles vxd i traversing node x during the time interval (τ i-1, τ i] are propagated forward on the successive arc a = σxd i and then added to the number of vehicles vyd j, with y = HD(a) and j ≥ i , proportionally to the measure of the intersection (τ j-1, τ j]∩(tai-1, tai].

Clearly, the number of vehicles vod

i

departing from each origin o must be initialized

consistently with the demand flows dod i. The procedure implementing the deterministic network loading map is described below, where L and Qi , i = 1, … , I+1, are ordered sets implementing the lists of nodes to be still examined and a node topological order for each instant i, respectively. The nodes are added to set L from the top and to the sets Qi from the bottom. function f = fD(c, t ; d) f=0,w=0,w=∞ for each d∈C QI+1 = {d} for i = I to 1 step -1 wdd i = 0 L = Qi+1 Qi = ∅ do until L = ∅ y = L(1) L = L \ {y} Qi = Qi ∪ {y} for each a∈BS(y) x = TL(a) ψ = cai + wyd(tai) if wxd i > ψ then σxd i = a wxd i = ψ if x∉L then L = L ∪ {x} if x∈Qi then Qi = Qi \ {x} end if next a loop next i

22

for each o∈C for i = 1 to I vod i = (dod i / μ) (τ i -τ i-1) next i next o for i = 1 to I for k = |N| to 2 step -1 x = Qi(k) a = σxd i y = HD(a) fai = fai + vxd i / (τ i -τ i-1) for each j | δ > 0 , δ = mis((τ j-1, τ j]∩(tai-1, tai]) vyd j = vyd j + vxd i ⋅ δ / (tai - tai-1) next j next k next i next d end function

Note that the temporal profiles of the node satisfactions and of the arc inflows yielded by the above procedure are only an approximation based on assumption (35) of the actual temporal profiles yielded by the implicit path network loading model presented in section 2.3 for given piece-wise linear arc performance temporal profiles as in (35.2). The effects of this approximation are empirically discussed in section 5.

4.3 Dynamic user equilibrium The implicit path fixed point formulations (29) and (34) are solved through the MSA as follows: function DUE k=0 , f=0 , y=ε do until ||f -y||∞ < ε or k > kmax k = k +1 [t, c] = γ(f) DETERMINISTIC NLM) y = fD(c, t ; d) | f = f +1/k ⋅(y -f) loop end function

PROBIT NLM)

y = fP(c, t ; d)

where ε and kmax are, respectively, the maximum flow difference and the maximum number of 23

iterations in the stop criterion, while the Probit NLM is implemented as follows: function y = fP(c, t ; d) y=0 for h = 1 to H ĉ = ĉ(c ; χ) x = fD(ĉ, t ; d) y = y +1/h ⋅(x -y) next h end function

5

NUMERICAL RESULTS

The network of Sioux Falls, consisting of 76 directed arcs and 24 centroids, has been considered to carry out some numerical experiments that aim at showing the efficiency and efficacy of the proposed method. To this end, the known daily demand has been distributed consistently with an arbitrary temporal profile simulating a morning peak within a 3-hours period of analysis, and the path choice has been assumed to be deterministic. Specifically, our time-continuous network loading algorithm (with time intervals of 5 minutes) is compared to the best possible time-discrete algorithm (with a time discretization of 10 second), described in Chabini (1998), in terms of the equilibrium temporal profiles of arc inflows and performances on one side, and computational times on the other. The graphs in figure 6 depict, respectively, the temporal profile of the sum, for all the arcs, of the absolute value of the differences between the inflows and the travel times yielded by the two methods, showing that the two equilibrium patterns are substantially identical. Instead the calculation time has been 3 sec with the time-continuous algorithm and 116 sec with the timediscrete algorithm. In view of the fact that the approximations induced by assumption (35) are not relevant in practice when the time intervals last only few seconds, these results also show that the effects

24

on the equilibrium temporal profiles of this approximation are rather weak also when the time intervals last several minutes.

Inflows (1000 veh/h)

500

Sum of inflows Sum of absolute inflow differences

450 400 350 300 250 200 150 100 50 0 0.00

0.30

1.00

1.30

2.00

2.30

3.00

Travel times (1000 sec)

16

Sum of travel times Sum of absolute travel times differences

14 12 10 8 6 4 2 0 0.00

0.30

1.00

1.30

2.00

2.30

3.00

Figure 6 - Difference between the time-continuous and the time-discrete models in terms of equilibrium temporal profiles.

6

CONCLUSIONS

In this work, DUE has been formulated and solved as a fixed point problem without introducing the DNL as a sub-problem, but extending to the dynamic case the concept of NLM, where the implicit path choice model can be either deterministic or stochastic (Probit through Montecarlo simulation). The proposed assignment algorithm reflects this approach as the temporal consistency of the flow pattern through the arc performance model is attained 25

only jointly with the equilibrium. Moreover, a new dynamic shortest path (minimum cost) algorithm is presented to evaluate the NLM in the deterministic case. The main feature of this algorithm is its capability to handle a continuous travel time pattern by approximating the temporal profiles with piece-wise linear functions of time, whose reference instants can define time intervals of several minutes. This allows to capture the features of road traffic dynamic that are relevant for most planning purposes, while keeping computational times to a minimum.

REFERENCES 1. Ahuja R.K., Orlin J.B., Pallottino S., Scutellà M.G. (2002) Minimum time and minimum cost path problems in street networks with traffic lights, Transportation Science 36, 326-336. 2. Bellei G., Bielli M. (1996) Dynamic traffic assignment in congested networks, in Advanced Methods in Transportation Analysis, ed.s. L. Bianco, P. Toth, Springer Verlag, Heidelberg, Germany, 263-298. 3. Bellei G., Gentile G., Papola N. (2002) A within-day dynamic traffic assignment model for urban road networks, submitted to Transportation Research. 4. Bellei G., Gentile G., Meschini L., Papola N. (2002) A demand model with departure time choice for within-day dynamic traffic assignment, to be published in EJOR. 5. Bellei G., Gentile G., Meschini L. (2003) Un nuovo algoritmo per il problema dell’equilibrio dinamico deterministico, accepted for publication in Metodi e Tecnologie dell’Ingegneria dei Trasporti. Seminario 2002, ed.s G. Cantarella, F. Russo, Franco Angeli s.r.l. , Milano, Italy. 6. Bertsekas D.P. (1993) A simple and fast label correcting algorithm for shortest paths, Networks 23, 703-709. 7. Cantarella G.E. (1997) A general fixed-point approach to multimode multi-user equilibrium assignment with elastic demand, Transportation Science 31, 107-128. 8. Cascetta E. (2001) Transportation systems engineering: theory and methods, Kluwer Academic Publisher, Dordrecht, The Netherlands. 9. Chen H.K., Hsueh C.F. (1998) A model and an algorithm for the dynamic user-optimal route choice problem, Transportation Research B 32, 219-234. 10. Chabini I. (1998) Discrete dynamic shortest path problems in transportation applications: complexity and algorithms with optimal run time, Transportation Research Record 1645, 170-175. 11. Daganzo C.F. (1997) Fundamentals of transportation and traffic operations, Pergamon, Oxford, UK, 116. 12. Gentile G. , Meschini L. , Papola N. (2003) Macroscopic arc performance models for within-day dynamic traffic assignment, submitted to Transportation Research.

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13. Heydecker B.G., Addison J.D. (1996) An exact expression of dynamic traffic equilibrium, in Proceedings of the 13th International Symposium on Transportation and Traffic Theory, ed. J.B. Lesort, Elsevier, Amsterdam, 359-383. 14. Huang H.J., Lam W.H.K. (2002) Modelling and solving the dynamic user equilibrium route and departure time choice problem in network with queues, Transportation Research B 36, 253-273. 15. Janson B.N., Robles J. (1993) Dynamic traffic assignment with arrival time costs, in Transportation and Traffic Theory, ed. C.F. Daganzo, Elsevier, Amsterdam, 171-18. 16. Kaufman D.E., Smith R.L., Wunderlich K.E. (1997) User-equilibrium properties of fixed points in dynamic traffic assignment, submitted to Transportation Research. 17. Pallottino S., Scutellà M.G. (1998) Shortest path algorithms in transportation models: classical and innovative aspects, in Equilibrium and advanced transportation modelling, ed.s P. Marcotte, S. Nguyen, Kluwer Academic Publisher, Dordrecht, The Netherlands, 245-281. 18. Papageorgiou M. (1990) Dynamic modeling, assignment and route guidance in traffic networks, Transportation Research B 24, 471-495. 19. Smith M.J. (1993) A new dynamic traffic model and the existence and calculation of dynamic user equilibria on congested capacity-constrained road networks, Transportation Research B 27, 4963. 20. Tong C.O., Wong S.C. (2000) A predictive dynamic traffic assignment model in congested capacity-constrained road networks, Transportation Research B 34, 625-644. 21. Wie B.W., Tobin R.L., Friesz T.L., Bernstein D. (1995) A discrete time, nested cost operator approach to the dynamic network user equilibrium problem, Transportation Science 29, 79-92. 22. Wisten M.B., Smith M.J. (1997) Distributed computation of dynamic traffic equilibria, Transportation Research C 5, 77-93. 23. Wu J.H., Chan Y., Florian M. (1998) The continuous dynamic network loading problem: a mathematical formulation ad solution method, Transportation Research B 32, 173-187. 24. Xu Y.W., Wu J.H., Florian M., Marcotte P., Zhu L.H. (1999) Advances in the continuous dynamic network loading problem, Transportation Science 33, 341-353.

27

ABSTRACT In this paper the Dynamic User Equilibrium is formulated and solved as a fixed point problem in terms of time-continuous real valued temporal profiles of arc inflows and arc performances, where travel demand is specified both through a deterministic and a stochastic implicit path choice model. By extending to the dynamic case the concept of Network Loading Map, yielding arc flows for given demand flows consistently with certain arc performances, the equilibrium is formulated without introducing the Dynamic Network Loading as a subproblem. The proposed assignment algorithm reflects this approach as the temporal consistency of the flow pattern through the arc performance model is attained only jointly with the equilibrium. In addiction, a new dynamic shortest path algorithm is presented to evaluate the Network Loading Map in the deterministic case, which can also be utilized within a Montecarlo simulation to evaluate it in the stochastic case. The main feature of this algorithm is its capability to handle a continuous travel time pattern by approximating the temporal profiles with piece-wise linear functions of time defined on long time intervals. Keywords: within-day dynamic traffic assignment, dynamic user equilibrium, fixed point problem, time-continuous temporal profiles, deterministic network loading map, implicit path choice model. 1

1

INTRODUCTION

This paper focuses on within-day dynamic traffic assignment regarded as a Dynamic User Equilibrium (DUE), or predictive assignment. Extending to the dynamic case Wardrop’s first principle, the deterministic DUE is defined as the state where no user can reduce his travel cost by unilaterally changing path (Smith, 1993; Heydecker and Addison, 1996). The stochastic DUE is defined analogously referring to the case where the costs perceived by users are modelled as random variables. It is assumed that users associate to each path its actual cost, given by the sum of the relative arc costs each one evaluated at the actual arc entering instant (that is the cost that would be actually experienced while travelling along the path), and then choose a minimum cost path, based on the hypothesis that they are rational decision makers. Note that DUE is different from the Dynamic User Optimum (DUO), or reactive assignment, where users are assumed to choose on the basis of the instantaneous path costs, given by the sum of the relative arc costs all evaluated at the departure instant from the origin. Due to the kind of temporal consistency that is to be ensured, DUE is more difficult to formulate and solve than DUO. In the literature, the deterministic DUE is mainly formalized as a finite-dimensional variational inequality problem in terms of, either path inflows and costs (Smith, 1993; Wie et al., 1995; Huang and Lam, 2002), or arc inflows and costs (Chen and Hsueh, 1998), based on the classical non-linear complementarity formulation of Wardrop’s first principle. However, other approaches exists: Janson and Robles (1993) formulate DUE as a bi-level nonlinear program in terms of arc inflows; Papageorgiu (1991) and Wisten and Smith (1997) propose an optimal control problem in terms of arc conditional probabilities; Bellei and Bielli (1996) and Kaufman, Smith and Wunderlich (1997) present fixed point formulations in terms, respectively, of arc inflows and arc conditional probabilities. With regard to the stochastic 2

DUE, a fixed point formulation in terms of arc inflows can be found in Cascetta (2001). In all these fixed point formulations, the Dynamic Network Loading (DNL), which consists in the seeking for given path flows of an arc flow pattern consistent with the travel time pattern through the arc performance model (Wu, Chan and Florian 1998; Xu et al., 1999), is introduced as a sub-problem of DUE. The algorithms proposed in literature to solve DUE are usually based on diagonalization methods implementing a bi-level approach, where iteratively an Upper Problem (UP) is solved for a given solution of a Lower Problem (LP) which is then consistently updated. In most cases, the UP consists in the non-linear problem of determining arc flows and costs resulting from a supply-demand equilibrium for a given travel time pattern, whose objective function is the classical integral sum employed in the formulation of traffic assignment in the static case, while the LP consists in the linear problem of updating the travel time pattern for given arc flows; the UP is solved with the Frank-Wolfe algorithm (Janson and Robles 1993; Chen and Hsueh, 1998), or with heuristic methods (Algorithm D, Wisten and Smith, 1997; swapping process, Wie et al., 1995). Alternatively, the UP is the path choice model, while the LP coincides with DNL problem; the latter is solved in Bellei and Bielli (1996) through the Method of Successive Averages (MSA), and in Tong and Wong (2000) using a traffic simulator. In a recent paper (Bellei, Gentile and Papola, 2002), a new fixed point formulation of the stochastic DUE in terms of time-continuous real valued temporal profiles of arc flows and arc performances (travel times and generalized costs) is presented, where the concept of Network Loading Map (NLM), yielding arc flows for given demand flows consistently with certain arc performances (Cantarella, 1997), is extended to the dynamic case, thus avoiding to introduce the DNL as a sub-problem. An implicit path algorithm is also proposed for the Logit case, 3

where the temporal consistency of the flow pattern through the arc performance model is attained only jointly with the equilibrium. The aim of this work is to extend this approach to the case where travel demand is specified through the deterministic or the Probit path choice model. Furthermore, a new dynamic shortest path (minimum cost) algorithm is presented to evaluate the Network Loading Map in the deterministic case, which can also be utilized within a Montecarlo simulation to evaluate it in the Probit case. The main feature of this algorithm is its capability to handle a continuous travel time pattern by approximating the temporal profiles with piece-wise linear functions of time. Our approach permits to avoid the thick time discretization (time intervals of few seconds), which is necessary to exploit the acyclic property resulting from the assumption of time-discrete temporal profiles (Pallottino and Scutellà, 1998). The reference instants on which the piecewise temporal profiles are based can define time intervals of several minutes. This, on one side, allows to capture the features of road traffic dynamic that are relevant for most planning purposes (some real-time application require by its nature a thick time discretization), and on the other side, results in very short computational times. In the numerical section of this paper a comparison has been carried out between our time-continuous approach with piece-wise linear approximated temporal profiles (with time intervals of 5 minutes) and the time-discrete approach described in Chabini (1998) (with a time discretization of 10 second), in terms of the equilibrium temporal profiles of arc flows and performances on one side, and computational times on the other. Considering one destination at the time, the algorithm computes in reverse chronological order the minimum cost from each node to the destination by means of a label correcting procedure, and then propagates in chronological order and in reverse topological order the 4

flows on the network toward the destination, regardless of their origin and departure time. This differs substantially from the approach of taking explicitly into account the trajectories on the network made by the demand flows, which has been adopted in Bellei, Gentile and Meschini (2003). Moreover the latter cannot handle generalized arc costs; on the contrary, another feature of the proposed dynamic shortest path algorithm is its capability to handle the case where the path costs are not proportional to travel times, thus allowing for effective modelling of toll pricing strategies in a dynamic framework. To focus our attention on the innovations here introduced and to simplify notation, a single user class and a fixed origin-destination demand will be considered, regarding only path choice as elastic. The case of elastic demand is addressed in Bellei et al. (2002), where a departure time choice model particularly suitable for DUE is also presented. On the supply side, the arc performances are evaluated by means of a simple link-based macroscopic model, which permits to represent vehicle queues explicitly by introducing a bottleneck (i.e. a capacity constraint) at the final section. This model is non-separable with respect to time, because the performances at a given instant depend on the previous history of inflows, but is separable with respect to space (e.g. the spillback phenomenon is not represented).

2

THE MATHEMATICAL MODEL

User trips on the road network are modelled through an oriented graph G = (N, A), where N is the set of the nodes and A is the set of the arcs. Each arc a is identified by its initial node TL(a), referred to as tail, and by its ending node HD(a), referred to as head, i.e. a = (TL(a), HD(a)). The origins and the destinations of user trips constitute a subset C of nodes, referred to as centroids. In the following it is assumed that, when travelling from node

5

x to destination d, users consider the finite set Kxd of paths from x to d with an opportune maximum number of cycles, ensuring that Kxd includes the shortest paths. Graph G is assumed to be strongly connected, i.e. Kxd ≠ ∅. The sets of the arcs exiting and entering, respectively, a given node x are referred to as forward star and backward star, and denoted FS(x) = {a∈A: TL(a) = x}, BS(x) = {a∈A: HD(a) = x}. Path topology is described through the following set notation: KaTL(a)d

sub-set of the paths belonging to KTL(a)d that begin with arc a ;

Akxd

set of the arcs constituting path k∈Kxd ;

Akxda

set of the arcs constituting the sub-path of path k∈Kxd between x and the tail of a∈Akxd .

When a cyclic path comprises a given arc more than once, in the above notation the arc a must be regarded with its order of appearance in the path. As the analysis is carried out within a dynamic context, the model variables are temporal profiles, here represented as piecewise C1 real valued functions of the continuous time variable τ. With reference to the flow pattern we use the following notation: dod(τ)

demand flow of users that travel from origin o to destination d ≠ o departing at time τ ;

d

temporal profile of the (D × 1) demand flow vector, where D = |C|⋅(|C|-1) ;

fa(τ)

flow of vehicles entering arc a at time τ ;

f

temporal profile of the (|A| × 1) arc inflow vector ;

Pkxd(τ)

probability to choose path k∈Kxd for users that travel from node x to destination d departing at time τ ;

pad(τ)

probability of using arc a, conditional on being at node TL(a) at time τ, when travelling to destination d ;

p

temporal profile of the (|A|⋅|C| × 1) arc conditional probability vector. 6

In the following the temporal profiles of the demand flows, referred to the departure time from the origin, are assumed to be fixed, bounded from above, and null outside the interval [0, Ω]. With reference to the performance pattern we use the following notation: ca(τ)

cost of arc a for users entering it at time τ ;

c

temporal profile of the (|A| × 1) arc cost vector ;

ta(τ)

exit time from arc a for users entering it at time τ ;

ta-1(τ)

entrance time in arc a for users exiting it at time τ ;

t

temporal profile of the (|A| × 1) arc exit time vector ;

ma(τ)

monetary cost of arc a for users entering it at time τ ;

Ckxd(τ)

actual cost of path k∈Kxd for users leaving node x at time τ ;

Tkxda(τ)

entrance time in arc a∈Akxd for users that follow path k∈Kxd leaving node x at time τ ;

wxd(τ)

minimum actual cost to reach destination d leaving node x at time τ ;

w

temporal profile of the (|N|⋅|C| × 1) minimum cost vector.

The supply model is founded on a link-based macroscopic flow model, where vehicles are represented as particles of a mono-dimensional partly compressible fluid, and capacity constraints are introduced in order to represent vehicle queues explicitly. In this context the First In First Out (FIFO) rule holds true, and the travel time temporal profiles are continuous (Cascetta, 2001). Then, if the inflow is positive between two given instants τ΄ < τ˝ , we have: ta(τ΄) < ta(τ˝) .

(1)

The monotonicity expressed by (1) ensures that the temporal profiles of the exit times are invertible. On the contrary, the flow temporal profiles are inherently discontinuous because, whenever the queue vanishes on an arc, the outflow reduces instantaneously from the capacity to a value related to the inflow. 7

Despite the FIFO rule, due to the presence of monetary costs (e.g. tolls), whose temporal profiles may be highly discontinuous, it may be convenient to wait at nodes in order to enter a given arc later (Ahuja et al. ,2002). In the following we assume that vehicles are not allowed to wait at nodes, so that cyclic paths may be convenient. We assume also that the arc costs are not smaller than a strictly positive constant (ca(τ) ≥ c > 0), so that the shortest paths include at the most a finite number of cycles. Clearly, if the departure time choice is included in the demand model, the motivation to consider cyclic paths becomes less relevant. The main variables of the mathematical model have been introduced; in the following subsections we focus on the relations among them.

2.1 Arc performance model The congestion phenomena are formally represented assuming that the temporal profiles of the arc exit times and of the arc costs depend on the temporal profiles of the arc inflows: t = t( f ) ,

(2)

c = c( f ) .

(3)

The performance functionals (2) and (3) are here specified with reference to a simple linknode model, where the generic arc a terminates with a bottleneck having a constant vehicle capacity Qa > 0 and is characterized by a constant under-saturated travel time ua , which takes into account the uncongested link travel time including the delay due to the final intersection. When the inflow exceeds the bottleneck capacity, an over-saturated queue occurs, causing a congestion delay. The resulting exit times are implicitly expressed by the following equation, which is based on vehicle conservation and FIFO rules: 8

Fa(τ) = Ea(ta(τ)) ,

(4)

where Fa(τ) and Ea(τ) are, respectively, the cumulative inflow and outflow at time τ. The former, by definition, is given by: τ

Fa(τ) = 0∫ fa(σ)⋅dσ ;

(5)

while the latter, based on the Newell-Luke minimum principle (Daganzo, 1997), is given by: Ea(τ) = min{Fa(τ -ua), Fa(σ -ua) +(τ -σ)⋅Qa : 0 ≤ σ ≤ τ} ,

(6)

which expresses the fact that the cumulative outflow cannot increase faster than the capacity as depicted in Figure 1. A numerical solution of the system of equations (5)÷(4) will be provided in section 4.1 .

ua Fa

Ea Qa

σ

1

τ

σ

2

time ta(τ)

Figure 1 - Newell-Luke minimum principle applied to the link-node model.

Alternatively, the more sophisticated models proposed in Gentile, Meschini and Papola (2003) can be adopted without difficulties. The arc cost for users entering at time τ is simply assumed to be: ca(τ) = η⋅(ta(τ)-τ) + ma(τ) ,

(7)

where η is the value of time.

9

2.2 Deterministic path choice model Assuming that the costs perceived by users are additive, the actual cost of path k∈Kxd at time

τ is the sum of the costs of its arcs a∈Akxd, each of them referred to the time Tkxda(τ) when the users leaving node x at τ reach node TL(a), i.e. Ckxd(τ) = ∑ a∈Akxd ca(Tkxda(τ)) .

(8)

Under the assumption that users are perfectly informed rational decision-makers, the resulting behaviour is such that only minimum cost path are utilized. The deterministic path choice model can then be formulated based on the following extension to the dynamic case of Wardrop’s first principle: - if the generic path k∈Kxd is used at time τ, that is its choice probability Pkxd(τ) is positive, then its actual cost Ckxd(τ) is equal to the minimum actual cost wxd(τ) to travel between x and d at the same time; - vice versa, if path k is unused, that is its choice probability is null, then its actual cost may not be smaller than the minimum actual cost. Formally, we have: wxd(τ) = min{Ckxd(τ): k∈Kxd} ,

(9)

Pkxd(τ) ⋅ (Ckxd(τ) - wxd(τ)) = 0 .

(10)

Clearly, the choice probabilities must satisfy the following conditions: ∑ k∈Kxd Pkxd(τ) = 1 ,

(11)

Pkxd(τ) ≥ 0 .

(12)

To be notice that, when there is more than one minimum cost path between any o/d pair, the choice probability pattern solving the system (10)÷(12) is not unique.

10

2.3 Implicit path formulation in the deterministic case We now extend to the deterministic case the implicit path formulation presented in (Bellei, Gentile and Papola, 2002), which is based on the concepts of node satisfaction and arc conditional probability. The main idea underlying the implicit path formulation is to decompose the path choice in a sequence of arc choices taken at each node among the arcs of its forward star, while travelling towards the destination, as depicted in Figure 2. In general, this decomposition is possible only if the arc conditional probabilities at each node are equal for all users directed to the same destination regardless of the sub-path so far utilized.

τ

tb(τ)

time d∈C

x ≡ TL(a)∈N

a∈A

b∈FS(x) cb(τ) -wHD(b)d(tb(τ)) Figure 2 - The arc choice.

Because the paths in Kxd can be partitioned on the basis of their first arc, equation (9) can be written as:

wxd (τ ) = min {Ckxd (τ ) : a ∈ FS ( x ) , k ∈ K axd } .

(13)

As the paths in KaTL(a)d coincide with the paths in KHD(a)d but for a, on the basis of (8) the right 11

hand side of (13) becomes:

{

}

( a )d min ca (τ ) + C HD ( ta (τ ) ) : a ∈ FS ( x ) , j ∈ K HD(a )d , j

which, because minimization is associative, can be written as:

{ {

}

}

( a )d min min ca (τ ) + C HD ( ta (τ ) ) : j ∈ K HD(a )d : a ∈ FS ( x ) . j

Because the term ca(τ) is common to all the terms of the inner minimization, based on (9) equation (13) becomes:

{

}

d wxd (τ ) = min ca (τ ) + wHD ( a ) ( ta (τ ) ) : a ∈ FS ( x ) .

(14)

Note that (14) coincides with the opposite of the deterministic specification of the satisfaction associated to the arc choice at the generic node x. For this reason the minimum actual costs can be referred to as node satisfactions. This recursive equation can be formally expressed by the functional: w = w(c, t) .

(15)

By definition, the choice probability Pkxd(τ) of the generic path k∈Kxd at time τ is equal to the product of the conditional probabilities of its arcs a∈Akxd, each of them referred to the time Tkxda(τ) when users departing from x at τ reach TL(a), i.e. Pkxd (τ ) = ∏ a∈Axd pad ( Tkxd a (τ ) ) .

(16)

k

With reference to the deterministic case, it is clear that, if a sub-path connecting a given node x to destination d has minimum cost, then it may be used by some traveller, and vice versa it may not, regardless of the sub-path utilized to reach x. However, when there is more than one minimum cost path from x to d, the conditional probabilities of the arcs exiting node x depend, in general, on the sub-path so far utilized, because different demand components may 12

split differently among minimum sub-paths. Yet, for a given cost pattern, there is always some solution point of the system (10)÷(12) consistent with the assumption that the arc conditional probabilities at each node are equal for all users directed to the same destination regardless of the sub-path so far utilized. Then, referring to these points, based on (16) it is: Pkxd (τ ) = pad (τ ) ⋅ PjHD( a )d ( ta (τ ) ) ,

(17)

where a is the first arc of path k∈Kxd and path j coincides with k but for a. Moreover, in order to determine the arc conditional probability of the generic arc a , we can refer to the users that depart from its tail, thus obtaining: pad (τ ) =

∑

TL a d k∈K a ( )

TL( a )d

Pk

(τ )

.

(18)

Multiplying both sides of (18) by wTL(a)d(τ), on the basis of (10) we have: d pad (τ ) ⋅ wTL ( a ) (τ ) =

∑

TL( a )d

TL a d k∈K a ( )

Pk

(τ ) ⋅ CkTL(a )d (τ )

.

(19)

As the paths in KaTL(a)d coincide with the paths in K HD(a)d but for a, on the basis of (17) and (8) the right hand side of (19) becomes:

∑

j∈K

HD ( a ) d

pad (τ ) ⋅ Pj

HD ( a )d

( t (τ ) ) ⋅ ( c (τ ) + C a

a

HL( a )d j

( t (τ ) )) , a

which can be written as:

⎛ HD a d pad (τ ) ⋅ ⎜ ca (τ ) ⋅ ∑ Pj ( ) ( ta (τ ) ) + ⎜ HD a d j∈K ( ) ⎝

∑

j∈K

HD ( a ) d

HD( a )d

Pj

( t (τ ) ) ⋅ C a

HL( a )d j

⎞

( t (τ ) ) ⎟⎟ . a

⎠

Based on (11) and (10), from (19) we then have:

(

)

d d pad (τ ) ⋅ ca (τ ) + wHD ( a ) ( ta (τ ) ) − wTL( a ) (τ ) = 0 .

13

(20)

By definition, the arc conditional probabilities must also satisfy the following conditions:

∑

a∈FS ( x )

pad (τ ) = 1 ,

(21)

pad (τ ) ≥ 0 .

(22)

Note that the system (20)÷(22) coincides with the deterministic specification of the probabilities associated to the arc choice at the generic node x, given the satisfactions of the nodes HD(a), a∈FS(x). Note that, when there is more than one arc choice alternative with minimum cost from any node, the arc conditional probability pattern solving the system (20)÷(22) is not unique. It is then formally expressed through the following point to set map: p ∈ p(w, c, t) .

(23)

Clearly, for a given performance pattern, the set of path choice probabilities obtained by composing the maps (23) through equation (16) is only a subset of the solution set of the system (10)÷(12). However, because when solving the assignment problem we just need to determine any point of the choice map, this circumstance does not constitute a relevant limitation of the implicit path formulation. The flow fad(τ) directed to destination d entering arc a at time τ is given by the arc conditional probability pad(τ) multiplied by the flow entering node TL(a) at time τ ; the latter is given, in turn, by the sum of the outflow fbd(tb-1(τ))⋅dtb-1(τ)/dτ of each arc b entering TL(a), and of the demand flow dTL(a)d(τ) from TL(a) to d, which is null if TL(a)∉C; i.e. ⎡ d TL( a )d (τ ) ⎛ d −1 dtb −1 (τ ) ⎞ ⎤ f (τ ) = p (τ ) ⋅ ⎢ + ∑ b∈BS (TL( a ) ) ⎜ f b ( tb (τ ) ) ⋅ ⎟⎥ , μ dτ ⎠ ⎦⎥ ⎝ ⎣⎢ d a

d a

(24)

where μ is the vehicle occupancy coefficient and the terms dtb-1(τ)/dτ stem from applying the FIFO and the vehicle conservation rule (Cascetta, 2001, p. 376). The total flow entering arc a 14

at time τ is then: f a (τ ) = ∑ d∈C f ad (τ ) .

(25)

Based on (24), (25) is expressed formally by the following functional: f = ω(p, t ; d) .

(26)

Note that, because the graph is strongly connected, the link capacities are strictly positive and the demand is bounded, then there exists a time Θ, with Ω < Θ < ∞, within which all the trips have been concluded, so that the arc inflow temporal profiles are null outside the period of analysis [0, Θ].

3

FORMULATION OF THE DYNAMIC USER EQUILIBRIUM

The implicit path formulation of the DUE model presented in the previous section is synthetically depicted in Figure 3, where the dotted arrow denotes any point of the corresponding point to set map.

network loading map

d

p(w, t, c) p

w

ω( p, t ; d)

w(c, t)

f

t

c

t( f ) c( f ) arc performance model

Figure 3 - Implicit path formulation of DUE. 15

In analogy with the static case, the NLM is a functional relation yielding an arc inflows f for given demand flows d consistently with certain arc performances t and c through the path choice model, which has been here formulated implicitly by introducing the node satisfactions w and the arc conditional probabilities p ; while the arc performance model yields the arc

costs c and the arc exit times t corresponding to certain arc inflows f . The system of the NLM and of the arc performance model formulates DUE as a fixed point problem.

3.1 The deterministic case The combination of (15) and (23) with (26) yields the implicit path formulation of the NLM: f ∈ ω(p(w(c, t), c, t), t ; d) .

(27)

Then, the deterministic NLM may be formally expressed through the following point-to-set map: f ∈ fD(c, t ; d) .

(28)

By combining (2) and (3) with (28), the deterministic DUE can be formalized as a fixed-point problem in terms of arc inflow temporal profiles: f ∈ fD(c( f ), t( f ); d) .

(29)

3.2 The Probit case The Probit path choice model, who’s formulation can be found in Cascetta (2001), is based on the random utility theory, where the travel costs are not known with certainty and thus are regarded as random variables. We assume that the perceived arc cost temporal profile ĉa of the generic arc a is equal to the sum of the arc cost temporal profile ca yielded by the arc performance model and of a random error temporal profile whose value at time τ is distributed as a normal variable, with null mean and variance proportional, through a

16

coefficient ξ, to a given cost term χa(τ). In order to improve numerical stability, we assume that the temporal profile χa is independent of congestion; however, χa is not necessarily constant in time, so that the level of uncertainty may vary during the period of analysis. The arc flow pattern resulting from the evaluation of the Probit NLM for given arc performances is accomplished through the well-known Montecarlo method, as follows. a) Get a sample of H perceived arc cost patterns: ĉah(τ) = ca(τ) + ψah⋅(ξ⋅χa(τ))0.5, ĉ h = ĉ(c; χ), h = 1, … , H ,

(30)

where each ψah is extracted from a standard normal variable N[0,1]. b) For each perceived arc cost pattern of the sample, determine through the deterministic NLM (27) a consistent arc inflow pattern; formally: y h ∈ ω(p(w(ĉ h, t), ĉ h, t), t ; d) , h = 1, … , H .

(31)

c) Calculate the average of the deterministic arc inflow patterns y h, thus obtaining an undistorted estimation of the Probit arc inflow pattern; formally: f = 1/H ⋅∑ h = 1, … , H y h .

(32)

Note that, based on (30), the same outcome ψah of the standard normal variable is used to perturb the whole temporal profile ĉah, so that the latter remains a continuous function of time. The Probit NLM, evaluated through (30)÷(32), can be formally expressed as: f = fP(c, t ; d) ,

(33)

and combined with (2) and (3) yields a formulation of the stochastic DUE: f = fP(c( f ), t( f ); d) .

(34)

Note that, because the Probit NLM is a point-to-point map, the resulting fixed point problem is based on a one-valued function. 17

4

SOLUTION ALGORITHM

In order to implement the proposed DUE model, the period of analysis [0, Θ] is divided into I time intervals identified by the sequence of instants (τ 0, … , τ i, … , τ I ), with τ 0 = 0 and τ I = Θ. In the following we assume to approximate the generic temporal profile x through either a piece-wise constant or a piece-wise linear function defined by the values taken at such instants, so that for the two cases we have, respectively: x(τ) = x i , τ∈(τ i-1, τ i] , i = 1, … , I ,

(35.1)

x(τ 0) = x0 , x(τ) = x i-1 + (τ -τ i-1) ⋅(x i -x i-1)/(τ i -τ i-1) , τ∈(τ i-1, τ i] , i = 1, … , I .

(35.2)

Specifically, the flow temporal profiles are assumed piece-wise constant, while the performance temporal profiles are assumed piece-wise linear. Based on this hypothesis, the generic temporal profile x can be numerically represented through the (1 × I+1) row vector x = (x 0, … , x i, … , x I ).

The initial and final state of the network is assumed to be known; here, without loss of generality the network is considered to be unloaded at times τ 0 and τ I. In the algorithm described below, the reference instants can define time intervals of several minutes, thus allowing to capture the features of road traffic dynamic that are relevant for most planning purposes, while keeping computational times to a minimum.

4.1 Arc performances Based on the assumption that the inflow temporal profiles are piece-wise constant as in (35.1), the temporal profiles yielded by the arc performance model presented in subsection 2.1 are actually piece-wise linear and their values at the reference instants can be obtained through the following procedure (Bellei, Gentile and Papola, 2002): 18

function [t, c] = γ(f) for each a∈A ta0 = ua for i = 1 to I tai = max{tai-1 +(τ i -τ i-1) ⋅fai/Qa , τ i+ua} cai = η⋅(tai-τ i) +mai next i next a end function

The only approximation affecting the corresponding performance temporal profiles (35.2) concerns the instants when the queues vanish, which do not coincide in general with any reference instant.

4.2 Network loading In this section we present a new dynamic shortest path algorithm to evaluate the deterministic NLM. The main feature of this algorithm is its capability to handle the continuous piece-wise linear temporal profiles of the travel time defined in (35.2) and obtained through the above procedure. In the following we aim at determining a specific value of the NLM, such that all users travelling toward a same destination d and traversing a given node x during the generic time interval (τ i-1, τ i] are propagated forward on one arc only, called the successive arc and denoted by σxd i, consistently with the assumption (35.1) referred to the arc conditional probabilities. Under this assumption, the flow traversing node x directed toward d during the time interval (τ i-1, τ i] utilizes arc a = σxd i and traverses node HD(a) during the interval (tai-1, tai]. Consistently with the deterministic path choice model, the successive arc σxd i shall belong to a minimum actual cost path connecting x to d at time τ i and can be determined coherently with equation (14) based on the performances of the arcs belonging to the forward star of x

19

and their head satisfactions. Note that by hypothesis the successor of each node is one arc only, the graph is strongly connected and the arc costs are non-negative; then, referring to any given instant τ i, the arcs σxd i, x∈N, constitute a tree. Beside the arc performances which are known at this stage, the node satisfaction (minimum actual cost) at a given reference instant depends on the satisfactions of other nodes at later reference instants and possibly at that same instant, but not on the node satisfactions at previous reference instants. In fact, based on (35.2), when a certain node satisfaction wyd i is improved, the Bellman triangular relation wTL(a)d i ≤ cai + wyd(tai) shall be verified for each arc a∈BS(y) such that tai < τ i+1, and, if the latter is not satisfied, the satisfaction wTL(a)d i shall be

consistently decreased and the successive arc σTL(a)d i possibly updated (see figure 4 below).

wyd i+1

TL(a) a

wyd(tai)

y

wydi

τi

ta i

τ i+1

d

Figure 4 - Dynamic shortest paths.

For this reason, it is convenient to perform the calculation of the node satisfactions and of the successor arcs in reverse chronological order, while it is necessary to employ either a label setting or a label correcting strategy (see for example: Bertsekas, 1993) for processing the nodes, where only the node satisfactions and the successor arcs of the current reference instant 20

are modified (in the case where the shortest arc travel time is greater than the longest time interval this is patently useless). Here we have adopted a label correcting approach, where the nodes whose satisfaction is updated are inserted at the top of a list of nodes to be still examined, while at each iteration the first node of the list is extracted and the Bellman relation is verified for its backward star. In order to enhanced the algorithm performance the list of nodes to be still examined is initialized with all the nodes in the order that they exit the list for the last time when determining the successive arcs for the next reference instant, so as to exploit the commonly experienced continuity over time of the path choice. Note that the latter is a node topological order, that is an ordering of nodes consistent with the tree topology of the successor arcs (for each successor arc, the order of its tail is greater than the order of its head).

vyd i vyd i+1 vyd i+2 tai-1

y

x

fa i

ta i

vxdi

τ i-1

τi

τ i+1

τ i+2

Figure 5 - Network flow propagation.

Note that the vehicles traversing a certain node at a given time instant have necessarily entered at a previous instant an arc of its backward star whose tail has a higher topological order. On this basis, to perform the network loading, the algorithm propagates in 21

chronological order the vehicle flows on the network toward the destination, regardless of their origin and departure time. Specifically, proceeding in reverse node topological order, the number of vehicles vxd i traversing node x during the time interval (τ i-1, τ i] are propagated forward on the successive arc a = σxd i and then added to the number of vehicles vyd j, with y = HD(a) and j ≥ i , proportionally to the measure of the intersection (τ j-1, τ j]∩(tai-1, tai].

Clearly, the number of vehicles vod

i

departing from each origin o must be initialized

consistently with the demand flows dod i. The procedure implementing the deterministic network loading map is described below, where L and Qi , i = 1, … , I+1, are ordered sets implementing the lists of nodes to be still examined and a node topological order for each instant i, respectively. The nodes are added to set L from the top and to the sets Qi from the bottom. function f = fD(c, t ; d) f=0,w=0,w=∞ for each d∈C QI+1 = {d} for i = I to 1 step -1 wdd i = 0 L = Qi+1 Qi = ∅ do until L = ∅ y = L(1) L = L \ {y} Qi = Qi ∪ {y} for each a∈BS(y) x = TL(a) ψ = cai + wyd(tai) if wxd i > ψ then σxd i = a wxd i = ψ if x∉L then L = L ∪ {x} if x∈Qi then Qi = Qi \ {x} end if next a loop next i

22

for each o∈C for i = 1 to I vod i = (dod i / μ) (τ i -τ i-1) next i next o for i = 1 to I for k = |N| to 2 step -1 x = Qi(k) a = σxd i y = HD(a) fai = fai + vxd i / (τ i -τ i-1) for each j | δ > 0 , δ = mis((τ j-1, τ j]∩(tai-1, tai]) vyd j = vyd j + vxd i ⋅ δ / (tai - tai-1) next j next k next i next d end function

Note that the temporal profiles of the node satisfactions and of the arc inflows yielded by the above procedure are only an approximation based on assumption (35) of the actual temporal profiles yielded by the implicit path network loading model presented in section 2.3 for given piece-wise linear arc performance temporal profiles as in (35.2). The effects of this approximation are empirically discussed in section 5.

4.3 Dynamic user equilibrium The implicit path fixed point formulations (29) and (34) are solved through the MSA as follows: function DUE k=0 , f=0 , y=ε do until ||f -y||∞ < ε or k > kmax k = k +1 [t, c] = γ(f) DETERMINISTIC NLM) y = fD(c, t ; d) | f = f +1/k ⋅(y -f) loop end function

PROBIT NLM)

y = fP(c, t ; d)

where ε and kmax are, respectively, the maximum flow difference and the maximum number of 23

iterations in the stop criterion, while the Probit NLM is implemented as follows: function y = fP(c, t ; d) y=0 for h = 1 to H ĉ = ĉ(c ; χ) x = fD(ĉ, t ; d) y = y +1/h ⋅(x -y) next h end function

5

NUMERICAL RESULTS

The network of Sioux Falls, consisting of 76 directed arcs and 24 centroids, has been considered to carry out some numerical experiments that aim at showing the efficiency and efficacy of the proposed method. To this end, the known daily demand has been distributed consistently with an arbitrary temporal profile simulating a morning peak within a 3-hours period of analysis, and the path choice has been assumed to be deterministic. Specifically, our time-continuous network loading algorithm (with time intervals of 5 minutes) is compared to the best possible time-discrete algorithm (with a time discretization of 10 second), described in Chabini (1998), in terms of the equilibrium temporal profiles of arc inflows and performances on one side, and computational times on the other. The graphs in figure 6 depict, respectively, the temporal profile of the sum, for all the arcs, of the absolute value of the differences between the inflows and the travel times yielded by the two methods, showing that the two equilibrium patterns are substantially identical. Instead the calculation time has been 3 sec with the time-continuous algorithm and 116 sec with the timediscrete algorithm. In view of the fact that the approximations induced by assumption (35) are not relevant in practice when the time intervals last only few seconds, these results also show that the effects

24

on the equilibrium temporal profiles of this approximation are rather weak also when the time intervals last several minutes.

Inflows (1000 veh/h)

500

Sum of inflows Sum of absolute inflow differences

450 400 350 300 250 200 150 100 50 0 0.00

0.30

1.00

1.30

2.00

2.30

3.00

Travel times (1000 sec)

16

Sum of travel times Sum of absolute travel times differences

14 12 10 8 6 4 2 0 0.00

0.30

1.00

1.30

2.00

2.30

3.00

Figure 6 - Difference between the time-continuous and the time-discrete models in terms of equilibrium temporal profiles.

6

CONCLUSIONS

In this work, DUE has been formulated and solved as a fixed point problem without introducing the DNL as a sub-problem, but extending to the dynamic case the concept of NLM, where the implicit path choice model can be either deterministic or stochastic (Probit through Montecarlo simulation). The proposed assignment algorithm reflects this approach as the temporal consistency of the flow pattern through the arc performance model is attained 25

only jointly with the equilibrium. Moreover, a new dynamic shortest path (minimum cost) algorithm is presented to evaluate the NLM in the deterministic case. The main feature of this algorithm is its capability to handle a continuous travel time pattern by approximating the temporal profiles with piece-wise linear functions of time, whose reference instants can define time intervals of several minutes. This allows to capture the features of road traffic dynamic that are relevant for most planning purposes, while keeping computational times to a minimum.

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