Jean-Patrick Lebacque

Mostafa Ameli

Ludovic Leclercq

Stochastic departure time user equilibrium with heterogeneous trip profile

Keywords: traffic congestion, peak-hour traffic dynamics, macroscopic model, stochastic user equilibrium, generalized bathtub model

Recently, the generalized bathtub model extended the classic bathtub model to capture various distributions of the trip length by introducing a new state variable: the number of active trips with remaining distances greater than or equal to a threshold. The traffic dynamics are reformulated by four partial differential equations that track the distribution of the remaining trip lengths. This study aims to formulate and solve stochastic user equilibrium for the dynamic departure time choice model based on the generalized bathtub model with heterogeneous trip attributes. In particular, the proposed framework is able to address any distribution for desired arrival time and trip length. We first formulate the problem in continuous form as a fixed-point problem. Then we apply a discretization method to address the trip-based setting. The proposed framework is applied to the real demand profile of the large-scale network of Lyon North for the morning peak hour.

INTRODUCTION

Rapid growth in the demand for the urban transportation networks has given rise to the critical need for mature management of the services in the transportation networks. In this respect, a deep understanding of users' travel behavior in the system is crucial. In order to demonstrate the users' behavior, we can assume that users are rational and tend to optimize their own utilities. However, the discrepancy in the various goals that users have in comparable situations in the network can be attributed to the violation of the user equilibrium (UE) principles, defined by [START_REF] Wardrop | Road paper. some theoretical aspects of road traffic research[END_REF]). This heterogeneity in the decision-making process of users can be addressed by defining stochasticity in the mathematical model [START_REF] Kawakami | Dynamic stochastic equilibrium model in multiple vehicle type transportation networks[END_REF]. In the context of departure time choice models, the solution is defined as stochastic user equilibrium (De Palma, Ben-Akiva, [START_REF] De Palma | Stochastic equilibrium model of peak period traffic congestion[END_REF]). This study focused on modeling and solving the stochastic departure time UE problem.

In addition to the users' decision-making process, the traffic congestion should be captured based on the characteristic of the transportation network. Macroscopic models aim to rule out urban traffic congestion by one main assumption that states the speed is fixed for all internal users of a single traffic zone at each time step with respect to its traffic density and topological characteristics [START_REF] Mahmassani | Dynamic user equilibrium departure time and route choice on idealized traffic arterials[END_REF]. This assumption is validated by the study of [START_REF] Geroliminis | Existence of urban-scale macroscopic fundamental diagrams: Some experimental findings[END_REF], wherein they showed that single-unit macroscopic models could represent the congestion of an urban area independently of the demand. One of the advanced macroscopic models is the generalized bathtub model, recently proposed by [START_REF] Jin | Generalized bathtub model of network trip flows[END_REF]. The main advantage of this model compared to the classic models such as bottleneck models [START_REF] Vickrey | Congestion in manhattan in relation to marginal cost pricing[END_REF] is that it can capture the traffic dynamics for demand profiles with any distribution of trip length by introducing a new state variable: the number of active trips with remaining distances greater than or equal to a threshold. [START_REF] Ameli | Departure time choice models in congested transportation systems based on mean field games[END_REF] discussed the other advantage of the generalized bathtub model compared to the other macroscopic models.

The SUE model is addressed on the other macroscopic models (see, e.g., [START_REF] Lim | Dynamic departure time and stochastic user equilibrium assignment[END_REF][START_REF] Wang | Dynamic traffic assignment: A review of the methodological advances for environmentally sustainable road transportation applications[END_REF][START_REF] Long | Departure time choice equilibrium and tolling strategies for a bottleneck with stochastic capacity[END_REF][START_REF] Lamotte | Congestion and departure time choice equilibrium in urban road networks[END_REF]). In contrast, SUE is not yet formulated by the generalized bathtub model. The main advantage of using the generalized bathtub model will be the ability to consider any type of heterogeneity for the demand profile. Therefore, we aim to formulate and solve SUE for the departure time choice model based on the generalized bathtub model with heterogeneous trip attributes. In particular, we propose continuous and discrete models that are able to address any distribution for desired arrival time and trip length.

METHODOLOGY

The model.

Let us recall the main elements of the model.

• H(t): the number of travellers in the system at time t, t ∈ [O, T ] ⊂ T .

• This number is dis-aggregated wrt to desired arrival time t a ∈ T a and remaining travel distance x ∈ X . Thus K(t a , x,t) denotes the density at time t wrt to desired arrival time t a ∈ T a of travellers with remaining travel distance greater than x ∈ X . By this definition

H(t) = X a K(t a , 0,t) dt a (1)
• h(x): the initial number of travellers with remaining travel distance greater than x ∈ X ) and k(t a , x) (the initial distribution K) denote the initial conditions for H and K. Thus:

h(x) = X a k(t a , x) dt a (2)
• The total demand for trips has a density m(t a , x)dt a dx. A standing hypothesis is that the distribution of travellers wrt to trip length x is defined by an integrable bounded density function. On the other hand there may be a finite number of desired arrival times.

• The traffic assignment yields the distribution of traveller demand, which is denoted as f (t a , x,t) dt a dx dt, and expresses the departure time choice t of travellers, given their desired arrival time t a and their trip length x. f constitutes the main unknown of the assignment problem. f satisfies to the following convex set of constraints (K ):

(K ) T f (t a , x,t)dt = m(t a , x) f (t a , x,t) ≥ 0 (3)
• The speed of travellers v(t) is uniform in the network, time-dependant and a function of the total number of travellers H(t). Thus

v(t) = V (H(t)) (4)
A standing hypothesis of the model is that the fundamental diagram V is bounded from above and below: 0

< V min ≤ V ≤ V max .
Let us now describe the dynamics of K. They are described by the following system

z(t) de f = t 0 dt V (H(t)) (5.1) K(t a , x,t) = k(t a , x + z(t)) + t 0 ds F(t a , x + z(t) -z(s), s) (5.2) F(t a , x,t) = ∞ x dξ f (t a , ξ ,t) (5.3) (5)
(5.2) expresses that K has initial value k, and has inflow rate at time t

∞ x dx f (t a , x,t).
The outflow of the system results from travellers terminating their trip, when their remaining trip length is zero. By differentiating (5) the following system results:

∂ t K -v(t)∂ x K = f (6.1) v(t) = V (H(t)) and H(t) = T a dt a K(t a , 0,t) (6.2) K(t a , x, 0) = k(t a , x) the given initial condition (6.3) (6)
Let us now describe the dynamics of H. They are described by the following system

z(t) = t 0 dt V (H(t)) (7.1) H(t) = h(z(t)) + t 0 ds F(z(t) -z(s), s) (7.2) F(x,t) = T a dt a f (t a , x,t) (7.3) (7)
(7) can be deduced from ( 5) by setting x = 0 in the latter and integrating it over t a . It can be shown, adapting the classical Picard argument (refer to [START_REF] Ameli | Departure time choice models in congested transportation systems based on mean field games[END_REF]) that ( 7) admits a unique solution in z and H which depends continuously on the data and initial conditions. The reason for which it is possible to state an equation for H(t) is that the speed of traffic is considered uniform following (4). The solution of ( 7) is denoted

(7) ⇐⇒ H = H ( f , h) z = Z ( f , h) (8)
In the space of continuous functions, H and Z are Lipschitz continuous wrt h and weak-continuous wrt f (recall that f is a bounded positive measure). (refer to to [START_REF] Ameli | Departure time choice models in congested transportation systems based on mean field games[END_REF]).

Once (7) has been solved, it is straightforward to calculate K by evaluating the integrals (5.2) and (5.3).

Stochastic dynamic user equilibrium.

The time at which a traveller entering the system at time t with trip length x can exit the system is denoted as TA(x,t) (arrival time) and is given by:

TA(x,t) = z -1 (x + z(t)) (9)
Indeed z (TA(t))-z(t) = x. Thus the travel time is given by TA(x,t)-t. Given that v(t) ≥ V min > 0, it follows that z -1 is Lipschitz continuous and thus TA is also Lipschitz continuous.

The travel cost perceived by a traveller entering the system at time t, with trip length x and desired arrival time t a , is given by:

J(t a , x,t) = α (TA(x,t) -t) + φ (TA(x,t) -t a ) ( 10
)
where φ denotes the penalty. It is a convex function of the difference between the actual and the desired arrival time. A frequently used expression for the penalty φ is the following:

φ (ϑ ) = β -ϑ + + γ ϑ + (11)
with α > β . Note that (y) + de f

= max{y, 0}. J is a function of f and h via (10), ( 9) and ( 8):

J = J ( f , h) (12)
The Logit stochastic user equilibrium specifies that the demand m(t a , x) is disaggregated wrt t proportionally to exp [-ς J(t a , x,t)]:

f (t a , x,t) = m(t a , x) exp [-ς J(t a , x,t)] T ds exp [-ς J(t a , x, s)] (13)
ς denotes the sensitivity coefficient of the model. For numerical evaluation ( 13) is better reformulated as:

f (t a , x,t) = m(t a , x) T ds exp -[ς (J(t a , x, s) -J(t a , x,t))] (14)
(13) or ( 14) express f as a function of J:

f = F (J) (15)
Note that ( 13), ( 14) or ( 15) implies that f satisfies the demand constraints (K ).

The SDUE problem can be formulated as the following fixed point problem:

f = F (J) J = J ( f , h) (16)
If m is a bounded integrable function (i.e. m ∈ L ∞ (T a × X ) then it can be shown that ( 16) admits a fixed point in L ∞ (T a × X × T ). The idea is to show the result first in L 2 by a Brouwer-type argument based on the fact that the fixed point problem is set in (K ) which is convex and closed.

A similar result can be shown if m is distributed over a finite number of values of t a . Then (16) admits a fixed point which is in L ∞ (X × T ) for all possible values of t a .

A final point: care must be taken with boundary effects. To be specific, let us consider that the fixed point problem ( 16) is set in a departure time interval [0, T ]. Then the travel cost of travellers must be evaluated in the time interval [0, T + max x∈X (TA(x, T ))]. Since the travel time of a traveller leaving at time T depends on the velocity of travellers during the time-interval [T, T + max x∈X (TA(x, T ))] which in turn depends on the inflow into the system during that interval. This inflow must be specified independently of f . Typically the corresponding demand will be assumed to be 0.

Discretization

The discretization of the problem addresses the following tasks:

• to calculate H given f and the initial condition h i.e. to provide a numerical approximation of (8);

• to calculate J given H i.e. provide a numerical approximation of ( 9) and ( 12);

• to approximate numerically the Logit assignment formula (14) (i.e. the expression of f as a function of J (15)).

Three possible methods can be considered.

1. Discretize (6) directly. Indeed (6.1) can be recast as an advection type equation the discretization of which is simple because the wave speed is given by v(t) in (6.2). The size of the system can be reduced by integrating (6.1) over t a ∈ T a , thus solving with respect to KR(x,t)

de f

= T a dt a K(t a , x,t) which satisfies the same equation (6.1) as K but with one order lower dimensionality, since KR is only a function of x and t. Actually since it is possible to solve the dynamics wrt H only as in ( 7) we chose instead to solve ( 8) because H depends on t only.

2. Cell-wise discretization of ( 7). The idea is to discretize

T into cells [t n-1 ,t n ], n ∈ N , X into cells [x ℓ-1 , x ℓ ],
ℓ ∈ L , and for simplicity's sake we assume the set T a discrete. Then z and H are approximated by piecewise linear functions defined by the values z n = z(t n ) and H n = H(t n ). z -1 need not be explicitly calculated as we need essentially TA. TA is approximated by a piecewise linear function defined by its values TA ℓn = TA(x ℓ ,t n ). The disaggregated demand f is approximated by a piecewise constant function which takes the values f t a ℓn = xℓ xℓ-1 dξ tn tn-1 dτ f (t a , ξ , τ). The demand m is also approximated by a piecewise constant function. Thus (7) boils down to an explicit Euler scheme for the integration of z i.e. (7.1), whereas (7.2) can be solved by simply evaluating the right-handside integral. F is evaluated as a piecewise constant function by calculating the integral in (7.3). TA results from ( 9) and then J is deduced as a piecewise linear function with values J t a ℓn = TA ℓnt n + φ (TA ℓnt a ). Finally f follows from J by ( 14).

3. Particle discretization of ( 7). The main difference with the second approach is that the disaggregated demand f is described also by particles p ∈ P. The particle discretization is easily put in correspondence with a micro-simulation. Each particle is endowed with a departure time T D p (which results from f ), an arrival time TA p , a remaining trip length x p , and a desired arrival time t a,p . The treatment of the dynamics of the system ( 7) are now different. The particle p enters the system at time T D p which is a data, x p decreases at a rate v n = V (H n ), and the particle p exits the system when x p = 0 which defines TA p . The number of particles present in the system yields H at any time, thus yield v and z. The travel cost for particle p is given by J p = α(TA p -T D p ) + φ (TA pt a,p ). Finally the f t a ℓn values follow from J by averaging ( 14) for all particles in a cell.

RESULTS AND DISCUSSION

We implemented and applied our SUE model to the Lyon North network, which includes 1,883 nodes and 3,383 links. The network characteristics are presented in [START_REF] Mariotte | Calibration and validation of multi-reservoir mfd models: A case study in lyon[END_REF]. The demand profile includes 62,450 trips trips during the morning peak hours (6:30 AM to 10:40 AM). The data set of the demand profile is published in [START_REF] Ameli | Lyon north realistic trip data set during the morning peak[END_REF] and its characteristics considered in this study are detailed in [START_REF] Ameli | Departure time choice models in congested transportation systems based on mean field games[END_REF]. The parameters of the scheduling preferences are set based on the study of [START_REF] Lamotte | Congestion and departure time choice equilibrium in urban road networks[END_REF] and ς = 0.75 s -1 based on [START_REF] Long | A nonlinear equation system approach to the dynamic stochastic user equilibrium simultaneous route and departure time choice problem[END_REF].

To solve the SUE problem, we followed most of the studies in the literature and used a heuristic method based on the method of successive average. The MSA ranking, introduced by [START_REF] Sbayti | Efficient implementation of method of successive averages in simulation-based dynamic traffic assignment models for large-scale network applications[END_REF], is employed to calculate the equilibrium. The main advantage of this method is that it guarantees convergence as it relies on predetermined step size. The convergence results for all user classes of trips are presented in Figure 1(a). The convergence pattern of the algorithm shows that it is stabilized after the fifth iteration. The few variations of different classes are caused by the heuristic nature of the algorithm that explores the solution space. Note that the exploration rate of heuristic methods also depends on the step size (Ameli, Lebacque, & Leclercq, 2020).

CONCLUSIONS

This study proposed a new formulation for departure time stochastic dynamic user equilibrium problem based on the generalized bathtub model. Based on the analytical analysis, the continuous formulation of the model can consider any distribution for trip length and desired arrival time of the users. In addition, the particle discretization approach is used to extend the model to trip-based formulation in order to address realistic test cases. The application of the proposed framework to the large-scale test case showed that the model and the solution method are capable of representing and solving the stochastic dynamic user equilibrium problem with multiple desired arrival times and heterogeneous trip length for a large number of trips that was little addressed before. The

Figure 1

 1 Figure 1(b) illustrates the final equilibrium departure time and arrival time distributions. Each green diamond denotes the departure time, and each red circle represents the arrival time of a trip, with the duration denoted by a horizontal blue line between the departure and the arrival time.In each class, the trips are sorted based on their trip length. In Figure1(b), the deformations of the distributions for all classes show that non-regular sorting pattern which is expected for the stochastic UE. The final solution validates the good performance of the solution method.

Figure 1 :

 1 Figure 1: Convergence and the solution characteristics.

authors are currently preparing an analytical test case to investigate the features of the continuous model and provide some analytical proofs. The second direction is to benchmark the model with other SUE models for macroscopic and microscopic models.


Mostafa Ameli
email: mostafa.ameli@univ-eiffel.fr

Jean Patrick Lebacque

Ludovic Leclercq

Evolution of multimodal final user equilibrium considering public transport network design history

Introduction

Analyzing the properties of a network equilibrium (uniqueness and stability) can help to have a better view about network state, robustness, and the effect of any variation in the network. The issue of unicity for User Equilibrium (UE) has long been a subject of concern in the literature on traffic assignment problems [START_REF] Beckmann | Studies in the Economics of Transportation[END_REF]; [START_REF] Daganzo | The uniqueness of a time-dependent equilibrium distribution of arrivals at a single bottleneck[END_REF]; [START_REF] Mounce | Uniqueness of equilibrium in steady state and dynamic traffic networks[END_REF]; [START_REF] Iryo | On the uniqueness of equilibrated dynamic traffic flow patterns in unidirectional networks[END_REF]). Much research has been performed on the unicity of traffic assignment solutions with several assumptions and limitations on the traffic network model [START_REF] Beckmann | Studies in the Economics of Transportation[END_REF]; [START_REF] Dafermos | The general multimodal network equilibrium problem with elastic demand[END_REF]; [START_REF] Nagurney | Comparative tests of multimodal traffic equilibrium methods[END_REF]; [START_REF] Wie | The existence, uniqueness and computation of an arc-based dynamic network user equilibrium formulation[END_REF]; [START_REF] Florian | On uniqueness and proportionality in multi-class equilibrium assignment[END_REF]; [START_REF] Sun | Range of user-equilibrium route flow with applications[END_REF]; [START_REF] Ameli | Day-to-day multimodal dynamic traffic assignment: Impacts of the learning process in case of non-unique solutions[END_REF]). In practice, the strong mathematical assumptions for unicity (e.g., FIFO or monotonicity) simply do not hold [START_REF] Boyles | Diverge models and dynamic traffic equilibria[END_REF]. Therefore, the existence of multiple equilibria can be expected mathematically for real test cases [START_REF] Levin | Improving the convergence of simulation-based dynamic traffic assignment methodologies[END_REF]. There are few analytical studies [START_REF] Netter | Affectation de trafic et tarification au cout marginal social: Critique de quelques idees admises[END_REF][START_REF] Wynter | A convergent algorithm for the multimodal traffic equilibrium problem[END_REF] that showed the existence of multiple equilibria in small networks in the traffic assignment context. Most of the studies focused on check the uniqueness of equilibrium and not use this property to analyze and improve the system.

A key argument (sufficient condition) for unicity is a strictly monotone travel time function with respect to the number of travelers that use a path [START_REF] Smith | The existence, uniqueness and stability of traffic equilibria[END_REF]; [START_REF] Aashtiani | Equilibria on a congested transportation network[END_REF]; [START_REF] Florian | Network equilibrium models and algorithms[END_REF]). This condition is quite restrictive as monotonicity can be guaranteed at the link level for mono-modal flow but hardly at the path level because of the intersection functioning. Traffic assignment models address the network equilibrium problem, including the travel time calculation, mathematically.

In the real multimodal networks, the network design can be changed in the long term, e.g., several new transportation facilities are added to the system. There are few studies on Static Traffic Assignment (STA) and Dynamic Traffic Assignment (DTA) models by considering the evolution of the network design. However, none of them investigate the nonunicity and history of the network together. The scientific question of this study is, does the network converge to different equilibria with different network history and the same final network design? In other words, is the current network situation sufficient to grasp the real user distribution inside the network?

To investigate the multiple equilibria problem with multimodal settings, we consider a day-to-day convergence process as a projected dynamical system. When unicity holds, this process converges to the single equilibrium loading [START_REF] Zhao | Day-to-day assignment models and traffic dynamics under information provision[END_REF]. Here, we focus on what happens when multiple solutions can be reached, as we aim to highlight what may drive the system to one equilibrium rather than another. More specifically, when considering a long-term day-to-day process, the final network may not be built at once but results from the successive opening of additional facilities. In this study, we consider that the road network is stable over the entire time horizon and that the public transport network is subject to the regular new openings (bus or metro line). We define this opening process as the network history and investigate how it would affect the final equilibrium state of the network.

First, we perform an analytical investigation and demonstrate the existence of multiple equilibria with respect to the network history of day-to-day STA. Second, we address the same question through simulation for a more complex test case (large-scale network, dynamic traffic assignment, multiple configurations for the network history). The analytical study highlights the causes for the existence of multiple solutions and the influence of network history. The numerical study aims to provide results considering a realistic urban setting. The influence of network history is investigated with a different angle as we are more interested in the convergence of multiple equilibria, i.e., the description of the differences between final possible states. A specific finding is that several network history configurations lead to shorter total travel times for the system than others. This may be of interest when considering public transport planning.

Analytical investigation

Let us consider a network with three modes of transportation, which are referred to as car, bus, and train (metro) (figure 1). There are two bus lines between origin and destination. Paths 1 and 2 are shared between car and bus, and path 3 is the train line. Therefore, there are five feasible path choices to reach the destination: car by path 1, car by path 2, bus line on path 1, bus line on path 2, and train on path 3, figure 1.

Figure 1: A single origin-destination network of this study

Note that the total demand for public transportation D P T and cars D C are given. The demand for public transportation is disaggregated per mode bus and train. The cost function for motor vehicles (car and bus) depends on not only the path flow for this mode but also the path flow of other modes as all vehicles interact. For example, on paths 1 and 2, bus flow and car flow result from a global congestion level that influences both bus and cars travel costs. The cost of mode m ∈ {B: Bus, C: Car} on path a ∈ 1, 2 is as follow:

c m a = g m a + ∑ µ∈{B,C} h m,µ a f µ a , ∀m ∈ {B, C}, a ∈ 1, 2 (1)
Here, we assume that c m a is defined as a linear function where g m a is the free flow cost of mode m on path a and h m,µ a is the impact factor of the flow of mode µ on path a (f µ a ) on the cost of mode m. Using linear functions may look as a strong assumption compared to the more realistic polynomial or exponential shape, e.g., BPR function (Bureau of Public Roads, 1964) or considering path capacity constraints [START_REF] Beckmann | Studies in the Economics of Transportation[END_REF]. It is not here as our main purpose is to understand the causes for multiple equilibria to appear. What triggers this phenomenon is the cross-dependency of the cost among modes, which is properly accounted for here.

The cost function for path 3 is even much simpler. First, only a train option is available, as the cost should only depend on the demand for the train. Second, train service is only adjusted to the demand for the long run (every year or twice a year when a new timetable is proposed). So during the day-to-day STA process, the perceived cost by users is mainly fixed as a combination of the travel time and the cost. In the end: c T 3 = λ for all path flow values (f T 3 ), where λ is considered as a constant value.

According to the Wardrop equilibrium definition, the network is at the UE flow distribution if and only if for every path a:

f m a (c m a -c m ) = 0, ∀m ∈ M, a ∈ A (2)
Therefore, when the equilibrium is reached, the cost of all used paths of mode m is equal to c m . The costs are assumed to be asymmetric: the effect of cars on buses is not the same as the effect of buses on cars. In other words, in the network (figure 1) for one or more modes m 1 ̸ = m 2 :

∂c m 1 a ∂f m 2 a ̸ = ∂c m 2 a ∂f m 1 a ⇐⇒ h m 1 ,m 2 a ̸ = h m 2 ,m 1 a ∀a = 1, 2, ∀m 1 , m 2 = B, C (3)

Network history design

In this study, we are interested in investigating the equilibrium when intermediate changes in the network design occur. In other words, the final network layout is always the same, but it may result from different intermediate steps, see table 1. Scenario 1 has no train line at first place. Thus, an intermediate equilibrium state (UE) is first achieved through the day-to-day learning process before the train line is added. Then, the metro line is added, and the second convergence process proceeds. Scenario 2 assumes that there are only cars and trains, and no buses during the first convergence period. Then buses are added, and a second convergence process is initiated starting from the equilibrium obtained by the first process. Scenario 3 is when all modes are active from the beginning. We will calculate the final network equilibrium for all the scenarios in table 1 and examine why different equilibrium may be raised.

Table 1: The scenarios of network design for the mono-OD test case.

Non unicity of equilibrium states in multimodal STA

In order to calculate and visualize the intermediate equilibrium for Scenario 1, we first explore the non-uniqueness of the network without train (D T = 0), i.e., we analyze the equilibrium solution(s) for the initial equilibrium state of scenario 1 in table 1. According to equation 3, on each path, the impacts of two modes on each other are not the same, which represents the real interaction between modes. In other words, the Jacobian matrix of the car and the bus cost functions is not symmetric positive definite, which means the cost functions for path 1 and 2 are not monotonic [START_REF] Iryo | Properties of dynamic user equilibrium solution: existence, uniqueness, stability, and robust solution methodology[END_REF].

Let us express costs on paths 1 and 2 as functions of the flows f m 1 . In order to visualize the equilibrium, we consider f m 1 as an independent and f m 2 as a dependent variable. The flows f m 2 can be considered dependent variables. We draw the curves ∆c m = c m 2c m 1 = 0 of the two vehicular modes on the flow diagram of path 1 in the (f C 1 ,f B 1 ) plane. We perform a complete analysis of different configurations of h m,µ a and highlights in which conditions we will have more than one equilibrium.

The projected dynamical system of traffic assignment

In order to address the day-to-day process in the static case, We define the network equilibrium problem as a projected dynamical system. It has been proved by several studies (see e.g., [START_REF] Nagurney | Projected dynamical systems and variational inequalities with applications[END_REF], [START_REF] Smith | A new dynamic traffic model and the existence and calculation of dynamic user equilibria on congested capacity-constrained road networks[END_REF], [START_REF] Jin | A dynamical system model of the traffic assignment problem[END_REF], [START_REF] Lebacque | The cross-entropy field for multi-modal dynamic assignment[END_REF]) that projected dynamical systems find the equilibrium point(s) by producing the solution trajectory (mapping function) based on a fixed point theory. We define 2D vector fields for the intermediate equilibrium of Scenario 1 and 3D vector fields for the final state of all scenarios.

We also analyze the stability of the solution based on the projected dynamical system. In summary, the UE solution E is stable only if the day-to-day process brings the solution back to E when a small perturbation ϵ moves it from E in any direction of the feasible region (converging arrows to E). Figure 2 presents an example of a test case considered wherein we have multiple equilibria. It shows the trajectory of the different starting points in the solution space. Almost all of the initial path flow distribution converges to the two stable equilibria on the corners while there is a rare situation wherein the starting point is located on a line whereon the field lines direction is thorough to the unstable equilibrium (the intersection of ∆c C and ∆c B ). Figures 3a and3b present Scenario 1. We choose the uniform flow propagation for the initial solution. The intermediate equilibrium in the car-bus network is obtained based on the projected dynamical system (figure 3a). Then we add the train line to the system and calculate the final equilibrium (figure 3b). Figure 3c presents Scenario 2, wherein the intermediate equilibrium is the unique one from the car-train network. The results show that with a different order of mode activation, the system reaches to different equilibrium. 3. Multi-modal simulation-based day-to-day DTA

In this section, we address the question of network history and multiple multimodal user equilibria in a more realistic framework. Now, we resort to a dynamic traffic simulator for the network loading and focus on a real network. Although the setting is more complex, mechanisms similar to those described in the previous section apply and induce non unicity of equilibria, and dependence of equilibria to the order of activation of facilities.

Dynamic test case

Using a dynamic simulator permits us to consider the large-scale network of Lyon 6e + Villeurbanne (figure 4a) with 1,883 Nodes, 3,383 Links, 94 Origins, 227 Destinations and 54,190 trips. Walking, buses and private cars are initially available transportation modes in the network. Figure 4b presents 31 bus lines in the Lyon 6e + Villeurbanne network includes 176 bus station (figure 4c). There are three metro lines (A, B and C) and 25 metro stations in the network (figure 4d). The network is loaded with travelers of all ODs with a given departure time in order to represent 1.5 hours of the network with the demand level based on the study of [START_REF] Krug | Reconstituting demand patterns of the city of lyon by using multiple gis data sources[END_REF]. The goal is to analyze the final equilibrium solution obtained by a day-to-day DTA model with different settings corresponding to different successive introductions of the metro lines.

Experiment scenarios

For each scenario of opening metro lines, we run the day-to-day DTA for 300 days. A quarter of users only have access to the public transportation system (bus and metro) and the other three quarters have access to all transportation facilities (private car, bus, and metro) in the network. We can open three metro lines at the same time (called A&B&C scenario) and calculate the equilibrium or successively open one metro line every 100 days (e.g., ACB denotes a scenario wherein we first open metro line A, then after 100 days we open metro line C, and after on day 200, we open metro line B) and look for the final network state after 300 days. There are seven possible orders to activate the metro lines. All the scenarios are started by the final equilibrium solution of the network without metro lines. The initial assignment pattern of each step is the final equilibrium flow distribution of the previous step. For Scenario 1 (A&B&C)), all three metro lines are activated at the same time and once the day-to-day process is executed in order to equilibrate the system.

Numerical results

The full day-to-day process is conducted for all the scenarios, and we verify that all the simulations converge to a satisfactory UE solution, i.e., the average delay of all scenarios is less than 12 seconds, which shows good quality for the equilibrium in the large-scale network and given the demand level [START_REF] Ameli | Cross-comparison of convergence algorithms to solve tripbased dynamic traffic assignment problems[END_REF]. Next, we evaluate the final solution in order to investigate the unicity of the solution.

Table 2 presents the usage of metro lines at the equilibrium state for all scenarios. The number of users who take metro line A is between 2089 and 2983. This means if we open the metro line in the order ACB, we will have 42% more users that take metro line A than in the CAB scenario. This width of interval for metro lines B and C is 1250 and 640. The total travel time is standard criteria for evaluating traffic network performance. According to the results in table 2, by opening the metro lines in the order ABC, we can save 600 hours (3%) on average compared to the other scenarios. The total travel time values of the scenarios are in the range of [18559.33, 19701.43], which is the range of the potential equilibrium space. This shows that the intermediate state of the network has a significant impact on the final UE. We plan to measure public transportation systems' reliability and multimodal equilibrium stability to investigate which history of the network design can provide a more reliable transportation service.

Figure 2 :

 2 Figure 2: The trajectory of different initial solutions (red squares) on flow diagram of path 1

  (a) Intermediate equilibrium convergence of Scenario 1 with the initial solution: f m 1 = ( 3 4 D C , 3 4 D P T ). (b) Final equilibrium convergence of Scenario 1. The intermediate equilibrium comes from Figure 3a (c) Equilibrium convergence of Scenario 2.

Figure 3 :

 3 Figure 3: The impact of network design history on the final equilibrium.

Figure 4 :

 4 Figure 4: Multimodal traffic network of Lyon 6e + Villeurbanne

Table 2 :

 2 Primary results

	Scenario Sequence	Number of users used Metro Total travel time (hours) A B C
	1	A&B&C 2236 3057	3708	19454.22
	2	ABC	2826 2771	3316	18559.33
	3	ACB	2983 2636	3678	19070.50
	4	BAC	2419 3077	3138	18967.44
	5	BCA	2608 3117	3481	19199.75
	6	CAB	2089 3886	3564	19701.43
	7	CBA	2313 2977	3778	18644.19

Acknowledgement

This work is supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program. (Grant agreement No 646592 -MAGnUM project)