Mario G Carlos

Ómez Pati Ño

Christine Buisson
email: christine.buisson@univ-eiffel.fr

Mehdi Keyvan-Ekbatani

m/s 5 m/s parameters Mean SD % Mean SD % Mean SD % ∅ PBC 34,47 0,0 0,0 30,29 0,0 0,1

Keywords: Parameters variability, Car-following models, Newell's model, Capacity, Capacity distribution

We contribute to the vehicle-level analysis of two macroscopic features of the road traffic: capacity variability and capacity drop. In this paper, we focus only on the car-following behavior and leave the part related to lane-change maneuvers for the future research. In particular, we study a simplistic car-following model (Newell's with bounded acceleration) for a single-lane scenario. In this work, by introducing a speed limitation across a zone, a bottleneck with variable nominal capacity has been created. We use a continuous event-based numerical resolution method. Consequently, We are able to vary the three Newell's model parameters: maximal acceleration, minimal distance, reaction time. It has been shown that the variability of those car-following parameters (e.g., reaction-time, minimal-distance, and maximal-acceleration) has a strong impact on the pre-breakdown capacity variation and also on the queue discharge flow. It has been concluded that this parameters variability does impact the drop (provided that the maximal acceleration has a relatively high mean value). Various distribution shapes (uniform, truncated Gaussian, and Gamma) have been exploblack. It has been realized that this does not have any significant impact on the capacity distribution. Concerning the amplitude of the capacity distribution, we demonstrate that the reaction time is the parameter with the highest impact followed by the minimal distance. If all parameters vary with an amplitude of 30 %, we show that the capacity standard deviation, in this scenario without lane changes, is about half the experimental values reported in the literature.

Introduction

Most of the existing studies on macroscopic characteristics of freeway traffic flow are related to the traffic condition during congestion i.e. (i) bottleneck activation and the value of the capacity, especially the distributed nature of the capacity; (ii) capacity drop, namely the difference in capacity observed for the same location between the free flow and the upstream congested condition; and (iii) the traffic instabilities/ traffic oscillation (or stop and go (SG) waves).

Several works have shown that freeway capacity varies depending on external conditions such as traffic and geometric characteristics (1-3), specific traffic controls (4-7) and weather conditions [START_REF] Ibrahim | Effect of adverse weather conditions on speed-flow-occupancy relationship. 730[END_REF]. However, empirical observations reveal variations in capacity values from day to day in the same location with the same traffic control and similar weather [START_REF] Chung | Relation between traffic density and capacity drop at three freeway bottlenecks[END_REF][START_REF] Alexandra | Defining, Measuring, and Modeling Capacity for the Highway Capacity Manual[END_REF][START_REF] Hall | Freeway capacity drop and the definition of capacity[END_REF][START_REF] Lorenz | A Probabilistic Approach to Defining Freeway Capacity and Breakdown[END_REF][START_REF] Brilon | Reliability of Freeway Traffic Flow[END_REF].The aforementioned works suggest that these capacity variations can be due to physical differences in congestion occurrence mechanisms. Some researchers found that the observable congestion after the bottleneck activation can be generated with a lower or higher flow rate than the value accepted as the capacity. In this regard, [START_REF] Lorenz | A Probabilistic Approach to Defining Freeway Capacity and Breakdown[END_REF][START_REF] Brilon | Reliability of Freeway Traffic Flow[END_REF][START_REF] Xie | Modeling lane-specific breakdown probabilities at freeway diverge sections[END_REF] consider the traffic jam occurrence as a probabilistic event. As a result, we conclude that capacity is a distributed variable. Moreover, many studies have observed two different capacities:

25

• the maximal flow observed before congestion onset:

Pre-Breakdown Capacity (PBC), and • the maximal flow downstream of the queue, observed after the breakdown: Queue Discharge Flow (QDF).

The latter is always lower than the former, sometimes up to 30 a difference of 30 % [START_REF] Alexandra | Defining, Measuring, and Modeling Capacity for the Highway Capacity Manual[END_REF][START_REF] Hall | Freeway capacity drop and the definition of capacity[END_REF][START_REF] Yuan | Capacity drop: a relation between the speed in congestion and the queue discharge rate[END_REF][START_REF] Yuan | 760 Capacity drop: a comparison between stop-and-go wave and standing queue at lane-drop bottleneck[END_REF]. The difference in the value of the aforementioned capacities is called capacity drop (CD). This phenomenon was observed at fixed bottlenecks (e.g., merges, diverges, lane-drop, sags) [START_REF] Chung | Relation between traffic density and capacity drop at three freeway bottlenecks[END_REF][START_REF] Cassidy | Increasing the capacity of an isolated merge by metering its onramp[END_REF][START_REF] Srivastava | Empirical observations of capacity drop in freeway merges with ramp control and integration in a first-order model[END_REF][START_REF] Yuan | Capacity drop: a relation between the speed in congestion and the queue discharge rate[END_REF] or in random locations. Interestingly, in cases of random locations, [START_REF] Jorge | A mechanism to describe the formation and propagation of stopand-go waves in congested freeway traffic[END_REF] It has been reported that the bottleneck occurrence is due to spontaneous SG waves (see for example: (14-16)). The two capacities (PBC and QDF) may independently vary even with same external conditions, as reported by many researchers [START_REF] Alexandra | Defining, Measuring, and Modeling Capacity for the Highway Capacity Manual[END_REF][START_REF] Brilon | Reliability of Freeway Traffic Flow[END_REF].

(3) reported a variability of 5.5% to 10.5 % of PBC for three different sites on US freeways. In a similar study, [START_REF] Brilon | Reliability of Freeway Traffic Flow[END_REF] indicated that the PBC standard deviation and the mean values for 15 different German freeways are 6.6 % to 13.5 % and 9.4 %, respectively. Similar results are reported in [START_REF] Calvert | 780 Application of advanced sampling for efficient probabilistic traffic modelling[END_REF]. The capacity distributions, the value of the drop and stop and go wave occurrence are often explained by the emergence of collective effects originated in microscopic phenomena. Based on the findings in the relevant literature, both lane changes and car-following contribute to these phenomena. However in this paper the focus is given solely to car-following behavior in case of capacity distribution and capacity drop. More specifically, we concentrate on car following variability impact on the distribution of capacity and the value of the capacity drop, considering that all vehicles obey the same model. We chose the simplest car following model: Newell's model [START_REF] Newell | A simplified car-following theory: a 785 lower order model[END_REF], modified, according to [START_REF] Jabari | A probabilistic stationary speed-density relation based on Newell's simplified car-following model[END_REF] in order to include a finite acceleration. This allows us to limit the number of distributed parameters to three (i.e. maximal-acceleration, reaction-time, and minimal-distance).

The remainder of this paper is organized as follows. First, a thorough literature review is presented. This is followed by the description of the modeling framework. Then, an analysis of the influence of the distribution of the CF parameters in the exact event-based scheme is presented. The paper ends with a section devoted to discussion and conclusion.

Underlying microscopic reasons for macroscopic phenomena

Many researchers consider that the origin of capacity distribution, Capacity Drop (CD), and/or Stop and Go waves 70 (SG) lies at the inter-vehicular interaction level, but, there is no consensus on the microscopic behavior generating these macroscopic observations. A comprehensive study of the impact of the behaviors interaction combinations is difficult due to the strong relationships of behaviors between 75 longitudinal interactions and lane changes.

Before presenting the microscopic causes of the aforementioned macroscopic phenomena of traffic flow in congestion, we first detail the content of the literature on those observations and the way car-following (CF) and lane-changing (LC) are used to explain the capacity distribution, capacity drop, and oscillations triggering. This results in presentation of Table 1 where we classify the most relevant references in this regard. distinguishing the considered microscopic behavior and the approaches (e.g., observation, modeling-simulation, or both). Noticeably, very few papers were identified where an explanation of the capacity distribution is provided according to any of the microscopic phenomena.

Lane-changing approaches

As we can see in Table 1, some papers explain CD and SG by the impact of LCs, either by the influence of insertion into the target lane or exiting from the origin lane. For lane changer's inserting behavior, studies report that vehicles accept a short gap immediately after the insertion, then gradually relax to return to a normal spacing given by the speed-spacing relationship. That is known as the relaxation process [START_REF] Leclercq | Relaxation Phenomenon After Lane Changing: Experimental Validation with NGSIM Data Set[END_REF][START_REF] Laval | Microscopic modeling of the relaxation phenomenon using a macroscopic lane-changing model[END_REF][START_REF] Duret | Onset of Congestion 810 from Low-Speed Merging Maneuvers within Free-Flow Traffic Stream: Analytical Solution[END_REF][START_REF] Kim | Driver relaxation impacts 815 on bottleneck activation, capacity, and the fundamental relationship[END_REF][START_REF] Zheng | The effects of lane-changing on the immediate follower: Anticipation, relaxation, and change in driver characteristics[END_REF].

Similarly, the potential followers in the target lane can "anticipate" the maneuver just before the insertion and create a long gap (courtesy spacing). The anticipation process is significantly different from the relaxation process [START_REF] Zheng | The effects of lane-changing on the immediate follower: Anticipation, relaxation, and change in driver characteristics[END_REF]. These two processes create a disturbance in the traffic stream that can be recovered gradually in time. It involves thus the acceleration process after a deceleration process which can lead to oscillations apparition (or SG waves). In this way, several studies conjecture that the LCs increase the probability of SG wave apparition (16, [START_REF] Ahn | Freeway Traffic Oscillations and Vehicle Lane-Change Maneuvers[END_REF][START_REF] Mauch | Freeway traffic oscillations: observations and predictions[END_REF][START_REF] Zheng | Freeway Traffic Oscillations: Microscopic Analysis of Formations and Propagations using Wavelet Transform[END_REF][START_REF] Leclercq | Capacity drops at merges: New analytical investigations[END_REF].

Consequently, vehicles can be confronted with the limited acceleration capability in the acceleration process and it can create a long-spacing in front of them. The studies (16, 22, 24, 28-30) explain that the CD phenomenon is due to the bounded acceleration after an insertion event into the target lane. Moreover, the exit maneuver from the origin lane leads to a longer gap that can incite the immediate follower to accelerate (16, 29).

Car-following approaches

We noticed previously that LC impacts the driving behavior by insertion and exit maneuvers. Then, the systematic LCs events on the road can be a good explanation for the observed traffic phenomena. However, we consider that longitudinal behavior plays also an important role in these observations. We focus now on different characteristics of the CF behavior, which can explain macroscopic phenomena (see Table 1): Some papers consider the random noise in acceleration as the best approach to represent the human errors that are responsible for traffic instabilities independently of the unstable nature of CF models [START_REF] Yuan | A geometric Brownian motion car-following model: towards a better understanding of capacity drop[END_REF][START_REF] Treiber | The Intelligent Driver Model with Stochasticity -New Insights Into Traffic Flow Oscillations[END_REF][START_REF] Ngoduy | Langevin method for a continuous stochastic car-following model and its stability conditions[END_REF][START_REF] Laval | A parsimonious model for the formation of oscillations in car-following models[END_REF] (this was also included in trains motion modeling [START_REF] Corman | Stochastic process in railway traffic flow: Models, methods and implications[END_REF]). They incorporate acceleration noise into CF models leading to stochastic nature. This analysis can explain the oscillations observed in absence of LCs. Paper [START_REF] Treiber | The Intelligent Driver Model with Stochasticity -New Insights Into Traffic Flow Oscillations[END_REF] includes, in three different models (Intelligent Driver Model, Parsimonious CF model, and Full Velocity Difference Model), a white acceleration noise that 150 is constant over time and between vehicles. First, the study highlights the fact that the three CF models lead to the same general findings. Authors observe that the acceleration noise systematically conducts to overestimate the speed standard deviation of vehicles and they conjecture that this instability 155 mechanism alone is sufficient to reproduce the observations. The simulations demonstrate that the amplitude of the sub-critical fluctuations can increase strongly and thus can result in oscillations. Similarly, from the simulation findings, the study [START_REF] Laval | A parsimonious model for the formation of oscillations in car-following models[END_REF] argues that driver's error increases with roadway heterogeneity (e.g. upgrades) which can not only induce bottlenecks but also exacerbate oscillations growth. Paper [START_REF] Treiber | The Intelligent Driver Model with Stochasticity -New Insights Into Traffic Flow Oscillations[END_REF] reports the same results (bottleneck activation and oscillation growth) for acceleration noise and action points (e.g. deceleration areas). Besides, the differences in vehicles characteristics (e.g. maximal-acceleration, reaction-time) can be accentuated depending on the road segment specificity. This can also, without LC, generate spontaneous oscillations. An aggressive driver behavior leads to hysteresis loops which induce the upstream propagation of oscillations [START_REF] Chen | On the periodicity of traffic oscillations and capacity drop: The role of driver characteristics[END_REF]. The authors of [START_REF] Jorge | A mechanism to describe the formation and propagation of stopand-go waves in congested freeway traffic[END_REF] propose a model which considers an upgrade (in a single lane without LC) to capture the mechanism that triggers periodic oscillations. With the parameters calibrated using NGSIM data (US-101 site), the simulation results explain oscillations as a consequence of driver heterogeneous reactions to deceleration waves.

From observations, the study [START_REF] Zheng | Freeway Traffic Oscillations: Microscopic Analysis of Formations and Propagations using Wavelet Transform[END_REF] quantifies later the number of oscillations observed in two different sites, I-80 and US-101 of NGSIM data. The results suggest that oscillations can be instigated by different factors, the predominant trigger depending on the site characteristics. In the first location (waving area), the majority of the SG was originated from LCs. On the other one (uphill), 66 % of the SG were originated by longitudinal driving disturbances due to the limited perception of the road inhomogeneity. Similarly, paper [START_REF] Goni Ros | Car-following behavior at sags and its impacts on traffic flow[END_REF] shows that oscillations upstream of a sag in Tokyo (Japan) are created in most cases by spontaneous instabilities in longitudinal behavior, rather than by LC impact. Based on these findings, the authors assume that capacity reduction is due to changes in CF behavior caused by the sag. Indeed, the impact on the traffic dynamics depends on the sag characteristics (e.g. the value of the slope angle in a downhill-uphill can influence the congestion apparition and also the number and amplitude of SG waves [START_REF] Kaur | Modeling and simulation of driver's anticipation effect in a two lane system on curved road with slope[END_REF].

The acceleration error is not the only approach able to explain the traffic phenomena based on longitudinal interactions, also acceleration variability among drivers can be an explanation: for example, a long-spacing can be created between two successive vehicles if the follower's acceleration capability is lower than the leader's. In the same study [START_REF] Yuan | 760 Capacity drop: a comparison between stop-and-go wave and standing queue at lane-drop bottleneck[END_REF], the authors consider a dynamical reaction-time extension in the acceleration process that depends on the congestion level. That means a negative relation between reaction-time value and speed value in congestion. The numerical findings explain that the intra-driver mechanism and its variation (reaction-time evolution) yields to a relation between the speed in congestion and the Queue Discharge Flow, as found in empirical observations. We note also that this paper introduces the intra-driver variability notion to explain CD amplitude, which means that vehicles change their driving features according to traffic conditions. For example, authors consider a driver's tendency to take a larger headway downstream of an oscillation [START_REF] Yuan | 760 Capacity drop: a comparison between stop-and-go wave and standing queue at lane-drop bottleneck[END_REF][START_REF] Chen | On the periodicity of traffic oscillations and capacity drop: The role of driver characteristics[END_REF]. Oscillations affect the driver's behavior and this can result in a change of individual characteristics (e.g., aggressive drivers are less aggressive after passing the oscillation [START_REF] Zheng | Freeway Traffic Oscillations: Microscopic Analysis of Formations and Propagations using Wavelet Transform[END_REF] and tend to adopt larger response times). Thus, the reduction in bottleneck discharge rate ensues even in absence of LCs [START_REF] Yuan | 760 Capacity drop: a comparison between stop-and-go wave and standing queue at lane-drop bottleneck[END_REF]. In this regard, paper [START_REF] Chen | On the periodicity of traffic oscillations and capacity drop: The role of driver characteristics[END_REF] focuses on the mechanism of periodicity of traffic oscillations and finds that drivers' reactions to oscillations around a bottleneck have a profound impact on CD. Moreover, the amplitude of the oscillation could influence the acceleration/deceleration values (e.g. hysteresis level [START_REF] Chen | Capacity-drop at extended bottlenecks: Merge, diverge, and weave[END_REF]) and thus the discharge rate (16).

With a modeling approach, [START_REF] Zhang | A car-following theory for multiphase vehicular traffic flow[END_REF] proposes a CF model dedicated to explaining CD under different driving conditions. Driver's time headway is assumed to be a function not only of distance headway but also of traffic conditions. Although their results were not validated with observations, this work sheds light on different spacingspeed relationships that vehicles could adopt according to the individual characteristics and the impact of this on the FD. These characteristics of the vehicle-driver pairs explain the 235 difference between vehicles population including the desired speed, acceleration/deceleration capability, reaction-time and accepted minimal-distance. In the present paper, the first step in a larger work, we consider this inter-vehicle variability as independent of traffic conditions. 240 Some papers mention reaction-time and minimal-distance distributions as inter-vehicle variability [START_REF] Han | Stochastic modeling of breakdown at freeway merge bottleneck and traffic control method using connected automated vehicle[END_REF][START_REF] Yang | Variability of observed drivers' car-following behavior on expressway basic segment[END_REF][START_REF] Ossen | Driver Heterogeneity in Car following and Its Impact on Modeling Traffic Dynamics[END_REF][START_REF] Jorge | A mechanism to describe the formation and propagation of stopand-go waves in congested freeway traffic[END_REF][START_REF] Chen | A behavioral car-following model that captures traffic oscillations[END_REF]. From empirical trajectories, other studies [START_REF] Ossen | Driver Heterogeneity in Car following and Its Impact on Modeling Traffic Dynamics[END_REF][START_REF] Ossen | Heterogeneity in carfollowing behavior: Theory and empirics[END_REF] obtained different vehicles-drivers parameters for every specific model. They propose that inter-vehicle variability can be 245 classified into two types: i) different driving styles (captured by different CF models) and; ii) different inter-vehicle characteristics (captured by distributed CF parameters of the same model). The study [START_REF] Ossen | Driver Heterogeneity in Car following and Its Impact on Modeling Traffic Dynamics[END_REF] focuses on the analysis of the inter-vehicle variability in CF by comparing heterogeneous 250 and homogeneous platoons to study traffic stability via simulation. The simulation findings show that inter-vehicular heterogeneity influences asymptotic stability. It means that disturbance propagates quickly and can grow in magnitude from vehicle to vehicle. Indeed, the plots in Fig. 3 in [START_REF] Ossen | Driver Heterogeneity in Car following and Its Impact on Modeling Traffic Dynamics[END_REF] 255 show that the traffic jam appears earlier and leads to higher congestion level due to the CF parameters variability.

Similarly, the inter-drivers difference of responses to the flow disturbance can reveal the mechanism of the probabilistic breakdown occurrence at bottlenecks [START_REF] Yang | Variability of observed drivers' car-following behavior on expressway basic segment[END_REF][START_REF] Ossen | Heterogeneity in carfollowing behavior: Theory and empirics[END_REF]. 260 [START_REF] Yang | Variability of observed drivers' car-following behavior on expressway basic segment[END_REF] conjectures that some following drivers are not affected by the unnoticed driving changes of their leaders as much as others at sags. To model this phenomenon, they estimate different CF parameters distributions and propose a microscopic simulation. The results explain that the 265 congestion occurrence probability increases with the traffic demand. However, the estimated cumulative distribution function is not the same as the empirical curve. The authors link it to a potential relationship between a sub-part of the CF parameters.

270

The investigation in [START_REF] Han | Stochastic modeling of breakdown at freeway merge bottleneck and traffic control method using connected automated vehicle[END_REF] proposes a stochastic traffic breakdown model for freeway merge bottlenecks to reveal the breakdown mechanisms considering various traffic parameters (e.g. desired speed, merging speed, wave speed, merge flow rate, individual vehicle-driver's characteristics). 275 A Bayesian approach is developed to estimate the breakdown probability associated with the individual parameters. The analytical results show that the probability of traffic breakdown increases with higher values of drivers' reaction times and the standard deviation of headway distribution. 280 In this paper, the standard deviation of reaction-time has no impact on the probability curve.

Research gaps

Based on the previous literature summary, we conclude 285 that many authors consider that bounded acceleration after LC is responsible for the CD phenomenon. We note also that intra-driver variability (reaction-time evolution) is an important explanation of that phenomenon. However, several of those papers only focus on the Queue Discharge Flow (QDF) variability and the Pre-Breakdown Capacity (PBC) variability has been less analyzed. Moreover, the literature findings shed light on the impacts of acceleration variability on the CD phenomenon. Note that the inter-vehicle variability of reaction-time and minimal-distance in CF behavior were also mentioned as an important contribution to explain macroscopic observations. Consequently, for a complete analysis of macroscopic phenomena, we consider that a focus on the inter-vehicle variability between vehicles' behavior must be scrutinized, in particular in its impact on the macroscopic features of traffic like the PBC variability and CD amplitude.

The hypothesis of the present paper is that longitudinal interactions may well explain the macroscopic observations even in the absence of LCs. The aim of this analysis is to determine the link between the inter-vehicle variability of car-driver pairs and the macroscopic phenomena (PBC distribution and CD), explained only by the CF. A systematic exploration of the CF parameters variability (e.g. reactiontime, minimal-distance and maximal-acceleration) is the only way to explore the potential cumulative effects. We choose to explore a unique CF models: Newell's model [START_REF] Newell | A simplified car-following theory: a 785 lower order model[END_REF] because of its low number of parameters (modified to include a finite acceleration, according to [START_REF] Jabari | A probabilistic stationary speed-density relation based on Newell's simplified car-following model[END_REF]). We show that the reaction-time is the most important parameter influencing the two macroscopic variables capacity variability amplitude and capacity drop.

Microscopic Modeling Framework

Model definition

As mentioned earlier, we use Newell's model [START_REF] Newell | A simplified car-following theory: a 785 lower order model[END_REF] because of its simplicity and the fact that the parameters are translatable from the microscopic level to the macroscopic one. The equation below corresponds to the extended version of Newell's model presented in [START_REF] Jabari | A probabilistic stationary speed-density relation based on Newell's simplified car-following model[END_REF]. In this equation the first line corresponds to the congested regime, and the second line to the free flow regime, τ n , d n , and a n being respectively the reaction-time, the minimal-distance, and the maximalacceleration.

ẋn (t) = min min ∆x n-1,n (t) -d n τ n ; v n (t -τ n ) + a n • τ n ; v n f (1)
They are two differences between this version and the original one [START_REF] Newell | A simplified car-following theory: a 785 lower order model[END_REF]. First, we associate each vehicle-driver pair with its parameters, keeping them constant. This corresponds to our objectives as we focus on inter-vehicle variability, and we do not want to include intra-driver variability. Second, on the contrary to the original version where acceleration and deceleration are infinite, a maximal-acceleration parameter (a n ) permits to limit the acceleration in the free flow regime. Our motivation to include a n and its variability is the literature findings of the capacity drop phenomenon (see Table 1). Note that the modified Newell's model does not include deceleration in its equation 1 (positive acceleration only being limited). More generally speaking, taking into account deceleration in CF model certainly modifies locally the car behaviors in the upstream limit of the congestion, but we believe that this may not have a strong impact on the global scale.

We keep identical, over all vehicles, the maximal desired speed (v n = v f ) given that our study considers longitudinal circulation of vehicles in a single traffic lane. A slower vehicle in this single lane scheme could change the independence properties of the follower vehicles.

Simulation settings

We define the simulation settings in order to observe congestion onset without lane changes. Thus, we chose a onelane stretch of road with a zone where speed is limited to a value U l , lower than the free flow speed: U f . U l variation permits the exploration of different congestion levels. The entering time instants are defined according to:

• An input flow profile, which defines the demand in two periods: (i) the traffic flow increases linearly, from an initial flow value (q 0 ) to a maximal flow value (q max );

(ii) the traffic flow remains constant at q max .We set the maximal input flow to correspond to the value of the theoretical congested capacity associated to the maximal value of the limited speed, for an illustration see Fig. 1 (top);

• The classical exponential distribution of inter-vehicular headways at the simulated zone entrance.

We can thus observe the transition conditions from the free flow regime to the congested flow regime and therefore identify the congestion onset.

The PBC is defined as the entering flow at the moment this triggering vehicle enters the link. The QDF is the mean downstream flow observed during the remaining period after this vehicles exits the speed limitation zone. Applying the formulas proposed by [START_REF] Edie | Discussion of traffic stream measurements 975 and definitions[END_REF], we use the window size of ∆ t = 60 s and ∆ x = 100 m to measure density and flow rate during the time with a frequency of 20 s. Density and flow are used to construct the simulated fundamental diagrams. In this study, We don't consider the queue length as an output variable. It is indirectly linked to the PBC and can be inferred from its value. Note that for this test, we use an individual time-step numerical scheme (resolution based on individual reaction-time value). The code is available * . Table 2 presents macroscopic and car-following parameters, as well as the various scenarios.

• truncated Gaussian (to avoid negative values),

• gamma. The uniform distribution is chosen for its simplicity, as well as the truncated Gaussian. We selected also the Gamma distribution because in the literature, some authors found this shape is adequate for the reaction-time distribution [START_REF] Yang | Variability of observed drivers' car-following behavior on expressway basic segment[END_REF].

The Fig. 2 illustrates how the three distributions are adapted, for parameter p, to present the same (σ p /µ p ) ratio and the same µ p value.

For each parameter p (the reaction-time τ n , the minimaldistance d n and the maximal-acceleration a n ), the amplitude is defined by the ratio between the standard deviation (σ p ) value and the mean value (µ p ). We study three amplitudes for each distribution: 10 %, 20 %, 30 %.

The mean value of some parameters is directly determined from the fundamental diagram taking into account that µ τ = 1/k max * w and µ d = 1/k max . Table 2 presents the macroscopic and car-following parameters values used in the rest of the paper.

It is well known that in Newell's modeling framework, τ and d are not independent: w, the backward wave propagation speed links the two through the relation w = d/τ [START_REF] Chen | A behavioral car-following model that captures traffic oscillations[END_REF][START_REF] Chiabaut | From heterogeneous drivers to macroscopic patterns in conges-980 tion[END_REF]. Therefore, we can choose to vary d n and τ n independently or to vary one of them (the reaction-time τ n , for example) and to deduce the value of d with a constant w value. This is illustrated in Fig. 3, which represents the impact on various methods of parameters random generation on the two versions of the fundamental diagrams: the macroscopic (Q = f (K)) and the microscopic one (s = f ′ (v), where s is the spacing between the car and its leader, v is the speed). If w is constant (see 3.a), the capacity and the maximal density vary in the same proportion, this is not the case if d and τ vary independently.

In addition to the congestion level (defined by the limitation speed U l ), the parameter distribution function and the level of variability, each traffic scenario is characterized by the (set of) distributed parameters: • ∅: no variation;

• a: only the maximal-acceleration varies;

• d: only the minimal-distance varies (as a consequence, w varies), see Fig. The simulation considers 100 replications of each traffic scenario to obtain the distributions of PBC and QDF.

Car-following variability impact on macroscopic variables

Inter-vehicle variability impact on the fundamental diagram

To begin with, in Fig. 4, we present the impact of various levels of variability, with a standard deviation equals to 0 % up to 30 % of the mean values for all parameters. In this case, the U l values are randomly selected between U min l and U max l to explore the congested branch of the FD (see state B in Fig. 1). As expected, one observes a triangular shape, consistent with Newell's model. The spreading of the congested branch increases while the variability amplitude increases. Logically, for the free flow branch, there is no evolution as we keep the same U f without any variability thereof.

Q max w K max Flow SD (veh/min) (m/s) (veh/m) (

455

For each fundamental diagram, we fit a straight line which leads to the fitted values of the macroscopic parameters. For the case without variability (see Fig. 4-a), a small difference can be seen between the values of the fixed and fitted parameters, not greater than 3 %. This is due to the precision 460 of the measurement method (Edie's method).

The variability amplitude has not only an impact on the spreading of the points of the congested branch, but also on the fitted macroscopic parameters values. Those observations are summarized in table 3. We evidence that w decreases 14 % and K max increases 15 % as variability amplitude increases from 0 to 30 %, the Q max value being almost unchanged. Keep in mind that the mean of each individual parameter remains the same. This table presents also the standard deviation of the flow in the density range from 0.03 to 0.035 veh/m, i.e. between the theoretical critical density and twice this value. This quantifies the spreading of the fundamental diagram due to the inter-vehicle variability. We can conclude that the variability amplitude has a direct impact. the Pre-Breakdown Capacity (PBC) and Queue Discharge Flow (QDF) for three values of speed limits. The theoretical 480 capacity is mentioned as well as the speed limit value at the top of each column.

Dependence of capacity on varying parameters

The table 4 synthesizes those results with the mean values of the two capacities and their standard deviations.

A global observation can be done when comparing the 485 QDF (grey line) to the PBC (black line). The QDF mean Prepared using TRR.cls

p) q) r) (d, τ) All Var τ d a ϕ PBC QDF U l q max =15 m/s, =10 m/s, =5 m/s, U l U l =34.2 veh/min
q max =30.0 veh/min q max =21.8 veh/min Table 4. Impact of the parameters variation and of the speed limit on the Pre-Breakdown Capacity (PBC) and the Queue Discharge Flow (QDF) mean values (veh/min) and standard deviations (SD).% is the ratio between SD and mean. The variability amplitude is of 20 %, the parameters are distributed according to a truncated Gaussian distribution.

value is slightly lower than the mean PBC (the difference is of the order of one percent). This is due to the fact that we adopt a version of Newell's model with a limitation in maximal acceleration. But this difference is very low and the capacity drop value we observe, compared to what is reported in the literature about CD observations, is negligible.

For the no variability case (top row), the cdfs present the typical shape of a deterministic function: no congestion occurs below the theoretical capacity, whereas in 100 % of the cases, congestion occurs for a flow value above the theoretical capacity. All the next figures are drawn with a variability amplitude of 20 % (the impact of the variability amplitude will be studied later on).

The second row of Fig. 5 presents the impact of the variability of a n , the maximal-acceleration parameter. Its impact is negligible on the variability both of PBC and QDF, whatever the speed limit value (U l ). Next, we vary the two other parameters: d n and τ n . We chose first two variability configurations where w is modified 505 accordingly to the parameter variation, the other parameter (τ n if we study the impact of d n variability, d n when τ n varies) remaining constant.

The impact of the variability of the minimal-distance, d n (third row), is larger than the one of the maximal-acceleration a n . Table 4 confirms that the impact of the minimal-distance variability increases when the speed limit value decreases. Indeed, the variation of d n has logically a bigger impact for congestion with high-density values than for light congestion.

The individual parameters variability that has the larger impact is the reaction-time τ n . We can see on table 4 that for speed limit value of 15 m/s, the PBC variability ratio (standard deviation divided by the mean value) is of 2.3 %. Note that the variability ratio for the QDF is less than twice lower (1.0 %). Conversely to what was observed for d n variability, τ n variability has the larger impact when the speed limit is larger (see Fig. 5.j to 5.l).

The before the last row of this figure presents the (d, τ ) case where w is constant, the individual value of d n for a vehicle being directly determined by the one of τ n by the relationship d n = w × τ n (the variability amplitude of 20 % being applied to τ n ). We can see in this row (Fig. 5.m to .o) that the variability of the PBC and the QDF is higher in this case than in the case where τ n varies while d n remains constant (compare with Fig. 5.j to .l). One observes that when all parameters vary independently (Fig. 5.p to .r) the impact on the variability of the PBC is not amplified.

From what precedes, we conclude that when only one parameter is varying, the most impacting parameter, in terms of its variability, on the capacities distributions (both for the PBC and for QDF) is the reaction-time, the minimaldistance variability having a lower impact, and the maximalacceleration distribution having a lesser impact. The relative impact of τ n variability is larger when the speed limit is larger, the reverse effect is observed for the minimaldistance variability. When τ n and d n vary simultaneously, w remaining constant, the impact is maximal. This could be expected as we observe the highest spreading on the third row of Fig. 3, corresponding to this case. Combining the variability of all parameters does not significantly increases the widths of the capacities distributions.

Therefore, we consider that in our specific study case, where no lane change is possible, the maximal-acceleration variability has a very limited impact. This result must be compared with the one presented in [START_REF] Leclercq | Capacity drops at merges: New analytical investigations[END_REF] where the acceleration variability of the lane changer at merges is, by analytical calculation, proven to have a strong impact on the capacity drop. More generally, we establish that the experimental observations of the capacity drop amplitude are not linked to the distribution of the car-following parameters.

Prepared using TRR.cls

We prove, on the contrary, that CF parameters variability, per se has a strong impact on capacities distributions, especially when τ n and d n vary simultaneously, w remaining constant ((d, τ ) case.).

We will now examine the impact of the distribution shapes and amplitudes in two cases, when τ n and d n vary simultaneously, with w constant, and when all parameters vary independently. Note that the shapes and amplitude of the distributions are, in the (d, τ ) case, applied to the τ n parameter.

Dependence of capacity variability on the distributions shapes and amplitudes

The Fig. 6 presents in two cases the cdf of PBC and of QDF for the three distributions shapes: truncated Gaussian, Gamma and Uniform, with a speed limit of U l = 10m/s.

The table 5 presents the corresponding numerical values for the mean and the standard deviation. Although the shapes are significantly different (as can be seen from Fig. 3), the impact on the variability amplitude of the two capacities distributions seems negligible. The Fig. 7, as well as table 6, present the results for the truncated Gaussian distribution for three amplitudes: 10 %, 20 %, 30 %, with a speed limit of U l = 10m/s. There is clearly a direct impact of the parameters' variability amplitude on the capacities values and on their variability amplitude. First, the mean value of the PBC decreases as the variability amplifies (from 30.29 veh/min for the no variability case, down to 29.49 veh/min for a variability amplitude of 30 %, the decrease being of 2.6 %). The same phenomenon, with a lesser impact, is observed for the QDF 585 (from 30.03 to 29.53 veh/min, with a decrease of 1.6 %). Second, the larger the parameters amplitude the larger the capacities standard deviation. This result is coherent with what was expected.

Discussion and conclusions 590

The extensive literature review provided in this paper confirms that researchers are giving diverse and contradictory explanations for capacity distribution, capacity drop, and stop and go waves. Most of the studies in the literature address the macroscopic observations of the capacity drop 595 and only few studies devoted to stop and go waves and capacities distribution. When trying to link the phenomena to the microscopic behaviors of vehicles, some authors argue that lane changing is the main cause of capacity drop and stop and go waves -either the inserting or the exiting events-, 600 others consider that the random noise in acceleration is the key explanation for the capacity drop or for the stop and go waves.

We define a simple simulation test-bed to create a bottleneck by considering only car-following behavior. A speed limit has been imposed in a part of the one-lane stretch of road. We use the classical Newell's model [START_REF] Newell | A simplified car-following theory: a 785 lower order model[END_REF], expressed after [START_REF] Jabari | A probabilistic stationary speed-density relation based on Newell's simplified car-following model[END_REF] in terms of speed, chosen for its ease of interpretation and for its simplicity. We focused here on the following three parameters characterizing the individual behavior of the driver-vehicle pairs: minimal-distance d n , reaction-time τ n and maximal-acceleration a n .

We did not observe any impact of the car-following parameters on the capacity drop: the difference between the Pre-Breakdown Capacity and the Queue Discharge Flow is lower than 1 % in all the studied configurations. Note that we did chose a rather large mean value of the maximalacceleration of 2.5 m/s 2 .

Our results show that among those three parameters, the reaction-time variability has the larger impact on the capacities distribution. This can be explained by the figure 3, where we can see, on the second row, that the capacity variability is higher than in the first row where d varies.

630

Moreover, an uncertainty propagation calculus can confirm this. The flow associated to a congested regime whose speed is limited to u is expressed, with the Newell's model as:

C u = 1 (τ + d/u) (2)
Applying the uncertainty propagation method in the case of independence of parameters d and τ , one obtains the value of the relative variation of C u :

σ 2 Cu C u 2 = σ 2 τ + σ 2 d /u 2 (τ + d/u) 2 (3)
Since the variance σ 2 d for the parameter d is divided by the square of the speed limit u, the weight of this variance in the relative variance of the capacity is less than that of the parameter τ .

The impact is higher when all the three parameters vary, the maximal-acceleration a independently, and the minimaldistance parameter d, in conjunction with reaction-time τ , permitting to keep w, the backward wave propagation speed, constant. The maximal impact was obtained in the case where the variability amplitude of each parameter is of 30 %, which corresponds to the highest studied amplitude. We did not observe a significant impact of the distribution function on any macroscopic characteristic (capacity drop or capacities variability). We compared the most classical distributions: Gamma, truncated Gaussian, and uniform.

Compared to the capacity distribution observations reported in the literature, our results explain between an 1/2 and a 1/3 of the standard deviations (mean std of 7.2 % for (3) and of 9.4 % for [START_REF] Brilon | Reliability of Freeway Traffic Flow[END_REF]). This means that the car following parameters contribution to the capacity distribution is far from being negligible. We realize exactly the same study with the Gipps' car following model and obtained similar results [START_REF] Gomez-Patiño | Variabilité inter-individuelle du 985 comportement longitudinal de véhicules à conduite manuelle ou automatisée : impacts sur la capacité des voies[END_REF].

The key finding in this work is the significant impact of the distribution of the reaction-time τ n parameter. In many microscopic simulation tools, τ n is equal to the computation time-step and therefore uniform for all vehicles. From our results we can clearly conclude that any simulation study willing to scrutinize the variability among days concerning congestion creation, duration and amplitude, ignoring the variability of the reaction time will fail to reach its goals.

Moreover, microscopic simulation tools are often used to test traffic control methods. The simplification brought by the Prepared using TRR.cls using of a unique value for the reaction time is a potential source of erroneous estimation of the impact of these controls on the basis of microscopic simulation tools.

Future researches will be devoted to the study of lane changes impacts on the capacity drop and the capacity distribution, with a focus on the behavior variability among driver-vehicles pairs. 16. Oh, S. and H. Yeo.

Impact of stop-and-go waves and lane changes on discharge rate in recovery flow.

FlowFigure 1 .

 1 Figure 1. Description of the fundamental diagram and the capacities measurement method: a) Theoretical fundamental diagram with the physical meaning of the various macroscopic parameters with the variation range of the limited speed U l between 5 and 20 m/s, b) measurement method of PBC and QDF, c) speed evolution of some vehicles identified immediately after the congestion onset. V0 is the last vehicle not affected by congestion and V1 is the triggering vehicle (represented in red in Fig. b). Congestion is considered when at least 10 vehicles are in the queue.

Figure 2 .

 2 Figure 2. Parameters distributions according to the different function shape with theoretical variation amplitude of 20 %.

  3.a and 3.b; • τ : only the reaction-time varies (as a consequence, w varies), see Fig. 3.c and 3.d; • (d, τ ): in this configuration, τ and d variations are linked, w is constant; in addition, the acceleration varies, Fig. 3.e and 3.f; • All V ar: all parameters vary, independently, see Fig. 3.g and 3.h.

Figure 3 .

 3 Figure 3. Parameters variation impact on the shape of the fundamental diagram (FD). Left column: macroscopic FD (Q = f (K)); right column: microscopic FD (s = f ′ (v)). a-b) d varies and τ remains constant, c-d) τ varies and d remains constant, e-f) the variation of τ determines the variation of d, w remaining constant, g-h) τ and d vary independently, w may vary.

Fig. 5 Flow 18 DensityFigure 4 .

 5184 Fig. 5 presents the impact of each combination of parameters variability on the cumulative distribution function (cdf) of

  [veh/min] Flow [veh/min] Flow [veh/min] a) b

Figure 5 .

 5 Figure 5. Impact of the CF parameters inter-vehicle variability and of the speed limit on cumulative distribution function of the PBC and QDF. The different variability configurations are organized in the rows and the three speed limits values are organized in the columns. The variability amplitude is of 20 %, the parameters are distributed according to a truncated Gaussian distribution.

FlowFigure 6 .

 6 Figure 6. Impact of the shape of the distribution of the individual parameters on the capacities cumulative distribution functions. Variability amplitude if of 20 % and all the parameters vary. The speed is limited to 10 m/s.

Figure 7 .

 7 Figure 7. Impact of variability amplitude on the capacities cumulative distribution functions. The left column presents the case when τ n and d n vary simultaneously with constant w, the right column the case when all parameters vary independently. The three variability levels are organized in the rows. The speed is limited to 10 m/s.

Table 1 .

 1 Classification of the literature associating each of the main macroscopic traffic characteristics observed in congestion and microscopic behavior. Depending on the approach of every paper, a color code is applied: italic text: observation-based paper; bold text: modeling/simulation-based paper; blue text: both.

	• Random noise in acceleration caused by the drivers'
	lack of concentration in keeping a constant speed.
	• Maximal-acceleration spread due to drivers' responses
	variability, which can generate long gaps due to
	acceleration difference (e.g. low acceleration vehicle
	behind high acceleration vehicle).
	• Longitudinal driving instabilities due to external
	changes (e.g. sags, tunnels, roadside barriers, weather)
	leading to abrupt acceleration changes (e.g. overreac-
	tion).
	• Intra-driver variability that reflects the changes in
	driving characteristics of the same driver adopting
	another behavior in different traffic conditions (e.g.,

an aggressive driver can become less aggressive after passing a traffic oscillation (

27

)).

• Inter-vehicle variability that denotes different characteristics inside the vehicles population (e.g. passenger cars and trucks, younger and older drivers).

Table 2 .

 2 Macroscopic

	Macroscopic parameters Description	Values	Units
	U f	free flow speed	30	m/s
	w	backward wave propagation speed	6	m/s
	k max	maximal density	0.133	veh/m
	Q max	maximal capacity	40	veh/min
	CF variables
	x n (t)	position of vehicle n at time t		m
	ẋn (t) = v n (t)	speed of vehicle n at time t		m/s
	ẍn (t)	acceleration of vehicle n at time t		m/s 2
	∆x n-1,n (t)	distance of vehicle n with its leader n -1 at t		m
	∆v n-1,n (t)	speed dif. between vehicles n and n -1 at t		m/s
	CF parameters		Mean
	τ n	reaction-time of vehicle n	1.25	s
	d n	minimal-distance of vehicle n	7.50	m
	a n	maximal-acceleration of vehicle n	3.00	m/s 2
	v f = U f	maximal desired speed (identical for all n)	30.0	m/s
	Scenario definition
	U l	speed in the speed limit zone	5 m/s, 10 m/s, 15 m/s
		parameters distribution definition	uniform / T. Gaussian / Gamma
		variability distribution	no / one param. / all parameters
		variability amplitude	SD = 10 %, 20 % or 30 % of the mean

, car following variables and parameters and scenario definition of the simulation (T. Gaussian stands for "truncated Gaussian", chosen to avoid negative values)

Table 3 .

 3 Impact of the variability of all car-following parameters of the Newell's model on the macroscopic parameters resulting from the fitting of the fundamental diagrams of Fig.4. The quantity "Flow SD" is the standard deviation of the flow values in the density in the range [0.035;0.040].

Gustave Eiffel University, University of Lyon, ENTPE, Licit, Lyon, France

Complex Transport Systems Laboratory (CTSLAB), Department of Civil

Prepared using TRR.cls

https://github.com/carlosmgop/

Distributions of car-following parameters

The simulation attributes to each driver-vehicle pair its own parameters, selected according to a distribution. We study here three different distributions:

• uniform, The central columns present the case when τ n and d n vary simultaneously with constant w, the right columns the case when all parameters vary independently. The three variability levels are organized in the rows. The speed is limited to 10 m/s.

In the current work, we quantify the impact of the individual parameters variability of the car-following part 605 of the microscopic behavior of the vehicles. We choose this approach not because we neglect the impact of lane change. We believe the two aspects need first to be addressed separately. The combination of lane change behaviors and car following distribution is left for future research.


Sebastian Hörl

Exploring accelerated evolutionary parameter search for iterative large-scale transport simulations in a new calibration testbed

Keywords: transport simulation, calibration, parameter search, acceleration, mode share, MAT-Sim

Large-scale agent-based transport models of whole territories have become an important tool in research and planning of new services and policies. Yet, studies based on those tools are rarely reproducible due to the complexity of data sources and modeling processes. One important element towards fully replicable simulations is automatic calibration of behavioral and infrastructural model parameters. The present paper contributes to standardizing the calibration process by describing a consistent framework for benchmarking calibration objectives and optimization algorithms. Furthermore, the paper advances the current state of the art by exploring the integration of a search acceleration method for iterative simulators (opdyts) with sample-based evolutionary search algorithms. In a use case for Paris and the MATSim simulator, we demonstrate the applicability of the framework. We show that opdyts accelerates the parameter search process, although its comparative runtime benefits decrease with higher availability of computational resources.

INTRODUCTION

Agent-based transport simulations have gained in interest over the past years as they allow studying in detail the highly dynamic interactions between customers, mobility providers and system control measures in increasingly digitalized transport systems. By adding behavioral components to simulate the decision-making of people, agent-based simulations have been used to construct scenarios for future transport technologies, mobility services, and policies.

The present paper is embedded in a larger effort of making such large-scale agent-based transport simulations reproducible. [START_REF] Hörl | Synthetic population and travel demand for Paris and Île-de-France based on open and publicly available data[END_REF] propose a method to create synthetic travel demand data sets consisting of synthetic households, persons and their daily activity patterns from open and publicly available data with a reference use case for Paris and Île-de-France. By adding network data from OpenStreetMap and regional public transport schedules, the pipeline allows producing open and runnable simulations for the agent-based transport simulation framework MATSim (Horni, Nagel, & Axhausen, 2016). While the demand data can serve as input to any agent-based transport simulator, it is currently optimized to be used in MATSim's eqasim extension (Hörl & Balac, 2021a), which focuses on mode decisions which are governed by discrete choice models. In the full model set up, agents interact in a traffic simulation, perceive traffic conditions and perform mode decisions for the trips in their daily schedules after a one-day simulation. After, their updated daily schedules are simulated again, until the decision of the agents and the traffic conditions reach a stochastic equilibrium state. The process is iterative and dependent on input parameters that value travel time, monetary costs or line switches when comparing different mode alternatives.

While the choice model parameters can be estimated from (proprietary) survey data, key metrics such as mode shares or network flows will still show differences when compared with reference data sets. Hence, both demand-side (behavioral) and supply-side (network) parameters need to be adjusted to achieve a high correspondence with reference data. So far, the calibration process has been performed manually with a researcher proposing a set of parameters, running the simulation and comparing the resulting metrics with reference data. However, within the scope of reproducibility, we are interested in establishing an automated pipeline from raw data sets to the final calibrated simulation. In this context, the process of model parameter calibration needs to be automated. Having in mind that large-scale agent-based transport simulations have high demand in computation time, using efficient and adapted algorithms is a key requirement.

BACKGROUND

In the context of MATSim, some efforts of automatic calibration have been presented. [START_REF] He | A validated multi-agent simulation test bed to evaluate congestion pricing policies on population segments by time-of-day in New York City[END_REF] regard MATSim as a black box with input parameters and an objective value as output to calibrate network flows using the SPSA approach [START_REF] Spall | Implementation of the simultaneous perturbation algorithm for stochastic optimization[END_REF]. [START_REF] Agarwal | Calibration of behavioral parameters in an agent-based transport simulation[END_REF] describe a proof-of-concept for calibrating mode choice parameters using a search acceleration strategy proposed by [START_REF] Flötteröd | A search acceleration method for optimization problems with transport simulation constraints[END_REF]. Apart from those efforts, no readily useable calibration framework and benchmark for MATSim or similar simulators exist, which is an endeavor that will be backed by the present research.

A common approach to perform parameter optimization for stochastic functions without calculable derivatives is black-box optimization. Various known optimization approaches such as SPSA. A common class of approaches are covered by Evolutionary Search (ES) strategies. Those algorithms are usually population-based in that in every iteration, a set of potential candidates is generated. Usually, these candidates are generated by mutating or recombining well-performing parameter choices from previous generations. A frequently used approach is Covariance Matrix Adaptation ES [START_REF] Hansen | The CMA Evolution Strategy: A Comparing Review[END_REF] with CMA-(λ ,1)-ES being one specific version. Briefly summarized, the algorithm is based on a normal distribution with a mean and covariance matrix that is adapted, based on the performance of new proposed parameters. Specifically, the simple (λ ,1) variant prescribes to sample λ new candidates from the given distribution in each iteration and replaces the current mean with the most successful candidate if it is better than the current best. An implementation of CMA-(λ ,1)-ES in pseudocode can be found in [START_REF] Igel | A computational efficient covariance matrix update and a (1+1)-CMA for evolution strategies[END_REF].

Any black box algorithm can be used to calibrate model parameters, as long as the specific model can be packaged as a function with input parameters and a resulting output value. It is, hence, possible to calibrate a MATSim simulation using CMA-(λ ,1)-ES. However, speeding up the calibration process means exploiting specificities of the simulator. One of the major features of MATSim is that it progresses iteratively, and that the calculations in one iteration are only dependent on the previous state. The latter property especially means that the simulation state can be saved and restarted later on. Based on these properties, [START_REF] Flötteröd | A search acceleration method for optimization problems with transport simulation constraints[END_REF] proposes the search acceleration method opdyts that not only operates on full simulation runs, but takes into account transitional states. In brief, the algorithm requires the user to propose a set of N parameter candidates, which are then implemented in respective simulation runs indexed by k ∈ {1, ..., N}. Each run is then transitioned for T steps, which are represented, for instance, by the iterations of a MAT-Sim simulation. After, sequentially, one run k is selected based on the transitional performance of all existing runs and transitioned for another T steps. This process is repeated and aims at pushing forward runs with promising transitional states and ignoring others early on in the process. Once a full run completes, it is registered as the new best candidate. Subsequent opdyts iterations are [START_REF] Flötteröd | A search acceleration method for optimization problems with transport simulation constraints[END_REF] leaves mostly open how to generate parameter candidates initially or before an opdyts iteration. In the paper, an example of a city tax simulation is provided in which hourly fees are varied in discrete quantities of ±0.25. Hence, a full enumeration of changing the fee levels is proposed.

In more general cases, parameters may be continuous, a scenario for which no advice is given by [START_REF] Flötteröd | A search acceleration method for optimization problems with transport simulation constraints[END_REF]. Structurally, a mecahnism is required that (1) allows a user to sample N different parameter vectors from a proposal distribution, and (2) allows letting the user feed back exactly one winning parameter set and its objective value. These properties are fulfilled by many sampling-based black-box optimization algorithms, notably elitist ES such as CMA-(λ ,1)-ES.

METHODOLOGY

The larger aim of the research presented here is to systematically benchmark different calibration objectives, optimization algorithms and their hyperparameters to arrive at automatically calibrated versions of standard simulations within the MATSim framework, such as the eqasim use case for Île-de-France.

For that purpose, a calibration/optimization framework has been developed that provides implementations for a large range of existing black box algorithms and standardized components for MATSim that allow to extract information to calculate various calibration objectives.

Optimization loop

We propose a classic optimization loop in which an algorithm proposes one or multiple parameter vectors that are executed in a simulation environment. Once the simulation runs have finished, an objective value is calculated for each run which quantifies the mismatch between simulation and reference data. The objective values are fed back to the optimization algorithm, which enters into the next iteration. The process is visualized in Figure 1.

The process is implemented in a modular Python framework, which is available online1 and which can be easily applied to any (transport) simulator by exposing a number of actions to the framework:

• submit(id, parameters, information): A unique identifier (id) generated by the framework is passed to the simulation environment and asks it to start a new simulation with a vector of parameters chosen by the selected optimization algorithm. Furthermore, the algorithm may pass additional information. This field is especially useful for advanced • get(id): The framework requests the simulation environment to return information on the simulation run. This call should block the system until a run has finished and return the objective value.

• clean(id): The framework requests the simulation environment to clean up any results of a previous simulation run because it is not needed anymore.

Currently, the framework provides a range of commonly used algorithms such as various random sampling approaches, elitist and non-elitist CMA-ES and NES, or SPSA. Especially, samplingbased methods are defined through dedicated interfaces which can then be used directly in opdyts.

Stopping criterion

A crucial component that is implicitly part of the calibration loop is the stopping criterion that is used to decide when a simulation has finished. Today, MATSim simulations are usually run for an a priori fixed number of iterations. Often, researchers would simulate an initial baseline case, then wait until key metrics such as the overall mode share stabilize (as shown in Figure 2a) and note down the appropriate number of iterations. However, in an automated calibration loop, simulations need to be stopped such that simulations runs with different parameters (1) are as long as necessary to be comparable and not still in a transitional state when they are stopped, and (2) are as short as possible to save computational time. [START_REF] Hörl | Simulation of price, customer behaviour and system impact for a cost-covering automated taxi system in Zurich[END_REF]) describe a computational stopping criterion that is adapted here. We propose to examine the overall mode share as the major output of an eqasim simulation using the following equations. We denote the share of any mode in iteration i ∈ N as y i ∈ R and first smooth the value over a horizon S ∈ N both left and right of i:

s i = 1 2S i+S ∑ j=i-S y j (1)
We then approximate the derivative of the mode share signal with a two-sided difference equation with horizon H ∈ N: We stop the simulation as soon as for any d i three criteria hold. The horizon H is reused to check that the value d i at the center, as well as d i-H and d i+H must not exceed a predefined threshold

d i = s i+H -s i-H 2H (
T ∈ R + : | d i+u | ≤ T ∀ u ∈ {-H, 0, H} (3)
Hence, if a simulation has already advanced until iteration I ∈ N, valid values for d i+H are only available for i ≤ I -2H -S, which also means that at least 2H + S iterations of a simulation run need to be performed. In our experiments, we require that the condition be met for all transport modes. The process is visualized for one mode in Figure 2 with S = 20, H = 10, and a threshold of T = 0.5 × 10 -2 × 10 -2 requiring changes of less than 0.5% over a period of 100 iterations.

Calibration objectives

Three major objectives have been implemented in the MATSim-specific part of the framework that allow to compare the simulation with open and publicly available reference data for French use cases in terms of (1) mode shares, (2) network flows, (3) and travel times.

Information on network flows comes from the open data portal of the city of Paris, where flow information from all loop detectors are published on a daily basis. The reference network can be map-matched to the one used in the MATSim simulation, and the difference ∆ j between reference and simulated counts can be calculated for all matchable road links j.

For travel time comparison, data from Uber Movements can be used, which gives hourly travel time information between fine-grained zones. The data can be compared by aggregating car flows in the simulation on the same origin-destination (OD) zones to calculate the offsets in travel times as ∆ j for each OD pair j.

Finally, modal shares in the simulation can be compared with reference shares either from the National Household Travel survey (which is available as open data) or from the regional survey, which is available upon request. To avoid overfitting of the Multinomial Logit model that is used in our eqasim implementation with mode-specific constants, we are interested in shares by Euclidean distances. Figure 3 visualizes the reference data, which shows that there is naturally a strong dependency of mode use on distance, which we aim to recover in our simulations. To define the mismatch, we divide all trips in increasingly large distance bins and note down the mode shares in each distance bin. The difference between simulation and reference is denoted as ∆ j for each mode-distance bin j.

Finally, in all three cases, a single objective value J(∆) can be defined by using a norm of the mismatch vector, e.g.

J(∆) = ∑ j |∆ j | or ∑ j ∆ 2 j or max j ∆ j (4)

Parameters

To guide the simulation, we currently expose the following parameters in our eqasim simulations:

• Mode choice parameters describe the parameters of the integrated Multinomial Logit model. The model structure and (manually) calibrated parameter values have been documented in [START_REF] Hörl | Dynamic demand estimation for an AMoD system in Paris[END_REF] for a case study of automated taxis in Paris.

• Link capacities are initially defined based on information from the underlying Open-StreetMap (OSM) network and the pt2matsim2 converter. The calibration parameters are given as scaling factors that are provided per OSM road category.

• Link freespeeds are also initially based on OpenStreetMap and can be controlled through calibration factors per OSM road category.

While other parameters can be added, the ones listed above can easily be defined using the current state of the Python framework that is wrapped around MATSim.

USE CASE

The present use case has the aim to demonstrate a relevant application of the proposed framework and to perform a benchmark between the bare CMA-(λ ,1)-ES algorithm and its use within opdyts.

In both cases, the ES algorithm is configured with the same hyperparameters and such that it proposes λ = 8 samples. Opdyts is configured to perform transitions of T = 20 iterations in MATSim.

As the calibration objective, we choose mode share by Euclidean distance with L 1 norm, and we choose to calibrate the alternative-specific constants β ASC,car , β ASC,bicycle , and β ASC,walk , the marginal utility of travel time by car (β tt,car ), and a uniform capacity scaling factor s for all road links. For testing, we use a 1% sample of our synthetic population for Île-de-France and reset the initial parameter values to -0.1 For better tractability, β tt,car is rescaled by a factor of 10 and the capacity factor is initialized to 1.0.

Figure 4 shows the calibration run for the elitist CMA-ES approach. On the left, the progression of the objective value is shown, while on the right, the evolution of the current best parameter Since opdyts operates on partial simulations, only a comparison in terms of MATSim iterations makes sense. Figure 5 shows the direct comparison between the standard and accelerated case on the left. Clearly, opdyts is able to reduce the number of MATSim iterations, and, hence, the computational effort for the calibration by early dropping unpromising simulation runs. This behavior can be seen on the right in Figure 5 where each color represents one parameter candidate. The runs are plotted such that, from a common starting point after each opdyts iteration, the evaluated runs are displayed in parallel. One can see that, initially, always a clear candidate can be identified early on, while at low objective values often multiple runs are transitioned for a while before a clear winner is identified.

Finally, it should be mentioned that the results shown in Figure 5 are valid if we think of a sequential execution of both algorithms, i.e. all simulations are run after another, even when they are submitted in batch. However, the framework is able to perform MATSim runs in parallel if memory and computational power are sufficient. In fact, the experiments were performed on a machine that allows four MATSim runs in parallel. This means that the CMA-(λ ,1)-ES algorithm effectively executes the first four and the second four runs of one iteration in batch. In contrary, in opdyts, only the first T MATSim iterations of each run are performed in parallel (as we need to know their initial performance), while the remaining transitions are necessarily performed sequentially. While we do not provide a direct comparison in runtime, as the calibrations have been performed on different machines, Figure 5 compares the process when accounting for iterations that can be performed in parallel when either two, four, or eight runs fit into memory.

DISCUSSION

While the general applicability of the framework has been shown, particular aspects of the present approach need further discussion and research.

We have shown that the combination of opdyts and a sampling-based black-box optimization algorithm such as elitist CMA-ES is a viable way to speed up the calibration process. However, the sequential nature of opdyts can limit performance when computational resources are abundant. It would be interesting to reflect upon ways to further increase parallelism in opdyts, e.g. by running multiple, potentially communicating, chains of the algorithm in parallel.

Regarding the stopping criterion, the results in Figure 5 indicate that further work is needed. While simulations should be stopped when their objective value has stabilized, some runs still exhibit slight transitional dynamics. This behavioral results from the fact that the stopping criterion in MATSim (overall mode shares) is not congruent with the objective calculated in the optimization loop (mode shares by distance). The stopping criterion in MATSim should, hence, be adapted to represent the objectives calculated in the surrounding Python wrapper.

For future work, a systematic benchmark of calibration objectives and algorithms is planned to give structured advice to researchers on how to configure the framework for specific use cases. These cases should also include applications to other simulators, such as SUMO.

Finally, additional calibration techniques can be added. Promising pathways can be found in Bayesian Optimization and surrogate modeling. Furthermore, multi-fidelity approaches, which can adaptively request information either for full-scale or computationally efficient down-sampled simulations, may speed up calibration processes.

CONCLUSION

In conclusion, we present a functional and ready-to-use framework for automatically calibrating well-defined large-scale agent-based transport simulations in France based on the eqasim framework. While the system already provides value for these specific cases, the framework can easily be extended to other simulators. In the future, it can be used to benchmark different combinations of calibration tasks and algorithms.

As a second insight, we propose a consistent way of using the opdyts method developed by [START_REF] Flötteröd | A search acceleration method for optimization problems with transport simulation constraints[END_REF] in the context of continuous parameter calibration, by connecting it with common black-box optimization algorithms from the facility of Evolutionary Search approaches. Our initial experiments show that opdyts is improving the computational efficiency, but that its benefits decrease if computational resources are sufficient to perform large parallel batches of simulations.

Figure 1 :

 1 Figure 1: Structure of the calibration loop

Figure 2 :

 2 Figure 2: Mode shares in MATSim and stopping criterion

Figure 3 :

 3 Figure 3: Mode shares by Euclidean distance between simulation output (solid) and reference data (dotted)

Figure 4 :

 4 Figure 4: Objective and parameter values for elitist CMA-ES

Figure 5 :

 5 Figure 5: Comparison with accelerated calibration

https://github.com/sebhoerl/boptx

https://github.com/matsim-org/pt2matsim

ACKNOWLEDGMENT

This research work has been carried out in the framework of IRT SystemX, Paris-Saclay, France, and therefore granted with public funds within the scope of the French Program "Investissements d'Avenir".