Mahendra Paipuri
email: mahendra.paipuri@entpe.fr

Ludovic Leclercq
email: ludovic.leclercq@entpe.fr

Jean Krug
email: jean.krug@entpe.fr

Validation of MFD-based models with microscopic simulations on real networks: Importance of production hysteresis and trip lengths estimation

Keywords: Macroscopic Fundamental Diagram, production hysteresis, trip length estimation, accumulationbased model, trip-based model, micro-simulation, validation

Validation of MFD-based models with microscopic simulations on real networks: Importance of production hysteresis and trip lengths estimation

INTRODUCTION

There had been plenty of developments in employing Macroscopic Fundamental Diagrams (MFD) to predict the traffic state dynamics at the network level in the recent past. The MFD relates the density of vehicles to the mean flow in the network. This relationship was first introduced by Godfrey (1) and then reintroduced by Daganzo (2) to formulate new urban model. The existence of MFD under certain regularity assumptions is verified by Geroliminis and Daganzo [START_REF] Geroliminis | Existence of urban-scale macroscopic fundamental diagrams: Some experimental findings[END_REF]. Since then, several applications like traffic state estimation (see e.g. Knoop and Hoogendoorn, Yildirimoglu and Geroliminis, 4, 5), perimeter control (see e.g. Keyvan-Ekbatani et al., Haddad and Mirkin, Ampountolas et al., [START_REF] Keyvan-Ekbatani | Exploiting the fundamental diagram of urban networks for feedback-based gating[END_REF][START_REF] Haddad | Coordinated distributed adaptive perimeter control for large-scale urban road networks[END_REF][START_REF] Ampountolas | Macroscopic modelling and robust control of bi-modal multi-region urban road networks[END_REF], cruising-for-parking (see e.g. Cao and Menendez, Leclercq et al.,[START_REF] Cao | System dynamics of urban traffic based on its parking-relatedstates[END_REF][START_REF] Leclercq | Dynamic macroscopic simulation of on-street parking search: A trip-based approach[END_REF], etc. are proposed based on MFD approach.

Even though Geroliminis and Daganzo (3) reported a well-defined MFD for the city of Yokohama, it is to be noted that the empirical data from the traffic network of Yokohama approximately satisfies the regularity requirements proposed in Daganzo and Geroliminis [START_REF] Daganzo | An analytical approximation for the macroscopic fundamental diagram of urban traffic[END_REF]. Some of them are homogeneous link distributions, slow varying demand, etc. Buisson and Ladier [START_REF] Buisson | Exploring the Impact of Homogeneity of Traffic Measurements TRB 2019 Annual Meeting Original paper submittal on the Existence of Macroscopic Fundamental Diagrams[END_REF] first reported a bimodal MFD curve for the city of Toulouse using the empirical data of the traffic network. A clockwise hysteresis-like loop is observed, which is characterized by higher flow during loading and lower flow during unloading. Gayah and Daganzo [START_REF] Gayah | Clockwise hysteresis loops in the Macroscopic Fundamental Diagram: An effect of network instability[END_REF] provided a deeper analytical investigation into the phenomenon of clockwise hysteresis and concluded that uneven congestion and drivers inability to re-route during the congestion peaks can be possible reasons for the hysteresis-like loops in MFD. Geroliminis and Sun [START_REF] Geroliminis | Hysteresis phenomena of a Macroscopic Fundamental Diagram in freeway networks[END_REF] showed a similar hysteresis-like loop in MFD based on the empirical data of freeway networks. Their work attributed the cause of hysteresis phenomenon to the different degree of spatial heterogeneity in density during onset and offset of the congestion period. Ramezani et al. [START_REF] Ramezani | Dynamics of heterogeneity in urban networks: aggregated traffic modeling and hierarchical control[END_REF] proposed a parametrization model of production MFD (p-MFD) based on heterogeneity of link density in the network. Another factor that influences the shape of MFD is the demand pattern as shown in Leclercq et al. [START_REF] Leclercq | Macroscopic traffic dynamics with heterogeneous route patterns[END_REF]. Mahmassani et al. [START_REF] Mahmassani | Urban network gridlock: Theory, characteristics, and dynamics[END_REF] showed that higher demand during congestion period results in the larger hysteresis loop in the MFD. Recently, Leclercq and Paipuri (18) proposed a deeper investigation of clockwise hysteresis phenomenon in the p-MFD by deriving the LWR solutions to an arterial case with internal bottleneck. They showed that when the network state is close to saturation, the congestion dynamics caused by bottlenecks with unequal shockwave speeds triggers the hysteresis shape in p-MFD. Following the empirical and analytical findings on production hysteresis in the literature, the importance of including hysteresis phenomenon in p-MFD for accurate description of network state dynamics is evident. Hence, first contribution of the present work is to include the hysteresis phenomenon in MFD-based simulations.

Another important question in formulating an accurate MFD simulator is definition of macroscopic trip lengths. Geroliminis and Daganzo [START_REF] Geroliminis | Existence of urban-scale macroscopic fundamental diagrams: Some experimental findings[END_REF] showed the existence of a linear relation between network production and trip completion rate and proposed the proportionality constant to be inverse of average trip length. However, Yildirimoglu and Geroliminis (5) compared the results of micro-simulation to the MFD-based simulations and concluded that using constant time invariant trip length to compute outflow has significant impact on the accuracy of the MFD-based simulation. Kouvelas et al. (19) also used constant trip length hypothesis in computing the outflow for their multi-reservoir simulations in the context of perimeter control. However, the authors stated that this assumption needs further investigation as strong fluctuations in demand and route choices can have an affect on outflow approximation. Leclercq et al. [START_REF] Leclercq | Macroscopic traffic dynamics with heterogeneous route patterns[END_REF] showed that the internal trip patterns not only depend on the OD matrix, but also vehicle routing strategy inside the reser-TRB 2019 Annual Meeting Original paper submittal voir. Therefore, the second contribution is to study the importance of level of description of trip lengths in a single reservoir setting in MFD-based simulation.

The accumulation-based MFD model is proposed by Daganzo (2) in the framework of single reservoir system. Later, this framework is extended to consider multiple trip lengths inside the reservoir in works of Geroliminis, Yildirimoglu et al. [START_REF] Geroliminis | Dynamics of peak hour and effect of parking for congested cities[END_REF][START_REF] Yildirimoglu | Equilibrium analysis and route guidance in large-scale networks with MFD dynamics[END_REF]. The main advantage of this model is being simple in terms of numerical resolution and computational complexity. Another MFD-based model, which gained significant attention in the recent past is the trip-based formulation. Originally based on idea proposed by Arnott [START_REF] Arnott | A bathtub model of downtown traffic congestion[END_REF], this approach is revisited by Leclercq The paper is organized as follows: section 2 presents the Lyon 6 network description, 2a presents the demand that is estimated from the loop-detectors data that is normalized by the total demand over 24 hr for the three different scenarios. The free-flow demand (in blue) is used for free flow scenario, whereas the network saturation case (in red) is used for congestion scenario with original and modified OD matrices. Figure 2b shows the respective actual demand from all different trips aforementioned inside the Lyon 6 network for two different demand levels. Time-dependent OD matrix is estimated from the empirical data of the Lyon city network. The estimated demand is only applicable to cars and there is no reliable data available for trucks. Hence, the truck demand is assumed to be 5% of the car demand in the present simulations.

Based on the OD matrix and route definitions, there are 19 080 different trips using the original OD matrix and network saturation demand pattern inside the Lyon 6 network and their corresponding distribution is shown in Fig. 2c and mean trip length is 1 505 m. For the case of modified OD matrix, the trip length distribution is presented in Fig. 2d with mean trip length of 1 652 m.

Micro-simulation settings

A triangular fundamental diagram is assumed with identical parameters for each vehicle category.

Two classes of vehicles are considered, namely cars and trucks. The parameters for cars used are: free-flow speed, u = 25 m/s, wave speed, w = 5.88 m/s and jam density, κ = 0.17 veh/m, where as for trucks: free-flow speed, u = 22 m/s, w = 5.88 m/s and κ = 0.075 veh/m. It is to be noted that the u is maximum free-flow speed and all vehicles will adjust the free-flow speed to the link speed limits, which are given by the network data.

Unimodal and bimodal MFD fits

Production MFD data is first calibrated by loading the considered network with different levels of static demand until a steady state is obtained in the micro-simulations. Figure 3b presents the data points on accumulation-production plane obtained for different demand levels. The network loading in the free-flow regime results in a good steady state approximation where changes in both production and accumulation are negligible with time. However, close to the network saturation scatter in the MFD data can be noticed, which is the consequence of pseudo steady states. A conventional unimodal fit is computed for the steady state MFD data which relates the mean accumulation with mean production in Fig. 3b. Note that the unimodal MFD fit cannot account for the scatter of MFD data close to network saturation. This computed unimodal fit can be expressed as follows,

P um (n) = -0.0024 n 2 + 5.9160 n, (1)
where P um (n) is unimodal fit of the p-MFD.

Besides the conventional unimodal fit, this work proposes the bimodal MFD fit to distinguish between network loading and recovery phases. To accomplish the task, a micro-simulation Hence, in the present work a bimodal MFD is computed using the hysteresis loop of the dynamic simulation and as stated earlier, unimodal p-MFD fit is estimated from static demand loadings.

It is noticed that until n ≤ 540 veh, the scatter of the MFD data in dynamic loading case is very low and hence, it is possible to represent this data by a unique parabolic fit. For n > 540 veh, two parabolic curves are fitted that follow the loading and recovering MFD points as shown in Fig. 3c. The relation between production and accumulation can be expressed as follows,

P bm (n) =          -0.0021 n 2 + 5.72 n n ≤ 540 -0.0020 n 2 + 5.89 n n > 540 & ∆n n ≥ 0 (Loading) -0.0025 n 2 + 5.55 n n > 540 & ∆n n < 0 (Recovery).
(

) 2
In order to avoid the discontinuity at n = 540, the curves are joined using a smoothening function.

In the present work, the trigonometric function tanh x is used to join the curves. Therefore, during the loading of the network, loading fit in Fig. 3c is used to maximize the network performance.

Similarly, during unloading phase, recovery fit is used to reproduce the hysteresis phenomenon observed in micro-simulations. The critical accumulation, n c , and the corresponding critical production, P c , are 900 veh and 3 680 veh m/s, respectively. Using the data points on the congestion part of the MFD, jam accumulation, n j , is extrapolated to 3 300 veh. In the implementation,

∆n(t) = n(t) -n(t -60)
, where t is time in seconds and a tolerance is used for ∆n(t) n(t) to avoid local oscillations.

Trip lengths estimation

The total number of trip lengths vary depending on the demand pattern and OD matrix. There are a total of 22 226 trips in the Lyon 6 network corresponding to the free-flow demand pattern shown in Figs. 2a and2b. As stated earlier, there are total of 19 080 and 26 120 trips for the case of network saturation scenario with original and modified OD matrices, respectively. The reason for having more trips in free-flow scenario than the network saturation with original OD matrix is that demand is kept at nominal level after the peak in the case of free-flow, while demand is reduced to a low value in the case of network saturation as shown in Fig. 2b.

In order to demonstrate the importance of level of description of trip lengths, four different methods of trip length estimation is proposed in this work.

Mean trip: Only one trip length value is considered inside the reservoir for all trips. It is defined as arithmetic mean of all trip lengths. Hence, the mean trip length depends on the scenario under consideration.

MFD-BASED SIMULATION TECHNIQUES Accumulation-based model

The following expression governs the dynamics in a single reservoir context with multiple trip lengths based on the conservation equation (see e.g. Daganzo, 2),

dn i dt = q in,i (t) -q out,i (t) for i = 1, . . . , ntrips, (3)
where n i is the partial vehicle accumulation for the trip i, q in,i and q out,i are the inflow and outflow, respectively. The computation of effective inflow and outflow is discussed in-detail in (26, 30).

The outflow of the accumulation model is governed by outflow demand function, O i (n i , n), which is defined as,

O i (n i , n) =      n i n P(n) L i n < n c n i n P c L i n ≥ n c , (4)
where n is total accumulation on all trips, i.e., ∑ ntrips i=1 n i , L i is the trip length of trip i and P(n) is the production computed from MFD. In accumulation-based model, outflow or trip-completion rate, G(n), is approximated as P(n) L and hence, it is also referred as PL (production over trip length) model.

Hence, the conservation equation ( 3) can be rewritten using eq (4) as follows,

dn i dt = q in,i (t) -O i (n i , n) for i = 1, . . . , ntrips. (5)
The Ordinary Differential Equation (ODE) in eq ( 5) is numerically resolved using first-order explicit Euler method as follows,

n t+∆t i -n t i ∆t = λ t i -O(n t i , n t ), (6)
where λ t i is the demand and ∆t is the time step. In the present work, a time step of 1 sec is used in all computations. Depending on the demand level for a given route, there can be as few as V (n(s)) ds.

The mean speed V (n) is computed from p-MFD, i.e., V Hence, along with other cases of trip lengths described earlier, individual trip lengths are also considered for event-based formulation.

VALIDATION RESULTS FOR THE REFERENCE SCENARIOS

Free flow traffic state scenario Firstly, a free flow scenario is considered where the peak demand is less than that of the network saturation state. Figure 4 shows the different state variables like accumulation, mean speed, outflow, etc. The normalized demand curve shown in Fig. 2a (in blue) is given to each OD matrix in micro-simulation. Since, the flow between an OD pair that transverse through Lyon 6 with origin/destination outside Lyon 6 cannot be predicted a priori, demand is computed from microsimulation data rather than the actual OD matrix data. In the case of trip-based approach, the starting times of each trip is the input and therefore, the micro-simulation data can be used directly.

However, in the case of accumulation-based approach, the demand per each trip is computed by taking the first derivative of cumulative curve of entering vehicles per trip. Since the exact trip starting times are known a priori from the micro-simulation data, entry supply function is not considered in the present work. This is done to avoid the discrepancies from the entry flow function, as the primary objective of the work is to study the accuracy of models with respect to p-MFD and trip lengths calibration. However, in the context of multi-reservoir settings, entry supply function must be defined and it is out of the scope of present work.

Network saturation traffic state scenario

In this section, a demand profile is considered such that the network is loaded close to the saturation. Figure 2a shows the normalized demand (in red) given to each OD pair in the microsimulation. As explained earlier, demand for MFD simulators is computed from the inflow cumulative curve of micro-simulation in Lyon 6 area. Figure 5a shows the accumulation evolution with time for accumulation-based, trip-based with both unimodal and bimodal p-MFD fits and microsimulation data. It can be noticed that the peak accumulation exceeds the critical value (n c = 900) and network is saturated. The trip-based unimodal p-MFD approach over-predicts the peak accumulation by 91 veh as presented in Table 1. This is due to use of mean p-MFD fit, which results in lower mean speeds and higher accumulation. Note that the accuracy of the unimodal trip-based approach improves both in L 2 and L ∞ norms as the description of trip lengths is refined. On the other hand, accumulation-based model with unimodal p-MFD fit yields results that are closer to micro-simulation ones than the bimodal case. Using the unimodal p-MFD, production is estimated incorrectly in accumulated-based model, however outflow is well predicted, see Figs. 5d and5c.

Since, the key element of the accumulation-based model is outflow, accumulation evolution is well captured as shown in Fig. 5a. The comparison of L 2 norms of accumulation-based and trip-based models with unimodal fit for corresponding trip length estimation infers that the models are very close in terms of accuracy. Besides, the evolution of mean speed and outflow presented in Figs. 5b and5c, respectively, are very similar for both MFD models with unimodal approach. As shown in Fig. 5d the hysteresis phenomenon cannot be reproduced using unimodal p-MFD fits for both MFD models. In the case of accumulation-based approach with unimodal fit, errors increase as the trip length description is refined. However, errors in outflow are very similar for all trip length estimation methods, which suggests that there is significant error in production compared to trip it can be observed that trip-based approach with bimodal fit is more closer to micro-simulation than its counterpart. The most accurate solution is obtained using the clustered trip lengths estimation method using trip-based approach with bimodal p-MFD fit. Using a single trip with mean trip length gives the least accurate estimation of accumulation in the case of trip-based approach with bimodal fit. Figure 2c presented earlier shows that the trip lengths vary from 1 m to 4 500 m. By taking a single trip with mean trip length, vehicles may travel longer distances, which results in higher accumulation. This can be improved by taking a weighted mean of all trip lengths based on demand per each trip. However, this data will not be readily available in the practical applications.

||n -n m || L 2 ||n m || L 2 ||V -V m || L 2 ||V m || L 2 ||O -O m || L 2 ||O m || L 2 ||n -n m || L ∞ ||V -V m || L ∞ ||O -O m || L ∞ Acc-
This conclusion complies with the formulation of trip-based approach when the trip lengths are distributed over wide range. Just like in the case of unimodal p-MFD, the accuracy of trip-based solutions with bimodal fit improves as trip length description is refined except for the case of individual trip lengths. The L 2 and L ∞ error norms of trip-based model with bimodal fit with individual trip length case are larger than clustered trips. This might be due to errors from meanspeed approximation is the dominant compared to trip-length distribution in the case of individual trip lengths. A similar trend is observed in both outflow and mean speed for the case of tripbased approach with bimodal fit. On the other hand, error in accumulation in the case of bimodal accumulation-based approach increases as the trip-lengths are refined. It can be concluded that in the case of bimodal accumulation-based approach, considering a single trip with mean trip length produces satisfactory results. However, it is worth noting that the errors of outflow decreases with increasing the refinement in trip lengths. It suggests that production is well estimated and dominant errors in outflow are due to the approximation of trip lengths. It infers that refining the trip lengths results in better estimation of outflow. The reason for the opposite trend in accumulation might be due to the presence of numerical dissipation in inflow computations. The L 2 norm of error for inflow cumulative curve between accumulation-based model and micro-simulation for mean trip case is 6.1 × 10 -4 , which is two orders lower than errors obtained in accumulation. However, as the number of trip lengths increase with fewer trips on each trip length in the reservoir, the error in inflow cumulative curve can influence the error in accumulation evolution. Finally, Fig. 5d shows that the clock-wise hysteresis pattern is obtained by both accumulation-based and trip-based models. The size of the hysteresis loop in accumulation-based is smaller than the micro-simulation one because of the under-prediction of peak accumulation.

Overall, comparison of different models infers that trip-based with bimodal fit, similar trips and accumulation-based with unimodal p-MFD fit, mean trip models are very close to the microsimulation results and gives a good estimation of accumulation evolution for the real network of Lyon 6.

Validation when changing the OD matrix

The next part of the study is to compare the MFD models to the micro-simulation when OD matrix is modified, which changes the internal trip patterns. The estimated trip lengths from microsimulation is now shown in Fig. 2d. As stated earlier, the micro-simulations are carried out using static assignment by predefining the routes and their corresponding assignment coefficients. The OD matrix of Lyon 6 is changed artificially by increasing the flow between OD pairs which have longer trip lengths (2 000 m -3 000 m) and decreasing the same amount of flow between OD pairs TRB 2019 Annual Meeting Original paper submittal which have smaller trip lengths (1 000 m -2 000 m). The idea is not to obtain a realistic scenario, but to create enough modifications in the trip patterns to have a significant differences from the reference scenarios. The accumulation-based MFD is applied with a unimodal p-MFD fit, while the trip-based is applied with a bimodal one.

Without re-calibration of p-MFD fit and trip lengths

In this part, the results are presented using the same p-MFD fits and trip lengths proposed for the original OD matrix. Such a situation arises when the modeler does not consider the OD matrix changes and uses the previous calibration settings. The normalized demand pattern is same as the network saturation case as shown in Fig. 2a, however the actual demand is slightly different from the case of original OD matrix as shown in Fig. 2b. The results of accumulation-based with unimodal p-MFD fit, mean trip and trip-based with bimodal fit, OD trips are presented. Figure 6 presents the results of MFD models and micro-simulation. It can be observed that evolution of accumulation and mean speed are inaccurate for MFD models, especially during the network recovery phase. The hysteresis loop obtained in the p-MFD from micro-simulation is comparatively bigger than MFD models as shown in Fig. 6d. Figure 6c shows that both accumulation-based and trip-based models estimate the outflow evolution with a good accuracy. Since, the inflow for both models are equal, albeit the numerical errors, the difference in the accumulation is due to the inconsistencies in p-MFD fits and trip lengths.

With re-calibration of only trip lengths

This part shows the results with re-calibration of trip lengths according to the modified OD matrix, however using the same p-MFD fit for the original scenario. Figure 7 presents the results for different state variables for both MFD and micro-simulation models. The first noticeable different between previous results in Fig. 6 and the present one is that the peak accumulation in the accumulation-based model is over-predicted by about 50%. The reason is that the mean trip length in the modified OD matrix case is 1 652 m compared to 1 505 m in the original OD matrix scenario. Hence, by using the same p-MFD fit as the original OD matrix case results in the smaller outflow and higher accumulation. The reduction of outflow can be observed in Fig. 7c for the accumulation-based model. On the other hand, the accuracy of the trip-based model is improved close to the peak accumulation using the re-calibrated trip lengths. However, the production hysteresis in the trip-based model is still not close to micro-simulation.

With re-calibrated p-MFD fit and trip lengths Hence, it can be concluded from this discussion that it is crucial to re-calibrate the p-MFD fits and trip length distributions when the OD matrix is changed. The results infer that changes in OD matrix do not effect the network loading significantly, as noticed in Fig. 7d, where original p-MFD fit with calibrated trip lengths captured micro-simulation trend quite reasonably.

However, re-calibration is necessary during the network recovery, as changes in OD matrix can have significant impact on the network unloading.

CONCLUSIONS

This work presents the calibration of p-MFD shape and trip length estimation of MFD-based approaches using validation of micro-simulation results on real network of Lyon 6. A reference free flow scenario and a network saturation scenario are presented to validate the MFD-based approaches. In addition, an additional case by changing the OD matrix is considered to study the impact of changes of OD matrix on accuracy of MFD simulations.

In the first case of free flow scenario, both accumulation-based and trip-based approaches gave satisfactory results using a unimodal p-MFD. Since, the network is largely in free flow regime and network unloading is slow, production hysteresis is negligible in this scenario. This test case is only used to benchmark the MFD-based approaches using micro-simulation data. In the following case of network saturation, clockwise hysteresis in the p-MFD is noticed from micro-simulations.

The importance of considering a bimodal fit for p-MFD to capture the hysteresis pattern is demon-

  et al., Daganzo and Lehe, Lamotte and Geroliminis[START_REF] Leclercq | Dynamic macroscopic simulation of on-street parking search: A trip-based approach[END_REF][START_REF] Daganzo | Distance-dependent congestion pricing for downtown zones[END_REF][START_REF] Lamotte | The Morning Commute in Urban Areas: Insights from Theory and Simulation[END_REF].Mariotte et al. (25) refined this idea to propose the so-called event-based model for a single reservoir system in the framework of trip-based MFD models. The main assumption of this approach is that all the vehicles travel at the same mean speed given by the MFD at a given time and exit the reservoir after they finish their individually assigned trip lengths. This model is computationally more demanding compared to its counterpart. However, trip-based model addresses few limitations of accumulation-based model which can be found in Mariotte et al. (25). More recently, Mariotte and Leclercq (26) extended the trip-based framework to multiple reservoirs systems that can have multiple trip lengths in each reservoir. Their work proposed a novel way to model the congestion spill-backs in the trip-based formulation. However, Leclercq and Paipuri (18) showed that no model is perfect and the tripbased exhibits inconsistent outflow patterns close to saturation. This can be avoided by monitoring the outflow, however the travel times in the reservoir are no longer consistent with trip lengths and mean speed. There have been complex formulations proposed in the MFD-based simulation approaches in the literature. The inclusion the production hysteresis and definition the trip lengths inside the reservoir of MFD models are still ongoing research questions. Most of the MFD-based simulation approaches are applied to idealized networks and there are only very few detailed validations on real networks. Hence, the contribution of this study is two-fold namely, a detailed investigation into MFD calibration and trip length estimation and a thorough validation of the MFD-based simulations on real network of 6 th district of Lyon city (Lyon 6), France. A conventional unimodal MFD and a bimodal MFD with hysteresis patterns are computed from micro-simulation data. This work is the first to consider the clockwise hysteresis-like loop in the p-MFD for both accumulationbased and trip-based models. Another contribution of this work is to establish the importance of the level of description of trip length distributions in reservoir simulation. The individual trip lengths are known a priori from the micro-simulation data and therefore, an accurate model can be built by considering each individual trip length in the MFD simulators. Apart from the individual trip lengths, other definitions like single mean trip, trip based on OD, etc. are considered in the present work. The accuracy of different approaches of trip lengths are presented. The given OD matrix is modified artificially to study the sensitivity of MFD-based simulations on the changes in OD pattern.

  (a) Map of the Lyon 6 ©Google Maps 2018. (b) Link level representation of the Lyon 6.(c) Link level representation of Lyon 6, Lyon 3 and Villeurbanne networks. Lyon 6 is highlighed in blue.

FIGURE 1 :

 1 FIGURE 1 : Lyon 6 network: map of the area, its link level description and whole network considered in micro-simulation.

FIGURE 2 :

 2 FIGURE 2 : Lyon 6 network: map of the area, its link level description, demand for different cases and trip lengths distribution.

FIGURE 3 :

 3 FIGURE 3 : Lyon 6 network: influence of aggregation period, calibration of p-MFD with static and dynamic loadings and their corresponding unimodal and bimodal MFD fits.

  with dynamic demand corresponding to network saturation shown in Fig.2bis carried out and the corresponding p-MFD data is plotted in Fig.3cin blue circles. It can be observed that the values of production in network loading and recovery are different owing the phenomenon of hysteresis. TRB 2019 Annual Meeting Original paper submittal

  OD trips: Depending on the origin and destination of each trip with respect to Lyon 6 area, four different types of trips can be identified: Trips starting outside and ending outside, Trips starting inside and ending inside, Trips starting outside and ending inside and Trips starting inside and ending outside. The mean trip length per trip type is computed and given as length to the respective trip. Similar trips: Several trips are clustered into bins based on the range of trip lengths. The mean trip length inside each bin is given as trip length to the corresponding trip. TRB 2019 Annual Meeting Original paper submittal Individual trips: Each individual trip is considered and the corresponding trip length is assigned to each trip in the MFD simulations. Hence, from the aforementioned definition of trip length estimations methods, it can be observed that the level of description of trip lengths increases from single mean trip to individual trips. In other words, trip lengths are exact in case of individual trips and least accurate in case of single mean trip. In the numerical results, different methods of trip length estimation are compared to demonstrate the influence of trip lengths description on accuracy of traffic state dynamics in MFD approaches.

1

 1 veh per trip during whole simulation time. Considering each individual trip in accumulation-based model can add significant numerical diffusion into the scheme. Hence, the case of individual trips is not considered for accumulation-based model. TRB 2019 Annual Meeting Original paper submittal Trip-based model The trip-based approach (see e.g. Arnott, 22) is based on the principle that all vehicles travel at the same speed at any given time. The vehicles leave the reservoir once they finish their assigned trip length. If a vehicle entered at time t traveled a distance L in time T (t), the trip-based model can be mathematically expressed as,

  n) = P(n)/n. In the present work, eventbased resolution proposed in Mariotte et al.,Lamotte and Geroliminis (25,[START_REF] Lamotte | The morning commute in urban areas with heterogeneous trip lengths[END_REF] is used in the trip-based formulation. In the event-based formulation, the entry and the exit of each vehicle is considered as an event and network variables like accumulation, mean speed etc., are updated for each event. As mentioned earlier, entry times of each vehicle is known a priori from microsimulation and hence, it is an input to event-based formulation. Each vehicle travels with mean speed that evolves based on traffic dynamics. Once the vehicle finishes its assigned trip length, the considered trip is completed and vehicle is removed from the reservoir. As proposed in Leclercq and Paipuri, Mariotte and Leclercq[START_REF] Leclercq | Macroscopic traffic dynamics in reservoirs under fast-varying demand profiles[END_REF] 26), the outflow (or trip completion rate) is bounded by the maximum capacity and to sustain the outflow to maximum capacity when network reaches saturation to avoid causality effect (see e.g. Merchant and Nemhauser,Friesz et al.,[START_REF] Merchant | A Model and an Algorithm for the Dynamic Traffic Assignment Problems[END_REF][START_REF] Friesz | Dynamic Network Traffic Assignment Considered as a Continuous Time Optimal Control Problem[END_REF]. The maximum capacity of the reservoir is computed from the micro-simulation results. Even though different trip lengths are considered, a single queue of vehicles is monitored during the simulation and maximum outflow limitation is applied to the single queue. This avoids the need of defining the maximum outflow for each trip length defined. Owing to the formulation of event-based scheme, it is possible to take all the different trip lengths into account while computing the traffic dynamics.

Figure

  Figure4ashows the evolution of accumulation with time for both MFD simulators with unimodal p-MFD fit along with the comparison to micro-simulation data. Bimodal p-MFD fit is not considered in this case as the considered demand peak is not high enough to produce hysteresis

FIGURE 4 :

 4 FIGURE 4 : Results of MFD-based approaches and micro-simulations corresponding to free flow demand scenario. OD trips estimation method is used in MFD-based models.

  lengths. The L 2 norms of mean speed and outflow for both accumulation-based and trip-based TRB t (hr) Accumulation, n(t) (veh) Acc-based(unimodal) Trip-based(unimodal) Acc-based(bimodal) Trip-based(bimodal) Microsimulation (a) Evolution of accumulation with time. t (hr) Mean speed, V(t) (m/s) Acc-based(unimodal) Trip-based(unimodal) Acc-based(bimodal) Trip-based(bimodal) Microsimulation (b) Evolution of mean speed with time.

FIGURE 5 :

 5 FIGURE 5 : Results of MFD-based approaches and micro-simulations corresponding to saturation flow demand scenario. OD trips estimation method is used in MFD-based models.

  with unimodal p-MFD fits are comparable for all trip length estimation methods without a specific trend. As stated already, this can be due to larger error in p-MFD estimation. Now comparing the bimodal accumulation-based and trip-based approaches, from Fig.5a,

Figure 8

 8 Figure 8 shows the evolution of accumulation, mean speed, outflow with time along with p-MFD obtained from MFD models and micro-simulation using re-calibrated trip lengths and p-MFD fits. It can be noticed that the hysteresis loop in the p-MFD is improved for the trip-based model compared to Figs. 6d and 7d. The p-MFD during the recovery phase of network in trip-based model is following the micro-simulation results. The consequence of this can be noticed in the evolution of accumulation, where accumulation values are higher during the recovery phase in Fig 8a compared to Fig. 7a. This highlights the importance of re-calibrating the p-MFD fit to accurately predict the transient state, especially during the network recovery. It can also be noticed that the evolution of accumulation during loading phase with both re-calibrated p-MFD fit and original p-MFD fit is very close. Even though, the peak accumulation in both accumulation-based model with unimodal fit and bimodal trip-based models are very close, it can be observed that the accumulation-based

FIGURE 6 :

 6 FIGURE 6 : Results of MFD-based approaches and micro-simulations corresponding to saturation flow demand scenario with modified OD matrix and without re-calibration of p-MFD fit and trip lengths. OD trips estimation method is used in MFD-based models.

FIGURE 7 :

 7 FIGURE 7 : Results of MFD-based approaches and micro-simulations corresponding to saturation flow demand scenario with modified OD matrix and with re-calibration of only trip lengths. OD trips estimation method is used in MFD-based models.

FIGURE 8 :

 8 FIGURE 8 : Results of MFD-based approaches and micro-simulations corresponding to saturation flow demand scenario with modified OD matrix and with re-calibration of p-MFD fit and trip lengths. OD trips estimation method is used in MFD-based models.

  strated for the trip-based model. In the case of accumulation-based model, a good estimation of outflow and accumulation evolutions is obtained with unimodal p-MFD fit. The comparison of MFD-based approaches to micro-simulation results suggests that the MFD simulations can estimate the evolution of accumulation, mean speed and outflow to a good accuracy. It is concluded that the trip-based approach with bimodal p-MFD gives good estimates of state variables, however a more refined description of trip lengths results in more accurate results. Finally, the influence of changing OD matrix on the MFD simulations is studied. It is concluded that the re-calibration of p-MFD fit and trip lengths are required to accurately predict the dynamics of the network.TRB 2019 Annual MeetingOriginal paper submittal

TABLE 1 :

 1

	L 2 and L ∞ norms of errors for state variables with different trip length estimation methods for accumulation-based and	trip-based approaches for original OD matrix.	MFD model (p-MFD fit) Trip length type

ACKNOWLEDGMENT This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No 646592 -MAGnUM project). The authors thank the reviewers for their insightful comments, which enhanced the quality of the paper. CONTRIBUTION STATEMENT The authors confirm contribution to the paper as follows: study conception and design: M. Paipuri, L. Leclercq; micro-simulation design and settings: M. Paipuri, J. Krug; analysis and interpretation of results: M. Paipuri, L. Leclercq; draft manuscript preparation: M. Paipuri. All authors reviewed the results and approved the final version of the manuscript.

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