Nicolas Chiabaut
email: nicolas.chiabaut@entpe.fr

Evaluation of a multimodal urban arterial: 1 the passenger Macroscopic Fundamental Diagram 2 3 4

Evaluation of a multimodal urban arterial: the passenger macroscopic fundamental diagram

Nicolas

The mode choice of travelers should be considered as well. It is intuitive that the effect 83 of the ratio of people using public transport rather than individual cars will impact the 84 performance of the transportation network. The p--MFD makes it possible to address this 85 issue and to understand how traffic conditions are modified by the mode choice of 86 passengers. Different equilibriums can be investigated, notably user and system 87 optimums. The ultimate goal of research towards this direction is to develop a strategy 88 that makes people switch from a mode to the other. 89 90

The paper is organized as follow: Section 2 introduces the notion of passenger 91 macroscopic fundamental diagram (p--MFD). Section 3 deeply investigates the impacts of 92 modal choice and makes it possible to analytically compare user and system optimums. 93 Section 4 focuses on the application of the p--MFD to network transportation services 94 optimization. Finally, Section 5 proposes a discussion. 95

As mentioned earlier, recent works suggest that there is a consistent relationship 110 between the average network vehicle density and average network flow. Such a 111 relationship is called a MFD. Consequently, we consider in the remainder of the paper an 112 idealized city. Roads shape a very meshed urban network with signalized intersections 113 (see Figure 1a). We also assume that the flows are uniformly distributed among origins 114 and destinations [START_REF] Leclercq | Macroscopic fundamental diagram: a cross--597 comparison of estimation method[END_REF]. Under this assumption, car traffic dynamics is 115 well reproduced by a MFD q(k) giving the space--mean flow of cars qc=q(k) [veh/h] on 116 each link as a function of the space--mean density of the links within the city k [veh/km] 117 (see Figure 1b). Notice that the MFD can now be easily estimated accounting for the 118 effect of buses and control strategies [START_REF] Geroliminis | The effect of variability of urban systems characteristics 569 in the network capacity[END_REF]Leclercq and 119 Geroliminis, 2013;[START_REF] Chiabaut | Performance analysis for different designs of a 559 multimodal urban arterial[END_REF]. It is also worth noticing that the lower case 120 letters refer to variables expressed in terms of vehicles whereas the upper case letters 121 correspond to variables expressed in terms of passengers. We also assume that the 122 maximal occupancy of a car is ρc [pax/veh]. Notice that, for a realistic purpose, we 123 consider that the maximal occupancy is equal to the observed average occupancy (1.2 124 pax/veh) rather than the effective maximal occupancy (5 pax/veh). We first assume a 125 trapezoidal car MFD for the sake of simplicity. This form is convenient to coarsely mimic 126 the influence of traffic signals. The parameters are the free--flow speed u [km/h], the 127 critical speed uc, the maximal flow capacity qx and the jam density kx. The congested wave speed is denoted w=u.qx/(u.kx--qx). We assume that all the links are composed of n 129 lanes (n=2). 130 131 132 We first consider that the public transport system is only composed of buses that share 135 the same roads as the car traffic. We also assume that trips of users can always be 136 realized either by individual car or public transport system. The transit system is 137 characterized by the bus time--headway h [h] and the maximal speed of the buses ut. We 138 also assume that the maximal occupancy of a bus is ρt [pax/bus] and that the buses are 139 mixed with car traffic and no lanes are dedicated to them. Moreover, we consider that 140 the average occupancies of both modes and the number of buses in operation do not 141 depend on the traffic conditions and the mode choice. It turns out that, for a given time--142 headway h the number of buses nbus in operation is nbus = L/(h.vt) (Hans et al., 2014) 143 where L [km] is the length of the bus lines of the transportation network and vt the 144 average speed of the transit system. Physically, it corresponds to static timetables. 145 To introduce the p--MFD, the flow is now expressed in terms of passengers per time 154

u c# q#[veh/h]# k#[veh/km]# Q(k)# (b), (a), u # q x# k x# k 1# k 2#
[pax/h]. Let P denote this flow. P is equal to the sum of passengers using cars Fc and 155 passengers using transit system Ft. It is worth noticing that Fc directly derives from the 156 car MFD q(k) while Ft must be obtained from the characteristics of the transit system. 157 Moreover, as in the classical definition of the MFD, it is thus really appealing to link the 158 flow to the average density of passengers in the city K. However, this is not trivial 159 because K has to be expressed in terms of passengers per space [pax/km]. Moreover, it 160 turns out that the mode choice of passengers between cars and buses has an impact on 161 the flow and density. This ratio is denoted τ and is equal to Kc/K where Kc is the density 162 of passengers using cars. 163

In this section, τ is considered as static, i.e. constant in time and independent of traffic 165 conditions. Consequently, τ is exogenously given. Physically, it corresponds to an 166 equilibrium situation where day after day the same demand and traffic conditions occur. 167

Notice that these assumptions are relaxed and studied in the forthcoming section. 168 Consequently, the total average flow and the total average density are: 169

𝑃 𝐾 = 𝐹 ! 𝐾 ! + 𝐹 ! 𝐾 ! 𝐾 = 𝐾 ! + 𝐾 ! (1) 170
Where Kc and Kt are respectively the average density in passengers of cars and buses. 171

It is thus appealing to link P with K and to correctly determine the function P(K) and 172 especially its shape. Consider a given density of passengers K. According to the 173 definition of the mode ratio τ, the density of passenger in cars is Kc= τ. K [pax/km]. It is 174 worth noticing that this density is also equal to the density of cars weighted by the 175 average occupancy, i.e. kc= Kc/ρc where kc is the density expressed in [veh/km]. It makes 176 it possible to calculate the average flow in terms of passengers: Fc=ρc.q(kc)= ρc.q(Kc/ρc) 177 where q is the car MFD. It comes that the associated average speed of passengers using 178 cars is vc=qc/kc. 179 180 Concerning the transit system, the density of bus passengers is given by Kt=(1--τ).K and 181 cannot exceed ρt.nbus/L the maximal density of the transit system. Consequently, the 182 associated flow is thus equal to 𝐹 ! = min 1 -𝜏 𝐾, 𝜌 ! 𝑛 !"# /𝐿 ℎ . Because we assume 183 that the bus fleet size nbus is constant, i.e. it does depend on traffic conditions, h directly 184 depends on the average speed of the transit system: h= L/(nbus.vt). Consequently, when 185 the car traffic does not constrain the buses, bus speeds are always equal to the free flow 186 speed ut and the headway h is not degraded. However, in the case of mixed traffic, 187 congestion may impact the bus system when the speed of the car vc is lower than the 188 maximal speed of the bus ut. It comes that ℎ = 𝐿 𝑛 !"# . min (𝑢 ! , 𝑣 ! ) where vc is the speed 189 of the cars: vc=qc/kc. 190 191 Finally, equation (1) can be expressed as: 192

𝑃 𝐾 = 𝜌 ! 𝑞 !" ! ! + ! ! . min 1 -𝜏 𝐾. 𝐿, 𝜌 ! 𝑛 !"# . min 𝑢 ! , 𝑣 ! (2) 193
Figure 2 presents the results. The thick gray line corresponds to the car MFD whereas 195 the thin black lines are the different p--MFDs. It is thus appealing to observe the impacts 196 of the bus systems characteristics on the p--MFD shape. To this end, Figure 2a pinpoints 197 the sensitivity of the p--MFD to the bus time--headways h. For this application, all the 198 variables are assumed constant except K (and P) and h that ranges between 3 min and 199 12 min. It turns out that a maximal capacity can be reached for a specific value. Notice 200 that the effect of the bus system on the car traffic dynamics is not accounted here, i.e. the 201 formulation of the car MFD does not depend on the time--headway h. For a given car 202 MFD, it is thus possible to determine the optimal h to maximize the transportation 203 system performance. In the same vein, Figure 2b highlights the impacts of τ on the p--204 MFD shape. Consequently, the only variables that change are K (and P) and τ that ranges 205 between 0.1 and 1. It is not surprising that the performance can also be optimized for a 206 specific value of τ. It is thus appealing to study in details the impacts of τ and to focus on 207 dynamic τ. This will be investigated in the next section. 208 209 210 Now that we have introduced the concept of p--MFD, this section focuses on the effects of 216 the mode choice and studies the associated equilibrium of the transportation network. 217

Notably, system and user optimums are investigated. To this end, we now consider that 218 the mode choice is based on the utility of the mode. The utility is expressed as the travel 219 cost of using that mode at the beginning of the trip. The travel cost of the whole route 220 We first focus on the system optimum. It corresponds to the second principle of 226 Wardrop. In such an equilibrium situation, the average journey time is minimum, i.e. the 227 average speed of passengers is maximal. It is worth noticing that the speed is equal to 228 the ratio of the demand with the density. Consequently, for a given density K, the 229 associated flow P must satisfy the following equation to maximize the average speed: 230

𝑃(𝐾) = max ! 𝜌 ! 𝑞 !" ! ! + ! ! . min 1 -𝜏 𝐾. 𝐿, 𝜌 ! 𝑛 !"# . min 𝑢 ! , 𝑣 ! (3) 231
Based on this equation, we are now able to determine the function P for all the possible 232 traffic conditions, i.e. all the possible values of K. 233 234

Free-flow conditions 235 236

The free--flow conditions correspond to the situations where the total passenger demand 237 is satisfied by the system. The p--MFD is directly obtained by solving equation (3). Figure 238 3a presents the resulting p--MFD in case of a trapezoidal car MFD. It turns out that the 239 passenger mode allocation τ is not constant. This is confirmed by Figure 3b that shows 240 the evolution of τ with respect to the passenger density level. Car is the unique mode 241 until the density reaches the critical density k1 (see Figure 1b), i.e. the demand will 242 exceed the maximal car capacity ρc.qx. Then passengers have to switch from cars to the 243 transit system. Note that this corresponds to the optimal situation where passengers are 244 ready to change mode rather than to degrade the traffic conditions. We do not focus in 245 the paper on the possible policies to make users change mode. However, incentive or 246 congestion pricing, traffic information, prescriptive management could be efficient and 247 innovative solutions. 248 This method can be applied for any shape of MFD. However, calculations are more 255 complicated. We now assume that the car MFD can take any concave shape. We consider 256 here, for the example, a function composed of a parabolic and linear part (see Figure 3c): 257

𝑄 𝑘 = 𝑎𝑘 ! + 𝑏𝑘 for 𝑘 < 𝑘 ! (4a) 258 𝑄 𝑘 = 𝑤(𝑘 ! -𝑘 ! ) otherwise. (4b) 259
Where, a=--u 2 /(2.qx), b=u and kl=--(w+b)/(2a). Such a formulation ensures to maintain the 260 same free--flow speed than the triangular car MFD. 261

To determine the associated p--MFD in case of SO, equation (3) has to be re--written as: 262

𝐾(𝑃) = min ! 𝐾 ! + 𝐾 ! (5) 263
It is also worth noticing that Kc and Kt (respectively) can be expressed as a function of Fc 264 and Ft (respectively). Let us consider 𝐹 ! * and 𝐹 ! * to be the optimal solution of (5) for a 265 given total density K * . A small increment 𝛥𝐹 ! and 𝛥𝐹 ! of the flows Fc and Ft will change 266 the density value. This change can be approximated by a first order Taylor expansion: 267

ρ t .k t ρ t .k t +ρ t .n bus /L τ* Density K [pax/km] τ (d) 𝐾 𝐹 ! * + Δ𝐹 ! = 𝐾 ! 𝐹 ! * + 𝐾 ! ! 𝐹 ! * . Δ𝐹 ! (6a) 268 𝐾 ! 𝐹 ! * + Δ𝐹 ! = 𝐾 ! 𝐹 ! * + 𝐾 ! ! 𝐹 ! * . Δ𝐹 ! (6b) 269
Thus, the total variation of density is: 270

Δ𝐾 = Δ𝑃 𝐾 ! ! 𝐹 ! * . !! ! !! + 𝐾 ! ! 𝐹 ! * . !! ! !! (7) 271
This equation can be simplified because if we consider that buses are not affected by 272 traffic congestion, thus:

𝐾 ! ! 𝐹 ! * = 1 𝑢 ! because 𝐾 ! = 𝑛 !"# . 𝜌 ! 𝐿 , 𝐹 ! = 𝜌 ! ℎ and 𝑛 !"# = 273 𝐿 ℎ. 𝑢 !
. This is true because we only consider free--flow situations in this section of the 274 paper. It comes: 275

Δ𝐾 = Δ𝑃 !! ! !! ! 𝐹 ! * . !! ! !! + ! ! ! . !! ! !! (8) 276
Δ𝐾 is given by the combination of

!! ! !! and !! !
!! that minimizes the RHS of equation ( 8). 277

This quantity admits a lower bound that is equal to

!! ! !! ! 𝐹 ! * when !! ! !! ! 𝐹 ! * < ! ! ! or to ! ! ! 278
otherwise. Thus, it appears that when the total flow varies from Δ𝑃 the optimal solution 279 of ( 8) is only modified for the car flow if

!! ! !! ! 𝐹 ! * < ! ! !
and for the transit flow otherwise. 280 Consequently, passengers have to shift of mode when

!! ! !! ! 𝐹 ! * = ! ! ! to reach SO. 281 282
Based on these results, the p--MFDs for SO in free--flow conditions are highlighted in 283 Figure 3c. The evolution of τ with respect to the demand level is slightly different from 284 the trapezoidal MFD case, see Figure 3d. To obtain the SO solution, car is the unique 285 mode until a certain car density value that corresponds to kt such as 𝜕𝑞(𝑘 ! ) 𝜕𝑘 = 𝑢 ! ; 286 then passengers switch to the transit system until all the buses are full; finally, the 287 remaining car capacity is used until the system's capacity is reached. 288 289

Congested conditions 290 291

We now aim to determine the p--MFD when traffic is congested. For cars, the MFD 292 directly accounts for this capacity reduction. For buses, they are not impacted for small 293 congestion, i.e. when Fc(Kc)/Kc > ut. However, when Fc(Kc)/Kc < ut, the buses are slowed 294 down by the queues. Characteristics of the transit system have to be dynamically 295 modified according to ℎ = 𝐿 𝑛 !"# . min (𝑢 ! , 𝑣 ! ) . It makes it possible to account for 296 congestion in the expression of Ft. Because car--MFD expressed in terms of passengers, i.e. Fc, also reproduces congested states, the congested part of the p--MFD can be 298 determined according to equation (3). It turns out that equation ( 3) is only a 299 maximization process that depends on the formulation of Fc and Ft. 300 301 302 Figure 4 presents the associated p--MFD. Figure 4a highlights the influence of the bus 306 time--headways h for a car trapezoidal MFD, i.e. all variables are constant except h that 307 ranges between 3 min and 12 min. It is worth noticing that the transformation is always 308 homothetic because of the linearity of the car MFD shapes. Indeed, for other concave 309 shapes, Figure 4b shows very different results. It is interesting to notice that a specific 310 combination of bus time--headways and car MFDs leads to optimal situations. It is thus 311 appealing to use the p--MFD to determine the optimal bus time--headway for a given 312 situation. It will be studied in Section 4. 313 314

User optimum 315 316

We now aim to assess the effect of the mode assignment equilibrium on the associated 317 p--MFD. Indeed, we have only focused on the system optimum since the start of this 318 paper. This assumption is now relaxed to study other modal choice assignment models. 319

Especially, we focus on the Wardrop's first principle and the Logit model. 320 321 Each user non--cooperatively seeks to minimize his cost of transportation. Consequently, 325 the speeds in all modes actually used equal between them and are higher than those that 326 would be experienced by a single traveller on any unused mode. This principle is 327 referred to as user equilibrium (UE) in the remainder of the paper. Consequently, when 328 the transportation network satisfies the UE, passengers either use only the car or they 329 use both modes. 330 331

Wardrop's first principle

Figure 5a shows the associated p--MFD in the case of car trapezoidal MFD. It turns out 332 that is still an all--or--nothing situation. Car is the preferred mode until the arterial 333 becomes congested and the speed of the cars becomes the same as the speed of the 334 buses. Then, the passenger demand is split in both modes according to a ratio τ. It is also 335 worth noticing that the difference in terms of flow between the UE and SO is very high 336 (see SO p--MFD in blue). Moreover, these differences occur in free--flow and lead to a 337 capacity reduction in case of UE. It means that the performance of the transportation 338 network can be deeply optimized by managing the demand rather the self--organization 339 situation. Traffic management strategies must focus on free--flow situations that are 340 really close to the system capacity. 341 342 Figure 5b highlights the p--MFD for a parabolic--linear car MFD. For this specific shape of 343 car MFD, differences between black and blue lines are smaller but the network 344 performance can still be increased by changing the equilibrium from user to system 345 optimum. 346 347

Density K [pax/km] Flow P [pax/h] (a) UE SO q(k) Density K [pax/km] Flow P [pax/h] (b) UE SO q(k)

The Logit model 351 352

In the vein of [START_REF] Leclercq | Estimating MFDs in Simple Networks with Route 595 Choice[END_REF], the Logit model can also be easily 353 adapted to a multimodal traffic. The associated situation is referred to as stochastic user 354 equilibrium (SUE) in the remainder of the paper. We now assume that the ratio of flows 355 between cars and buses only depends on the difference in travel times, i.e. speed, 356 between both modes: 357

! ! ! ! = 𝑒 !! ! ! ! ! ! ! ! (9) 358 359
Where θ is the parameter of the Logit model and L the average trip length. 360 361

We have now all the equations to formulate a parametric expression of the flow P for the 362 arterial with respect to vc. This defines a simple method to calculate the p--MFD for the 363 Logit model. Figure 6 presents the resulting p--MFD. Notice that all the variables are 364 constant except θ. Therefore, we have tested several θ values (from 0.05 to 0.9 with an 365 increment of 0.05). It turns out that p--MFDs fall between UE and SO. Moreover, UE, SUE 366 and SO only differ at the network level in free--flow situations when average speed is 367 comprised between u and ut for trapezoidal MFD and us and ut for curved MFD (us such 368 as

!! ! !! ! = 𝑢 ! )
. This is not surprising because in congestion both modes have the same 369 speed. The mode ratio is then constant. 370 371 372

Flow P [pax/h] (b) SO UE SUE u t# u t# u s#

Equilibriums comparison 376 377

Even if rough comparisons between UE and SO are proposed by Figure 5a, it is still 378 appealing to study these difference in more detail. To this end, we now focus on the 379 marginal costs of UE and SO. This will give useful insights on the pricing policy that can 380 be implemented to adapt the costs of each mode and switch the transportation network 381 from UE to SO. 382 383

The marginal cost of a mode defines the increase of the cost of this mode if one more 384 passenger chooses to use this mode. The marginal cost of mode i is obtained by 385 differentiating the cost function ci(Ki): 386

𝑀𝐶 ! 𝐾 ! = 𝑐 ! 𝐾 ! + 𝐾 ! 𝑐 ! ! (𝐾 ! ) (10) 387
Here the cost of a mode is characterized by the average speed, i.e. ci= Fi(Ki)/Ki. Thus, the 388 derivative c ' i can be formulated as: 389

𝑐 ! ! (𝐾 ! ) = ! ! ! ! ! ! !!(! ! ) ! ! ! (11) 390
Finally, the marginal costs of each mode are equal to: 391

𝑀𝐶 ! 𝐾 ! = 𝐹 ! ! 𝐾 ! = !! ! (! ! ) !! ! 𝑀𝐶 ! 𝐾 ! = 𝐹 ! ! 𝐾 ! = !! ! (! ! ) !! ! (12) 392
With this pricing, SO is reached by letting the system self--organized, i.e. Wardrop's user 393 equilibrium. The conditions of SO highlighted in the previous section can be identified. It 394 turns out that the demand is entirely assigned to the individual vehicle mode until 395

!! ! (! ! ) !! ! = !! ! (! ! ) !! !
. It is also worth noticing that dynamic pricing is required to reach system 396 optimum because

!! ! (! ! ) !! ! changes with Kc.
Especially, the pricing is lower when traffic 397 becomes saturated due to the concave shape of 𝐹 ! (𝐾 ! ). 398 399

APPLICATION TO TRANSPORTATION NETWORK SERVICES OPTIMIZATION

401

The section aims to use the p--MFD as a powerful tool to compare and design optimal 402 control strategies. Firstly, we assume that τ can depend on the passenger demand, i.e. τ 403 is dynamic. According to this assumption, it makes it possible to compare the upper 404 envelope of the associated p--MFD. Secondly, p--MFDs are used to investigate how DBLs impact the transportation system performance and may be a powerful tool to increase 406 capacity even in UE case. 407 408

Optimal bus headway

410

When buses and cars share the same network, buses tend to constrain traffic flow 411 because of their lower speed. They may act as moving bottleneck that create local 412 capacity reductions. Intuitively, the occurrence of these reductions increases with the 413 frequency of the bus system [START_REF] Xie | Macroscopic Fundamental Diagram for Urban 607 Streets and Mixed Traffic: Cross--comparison of Estimation Methods[END_REF]. Consequently, an acceptable trade--off 414 between the bus frequency and the impact on the capacity of the car--MFD must be 415 determined to ensure an efficient performance of the transportation network. 416 417

To cope with this issue, the p--MFD is an adapted tool to calculate the optimal bus time--418 headway to reach the system optimum. To this end, the car MFD now accounts for the 419 effects of buses. q(k) is parameterized by the bus system characteristics. Figure 7a 420 highlights the influence of the bus time--headways on car MFD. As previously mentioned, 421 an increase of hbus reduces the capacity for cars. To mimic this influence, the maximal 422 capacity of cars qx now depends on the headway: 423

𝑞 ! ℎ = 𝑛 -𝑒 !! ! ! ! . 𝑞 ! /𝑛 (13) 424
where hm is the minimal acceptable headway (here hm=1min). Notice that we can obtain 425 more realistic MFD estimates by extending the work of Boyaci andGeroliminis (2011) 426 and[START_REF] Xie | Macroscopic Fundamental Diagram for Urban 607 Streets and Mixed Traffic: Cross--comparison of Estimation Methods[END_REF] to the network level. However, this section only aims to introduce 427 the general methodology to determine optimal bus time--headways. 428 429 Car MFDs are now directly related to the value of the bus time--headway h, see dotted 430 lines in Figure 7a. These lines correspond to h ranged between 1 min and 30 min. Notice 431 that the number of bus in operation nbus changes with h but remains independent to the 432 traffic conditions for a given h. Moreover, Figure 7a also shows the associated p--MFDs in 433 case of SO situation. This is clearly the most pertinent case to address for a city or transit 434 manager. For a given passenger demand, managers seek to maximize the average speed 435 on the transportation network, i.e. to minimize the average density. The upper envelope

𝑈 ! 𝐾 = max ! [𝑃 𝐾 ]
(13) 438 439

It ensures that the average speed is always maximal. Figure 7a highlights in red this 440 upper envelope. Consequently, p--MFD provides the optimal bus time--headway with 441 regard to the passenger demand. Figure 7b depicts the evolution of the optimal bus 442 time--headway with the passenger density. It turns out that very high bus frequencies 443 are required to reach high capacities. Unfortunately, such frequencies are very difficult 444 to maintain in practice. It is also worth noticing that the assumptions made to account 445 for the effects of bus in car MFD formulation strongly impacts the results. However, the 446 methodology proposed here can also be applied for a more realistic car MFD coming 447 from simulation as in [START_REF] Chiabaut | Investigating Impacts of Pickup--Delivery Maneuvers on traffic flow 557 dynamics[END_REF]. 448 449 450 The creation of DBL within the network engenders a capacity reduction for the cars. To 460 make this phenomenon explicit, consider here that α is the ratio of lanes fully dedicated 461 to a rapid public transport mode such as buses or trams. As Gonzales and Daganzo 462 As previously mentioned, the works of [START_REF] Xie | Macroscopic Fundamental Diagram for Urban 607 Streets and Mixed Traffic: Cross--comparison of Estimation Methods[END_REF] can be easily adapted to 467 estimate a more accurate MFD. Because these considerations are out of the scope of the 468 paper, the maximal capacity for cars is now equal to α.qx and the critical speed uc 469 remains constant for both cases of trapezoidal and curved car MFD. 470 471

We can theoretically segregate the public transport system into two parts: (i) a rapid 472 transit system that can use the DBL network and (ii) the remainder of the fleet. Notice 473 that the fleet size is equal for both studied cases. To mimic the effects of DBL on the 474 transit system, we consider that the average speed of vehicles (buses or trams) using the 475 DBL is increased. From a macroscopic lens, they have an average speed 𝑢 ! ! > 𝑢 ! . The 476 fleet of the buses that cannot use the DBLs keeps an average speed equal to ut. 477 478

We apply equation (3) to determine the associated p--MFD in the case of UE. Figure 8 479 presents the results. Notice that the fleet size is assumed constant and the only changing 480 variables are K (and P) and the α ratio that ranges between 0.5 and 1. Figure 8a 481 highlights the main difference between DBL and mixed cases (α is equal to 0.8). It is not 482 surprising to observe that rapid system is competitive before the remainder of the 483 transit system because of their higher average speed. Thus individual car is the only 484 used mode until the speed is reduced to the average speed of rapid system 𝑢 ! ! . The 485 remaining of the bus fleet becomes advantageous when the situation is enough 486 congested, see the DBL optimal area in Figure 8a. Notice that the switching traffic 487 conditions are directly given by the speed of the different modes. This process can be 488 extended to any number of modes. It is also worth noticing that, even in a very 489 congested situation, the flow of the p--MFD associated to the DBL case is never null. 490 Indeed, we have assumed that the DBLs are never blocked by traffic congestion. 491 493 Finally, we can identify the optimal domains of application in case of UE. As previously 497 mentioned, optimal domains are determined by identifying solutions that maximize the 498 flow for a given density, i.e. maximize the average speed. The red curves depict the 499 upper bound of p--MFD calculated for α--values comprised between 0.5 and 1. In the 500 simplest case of trapezoidal car--MFD, Figure 8b highlights that the creation of DBLs can 501 increase the capacity in free--flow conditions. It is also worth noticing that p--MFD for 502 DBL case never reached a null--flow in congested situations. Indeed, DBL ensures that the 503 bus system can still operate even in very congested states. Similar observations can be 504 formulated in the case of curved--MFD, see Figure 8c. It turns out that DBL can be an 505 optimal strategy in case of UE. This is very promising because global transportation 506 network performance can be increased by promoting public transport even in the case 507 of UE. This paper developed tools to analytically assess the performance of a multimodal 513 transportation network. To this end, the paper extends the MFD definition to account for 514 the average number of passengers in each mode. The objective is to obtain a unique 515 function to determine the domains of relevance of different transit strategies, where the 516 system cost is minimized. 517 518 First analytical considerations introduce the concept of p--MFD and study its sensitivity 519 to the bus system characteristics in case of a static mode choice. Then, the assumption is 520 relaxed to unveil the impacts of the mode choice on the transportation network 521 performance. Consequently, the user equilibrium case can be compared to the system 522 optimum situation. 523 524 This theoretical canvas can then be used to cross compare different transit strategies 525 and to design the optimal bus system characteristics. Especially, the paper focuses on 526 determining the more efficient bus time headway in case of mixed traffic. Then, the 527 study is devoted to the introduction of DBL. The p--MFD permits to determine the 528 optimal domains of application of DBL. 529 530

We acknowledge that the approach proposed in the paper is highly conceptual and 531 applied to a very idealized network. However, such an approach makes it possible to 532 provide a general modeling framework that can then be adapted to a large range of 533 situations. Nonetheless, this idealized analysis provides insights into how to assess the 534 global performance of a multimodal transportation network and how to compare 535 different traffic management strategies. 536 537 Finally, the results of this paper can be generalized for any design of the network. One of 538 the next extensions is to deal with a spatial distribution of traffic conditions on the 539 network. Indeed, the assumption of uniform distribution of flows can be relaxed 540 allowing for heterogeneous OD demands and mode choice ratio. Moreover, the work can 541 be extended to account for the feedback on the multimodal demand. Indeed, a fixed 542 demand has been considered in the paper but traffic conditions may induce less or more demand that have to be accounted for when calculating the p--MFD. More realistic car 544 MFD formulation can also be considered by resorting to simulation as in (Chiabaut et al., 545 2014) or more sophisticated estimation method (Hans et al., 2014a). Finally, a last step 546 will be to estimate the p--MFD from field data. This task clearly requires very detailed 547 data (passenger counts, vehicle occupancies, OD matrix, etc.). Urban mobility simulation 548 software may provide synthetic but insightful measurements to estimate more realistic 549 p--MFD. 550

Figure 1 :

 1 Figure 1: (a) a meshed urban network and (b) its associated car-MFD 133 134We first consider that the public transport system is only composed of buses that share 135

Figure

  Figure 2: p-MFD for static modal ratio (a) sensitivity to bus time-headway h

  time for cars and buses. This very naïve assumption can be refined 221 without changing the general methodology presented hereafter.

Figure 3 :

 3 Figure 3: (a) free-flow part of the p-MFD for a trapezoidal MFD and (b) evolution of τ with K

Figure 4 :

 4 Figure 4: Complete p-MFD in mixed traffic for (a) a car trapezoidal MFD

Figure 5 :

 5 Figure 5: Comparison of UE and SO in case of (a) a car triangular MFD (b) a parabolic-linear MFD

Figure 6 :

 6 Figure 6: Comparison of SUE, UE and SO in case of (a) a car trapezoidal MFD

Figure 7 :

 7 Figure 7: (a) Impact of the bus system on car MFD (b) upper bound of the transportation system

  , we assume that the car MFD is homogenously reduced of α times its original 463 formulation: 464𝑞 !"# (𝑘) = 𝛼𝑞(𝑘) (14) 465 466

Figure 8 :

 8 Figure 8: (a) Impact of DBL creation in case of UE and optimal domains of application for (b) a car

  Finally, Table1provides a nomenclature for the different variables and main 146 parameters utilized in the paper. It also provides the values that have been used to draw 147

	148	illustrations.
	149
		Variable	Description	Value (if constant)
		q	Car MFD [veh/h]	--
		qc	Flow of cars [veh/h]	--
		kc	Density of cars [veh/km]	--
		ρc	Average occupancy of cars [pax/veh]	1.2

u

Free--flow speed of cars [km/h] 50

Table 1 : List of main variables and parameters

 1

	uc		Critical speed of cars [km/h]	7.7
	vc		Average speed of cars [km/h]	--
	qx		Maximal capacity of cars [veh/h]	4250
	kx		Jam density of cars [veh/km]	450
	k1, k2, kl	Specific densities of cars [veh/km]	79, 214, 100
	w		Congested wave speed [km/h]	18
	n		Number of lanes per link [lanes]	3
	h		Bus time--headway [h/bus]	--
	ρt		Maximal occupancies of bus [pax/bus]	40
	nbus		Number of bus in operation [bus]	--
	L		Length of the transit system [km]	10
	vt		Average speed of the transit system [km/h]	--
	ut		Free--flow speed of the transit system [km/h]	36
	P		Total flow of the system [pax/h]	--
	Fc		Flow of passengers using cars [pax/h]	--
	Ft		Flow of passengers using transit [pax/h]	--
	K		Total density of the system [pax/km]	--
	Kc		Density of passengers using cars [pax/km]	--
	Kt		Density of passengers using transit [pax/km]	--
	τ		Mode choice ratio [%]	--
	α		Ratio of the network dedicated to transit [%]	--
	150
	151
	152	2.2. The p-MFD
	153

3.1. System optimum

The Wardrop's first principle[START_REF] Wardrop | Some theoretical aspects of road traffic research[END_REF] can be adapted to our multimodal case.

Up of the calculated p--MFDs corresponds to the set of the optimal situations:

Haddad, J., Shraiber, A., 2014, Robust perimeter control design for an urban region,

Hans, E.,Chiabaut, N., Leclercq, L., 2014, Investigating the irregularity of bus routes:

results, Journal of Advanced Transportation, http://dx.doi.org/10.1002/atr.1275.

Zheng, N., Geroliminis, N., 2013, On the distribution of urban road space for multimodal

congested networks, Transportation Research Part B 57,[326][327][328][329][330][331][332][333][334][335][336][337][338][339][340][341]


Florian Marczak

Ludovic Leclercq
email: ludovic.leclercq@entpe.fr.

Christine Buisson

A macroscopic traffic model for freeway weaving sections

The objective of this paper is to propose an analytical traffic model adapted to freeway weaving sections. To this aim, a theoretical weaving section is considered at the macroscopic level as the superposition of two merges and two diverges. The model endogenously incorporates the capacity drop related to the weaving lane changes. It only depends on six parameters: three parameters of the fundamental diagram (the free-flow speed, the jam density and the wave speed in congestion), the vehicles' acceleration, the relaxation factor and the length of the anticipation zone. A sensitivity analysis proves that the acceleration rate and the relaxation parameter most influence the capacity drop. The analytical model agrees well with microsimulation results and empirical data collected on a weaving section in France.

INTRODUCTION

Weaving areas are the crossing of two or more traffic streams traveling in the same direction. An intense lane changing activity occurs therefore at weaving sections. The lane changes and the complex interactions between the weaving and non-weaving vehicles affect the operation and reduce the capacity of weaving sections in relation to their equivalent basic freeway segments.

Traffic in weaving sections has been examined through data analysis. Using oblique cumulative vehicle counts from loop detectors at two different weaving sections in the USA, Lee and Cassidy (2009) and Skabardonis and Kim (2010) show that the bottleneck activation may be triggered by disruptive lane changes from the ramp to the freeway. The effects of the weaving lane changes are further studied in [START_REF] Rudjanakanoknad | Mechanism of a freeway weaving section as typical traffic bottleneck[END_REF] with video data collected on a weaving section in Bangkok. The authors prove that the capacity of the studied weaving section fluctuates over time depending on the number of lane changes and their destination. They observed that an increase of the on-ramp demand induces lane changes from slow to fast lanes and raises consequently the total capacity. They pinpointed also that an augmentation of the off-ramp demand induces lane changes from fast to slow lanes and reduces the capacity of the studied weaving section. Such interactions between weaving vehicles are investigated at a microscopic level in [START_REF] Sarvi | Modelling freeway weaving manoeuvre[END_REF]. The authors exhibit that the acceleration-deceleration behavior and the decision to change lane of the weaving vehicles are influenced by the surrounding freeway vehicles. More recently, [START_REF] Marczak | Empirical analysis of lane changing behavior at a freeway weaving section[END_REF] analyzed detailed microscopic trajectory data collected on a weaving section in Grenoble (France). The authors prove that the lane changing behavior depends strongly on the prevailing traffic conditions. When the traffic is saturated the lane changes occur at the beginning of the weaving section independently of their direction. When the traffic conditions are more fluid, the lane changing positions are more distributed along the weaving section. But interestingly, the lane changes still occur at the same location independently of their direction.

The performance of weaving areas has been also estimated through analytical procedures. The model described in (Lertworawanich and Elefteriadou, 2003) expresses the capacity of a weaving section as a function of the proportions of origin-destination demands and the speeds of the weaving and non-weaving vehicles. The current HCM 2010 methodology (TRB, 2010) to design weaving sections is an update of the HCM 2000 methodology (TRB, 2000) incorporating the improvements developed as part of the NCHRP 3-75 project (Roess and Ulerio, 2009a,b). The capacity of weaving sections is directly linked to the number of lane changes expressed as the percentage of the vehicles that desire to make a weaving movement and the length of the weaving section. However, significant differences may exist between empirical observations of weaving sections capacity and estimates from analytical procedures [START_REF] Cassidy | Operation of major freeway weaving sections: recent empirical evidence[END_REF][START_REF] Rakha | Analytical procedures for estimating capacity of freeway weaving, merge, and diverge sections[END_REF]. Some authors consider microsimulation to be more suitable and reliable to estimate the capacity and predict the operation of various weaving sections. [START_REF] Skabardonis | Simulation of freeway weaving areas[END_REF] calibrates CORSIM with field data measured on eight weaving sections with different configurations. The calibrated CORSIM predicts accurately the operation of the different weaving sections for all configurations and demand patterns. [START_REF] Calvert | Generic freeway weaving section capacity estimation through microsimulation[END_REF] generalize the approach initiated originally in (Dijker and Schuurman, 2003). The authors test different weaving configurations with SIMONE, a microscopic traffic model, and derive, from the microsimulation results, the corresponding expression of the capacity as a function of the weaving flow rate. Some previous works deal specifically with the development of models to reproduce the interactions between the weaving vehicles. Those interactions are modeled using intelligent agent concepts (Hidas, 2005), generic continuous gas-kinetic traffic flows model [START_REF] Ngoduy | Derivation of continuum traffic model for weaving sections on freeways[END_REF] or gap-acceptance theory [START_REF] Bahm | A simple lane change model for microscopic traffic flow simulation in weaving sections[END_REF]. an accurate estimation of a weaving section effective capacity. To address this issue, the objective of the present research is to propose a macroscopic model for a freeway weaving section with an allocation scheme similar to Newell-Daganzo model for merges [START_REF] Daganzo | The cell transmission model, Part II: network traffic[END_REF]. Figure 1 is a particular representation of the Daganzo model where the capacity of both incoming links is equal and equal to the downstream capacity of the merge (denoted C theo ). The x-axis and the y-axis give the effective flows on the main road and the on-ramp, respectively. In the (F; F)-zone, the merge is free flowing because the demands are lower than the capacity.

✲ ✻ q 1 C theo q 2 C theo (F; F) (C; F) (C;C) (F;C) F: free-flow; C: congestion C a p a c i t y c u r v e α
In the (C; F)-zone all the demand from the on-ramp succeeds in merging while the main road is congested.

In the (F;C)-zone the on-ramp is congested while the main road is free flowing. Finally, in the (C;C)-zone the on-ramp and the main road are congested and the effective flows share the downstream available capacity according to a fixed merge ratio α.

The proposed model should also incorporate an endogenous expression of the capacity drop. It is postulated that either the slowdowns or the accelerations of the weaving vehicles create voids in the traffic stream that reduce the capacity of the weaving section. It is assumed moreover that there are no spillbacks from bottlenecks downstream of the weaving section. At the macroscopic level, we consider the weaving section as the superposition of two merges and two diverges. We extend the analytical model proposed in (Leclercq et al., 2011) for the merges by explicitly incorporating (i) the effects of the relaxation at a macroscopic level and (ii) the lane changers' behavior in relation to the prevailing traffic conditions on the target lane. We also adapt the model initially proposed in (Laval, 2009) to derive analytically the effective capacity of the diverges.

In [START_REF] Sarvi | Microsimulation of freeway ramp merging processes under congested traffic conditions[END_REF], it is reported that none of the most frequently used commercial tools can correctly reproduce the traffic behavior near discontinuities of the highway network, especially in congested situations. Previous analytical models mostly fit the outcome of extensive microsimulations. Their results are therefore questionable. Our model is built on a rigorous empirical analysis of the lane changing behavior. The previsions of the model are therefore more accurate because they integrate explicitly the real traffic behavior.

Unlike the microsimulation models, the proposed analytical model will require few parameters. Moreover, those parameters will have a physical interpretation, and hence can be easily calibrated with field data. The proposed model will provide a direct estimate for the effective capacity of a weaving section without requiring any complex simulation runs or elaborated calibration procedure [START_REF] Ngoduy | Kernel smoothing method applicable to the dynamic calibration of traffic flow models[END_REF]. It also has operational applications. It could be the basis of a tool for traffic road managers to forecast in real time the operation of a weaving section or evaluate dynamic traffic management strategies such as ramp metering (Zhang and Wang, 2013). The estimated capacities could be also incorporated in a dynamic route choice model [START_REF] Ng | A dynamic route choice model considering uncertain capacities[END_REF] to guarantee better previsions.

The remainder of the paper is structured as follows. A description of the assumptions and the modelling approach are given in Section 2. To quantify the influence of the model parameters, a sensitivity analysis is performed in Section 3. Section 4 finally presents a comparison between the model estimations, empirical observations and microsimulation results.

MACROSCOPIC MODELING OF A WEAVING SECTION

Preliminary assumptions and notations and modeling approach

We consider a one-sided weaving section with one lane on the main road and one auxiliary lane. Lane 1 is assumed to be the main road while lane 2 is assumed to be the auxiliary lane. Each lane obeys a triangular fundamental diagram with a free-flow speed u, a jam density κ and a wave speed in congestion w. The theoretical capacity of each lane is noted C theo . C theo is measured in vehicles/time and equals:

C theo = uwκ u + w (1)
The demand upstream of the main road (resp. the auxiliary lane) equals λ 1 (resp. λ 2 ). q 1 (resp. q 2 ) is the effective flow coming from the main road (resp. the auxiliary lane) and crossing the weaving section. β 1 and β 2 are respectively the percentages of weaving vehicles driving from and toward the main road. λ i and β i (i ∈ 1, 2 ) are the inputs of the model while q i (i ∈ 1, 2 ) are the outputs of the model, see Figure 2.

We also introduce a constant acceleration rate a. The headways are the time intervals between two successive vehicles. They are assumed to follow a shifted exponential distribution (Gattuso et al., 2005;[START_REF] Chevallier | A macroscopic theory for unsignalized intersections[END_REF] upstream of the weaving section. The headways density function on lane i is thus given by:

f i H (h) =        δ i e -δ i (h-h x ) if h x ≤ h 0 otherwise (2)
Where h x is the minimum safety time headway:

h x = 1 C theo (3)
and

δ i = λ i 1 -λ i h x (4)
At a macroscopic level, the weaving section is seen as the superposition of two merges and two diverges.

The global model will be elaborated by (i) specifying traffic behaviors for those local units and by (ii) sketching out its operation when introducing feedbacks between the different components of the overall model. To synthesize the operation of the overall model, Figure 2 is a scheme presenting the feedbacks between the different components which will be presented later in the chapter. Two mechanisms can dictate the operation of the weaving section when lane i is congested and lane j is in free-flow:

• Firstly -the weaving vehicles exiting lane i can anticipate their lane change and reduce their speed inside the anticipation zone. If the number of weaving vehicles coming from lane i is high enough, the operation of the weaving section is dictated by the model presented in subsection 2.4, see Block 1 in Figure 2. This model gives an effective flow q d i ;

• Secondly -the weaving vehicles coming from lane j and merging on lane i can also degrade the traffic conditions in lane 1. In this case, the model presented in subsection 2.2 describes the operation of the weaving section, see Block 2 in Figure 2. This model gives an effective flow q m i .

At the end, the output of the model is the effective flow q i on the congested lane i which is the minimum between q d i and q m i . The model is strictly symmetric when considering a situation for which only lane j is saturated.

When both incoming roads of the weaving section are congested, we assume moreover that the effective flows fairly share the available downstream capacity. . λ 2 is the demand upstream of the main road (lane 2 in Figure 3a), while λ 1 is the demand upstream of the on-ramp (lane 1 in Figure 3a).

v i r λ i λ j β i β j Diverge model lane i: C d i Merge model lane j: C m j q d i j q m i j q d i j = q m i j yes no q d i u, w, κ, D i j , f i j u, w, κ, D ji , f ji L ant η max , a Merge model lane i: C m i q m i q i = min(q d i , q m i ) α j (β i , β j ) α i (β i , β j ) Inputs Block 2 : merge model (subsec. 2.2) Block 1 : diverge + merge model (subsec. 2.4)

Merge component

λ 2 λ 1 q 2 q 1 η Lane 1 Lane 2 a) ✲ ✻ k q κ ηκ b) u -w
It is assumed in (Leclercq et al., 2011) that the demand on lane 2 is high enough to create congestion. The LL-model relies on the assumption that the lane-changers merge on lane 2 at a speed v 0 then accelerate at a constant acceleration rate a. According to the theory developed in (Laval and Daganzo, 2006), the accelerations create voids in the traffic stream that reduce the effective capacity of the merge. Leclercq et al. (2011) evaluate at a large time scale the number of vehicles between two successive insertions using the variational theory. The merging process reduces the effective capacity C m downstream of the merge to Eq.( 5).

C m = wκ + wηλ 1      w + v 0 a - 1 a (w + v 0 ) 2 + 2aw λ 1 + as 2 w 2 2 (w + v 0 ) 2 + 2aw λ 1 3/2      (5)
Assuming that the inserting positions and the inserting times are independent, s, the standard deviation of their

sum is s = s 2 h + 2s 2
x w 2 where s h and s x are the standard deviations of the headways distribution and the inserting positions distribution, respectively. Leclercq et al. (2011) distinguish two situations whether lane 1 is free flowing or not. If lane 1 is in freeflow, it is assumed that the mergers instantaneously adapt the mean speed of the main road. This assumption leads to v 0 = wq 2 wκq 2 and C m becomes a function of q 2 . As C m is shared by λ 1 and q 2 , one obtains Eq.6 and q 2 is numerically computed by solving this equation.

C m (q 2 ) = λ 1 + q 2 (6)
In case of queuing on both incoming links, the effective flows share the available downstream capacity according to a fixed merge ratio α [START_REF] Daganzo | The cell transmission model, Part II: network traffic[END_REF]. Although there is no general definition of the merge-ratio (Torné et al., 2014), in the present paper, α is the ratio between both incoming link capacities. The merge ratio holds thus q 1 = αq 2 . As the on-ramp is saturated, the merging speed is simply v 0 = wq 1 wκq 1 and C m becomes a function of q 1 . The effective flows are computed solving Eq. 7.

C m (q 1 ) = 1 + 1 α q 1 (7)
The relaxation phenomenon takes place when vehicles involved in a lane change accept shorter spacings then gradually adapt their speed to reach their equilibrium spacings. The merging vehicles in relaxation induce non-equilibrium traffic states that increase the effective flow. The LL-model does not take into account this phenomenon and, as a consequence, it underestimates the effective capacity of the merge. To address this issue we add a dynamic relaxation factor η to the fundamental diagram downstream of the merge, see Figure 3b. If the number of vehicles coming from the unsaturated lane is low, the relaxation process affects only a couple of vehicles and therefore has little impact at a macroscopic level. We assume that η is a linear increasing function of λ 1 which is the demand coming from the on-ramp. The relaxation function is given by a single parameter η max corresponding to the maximum allowed relaxation parameter. We have:

η(λ ) =        1 if λ ≤ λ c 1 η max -1 αq α -λ c 1 (λ -λ c 1 ) + 1 elseif (8)
Where q α is the flow on the main road when both incoming links of the merge are saturated.

Moreover Leclercq et al. ( 2011) expressed s x simply as a function of the on-ramp length. They suppose therefore that the merging behavior depends on the road configuration but not on the prevailing traffic conditions on the target lane. However, recent empirical studies proved that either the road configuration or the traffic conditions influence the merging behavior [START_REF] Daamen | Empirical analysis of merging behavior at a freeway on-ramp[END_REF]Marczak et al., 2013). We address this issue by adjusting the standard deviation according to the mean speed on the target lane v 2 . The positions of the lane changes, X , are assumed to follow a theoretical distribution D 12 that depends on a vector of parameters p 12 :

X ∼ D 12 (p 12 ) (9)
Where p 12 is a function of the effective flow on the target lane, p 12 = f 12 (q 2 ). D 12 and f 12 will be adjusted from empirical considerations later in the paper.

s h is a parameter of the original LL-model. To reduce the number of parameters of our model, we propose to adjust endogenously s h as a function of the demand on the on-ramp. We have assumed that the headways follow a shifted exponential distribution. s h can be directly derived from Eq.( 2):

s h (λ 1 ) = 1 δ 12 = 1 λ 1 - 1 C theo ( 10
)
Finally we have the following expression of the effective capacity:

C m = wηκ + wηκλ 12      w + v 0 a - 1 a (w + v 0 ) 2 + 2aw λ 1 + as(λ 1 ) 2 w 2 2 (w + v 0 ) 2 + 2aw λ 1 3/2      (11)
With

s = 1 λ 1 - 1 C theo 2 + 2var(X ) w 2 (12)
It is worth noting that the shape of the headways distribution is not important only the standard deviation is. We have performed extensive numerical analysis for admissible ranges of parameters for q 2 between 0 and C theo . It appears that the upper and lower bounds of Eqs. ( 6) and ( 7) are positive and negative, respectively. As Eqs.( 6) and ( 7) are continuous function of q 2 , they admit at least one solution. The trucks are assumed to drive at a speed v r , which is lower than u, inside the uphill segment. Assuming that the exiting vehicles anticipate their maneuver inside a L ant -long anticipation zone and slow down at a speed v r this framework was further elaborated in the case of a simple diverging section with bounded accelerations [START_REF] Marczak | Empirical analysis of lane changing behavior at a freeway weaving section[END_REF], see Figure Figure 4a. The headways between two successive slow moving vehicles are moreover assumed to be constant in [START_REF] Marczak | Empirical analysis of lane changing behavior at a freeway weaving section[END_REF] while there are random in (Laval, 2009).

Diverge component

a) λ 1 (1 -β 1 )C d β 1 C d C d L ant ✲ ✻ v r v k q λ 1 q v r C theo κ b)
In accordance with Newell's kinematic wave theory, the slow moving vehicles are considered as moving bottlenecks. If λ 1 is high enough, the moving bottlenecks introduce shockwaves that propagate upstream at a wave speed v and reduce the effective flow at the diverging point, see Figure 4b. The expression given in [START_REF] Marczak | Empirical analysis of lane changing behavior at a freeway weaving section[END_REF] underestimates the capacity of the diverging junction for lower rates of slow moving vehicles. We consider therefore the analytical model presented in (Laval, 2009). Considering the renewal theory, Laval (2009) expresses the distribution of the headways between two successive slow moving vehicles at the beginning of the anticipation zone. The headways follow an exponential distribution whose density function is:

f H (h) =        l 1 e -l 1 h if h ≤ τ(v r ) e τ(v r )(l 0 -l 1 ) l 0 e -l 0 h if h > τ(v r ) (13)
With

τ(v r ) = L ant v r + w v r w l 0 = β 1 uwκ u + w , l 1 = β 1 v r wκ v r + w ( 14
)
Where τ is the disturbance time introduced by a slow moving vehicle and l 0 and l 1 are the mean slow moving vehicles arrival rate at the beginning of the anticipation zone when the flow at this point is C theo or q v r , respectively, see Figure 4b.

Finally, the effective capacity of the diverge, C d is simply given by

C d = 1 β 1 H ( 15
)
Where H is the mean headway between two successive exiting vehicles. H can be immediately derived from Eq.( 13). Note that randomly drawn lane changing positions do not influence the capacity of the diverge. Indeed, we assume that all the exiting vehicles drive at v r . The drop in capacity occurs therefore at the beginning of the anticipation zone independently of the lane changing positions.

Combining one upstream diverge with the associated downstream merge

From the two previous steps we have the capacity of a merge C m and the capacity of a diverge C d . The next step is to combine a diverge on lane 1 with a merge on lane 2, see Figure 5a. The developments presented in this section will be identical when combining one merge on lane 1 and one diverge on lane 2. The main road of the merge is assumed to be lane 2. We assume moreover in this subsection that λ 2 is low enough to prevent congestion on lane 2 because the situation for which lane 2 is saturated can be analytically solved with the merge model presented in subsection 2.2. The lane changing vehicles coming from lane 1 temporarily drive at v r inside the anticipation zone. The demand on lane 1 is high enough to initiate shockwaves that reduce the effective flow at the diverging point.

a) There are two links, respectively denoted d 1 and d 2 , downstream of the diverge on lane 1, see Figure 5a. No capacity reduction occurs downstream of d 1 . Therefore, the effective capacity of d 1 equals C theo . Because of the merging process, the capacity on d 2 is reduced to C m , which is the capacity given by the merge model for a fixed λ 2 . The merge model estimates also the effective lane changing flow q m 12 coming from lane 2, see Figure 5b. But for a given v r , the diverge model proposes another lane changing flow q d 12 . As u, w, κ, L ant , η max and a are fixed, we propose to adjust v r to ensure equality between q m 12 and q d 12 . v r is found so that the following equation is satisfied:

λ 1 λ 2 q e 12 d 1 d 2 q 1 C d C m L ant b) v r λ 1 , β 1 λ 2 Inputs Diverge model: C d Merge model: C m q d 12 q m 12 q d 12 = q m 12 yes no q 1 u, w, κ, D 12 , f 12 L ant η max , a
λ 2 + β 1 C d (v r ) = C m (v r , s d (v r )) (16)
Once again, assuming that the headways between two successive lane changes and the lane changing positions are independent, one can write:

s d (v r ) = s 2 H + 2var(X ) w 2 (17)
Where s H is the standard deviation of the headways distribution given by Eq.( 13).

v r was a parameter of the diverge model presented in subsection 2.3. When combining a diverge and a merge v r is no longer a parameter of the model, but v r is adjusted endogenously as a function of λ 2 . This is the feedback between the different models. One again we have performed extensive numerical resolutions for an admissible range of parameters. It appears that Eq.( 16) always admits a solution. Interestingly, Figure 6 shows that v r is a decreasing function of λ 2 . When the flow is higher on lane 2, the drivers coming from lane 1 anticipate more their lane changes because the offered gaps on lane 2 are smaller and the lane changes are therefore more difficult.

The overall model

The goal now is to describe the complete analytical model for the weaving section. At a macroscopic level, the weaving section is represented as the superposition of two merges and two diverges.

When both incoming roads are congested, we assume moreover that the effective flows fairly share the available downstream capacity according to a fixed priority ratio which is equal to 1 and independent of the percentages of weaving flows β 1 and β 2 . As a consequence the merge-ratios α 1 and α 2 are dynamic and dependent on β 1 and β 2 . This is a strong assumption because usually α is supposed independent of the upstream demand. [START_REF] Bar-Gera | Empirical macroscopic evaluation of freeway merge-ratios[END_REF] show empirically that the merge-ratios for independent merges can be reasonably estimated by the ratios between the number of lanes on the main road and the number of lanes on the onramp. The findings of this study indicate that the merge-ratios depend on the geometrical configuration of the merge area and consequently on the infrastructure supply. But the authors pinpoint some residual differences between merge-ratios and lane-ratios. They suggest therefore that other factors influence the merge-ratios.

More recently, [START_REF] Reina | Prediction of merge ratio using lane flow distribution[END_REF] developed a formulation of merge-ratios using lane flow distribution (LFD). The authors showed that the proposed LFD-based model can capture variation in merge-ratios with respect to the traffic conditions. [START_REF] Chevallier | A macroscopic theory for unsignalized intersections[END_REF] considered in simulation a dynamic priority sharing ratio to reproduce the drivers' aggressiveness. Moreover these studies are for independent merges and not for weaving sections. Note that this assumption on α 1 and α 2 can be relaxed without changing the model structure when further and specific data for weavings will be collected.

In case of continuous queuing on both incoming links, one can guarantee the consistency of the effective flow estimated by both merge models by writing:

       q α 1 = α 2 q α 2 β 1 (a) q α 2 = α 1 q α 1 β 2 (b) (18)
Interestingly, if Eq. ( 18)(a) and (b) are verified they yield the following simple condition:

α 1 α 2 = β 1 β 2 (19)
One can assume moreover that α 1 and α 2 are proportional and that there is a constant µ such that :

α 2 = µα 1 (20)
One can write therefore

α 1 = β 1 β 2 µ , α 2 = µβ 1 β 2 (21)
The remaining question to adjust endogenously α 1 and α 2 is the estimation of µ. We also assume that the effective flows on lane 1 and lane 2 compete to fairly share the downstream capacity on a one-to-one basis. One should have consequently:

min q α 1 , α 2 q α 2 β 1 = min q α 2 , α 1 q α 1 β 2 (22)
One can easily estimate µ solving Eq.( 22).

Previous studies using micro-simulation to estimate the capacity of a weaving section, pinpointed that the length of the weaving section is a key factor influencing its capacity (Dijker and Schuurman, 2003;[START_REF] Calvert | Generic freeway weaving section capacity estimation through microsimulation[END_REF]. Our model does not include explicitly the length of the weaving section as a parameter.

It is, however, important to stress that the length is implicitly included in the distribution of the lane changing positions. A longer weaving section may indeed introduce different lane changing behaviors and therefore a different distribution of the lane changing positions. As the standard deviation of this distribution is a parameter of the model, the length of the weaving section is implicitly taken into account in the analytical expression of the capacity.

Sensitivity analysis

A sensitivity analysis is now performed to determine the contribution of the different parameters to the capacity drop. Let us denote C the effective capacity of the weaving section. We define C as the effective flow passing the weaving section when both lane 1 and lane 2 are saturated. As we assumed that the main road and the auxiliary lane obey the same fundamental diagram, we have simply C = 2C theo . The sensitivity to the parameters and the relative capacity drop are quantified with c the complement of the ratio between the effective capacity and the theoretical capacity (Leclercq et al., 2011). We focus the sensitivity analysis on the acceleration rate a, the maximum relaxation factor η max and the length of the anticipation zone L ant . The results of the sensitivity analysis are presented in Figure 7. The dotted lines represent c for the situation with randomly drawn headways between two successive lane changes and a fixed lane changing position. The continuous lines represent c for randomly drawn headways and lane changing positions. Figure 7 : Sensitivity analysis: influence on c, the relative capacity drop of the weaving section, of a) the acceleration rate a, b) the maximum relaxation factor η max , c) the length of the anticipation zone L ant , and d) the percentages of weaving flows β 1 and β 2 . We have chosen β 1 = 0.55 and β 2 = 0.59 in the first three cases while we have fixed a = 2 m/s 2 , η max = 1.2, L ant = 20 m, w = 5.4 m/s and κ = 0.15 veh/m to study the sensibility to β 1 and β 2

Figure 7a shows the influence of a on the relative capacity drop for different values of η max . Figure 7b presents the influence of η max on the capacity drop for different values of a. It appears that increasing a or η max increases the effective capacity and reduces consequently c. When a is high, the voids created by the acceleration process in the traffic stream are smaller. The effective flow is therefore higher. η max is a dilatation factor of the fundamental diagram. It increases the effective flow crossing the weaving section. Figure 7c focuses on L ant for different values of a. L ant hardly influences the effective capacity. It appears moreover that the capacity drop is lower when the lane changing positions are randomly drawn. The effective capacity of the weaving section is therefore higher. Eq. ( 12) proves that one increases the standard deviation when considering randomly drawn headways and lane changing positions. Leclercq et al. (2011) show that the standard deviation reduces the relative capacity drop.

The capacity of a weaving section varies with respect to the percentages of weaving flows (Lertworawanich and Elefteriadou, 2003). We analyze therefore the sensitivity of the model to β 1 and β 2 . Figure 7d is an illustration of the (β 1 , β 2 ) plane which is split in different regions according to the relative capacity drop c.

These regions are bounded by isolines of c. First of all, one can observe that c is not symmetrical with respect to β 1 and β 2 . The standard deviation of the lane changing positions is a parameter of the model. The position distribution of the lane changes towards the auxiliary lane differs from that towards the main road because their respective parameters are not the same, see Figure 9. One observes therefore asymmetrical effects on c because those distributions, and hence the standard deviations, are different. For the lower values of β 1 and β 2 , c is less than 5%. c increases with increasing β 1 and β 2 : the higher the number of weaving vehicles, the higher the drop in capacity. For the higher values of β 1 and β 2 , less than 45% of the theoretical capacity are used. First of all, we fit empirically the distribution D of the lane changing positions. The data have been aggregated using the lane changing speed. We assume that this speed gives an accurate indication of the prevailing traffic conditions on the target lane. The data have been fitted with different theoretical distributions inside each speed class. It appears that the Gamma distribution fits the best the empirical observations. Figure 9 presents the Maximum Likelihood Estimations for the parameters of the adjusted distribution. a i→ j and b i→ j are the shape parameter and the scale parameter, respectively, of the Gamma distribution. We also represent the confidence intervals for a i→ j and b i→ j at a 5% significance level. Then a i→ j and b i→ j have been adjusted as functions of the speed with the simplest polynomial. The solid lines in Figure 9 are the adjusted polynomials.

Model verification and validation

Comparison with empirical observations

Microscopic trajectory data allow us to trace each vehicle from its origin at the beginning of the weaving section to its destination at the end of the weaving section. We can therefore have an accurate estimation of β 1 and β 2 . However we do not have enough data for a given couple of β 1 and β 2 . Moreover, we cannot observe all the possible capacities because of the numerous combinations of percentages of weaving vehicles. To address this issue, we extracted from the video recordings some periods for which the weaving section is either free flowing or an an active bottleneck. The bottleneck is located at the start of the block line between the main road and the auxiliary lane. When the bottleneck is active, the congestion moves upstream (i) only on the rightmost lane on the main road or (ii) on the auxiliary lane and the rightmost lane. We have taken only the situations in congestion for which the mean speed is lower than 40 km/h (11 m/s). We have computed from the trajectories the time instants at which the vehicles pass through a fictive detector located at the start of the

Comparison with simulation results

A second option for questioning our analytical model is to compare it with simulation results. The simulated weaving vehicles will temporarily consider their leader on the adjacent lane if the mean speed on the actual lane is higher than the mean speed on the adjacent lane. The leader on the adjacent lane is the closest vehicle driving in front of the considered weaving vehicle. The car following rule is an extension to Newell ′ s car-following model [START_REF] Newell | A simplified car-following theory: a lower order model[END_REF] with a relaxation term (Laval and Leclercq, 2007) and bounded acceleration [START_REF] Chevallier | A macroscopic theory for unsignalized intersections[END_REF]. We redirect our readers to [START_REF] Chevallier | A macroscopic theory for unsignalized intersections[END_REF] for a more precise description of the car-following model. The lane changing model simply expresses the probability of changing lane as a function of the longitudinal position. The lane changing positions follow a Gamma distribution. The parameters of the distribution are adjusted according to the results presented in Figure 9. The demands upstream of the weaving section are constant and the headways upstream of the weaving section are assumed to follow a shifted exponential distribution whose density function is given by Eq.( 2). We run 1500 replications. For each replication, we simulate 15 minutes of data. Then, we construct the curve of cumulative vehicles count (CVC) 1000 m downstream of the weaving section according to the origin of the vehicles. Finally, the effective capacity is the long-term average, measured with the slope of the CVC during the last 6 minutes of the simulation.

Figure 11 : Comparison between simulation results and the analytical capacity curves

Figure 11 presents the comparison between the simulation results and the theoretical capacity curves for the extreme couples of β 1 and β 2 . The simulation results and the analytical curves are in high accordance.

Conclusions

This paper presents a parsimonious macroscopic model for a freeway weaving section with an allocation scheme similar to the Newell Daganzo model and an endogenous expression of the capacity drop. The paper proposes an explicit formulation of the relation between microscopic interactions related to the weaving activity and their impact on the capacity at a macroscopic level. It is postulated in our study that either the slowdowns or the accelerations of the weaving vehicles create voids in the traffic stream, and that these voids reduce the capacity of the weaving section. We simply consider a theoretical weaving section as the superposition of two merges and two diverges. We assume therefore that the operation of the weaving section is dictated either by the operation of a single merge or by the operation of a single diverge. A parsimonious formulation of the capacity that depends only on six parameters is proposed. The acceleration and the relaxation parameter strongly influence the effective capacity, whereas the length of the anticipation zone hardly influences the capacity. The analytical expression of the capacity accords well with empirical observations and simulation results.

The work presented in this paper assumed some simplifying assumptions. The theory was developed for a very simple weaving section with one lane on the main road and one auxiliary lane. The traffic upstream of the weaving section is consequently strictly FIFO. The proposed model has to be further developed for other weaving configurations with more lanes on the main road. The merge model should specifically be generalized for configurations with more lanes on the main road. Extensions to multi-lanes freeways have been discussed in (Leclercq et al., 2011). But it is assumed that the capacity drop occurs (i) only on the shoulder lane while the other lanes are free flowing or (ii) identically on all downstream lanes. The model gives a bound of the capacity drop but it cannot reproduce properly lane flow distributions because it does not consider the discretionary lane changes that may reduce that capacity on the main road [START_REF] Cassidy | Increasing the capacity of an isolated merge by metering its on-ramp[END_REF]. Research has to continue to integrate explicitly the discretionary lane changes in the analytical expression of a multilanes merge capacity. Similarly, we have considered a diverge with one lane of the main road. The traffic is consequently strictly FIFO. More research is needed to estimate the effective capacity of a non-FIFO diverge with more lanes on the main road. Then the enhanced merge and diverge models should be aggregated in order to build a generalized model for weaving sections.

The proposed model has been compared to a single dataset. Future research should be performed to collect field data on different weaving sections and compare the capacities predicted by the model to field-measured capacities. We have made a strong assumption on the priority ratio when both incoming roads are congested assuming that it equals 1, independently of the percentages of weaving flows. This is in accordance to what was observed on the studied weaving section but must be compared to reality in more cases. Future empirical researches should estimate how the weaving ratio is linked to the percentages of weaving flows. As in (Sun and Elefteriadou, 2012) an instrumented vehicle-based experiment could be also designed to observe the drivers' action during the lane changes and have a deeper understanding of the anticipation behavior during those maneuvers.

Figure 1 :

 1 Figure 1 : Global allocation scheme for the Newell-Daganzo's model

Figure 2 :

 2 Figure 2 : Feedbacks between the different components of the global model

Figure 3

 3 Figure 3 : a) Merge component and b) triangular fundamental diagram with the relaxation

Figure 4

 4 Figure 4 : a) Sketch of the theoretical diverging junction and b) illustration of the effect of a moving bottleneck driving at v r

Figure 5

 5 Figure 5 : a) Combining one merge and one diverge and b) focus on block 1 of Figure 2

Figure 6 :

 6 Figure 6 : Evolution of v r as a function of λ 2 for different accelerations

Figure 8 :Figure 9 :

 89 Figure 8 : Data collection site in Grenoble. The traffic moves from left to right.

Table 1

 1 presents all the variables included in the model and their definitions.

	Notation	Definition	Classification
	β i	Percentage of weaving flow from lane i (%)	Input variable
	λ i	Demand on lane i (veh/s)	Input variable
	q i	Effective flow on lane i (veh/s)	Output variable
	q m i	Effective flow on lane i given by the merge model (veh/s)	Aux. variable
	q d i	Effective flow on lane i given by the diverge model (veh/s)	Aux. variable
	C theo	Theoretical capacity (veh/s)	Aux. variable
	C m	Merge effective capacity (veh/s)	Aux. variable
	C d	Diverge effective capacity (veh/s)	Aux. variable
	h x	Safety time headway (s)	Aux. variable
	u	Free flow speed (m/s)	Parameter (m)
	w	Wave speed in congestion (m/s)	Parameter (m)
	κ	Jam density (veh/m)	Parameter (m)
	a	Acceleration (m/s 2 )	Parameter (c)
	η max	Maximum relaxation	Parameter (c)
	L ant	Length of the anticipation zone (m)	Parameter (c)
	τ	Disturbance time (s)	Aux. variable
	α i	Merge-ratio (merge model on lane i)	Aux. variable
	µ	Proportionality between α 1 and α 2	Aux. variable
	v r	Slowdown speed (m/s)	Aux. variable
	v 0	Inserting speed (m/s)	Aux. variable
	p i j	D i j parameters	Aux. variable
	s x	Inserting positions standard deviation (m)	Aux. variable
	s h	Headways standard deviation (s)	Aux. variable
	s	Standard deviation of the inserting process (s)	Aux. variable

D i j

Distribution of the lane changing positions from lane i to lane j Aux. variable

Table 1 :

 1 Notations, definitions and classification: (m) et (c) stand for the parameters that need to be measured and the parameters that need to be calibrated respectively

Acknowledgements

Our thanks are to Anne-Christine Demanny (IFSTTAR/LICIT) for her careful reading of the paper and the anonymous referees for helping to greatly improve this article.

block line. Then the data have been aggregated over a two-minute time interval to obtain homogeneous traffic state and to estimate the effective flows and β 1 and β 2 . Finally we have determined respectively the maximum and minimum observed couples of β 1 and β 2 . Figure 10 presents the comparison between the empirical observations and the theoretical capacity curves which are the continuous lines. We fixed η max = 1.13 and a = 2.5m/s 2 . The fundamental diagram has been calibrated on each lane of the studied weaving section during congestion. We have u = 20 m/s, w = 5.38m/s and κ = 0.15veh/m. The lower measured percentages of weaving flows is β 1 = 0.55 and β 2 = 0.59 while the higher couple is β 1 = 0.69 and β 2 = 0.67. We estimate that β 1 and β 2 have been measured with a precision of 15%. For each couple of β 1 and β 2 we construct therefore a lower bound of the capacity curve with 0. G. Zhang and Y. Wang. Optimizing coordinated ramp metering: a preemptive hierarchical control approach.

Computer-Aided Civil and Infrastructure Engineering, 28(1):22-37, 2013.


est destinée au dépôt

=> Controllers must apply a different working method for transfer data link aircraft (Data link was recommended for non time-critical controller orders, while voice was retained time for critical orders)

About Capgemini Capgemini, one of the world's foremost providers of consulting, technology and outsourcing services, enables its clients to transform and perform through technologies. Capgemini provides its clients with insights and capabilities that boost their freedom to achieve superior results through a unique way of working, the Collaborative Business Experience. The Group relies on its global delivery model called Rightshore®, which aims to get the right balance of the best talent from multiple locations, working as one team to create and deliver the optimum solution for clients. Present in more than 30 countries, Capgemini reported 2008 global revenues of EUR 8.7 billion and employs over 90,000 people worldwide. More information is available at www.capgemini.com.

About Capgemini

Capgemini, one of the world's foremost providers of consulting, technology and outsourcing services, enables its clients to transform and perform through technologies. Capgemini provides its clients with insights and capabilities that boost their freedom to achieve superior results through a unique way of working, the Collaborative Business Experience. The Group relies on its global delivery model called Rightshore®, which aims to get the right balance of the best talent from multiple locations, working as one team to create and deliver the optimum solution for clients. Present in more than 30 countries, Capgemini reported 2008 global revenues of EUR 8.7 billion and employs over 90,000 people worldwide. More information is available at www.capgemini.com.

First

  introduction of ACM CPDLC service on ODS: a new transfer tool for controller Laurent Lapasset, Benoit Hermeline To cite this version: Laurent Lapasset, Benoit Hermeline. First introduction of ACM CPDLC service on ODS: a new transfer tool for controller. INO 2009, 8th SESAR Innovative Research Workshop & Exhibition, Dec 2009, Brétigny-sur-Orge, France. hal-01511738 Space, Navigation & Defence unit SHOOT is not any more an atomic not interruptible action: beginning; end; and possible blocking according to pilot's answer ! Speed Aircraft ID >> Yet another tool? Synthesis of the states of the DL communication and the states of the flights: Synthesis of the communication state of the DL channel of in order to anticipate the time of transmission of messages, to plan the display of the presence of DL equipment on the flights (transition period with mixed traffic), as soon as possible, the only visualized states will be: : DL channel is available for controller (for immediate use) : DL channel is « notified center » or available but not activated The state the «Data Link connection errors » are not relevant to show. Delay in reception or loss of the response of the pilot could lead to an unsafe situation: its the operational state ASSUME (Transfer of Responsibility) which will use to (re)synchronize the two OLDI/SYSCO grounds systems of respectively sector giving and receiving sector. ACM CPDLC service on ODS: a new transfer tool for controller Towards a new reality, controller centered: a new HMI. Data Link (DL) can't exist without an HMI, any evaluation of DL can't be done without the evaluation of the associated HMI. For a successful introduction, the key points are:

  Laurent Lapasset laurent.lapasset@capgemini.com -Ingénieur R&D ATM Benoît Hermeline benoit.hermeline@capgemini.com -Manager ATM Unit Tel : 05 61 31 52 00