A Mesoscopic Simulation-Based Dynamic Traffic Assignment Model

A mesoscopic simulation based

dynamic traffic assignment model

Authors:

M.P. Linares1,2, C. Carmona1, J. Barceló1,2, O. Serch1

1 inLab FIB Universitat Politècnica de Catalunya – BarcelonaTECH Jordi Girona 1-3 08034 Barcelona Spain

2 Departament d’Estadística i Investigació Operativa Universitat Politècnica de Catalunya – BarcelonaTECH Jordi Girona 1-3 08034 Barcelona Spain (mari.paz.linares, carlos.carmona, jaume.barcelo, oriol.serch)@upc.edu Tel: +34 93 401 6941

Paper submitted for presentation and publication to 93st Transportation Research Board 2014 Annual Meeting Washington, D.C. July 2013

# 6500 WORDS: + 4 Figures = 7500

TRB 2014 Annual Meeting Original paper submittal - not revised by author

Linares, Carmona, Barceló, Serch

2

ABSTRACT 1

Nowadays urban traffic is one of the main challenges for urban areas in terms of congestion, 2

accidents and pollution. This problem has been stressed by the increasing mobility promoted by the 3

use of private vehicles. So, the time wasted on users' trips has an important impact on urban 4

economy. In this context, transport models become relevant for the development of new strategies 5

and policies for a sustainable urban transport. In particular, models for traffic forecasting like 6

dynamic traffic assignment models, providing temporal coherent representation of the interactions 7

among traffic choices, flows, time and cost measures. In this paper, we develop a dynamic traffic 8

assignment model based on the dynamic user equilibrium by solving a variational inequalities 9

formulation under a preventive approach. An iterative solution algorithm, which is a modification of 10

the Method of Successive Averages, explicitly considers the time and a variable traffic demand on 11

each path of the network within the flow propagation and assignment processes. The dynamics of the 12

reassigned flows at each iteration is simulated by a new mesoscopic multiclass multilane simulation 13

model accounting for lane changes and traffic control at signalized intersections. 14

Keywords: dynamic traffic assignment, dynamic user equilibrium, mesoscopic traffic simulation, 15

Method of Successive Averages. 16



3

INTRODUCTION 17

The Dynamic Traffic Assignment (DTA) problem, which has been receiving the attention of the 18

researchers since the early 80's, can be considered as an extension of the traffic assignment problem 19

that is able to describe how traffic patterns evolve over time and space in the transport network 20

(Mahmassani [1]). 21

The increasing market penetration of the technologies, that underlie Intelligent Transport Systems 22

(ITS), has been one of the mean reasons to promote research on the problems of DTA, since belong 23

to the core models for ITS: advanced traffic management and traffic information systems. 24

DTA is aimed at determining time-dependent links and path flows in a congested road network 25

satisfying a set of individual objectives and/or an overall system goal. 26

According to the behavioral hypothesis modeling the individual decisions taken during the trip, the 27

DTA problems can be classified as follows: 28

• Dynamic System Optimum assignment (DSO) 29

• Dynamic User Equilibrium assignment (DUE) 30

Despite the mathematical difficulty involved in treating the DTA problems, this aspect does not 31

imply an obstacle to the different associated solution approaches associated when used real world 32

applications. In any case they must always be able of realistically reproducing observed traffic 33

behaviors. 34

The proposed approaches to solve the DTA problem may be classified into four broad 35

methodological groups: 36

• Mathematical programming 37

• Optimal control 38

• Variational inequalities 39

• Simulation 40

The first three groups of approaches are usually classified in turn into the so-called analytical 41

approaches. In these cases, the DTA problem is formulated as a mathematical problem which is 42

solved directly using optimization techniques. 43

Because of the limitations of mathematical programming and optimal control theory in DTA context 44

and the advantages offered by variational inequalities, analytical DTA models have migrated towards 45

the variational inequalities approach. An extended overview of mathematical programming, optimal 46

control and variational inequalities can be found in Peeta et al. [2]. 47

On the other hand, simulation based formulations, looking for approximate heuristic solutions with 48

reasonable computational costs, focus on enabling practical deployment for real networks. 49

While analytical models are mathematically rigorous, they rely on simplified assumptions to account 50

for traffic dynamics that make them unsuitable for large-scale real-world applications requiring 51

models that capture the stochastic characteristics of traffic dynamics in detail, by estimating and 52

predicting OD flows, travel times, queues and spill-backs, and dealing properly with flow 53



4

interruptions at signalized intersections These capabilities are generally beyond that of existing 54

analytical models, but within the abilities of simulation approaches. 55

LITERATURE REVIEW 56

The preceding section classified the DTA approaches into two groups: analytical and based on 57

simulation. Taking into account the above considerations we will focus our attention only on 58

simulation-based DTA models. 59

There are three types of simulation-based DTA models, which are distinguished based on the level of 60

granularity with which they represent the studied system. They range from low fidelity to high 61

fidelity as follows: 62

• Macroscopic simulation models 63

• Mesoscopic simulation models 64

• Microscopic simulation models 65

To begin with, macroscopic models treat traffic in an aggregate manner, such as a uniform or 66

homogeneous flow, without considering each constituent particle (individual vehicles). They 67

approximate flow propagation throughout the network using physical concepts and analytical 68

methods. Microscopic approaches model individual entities, decisions and interactions with higher 69

degree of detail. Each vehicle maneuvers at a specific simulation time step based on estimations 70

derived from a set of behavioral models, such as car following, lane changing, merging and yielding. 71

Mesoscopic models combine elements from microscopic and macroscopic approaches, representing 72

activities and interactions of each vehicle with less detail but still enough to account for the essentials 73

of traffic dynamics 74

Macroscopic Simulation-based DTA Models 75

Macroscopic DTA models apply physical analogies usually based on hydrodynamic theory in 76

describing traffic dynamics. The evolution of traffic over time and space is represented by a set of 77

differential equations, the continuum equation, the fundamental diagram and other equations in the 78

case of second order models, which are solved by numerical methods that have limitations in 79

appropriately including interactions between vehicles and roads. This is why some macroscopic 80

DTA models resort to simulation. 81

Macroscopic simulation-based DTA models, such as METANET [3] or INDY [4], describe both 82

traffic flow propagation and trip-maker decisions at aggregate level i.e. traffic is considered to be a 83

continuum in steady state. 84

In order to be able to simulate route choice decisions, in INDY, traffic flows are disaggregated by 85

route: traffic is assigned to a specific route by a route choice model. In METANET, traffic flows are 86

not disaggregated by route. Single commodity flows are routed by splitting proportions at network 87

nodes. The routes followed in a random network loading are generally not consistent with the route 88

flow rates resulting from the route choice model, since split proportions, which correspond to a given 89

route flow set, are not easily determinable. 90

91



5

Mesoscopic Simulation-based DTA Models 92

Mesoscopic simulation-based DTA models can either move individual vehicles or packets of 93

vehicles at an aggregate level, while trip-maker decisions are made individually; i.e. a microscopic 94

level of representation of individual trip-maker decisions is combined with a macroscopic description 95

of traffic flow propagation or Dynamic Network Loading (DNL). 96

Computation times in mesoscopic models are significantly reduced compared to microscopic models 97

due to the aggregate description of traffic flow. Mesoscopic models can computationally succeed in 98

the analysis of medium-sized networks. These models typically carry out both equilibrium 99

assignments and en-route assignments. 100

One way to simulate traffic is to group vehicles into packets, and route these packets through the 101

network (CONTRAM [5]). The packet of vehicles acts as one entity, sharing the speed determined 102

from the speed density function established for each link, and the density on the link at the moment 103

of entry. CONTRAM is based on time-dependent queuing theory. 104

Another mesoscopic paradigm is the queue-server approach used by some models, such as DynaMIT 105

[6] and DYNASMART [1]. In this approach, road segments are modeled with two parts: queuing 106

and moving. The vehicles travel through the moving part with the speed calculated using a 107

macroscopic speed-density function. DYNASMART and DYNAMIT use traffic models based on 108

flow that propagate vehicles on links according to a modified Greenshields speed-density 109

relationship. As the vehicle moves through the link, a queue-server transfers the vehicles 110

downstream to another link, or constructs queues on the current link, representing congestion. 111

Another DTA models based on mesoscopic simulation are METROPOLIS [7] and MEZZO [8]. The 112

last was developed at the Royal Institute of Technology as the mesoscopic component of a hybrid 113

mesoscopic-microscopic simulation model. 114

This approach combines the advantages of dynamic disaggregated traffic modeling (since the 115

vehicles are modeled individually), with an easy calibration and the use of macroscopic speed-116

density functions. 117

Microscopic Simulation-based DTA Models 118

Microscopic simulation-based DTA models describe traffic flow propagation with higher degree of 119

detail at the level of individual vehicles. They model the movement of each vehicle and their 120

interactions with each other and with the infrastructure. Trip-maker decisions such as route choice 121

are represented at individual level as well. 122

Traffic flow propagation in microsimulation models combines mathematical car-following and gap 123

acceptance models, with heuristics representing driver behavior (e.g. lane-changing behavior, gap 124

acceptance). In addition, traffic control, including signal operation, location and traffic detectors, are 125

also modeled in detail. These models require the detailed calibration of model parameters, which is a 126

very time consuming, and computationally costly task. 127

In the microscopic TRANSIMS model [9], traffic propagation is based on a cellular automata 128

technique for car-following and lane-changing, enhanced by additional rules for elements such as 129

signals, weaving lanes, unprotected turns, etc. The lanes of all links of the network are divided into 130



6

cells of equal size, which are either empty or occupied by a single vehicle. Since there are less model 131

parameters, cellular automata models are easier to calibrate. 132

In the microscopic-based DTA model DTASQ [10], later the commercial software DYNAMEQ, 133

attempts to capture the effects of car-following, gap acceptance and lane changing with a minimum 134

number of parameters. This leads to a reduction in the calibration effort. The simulation, based on a 135

simplified car-following relationship (ultimately, somewhat closer to cellular automata models), is a 136

discrete-event procedure. This leads to a sharp reduction in computational effort as well, when 137

compared to microscopic discrete time approaches. Mahut, in his thesis, provides a detailed 138

description of this DNL model [11]. Though categorized as a microscopic simulation-based DTA 139

model, DYNAMEQ properties (such as calibration efforts or computation times), have more 140

commonalities with mesoscopic simulation approaches than with typically microscopic. 141

Most microscopic simulation models use an en-route approach, but Barceló et al. [12] showed that 142

the microsimulation model Aimsun (Barceló, 1992) could also be used in conjunction with an 143

iterative equilibrium-like assignment method. Recently, Aimsun included the possibility to use 144

mesoscopic simulation, instead of microsimulation, to perform the DNL process into a DTA scheme 145

under DUE behavior hypothesis. TRANSIMS and DYNAMEQ deal with equilibrium assignment as 146

well. 147

Other microscopic simulation-based DTA models share the conceptual approaches discussed here. In 148

1993, DRACULA [13] was developed at the University of Leeds; in 1983, Van Aerde developed 149

INTEGRATION [14] and, finally, the most recent, DynusT [15] developed at the University of 150

Arizona in 2011. 151

A NEW DYNAMIC TRAFFIC ASSIGNMENT MODEL 152

Fundamentals 153

The DTA model developed in this paper is based on the dynamic extension of the Wardrop's 154

Principle referred before to as Dynamic User Equilibrium (DUE). In order to achieve the dynamic 155

equilibrium, we solve the variational inequalities formulation by employing a preventive approach 156

based on an iterative solution algorithm which is a modification of the Method of Successive 157

Averages proposed by Sheffi [16] for a variational inequalities problem. 158

DTA models for predicting user equilibrium flows on traffic networks are often solved by an 159

algorithm that iterates between two main components until a convergence criterion is satisfied, 160

systematized by Florian et al. [10]. These two components are referred to as DNL and path flow 161

reassignment. The DNL procedure takes the allocation of travel demands to network paths at each 162

time step as given, loads these into the network over time and computes time dependent path costs 163

from that network loading. The path flow reassignment step then uses the time dependent costs 164

obtained in the DNL to adjust the allocation of traffic to paths to be used in the next DNL. 165

DTA models based on simulation use a traffic simulator to reproduce the complex traffic flow 166

dynamics. In our case, we develop a mesoscopic multiclass multilane DNL simulation model, which 167

is embedded in the proposed DTA framework in order to capture realistically the traffic dynamics 168

169



7

Proposed DTA Model 170

A description of the proposed DTA model is given in what follows. Its conceptual structure is 171

illustrated in FIGURE 1. 172

173

FIGURE 1 Structure of the proposed DTA model. 174

175

The main inputs to this DTA are: the network geometry and control plans at signalized intersections 176

and the demandtime-sliced into discrete departure time periods. Demand is represented by a set of 177

time-dependent origin-destination matrices that contains one matrix for each departure time interval 178

for each vehicles class. 179

At the initialization step a K-static shortest paths algorithm is executed in order to obtain the best set 180

of paths for each OD pair. The inputs to the K-static shortest path algorithm are the free flow travel 181

costs of each link of the network. 182



8

It is important highlight here that starting the flow assignment process with only one possible path 183

for each OD pair for an all-or-nothing assignment, the possibility of generating false congestion is 184

very high. If it occurs, all the main links of the network can present congestion, and consequently 185

very high costs. In this case, in the next step of the process, we will find a shortest path that will not 186

use these congested links, so we could go far from the equilibrium, and more iterations would be 187

needed to converge. To avoid this, the initialization step uses a set of paths of predefined size M=K 188

for each OD pair provided by the K-shortest paths, the same for all departure time intervals. The 189

estimation of the appropriate value for parameter M is determined as part of the calibration process. 190

The method used is the link-based shortest path algorithm of Lim and Kim [17] that provides 191

efficient alternative paths considering overlapping among paths. This algorithm builds new paths 192

based on both the degree of overlapping between each path and the travel cost. 193

After the initialization step, and while neither the maximum number of DTA iterations nor the 194

convergence criterion is achieved, the algorithm iterates between two main components: the flow 195

reassignment and the DNL components. 196

Firstly, we try to assign the demand of the different OD pairs at each departure time interval into the 197

corresponding paths of the initial set of M “best paths” (��). The first iteration of the flow 198

reassignment component is performed differently from the assignment of the rest of the iterations of 199

the procedure. In this first iteration, the flow assignment is inversely proportional to the paths cost. 200

For all OD pair��, for all departure time interval�, for all paths ∈ ��, the flow is assigned by 201

following the Equation (1). 202

�� = 1 ��⁄∑ �1 ��⁄ ��∈��

· ��(1)

where: 203

�� is the flow assigned to the path from origin � to the destination �departing at time 204

interval � at iteration �. 205

�� is the cost of the path from origin � to the destination � departing at time interval �. 206

�� is the number of travels of the corresponding time dependent OD matrix. 207

208

After this, the DNL method is used in order to determine how these paths flows (the output of the 209

flow reassignment component) are propagated along the corresponding paths, generating time 210

dependent traffic intensities on the network sections, links travel times, paths travel times, etc., for 211

the current iteration. This DNL flow propagation is carried out by the proposed new mesoscopic 212

traffic simulation model. 213

This mesoscopic traffic simulation provides the new costs of all links in the network. Obviously, the 214

cost of a given link can be different in the first simulation interval than in the next ones. Therefore 215

we will have different costs for each link of the network for each time interval. 216

217



9

The DTA procedure uses this information: 218

• To update the cost of the paths used in the previously performed DNL. 219

• To calculate the new shortest path for each OD pair for each departure time interval. Because 220

the link cost is different depending on the time when the vehicle arrives to this link, we need 221

to use a time-dependent shortest path to calculate the new best path. The method used is an 222

adaptation of the algorithm proposed by Ziliaskopoulos and Mahmassani [18]. This may be 223

considered to be a temporal extension of the label correcting algorithms used to compute 224

static shortest path algorithms. 225

In order to know if the algorithm has achieved a DUE, we use the relative gap measure (RGap) based 226

on the duality gap function proposed by Janson [19], which measures the difference between the total 227

travel costs of the used paths at iteration k, and the total travel costs if all trips would had used the 228

current shortest path (��), for each OD pair and each departure time, given by the equation (2): 229

!"� =∑ #�� $% ∙ �� − ��(�� )�,�,�∈��+,-

∑ �� ∙ ��(�� ,�(2)

If the relative gap is acceptable then, the DTA algorithm ends, otherwise the flow assignment is not 230

in equilibrium and we need to improve it and we move to the next iteration. 231

If the DTA algorithm reaches the pre-established maximum number of iterations before achieving 232

convergence then, it ends. In this case, the corresponding result is the last calculated flow 233

assignment, although it is not an equilibrium solution. If the performed iterations are less than the 234

maximum number of iterations, the algorithm continues by executing a new flow reassignment. 235

We propose a modification of the Method of Successive Averages (MSA) to determine the new time 236

dependent paths flows, through the experimented travel times on these paths in the previous iteration 237

as a result of the DNL. 238

This variant of MSA needs, in addition to the current paths set for each OD pair for each time 239

interval, the new time-dependent shortest path for each OD pair for each departure time interval 240

calculated on the previous DTA iteration by using the travel cost obtained on the previously 241

performed DNL. The proposed flow reassignment procedure acts in a different way depending if the 242

new shortest paths belong to the current set of paths for the same interval or not. In the second case, 243

we add this new path to the set. 244

Hereinafter, the proposed DTA algorithm proceeds identically than in the first iteration until the 245

convergence criterion or the maximum number of iterations are achieved. 246

Throughout the following sections, we present a specification of DNL and flow reassignment 247

components of the presented DTA scheme. 248

Dynamic Network Loading 249

FIGURE 2 shows our DNL classification inspired by the representation proposed by Astarita [20], 250

who proposed a classification of the DNL mechanisms based on whether each of the proposed 251



10

procedures discretizes or not the time, the space and the demand. FIGURE 2 represents the models 252

in a three-dimensional space where the axes x, y and z are the time, the space and the demand, 253

respectively. Thus, the value zero represents the continuous models, which do not make any type of 254

discretization; while advancing on one of the axes towards infinity represents a discretization of the 255

variable of that axis. 256

This classification identifies the following modeling approaches proposed to perform the DNL 257

procedure, in particular for carrying out the link flow propagation: microsimulation models (car-258

following models and cellular-automaton models); space-continuous time-discrete models, 259

continuous or discrete in time link models, models following a packet approach and macroscopic 260

simulation models. 261

262

263

FIGURE 2 Proposed classification of based-on simulation DNL models adapted from Astarita [20]. 264

265

Because the objective is to develop a new DNL model as one of the main components of the 266

proposed iterative DTA procedure, it is necessary to have an efficient process of DNL, that needs 267

computational times as short as posible, even for the case of working with medium-sized or large 268

traffic networks. And, it is also very important to try to reach a balance in regard to the number of 269

parameters without losing the ability of suitably describing the traffic flow dynamics. On one side it 270

is the handicap of the calibration: the more parameter values dependent on the network more difficult 271

will be found the most suitable to reproduce reality. On the other hand, without using any input 272

parameter probably be difficult to reproduce traffic behaviors that should not be overlooked and that 273

are about network characteristics, the geographical location, etc. 274



11

Taking into account the above requirements in the selection between different DNL approaches into 275

a DTA model, we can rule out models based on macroscopic simulation. These models, which are 276

very suitable in terms of execution times, do not reproduce traffic behaviors needed when the 277

proposed DNL model is embedded into a DTA scheme. Furthermore, microscopic simulation models 278

are also routinely discarded. Microscopic procedures usually require very high computational time, 279

besides requiring a large number of input parameters that describe traffic behavior, which in practice 280

are difficult to calibrate. 281

Thus, the most suitable option seems to raise the use of models that combine the best properties of 282

both: microscopic and macroscopic models, i.e., a mesoscopic model. Our scope is to develop 283

models whose foundations are close to those of microscopic models, which are the best reproducing 284

reality, but dispense with some of the microscopic details. So, we try to achieve better execution 285

times and we substantially reduce the number of input parameters needed simlifying the subsequent 286

calibration effort while not sacrificing the essentials of flow dynamics. 287

The proposed DNL considers time in a continuous form, staying away from the usual way of 288

working of microscopic traffic simulation models that discretize time in steps of predefined size. 289

Another difference between the proposed model and microsimulation models is given by the discrete 290

consideration of the space rather than continuous, except for the case of cellular automaton models 291

which work with cells. Thus, in our case, there is not explicit control of what happens with the 292

vehicles inside the links. The model only pays attention to specific points of the link that are essential 293

to define the network loading correctly. However, the proposed model keeps microscopic models 294

key feature: the demand discretization, that is considered vehicle to vehicle. Thus, we locate the 295

proposed model at the point (0, "0�1234�ℎ, 1) into the previously presented scheme classification, 296

as shown in FIGURE 2. 297

Like some microscopic models based on car-following theory or cellular-automaton, in the proposed 298

model vehicles move trying to maximize their speeds in the presence of certain constraints. These 299

constraints ensure that vehicle trajectories satisfy position, speed and acceleration bounds, in 300

particular, trying to avoid vehicular collisions. 301

Thus, in the DTA model presented in this work, the DNL problem is solved using this new traffic 302

simulation model presented in more detail in Linares et al. [21] and summarized in what follows. 303

This model is based on a mesoscopic scheme that considers continuous-time link-based approach 304

with a complete demand discretization. 305

The use of different vehicles classes in the problem is allowed by the disaggregated treatment of the 306

demand. In addition, this model also allows the longitudinal discretization of the links in lanes, as 307

one of the goals is to reproduce the traversal movements produced by vehicles changing its lanes that 308

can considerably augment the link congestion. Therefore the proposed DNL is a multilane multiclass 309

traffic simulation model. 310

The DNL problem is considered from a discrete demand point of view, so, it is formulated by 311

defining a function for each vehicle, which from a specific link position 6, returns the time which the 312

vehicle in question reaches that position (�789: (x)) and when leaves it (�789� (x)). 313

The objective of the DNL is to calculate traffic variables for each link from the time dependent flow 314

assignment at each of the network paths. This calculation pivots on the knowledge of the input and 315

output times of the vehicle at each of the links of the network. So, the proposed model considers at 316

each lane of a link two positions (the initial and the final) in which it should evaluate the functions 317 �789: (x) and �789� (x). 318



12

From the above considerations, the development of the DNL model always follows the same method. 319

Given a certain link of the network, the model calculates the following times for each vehicle: 320

• Arrival time to the link initial position (�789: (6;)) 321

�789: (6;) = �789� �6�(02<=�>?1=3�)� + �AB�;;C��8 + �AB�;;DEFG8HH • Departure time from the link initial position (�789� (6;)) 322

�789� (6;) = I"6J�789: (6;), �KB88LFE + M789N • Arrival time to the link final position (�789: (6�)) 323

�789: (6�) = I"6 O�789� (6;) + �AB�;;LFE + �LGP, �HQRST�U�� (6�) + LVQRSWV + M789X 324

• Departure time from the link final position (�789� (6�)) 325

�789� (6�) = I"6Y�789: (6�), �A:E�:;; + M789Z where: 326

6; Link initial position. 327 6� Link final position. 328 [H Speed in link \. 329 1789 Length of that vehicle. 330 M789 Reaction time of that vehicle. 331 \789A�U� Vehicle leader by departure lane. 332 �AB�;;C��8 Penalty time for transferring the node. 333 �AB�;;DEFG8HH Time that this vehicle takes to cross the link first cell lenght at the 334

maximum allowed speed in it. 335 �A:E�:;; Time when node allows vehicle to pass. 336 �AB�;;LFE Time that this vehicle takes to cross link \ without taking into account the 337

first cell. 338 �LGP Delay applied to the vehicle that will depend on the density of crossed 339

lanes and on the number of lanes that the vehicle needs to cross 340 �KB88LFE Time when the link has enough free space to accommodate the vehicle. 341

��]� Lane through the vehicle goes out of the link. 342

The model is solved using an algorithm that is a discrete-event procedure. During its 343

development, different events have been designed and implemented. Some of these 344

events are related with the different times when the vehicle arrives to or departs from 345

certain specific link positions (initial and end). Moreover, other events are directly 346

related with the traffic simulation (start or end) and with the entry and exit of a vehicle 347

in the network. Finally, some complementary events have been needed to facilitate the 348

implementation of the process and to improve the time computational results. 349

For a more detailed description of this mesoscopic model and its corresponding 350

validation experiments, we refer the reader to Linares et al. [21]. 351


13


Flow Reassignment 352

The objective here is to develop an efficient DTA model. To achieve this aim, it is 353

fundamental to implement a method of flow reassignment that converges efficient to the 354

DUE. First of all, it is important to note that not all flow reassignment algorithms 355

achieve the expected equilibrium. Only the methods that implicitly assume that the 356

network traffic conditions are predictable (preventive methods) are able to achieve it 357

(Friesz [22] and Chiu et al. [23]). On the other hand, it would be reactive algorithms that 358

assume that network traffic conditions are not predictable and where the users can make 359

decisions while they are en-route, based on their real-time information about the current 360

traffic conditions. Due to the objective of this work, we propose a preventive flow 361

reassignment algorithm which bases its decisions on its historical experience. 362

Among the different preventive approaches existing in the literature, in this paper we 363

use the iterative flow reassignment algorithm proposed in Linares et al. [24], based on a 364

modification of the Method of Successive Averages (MSA). This is one of the most 365

widely used methods for path flow reassignment component in a DTA scheme because 366

is one of the most simple and efficient. 367

The aim of the proposed modification is to overcome the limitations observed during 368

the state-of-the-art study about DTA with MSA. We focus our efforts on solving two 369

drawbacks: 370

• The expensive requirement of memory space of the usual implementation of the 371

MSA algorithm. 372

• The indiscriminate flow diversion from each used paths to the new best path, 373

typical of the standard MSA not taking explicitly into account travel time 374

conditions 375

In order to improve the currently available options to solve these two limitations, we 376

develop a new MSA that combines some of the literature modifications with the 377

addition of new ones. In order to solve the first problem, we consider the general 378

scheme formulated by Mahut [25], which is based on the idea of limiting the number of 379

paths for each origin-destination pair for each departure time interval ( �̂��). To 380

overcome the second limitation, we propose the use of a diversion factor (_��) in 381

order to perform the reassignment by taking into account the cost of the alternative 382

paths. This new factor is based on a logit distribution of demand flow according to 383

actual travel costs of the alternative paths. The method considers the costs based on the 384

links actual travel times obtained in the DNL of the previous iteration of the global 385

procedure. Thus, 386

_�� = exp(−��(��))∑ exp(−��(��))�c��

where �� is the cost of the path based on the actual travel times �� of the path. 387

The specific proposed algorithm is shown in the following scheme: 388

389


14


For all OD pair (�, �) for all departure time interval�: 390

• If the maximum number of paths is not achieved �d�� $%d < �̂��: 391

a. Time-dependent shortest path (��) calculation based on the links costs 392

according to the actual link travel times obtained in the last DNL. 393

b. Path Flow Reassignment: 394

- If the shortest path is new �� ∉ �� $%�: 395

i. Assign the flow �� ∀ ∈ �� $% and on �� following: 396

�� = O h · �� =� = ��(1 − h ) · �� · _�� =� ≠ �� ii. Update paths set: �� = �� $% ∪ �� 397

iii. Update number of paths. 398

- Else �� ∈ �� $%� 399

i. Assign the flow �� ∀ ∈ �� $% following: 400

�� = kh · �� + (1 − h ) · �� $% =� = ��(1 − h ) · �� $% =� ≠ ��

( 2 )

ii. Update paths set: �� = �� $% 401

• Else, the maximum number of paths is achieved �d�� $%d ≥ �̂��: 402

a. Identify the shortest path �� among those already used paths �� $%. 403

b. Path Flow Reassignment: 404

i. Assign the flow �� ∀ ∈ �� $% following: 405

�� = kh · �� + (1 − h ) · �� $% =� = ��(1 − h ) · �� $% =� ≠ ��

( 3 )

ii. Update path set: �� = �� $% 406

where: 407

� is the iteration count. 408

�̂�� is the maximum number of paths considered from origin � to destination 409 � entering to the network during the time interval �. 410

�� is the set of paths from origin � to destination � entering to the network 411

during the time interval � at iteration �. 412

�� is the time-dependent shortest path from origin � to destination � 413

entering to the network during the time interval �. 414


15


�� is the flow assigned to the path departing at time interval � at iteration 415

�. 416

�� is the demand from origin � to destination � entering to the network 417

during the time interval �. 418

h is the MSA parameter according to the corresponding iteration �. 419

_�� is the previously defined diversion factor. 420

COMPUTATIONAL EXPERIENCES 421

Real Urban Test Network 422

In order to test the behavior of the developed DTA model, it was applied to the network 423

of Amara Berri district in the city of San Sebastian (Spain). The proposed test network 424

(FIGURE 3) consists of 76 nodes, 192 links and 301 turns. The maximum allowed 425

speed goes from 35 Km/h to 60Km/h depending on the link type. Each link has its 426

corresponding longitudinal division by lanes among 1 and 5 lanes. The performed 427

experiment used a synthetic demand that assigned to the network the total flow of 428

12.116 trips. Four 15-min matrices provided the origin-destination demand data for 13 429

zones resulting in 80 OD pairs. Two vehicle classes were considered: the 90% of the 430

demand corresponded to light vehicles with an effective length of 5 meters, while the 431

remaining 10% of the demand corresponded to a heavy vehicle class with an effective 432

length of 9 meters. 433

Calibration and Validation 434

A common practice in testing most of the existing mesoscopic models has been to make 435

two types of comparisons: 436

437

• Check the ability to reproduce traffic flow behavior in terms for instance of 438

how do they fit the expected fundamental diagram for critical links, and 439

• Use some “ground truth” to get a global insight of how the predicted flows, 440

travel times and so on fit those taken as external reference. 441

In some cases (Astarita et al. [26]) this second type of tests has been conducted in a 442

synthetic experiment against a microscopic simulation model. This has been the 443

calibration and validation alternative used in our case using as reference a microscopic 444

Aimsun model [27]. 445

The ability of the new DNL model to suitably reproduce the fundamental diagram was 446

checked in testing exercises reported in Linares et al. [21], due to space limitation we 447

will refer here only to the second type of validation exercises. 448

A calibration procedure was conducted for the proposed DTA model, which 449

fundamentally consisted of adjusting the following DNL parameters: the lane change 450

penalty (�LGP), the cross node time penalty (�AB�;;C��8) and the reaction times of both 451

vehicle classes. The final adjusted values for the penalty parameters were: �LGP =452


16


0.2sec, �AB�;;C��8 = 0.6sec, while the considered reaction times were: light vehicle 453

class = 0.5?2� and heavy vehicle class = 0.75?2�. 454

In addition, at the initialization step, the proposed DTA process requires a set of shortest 455

paths for each origin to each destination. As we previously explained, the decision of 456

the number of paths for each OD pair is a key point on the way to achieve the 457

convergence. The best option to get right is to test different options during the network 458

calibration process. In our case for the test network the best and more stable results, 459

avoiding flip-flop oscillations between paths, were achieved with 3 alternative paths for 460

each OD pair. 461

462

FIGURE 3 Results of the validation process. 463


17


FIGURE 3 depicts the results of the validation process in terms of the comparison of 464

the link densities for the micro and meso models and the relationship between vehicles 465

per link at the bottom. The results obtained for both models (proposed and benchmark) 466

are very similar, excepting the congestion caused by roundabouts. Proposed model 467

overestimates congestion at the roundabouts, causing significant differences in some 468

adjacent links compared with the results obtained from the benchmark. The study of the 469

errors shows the correctness of the proposed model obtaining a RMSE = 8.84 and a 470

Normalized RMSE = 6.42%. 471

The developed DTA was initially run by using three initial paths for each OD pair for 472

each interval. The 60-min loading interval was divided into four 15-min time intervals 473

for the proposed DTA model, which was run successfully for 30 iterations, each 474

requiring roughly 1 minute. 475

The relative gap (RGap) was calculated in order to take into account the different 476

departure time intervals in the analysis of the results. FIGURE 4 shows the obtained 477

RGaps values for four departure intervals ranged from 2 to 3.5 percent at the last DTA 478

iteration. 479

480

FIGURE 4 Relative Gaps Results. 481


18


CONCLUSIONS 482

The DTA model presented in this paper is based on the conceptual approach splitting 483

the process in two components: a path flow reassignment and a Dynamic Network 484

Loading (DNL). The path flow reassignment has been implemented in terms of a 485

modified MSA algorithm designed to solve the variational inequalities formulation of 486

the dynamic equilibrium formulation of the assignment problem. The DNL follows a 487

new modeling approach to emulate the flow propagation through the network along the 488

paths defined by the path flow reassignment. 489

The model has been calibrated and validated through a set of synthetic experiments 490

taking as reference an Aimsun microscopic simulation model of a real network. The 491

validation shows the ability of the DNL model to acceptably reproduce the basics of the 492

flow dynamics, as well as the capability of the model to achieve a DUE measured in 493

terms of the Relative Gap function. The validation process has been complemented 494

checking the similitudes between the link flows provided by both models and the 495

analysis of their differences. 496

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A Mesoscopic Simulation-Based Dynamic Traffic Assignment Model