Chapter 4
Modeling and Control of Traffic Flow1
4.1. General introduction
In the White Paper — European Transport Policy for 2010: Time to Decide1, we are told that:
“Because of congestion, there is a serious risk that Europe will lose economic competitiveness. The most recent study on the subject showed that the external costs of road traffic congestion alone amount to 0.5% of Community GDP (gross domestic product). Traffic forecasts for the next 10 years show that if nothing is done, road congestion will increase significantly by 2010. The costs attributable to congestion will also increase by 142% to reach 80 billion Euros a year, which is approximately 1% of Community GDP.
With respect to sustainable surface transport, the following actions were envisaged… increasing safety, and avoiding traffic congestion (in particular in urban areas), through the integration of innovative electronics and software solutions and by means of the use of advanced satellite navigation systems and telematic solutions.”
The European Commission’s white paper emphasizes the fact that the limited capacity of road infrastructures available to users, and constantly increasing demands with regard to traffic2, goods, and people, are the source of numerous problems encountered by highly industrialized societies. These problems, related mainly to recurring or non-recurrent congestion phenomena in large metropolitan areas in particular, are an obstacle to the socioeconomic evolution of our societies and are resulting in ever-growing direct and indirect costs for the population. The solution to the phenomenon of congestion has become one of the main concerns of transport centers.
One of the most natural solutions is the building of new infrastructures. This is not an easy solution, however, as it requires a considerable financial investment. Public authorities are often faced with a lack of space available for the construction of new roads, or are confronted with resistance from the people directly affected by these building projects.
The necessity to create traffic flow models has quickly become apparent in order to better manage these situations. A wide range of simulation tools has been developed on the basis of these models, aimed at helping road network operators ensure better traffic management. Indeed, dynamic simulation tools allow us to evaluate the impact of an operation’s action on traffic conditions and to propose effective management strategies in order to optimize infrastructure yield. These tools aid in stochastic and operational decision-making.
4.1.1. Different models of road traffic flow
Lighthill and Witham [LIG 55] presented the first flow model, based on the similarity between road traffic flow and the circulation of fluids in conduits. This analogy gave rise to a group of macroscopic models. Generally speaking, macroscopic models are dedicated to the prediction and assessment of traffic and to the regulation at the overall network level. However, their aggregate character prevents them from taking into account the movement of individual vehicles and the interaction of these vehicles with their environment.
Other types of models are able to address the individual behavior of vehicles and their interactions. These are called microscopic models. This class of model is used more frequently in traffic simulation. However, the large number of variables and parameters to be taken into account makes their use in the representation and simulation of a large-scale road network very difficult and costly.
There is also another category of models that studies the behavior of vehicles without explaining their interactions. These are called mesoscopic models. In this category of models, vehicles are grouped into batches, the movements of which are calculated using a macroscopic model.
The main difficulty encountered in traffic modeling has to do with problems of scale. If we consider an urban network, it is generally made up of quick routes, which necessitate macroscopic modeling, and crossroads, which usually require microscopic modeling. Problems of scale are related to space and time. They can be simply illustrated by considering a large road network in which the fluidity of traffic is diversely represented according to the part being considered. A very long section of road can be in a state of congestion with travel times being measured in hours, while another, shorter section of road is in a fluid state characterized by travel times measured in seconds. This situation cannot be effectively represented using the aforementioned models, which cannot simultaneously take these different particularities of traffic into account.
To resolve these problems of scale and make up for the insufficiencies of the traffic models cited above, hybrid models have recently appeared. This type of representation must, however, guarantee the preservation and continuity of flow during changes from one model to another.
4.1.2. Classification criteria for road traffic flow system models
Road traffic flow models offer an effective method for describing the phenomena and behavior of the flow of vehicles within urban and interurban networks. The variety of models developed to date requires categorization in order to best judge their ability to adapt to the problem being addressed. Several criteria have been proposed to distinguish traffic models from one another [HOO 01]:
– representation of variables: traffic flow models describe dynamic and complex systems in which time is the main variable. Two categories of models exist side-by-side, depending on whether time is taken as a continuous or discrete variable;
– representation of the process: modeling is based on the utilization of a group of variables and parameters that may be random. Representation models can thus be stochastic or deterministic; and
– level of detail: the degree of granularity plays an essential role in the classification of traffic models. This criterion is generally used to distinguish between the different models. A high level of detail allows us to describe the individual behavior of vehicles, a characteristic of microscopic models. An intermediate level of detail corresponds to mesoscopic models. Finally, macroscopic models are characterized by a low level of detail.
Traffic models are interesting because they allow us to represent the behavior and phenomena of traffic flow, yet they also constitute an incontrovertible support in the creation of simulators for the study and evaluation of traffic system performance. The reader can find additional information in [LIE 02].
Below, we will present these models according to their level of detail: first microscopic models, and then macroscopic ones. Mesoscopic models are not addressed in this chapter; however, the reader can find additional information in [PRI 61, BUC 68, PRI 71, PAV 75, BRA 76, HEL 97, HOO 98, HOO 99, HOO 00].
4.2. Microscopic models
Road traffic is a complex system by nature. It is very difficult to establish a model that will precisely describe the behavior of each of the vehicles in circulation in a road network. Generally, the models proposed in the literature are relatively simple. They can be grouped into two main categories. The first category includes car-following models; the second includes cellular automata models. We will describe each of these categories below.
4.2.1. Car-following models
There are four types of car-following models: the safety distance model; the optimal speed model; the stimulus-response model; and the psychological model.
4.2.1.1. Safety distance model
The model based on safety distance describes the dynamic of a vehicle with regard to the behavior of the vehicle in front of it. The simplest model was conceived on the basis of the following rule: “The safety distance of a vehicle n driving at a speed of 16.1 km/h must be equal to or greater than the length of this vehicle” [PIP 53]. Using this rule, we can deduce the safety distance that separates the nth vehicle driving at speed v from the (n−1)th vehicle in front of it:
where Ln is the length of vehicle n. In the Pipes model, the safety distance is proportional to speed. The same principle was proposed by Forbes [FOR 58]. Another formulation was proposed by Leutzbach [LEU 88], who introduced the idea of overall reaction time τg. This parameter has to do with:
– perception time, which is the time necessary for the driver to collect information about the environment, and more precisely, information about the obstacles facing him or her;
– decision time; and
– the braking time necessary for the driver to slow down or stop.
Leutzbach defined braking time in terms of a sufficient distance for a vehicle to come to a complete stop. This definition takes into account the driver’s reaction time and the maximum deceleration of the vehicle, which depends on a friction variable μ and gravity g. The safety distance is defined in terms of the maximum braking distance, which is defined by the following relationship:
The Forbes model was improved by Jepsen [JEP 98], who introduced the idea of the minimum distance dmin that a vehicle must respect, and a risk factor due to speed v. The risk factor translates a driver’s capacity to avoid a collision, or at least to limit its impact. This capacity is shown by the increase in the security distance by a linear factor f in relation to speed v.
The minimum distance dmin is the distance that separates two completely stationary vehicles during a traffic jam.
4.2.1.2. Optimal speed model
Proposed by Newell [NEW 61], this model supposes that a driver adapts his/her speed to an optimal speed v0 depending on the distance separating him/her from the vehicle in front. This model is expressed by the following relationship:
In equation [4.4], the reaction time is replaced by the braking time Δt of the driver. Initiating a limited firstt-order development, Bando [BAN 95] proposed an extension of the Newell model, described by the relationship:
In equation [4.6], v0 and dc are two optimal speed parameters. Note that, in this model, small disturbances may arise in certain conditions that can lead to traffic congestion.
4.2.1.3. Stimulus-response models
This type of model is based on the existence of a relationship between an action (the driver’s behavior) and a stimulus in the following form:
In general, a stimulus can be the distance that separates two vehicles or the relative speed of two vehicles when one is following the other, and the response can be the braking or acceleration of the following vehicle after a certain amount of time Tp. One of the first models of this type was proposed by Chandler [CHA 58], which supposes that the acceleration of a vehicle n at position xn(t) following vehicle n−1 is given by the following relationship:
where vn(t) and an(t) are the speed and acceleration, respectively, of vehicle n at time t, and γ is sensitivity. In this case, the stimulus is expressed by the difference between the speeds of the two vehicles. Gazis [GAZ 61] expressed the sensitivity γ of the driver by the following relationship:
Thus, vehicle n adjusts its speed vn(t) with respect to the distance and the difference in speeds with a delay, Tp.
Though they describe the behavior of vehicles in a microscopic environment, stimulus-response models, particularly those utilizing the Gazis model, fall far short of representing a driver’s real behavior. These models do not take into account the variability of traffic flow or vehicles, which are all characterized by the same behavior.
4.2.1.4. Psychological models
Using equations [4.7] and [4.8], we can deduce that a driver will react to small changes in speed (vn−1(t) −vn(t)) even if the distance is very large. Likewise, the driver’s reaction is zero if the difference between speeds is zero, even if the distance between the two vehicles is very small.
To show that the driver is limited in his/her perception of the stimulus to which s/he must react, several improvements based on perceptive psychology have been made [TOD 64]. These extensions are based on the following rules:
– when the distance that separates two vehicles is very large, the driver’s reaction is no longer limited by the fluctuations in speed of the vehicle in front;
– when the distances are minimal, some combinations (speed and distance) between the vehicles do not affect the behavior of the following vehicle, since the movements are relatively small.
One of the first psychological models was proposed by Wiedemann [WIE 74], in which he makes the distinction between limited and unlimited driving. This distinction is based on the perception threshold and takes into account the laws of lane change. This psychological model has served as the basis for the development of several microscopic traffic flow models, notably another model proposed by Koppa [KOP 99] and Rothery [ROT 99], which supposes that the angular resolution of the human eye is limited.
4.2.2. The cellular automata model
The cellular automata model is an effective tool for describing the dynamic and complex behaviors of traffic in detail. The primary model is based on a one-dimensional vector with L cells. In general, the size of a cell is 4.5 meters [NAG 92]. The length of the cell is chosen so that the vehicle passes from one cell to the next in a single step in time. Each cell is either occupied by one vehicle or is empty. Each vehicle possesses a total speed between 0 and vmax, which represents the number of cells crossed at each step in time. The state of the cell is defined as equal to −1 if it is unoccupied, and to speed v if it contains a vehicle. The number of empty cells in front of a vehicle is written as Δx. The system is updated by following the stages below, applied simultaneously to all of the vehicles in the vector:
– if v < vmax then v = v + 1. This rule represents a linear acceleration of the vehicle until it has attained its maximum speed;
– if v > Δx then v = Δx.3 This rule ensures the deceleration of the following vehicles in order to avoid collisions; and
– if v > 0 and p < pbrake then v = v−1. A random generator is introduced in order to slow the vehicles down. This rule models certain random driver behaviors as delayed accelerations and fluctuations in stable speed.
Thanks to these simple rules of movement and the ability to carry out parallel implementation, cellular automata are considered to be rapid simulation tools. They have been used in simulation as well as in allocation and supervision. The first single-lane route model was developed by Nagel [NAG 92] and was broadened by the same author into a multilane and multiclass model [NAG 99]. In order to make cellular automata more realistic, Wu [WU 99] directed the rules of movement toward the law of pursuit. Though still not perfect, validation work carried out on German and American highways [NAG 99, WU 99, ESS 99] showed that cellular automata perfectly reproduced the large-scale macroscopic behavior of road traffic flow.
4.3. Macroscopic models
Macroscopic models are based on an analogy with fluid dynamics. These models liken traffic flow, which is supposed to be homogeneous and one-directional, to a liquid in a conduit.
In this context, liquid flow is characterized by the following three main variables:
– outflow q(x,t) expressed in the number of vehicles per unit of time;
– density ρ(x,t) in the number of vehicles per unit of length; and
– speed v(x,t) in units of length per unit of time.
Using these three variables, two equations have been established to describe the evolution of traffic flow. The first equation, [4.9], expresses the law of conservation of mass:
The second equation, [4.10], is a balance equation linking outflow, density, and average speed of flow:
These two equations must be completed by a third equation in order to completely describe the evolution of traffic. This equation is used to distinguish two categories of macroscopic traffic flow models. The first equation designates first-order models, called LWR-type models, and the second equation defines second-order, or superior-order, models.
The following section provides an overview of these two categories of macroscopic models.
4.3.1. LWR-type first-order models
One of the first macroscopic models was proposed by Lighthill, Witham, and Richard, from whom the name LWR is derived [LIG 55, RIC 56]. This model uses the conservation equation (equation [4.9]) as well as the relationship between outflow, density and average speed (equation [4.10]). These two equations are completed by a speed equilibrium equation with respect to density, veq= v(ρ(x,t)). This last equation supposes that traffic is always in equilibrium and that it evolves by passing from one state of equilibrium to another. The speed equation in the equilibrium state, called the fundamental diagram, is empirical and is used to characterize the infrastructure on which vehicles are circulating (Figure 4.1).
Greenshields [GRE 34] proposed the first analytical relationship (equation [4.11]) of a fundamental diagram:
where vf is the free speed of flow and ρmax is the maximum density. This relationship allows us to trace different representations of the fundamental diagram, as well as the dependence between outflow and density (Figure 4.1), between speed and density, or between speed and outflow:
– when traffic is fluid (i.e. density is close to 0), interactions between vehicles are minimal. The vehicles then drive at their maximum desired speed, which results in a limit on the speed of the stream, written as vf;
– density is limited by a value ρmax that designates the maximum number of vehicles at a complete stop that a road can contain; and
– the maximum value of outflow qmax designates the capacity of the section being studied.
Analysis of the fundamental diagram in Figure 4.1 shows that it is composed of two parts. The first part is typical of the fluid circulation system where, for low density values, the outflow is weak. Beyond a so-called critical density, ρcr, the density increases causing saturation of the section of road being studied. This saturation results in reduced outflow and thus reduced speed. It corresponds to the second part of the fundamental diagram, which is characteristic of a congested system. When the density attains the maximum value, the outflow and speed cancel each other out and the system is at a standstill.
The linear relationship proposed by Greenshields leads to a critical density equal to half of the maximum density, while experimental measurements show that the critical density value is generally between 0.3ρmax and 0.2ρmax. In order to address this inconvenience, other relationships have been proposed, notably by Buisson [BUI 96] and Papageorgiou [PAP 98]:
Where ρmax is the maximum density, 
are parameters where l > 0, and m ≥ l is used to obtain different fundamental diagrams.
Other models have been proposed that allow us to more or less faithfully reproduce the observations made of different sections of road, notably May [MAY 90], who proposed another relationship:
where a is a curve adjustment parameter.
In Figure 4.1, we can see that the real measurements at equilibrium are spread out, particularly in the congested zone. Models of fundamental diagrams, therefore, remain rough approximations of reality.
The solution to these equations can be found using an analytical approach or a numerical approach. When an analytical solution exists, it offers the advantage of being exact, but it is often difficult to implement. The numerical approach, on the other hand, is generally easier to find, but it is inconvenient because it only provides an approximate solution.
The analytical solution to traffic flow equations is based on the method using defined characteristics, such as lines of plane (x,t) and gradient, 
. The solution is obtained by drawing these lines in the plane (x,t) from initial conditions to form a density field. When these lines meet, a discontinuity is created, which gives rise to shock waves. The solution can be non-unique if the characteristics diverge. This is why we then have recourse to entropic solutions. These consist of only accepting shock waves for which the density upstream of the critical density is lower than that downstream. Any discontinuity, where the density upstream is greater than the critical density downstream, gives rise to a fan (for more details, we refer the reader to [GOD 90] and [ANS 90]).
There is also a third approach, called the “particulate solution” [LEC 02]. This consists of breaking down the flow into basic particles representing vehicles. These particles move towards a state of equilibrium, rather than following each other. Density is approximated at every point by the inverse of the distance that separates two vehicles that are following each other, 
, where d is the distance that separates two vehicles. In the case of a section with several lanes, the relationship is expressed by 
(for each lane i). This method of solution is very effective in the treatment of certain problems, such as the assessment of road noise [LEC 02].
It is interesting to emphasize the importance of studying the solution to the LWR model in the case of the Riemann problem. This allows us to illustrate the way in which shock waves propagate, and to introduce the ideas used in the numerical solution to these equations. The Riemann problem consists of solving the LWR model in the case of specific initial conditions. At t = 0, we are considering a section of road for which the density is defined by:
The solution to this problem shows that depending on the values of ρamont and ρaval, when x = 0 a fan or a shock wave can form:
First-order models present some advantages, such as:
– reliability: they guarantee that the variables q and ρ maintain consistency at all points and at all times with the physical limits of the network (qmax, ρmax);
– existence of the analytical solution. This enables the theoretical study of flow behavior by exactly and easily calculating solutions for simple scenarios; and
– existence of expansion. This type of model has attracted the attention of several researchers [DAG 02, LEB 98, NEW 98], etc., as part of the modeling of traffic behavior in urban environments.
These models also have major inconveniences:
– they do not correctly model transitional phases; and
– they do not consider speed as a fundamental variable (which, moreover, is not defined in the discretized version of the model).
In order to remedy these insufficiencies, research [ZHA 02] has been conducted that has led to the development of another type of model that is capable of addressing the phenomena that occur outside a state of equilibrium. These models are called second-order, or superior-order, models.
4.3.2. Superior-order or second-order models
Second-order macroscopic models were developed in order to address the inadequacies of first-order models, and to take into account the dynamic aspects of traffic [LEC 02].
From this perspective, several authors have proposed extensions based on the dynamic flow acceleration equation defined by:
where:
– A is a term of relaxation toward the speed of equilibrium; and
– B expresses the individual behavior of vehicles. The form of this parameter allows us to distinguish the different second-order macroscopic models from one another.
, the general from of supervisor-order models is:
The rewriting of this system of equations in a classic form leads to:
where:
When 
(c being the characteristic velocity of the traffic), we find the first model of this type, which was proposed by Payne [PAY 71]:
where τ is the reaction time.
Anticipation expresses the individual behavior of drivers. In fact, each driver adapts his/her acceleration in response on the density gradient. In this way, s/he can slow down or accelerate depending on whether the zone is dense or fluid.
Though the Payne model corrects some inadequacies of the LWR model, it suffers from several inconveniences itself, such as:
– the reaction time, obtained from empirical data, can have very large values (for example one minute) to enable a physical interpretation;
– the loss of reliability during changes in the geometry of the network. Measurements taken of traffic variables, density or outflow can lead to values greater than those imposed by the characteristics of the network being studied. Moreover, the Payne model predicts negative speeds in certain situations. This phenomenon is known as wrong-way travel; and
– the behavior of a driver is influenced not only by the cars in front of him/her, but also by those behind. This does not correspond to reality, since the flow of traffic is anisotropic; i.e. vehicles are only sensitive to the changes that occur in front of them. This is why the solutions to this model prove to be utopian in some cases. Daganzo has also proposed an example showing that vehicles stopped in a waiting line “draw back”.
However, this last item involves not only criticisms of the Payne model; it is at the origin of a new generation of superior-order anisotropic models. These models are often written in the form of a system of hyperbolic equations that offer the advantage of having known analytical solutions. The solutions depend on the eigenvalues λ1 and λ2 of the matrix A(U) (equation [4.20]), which represent the propagation speeds of the information.
The solution to these superior-order models (defined by equations [4.17] and [4.18]) is analogous to that of the LWR first-order models described by a system of scalar hyperbolic equations. However, the formulation of the Riemann problem leads to the study of the hyperbolic system without its second term, since its effect can be disregarded. Depending on the eigenvalues of matrix A(U), the solution to the Riemann problem leads to shock waves, fans or discontinuities. We can notice from the eigenvalues of matrix A(U): λ1= v − c(ρ) and λ2= v + c (ρ); that λ1 < v < λ2. This clearly shows that information spreads faster than vehicles in
these models, and this proves to be completely unrealistic [AW 01]. The reader will find a more in-depth analysis of these criticisms in the articles by Aw and Zhang [AW 01, ZHA 00].
In this context, in the same year three pieces of research were carried out simultaneously without any of the authors citing each others’ work. The matrixA(U) of these three models [AW 01, ZHA 02, JII02] has the following form:
Only the coefficient c(ρ) distinguishes the three models:
– Aw and Rascle: c(ρ) = −ρp′ (ρ), where p is a function of ρ;
– Zhang: c(ρ) = ρv′eq(ρ);
– Jiang, Wu and Zhu: c(ρ) = −c0.4
The eigenvalues of this matrix are λ1 =v and λ2=v + c(ρ) where c(ρ) is always negative, λ1 = v and λ2 < v. These models respect the anisotropy of the traffic flow. Note that the first model (Aw and Rascle) is a general form of the other models. Taking p(ρ) = αρα, we get another form of the model proposed by Aw and Rascle.
Second-order models present the advantage of addressing transitional phenomena during variations in traffic flow. By correcting the Payne model by taking into account the anisotropic character of traffic, they allow us to avoid the prediction of negative speeds and to ensure the proper propagation of information. Unlike first-order models, these models consider speed as a principal variable, which facilitates the study of vehicle kinetics.
However, second-order models have several inconveniences of their own. They do not allow us to study the transitional phases of acceleration and deceleration. Moreover, their complexity makes the analytical solution of their equations very difficult, even for simple cases of traffic modeling. In addition, the taking into account of network discontinuities (the appearance of obstacles, variations in the geometry of the network, etc.) is often misrepresented. Finally, Papageorgiou [PAP 98] confirms that second-order models do not improve on the LWR model in any way. LeRoux [LER 02] affirmed in 2002 that all attempts to improve the Payne model have been in vain, and that the problem of modeling traffic is still present today. We can therefore note the most significant inconveniences of second-order models. These are that:
– they do not allow us to study the transitional phases of deceleration and acceleration; and
– they are complicated: analytical solutions cannot be found easily, even for simple cases. Second-order models are difficult to control. They do not allow us to model network discontinuities, even though these discontinuities are numerous in urban environments. In fact, the discontinuities are due to frequent variations in the number of lanes, to the existence of obstacles, etc. Knowing that second-order models do not formalize the speed of equilibrium with respect to space and time, they are therefore incapable of taking these discontinuities into consideration.
Consequently, second-order models are more appropriate for predicting the appearance of phenomena on highways than for studying vehicle kinetics in an urban environment (even if they are built on the dynamic equation of flow particle acceleration).
4.4. General remarks concerning macroscopic and microscopic models
4.4.1. Links between models
Despite the great differences that seem to separate them, traffic flow models are related. Numerous publications have shown these links between the various models:
– in their article, Klar et al. [KLA 98] showed that by using a simple microscopic model it is possible to determine the equations of the kinetics of gases used in the macroscopic description of flow;
– Van Aerde [WAN 94] studied the derivation of a microscopic model using discretization per particle. This was applied in the numerical solution of a second-order macroscopic model [HOC 88]. Hoogendoorn [HOO 00] used the particle discretization method to achieve a microscopic model using gas kinetics equations [LET 97];
– we saw in the previous section that most second-order macroscopic models, including the Payne model, are constructed on the basis of pursuit models. Table 4.1 gives several examples of this;
– Yserentant [YSE 97] presented a particulate method in order to obtain macroscopic model equations from a compressible fluid;
– Nagel [NAG 98] showed that it is possible to construct the fundamental diagram from cellular automata; and
– the application of the moments method [LEU 88] allows us to obtain macroscopic flow equations using a mesoscopic model.
It is interesting at this point to emphasize the works of Del Castillo [DEL 96] and Franklin [FRA 61]. Franklin developed a microscopic (stimulus-response) model taking into account macroscopic flow data, such as shock waves. Del Castillo proposed a law of pursuit; three parameters of which were obtained directly from the density-speed diagram. Shock waves propagate in the same manner in Franklin’s model as they do in Del Castillo’s law of pursuit.
Table 4.1. Construction method of macroscopic models
| Name of model | Method of construction |
|---|---|
| Payne [PAY 71] |  |
| Zhang [ZHA 00] |  |
| Jiang [JIA 02] |  |
| Zhang [ZHA 02] |  |
4.4.2. Domains of application of macroscopic and microscopic models
Microscopic models are often used for an offline simulation, thus allowing us to test new infrastructures (entry/exit ramps, the elimination of a lane, etc.) or new automobile equipment (driver assistance systems), or to get an approximate idea of flow data that are difficult to obtain empirically. The application of microscopic models in the real-time regulation of flow is very limited, given the enormous calculation times such models require, and the absence of an explicit model describing the relationship between entry and exit data. Moreover, they are incapable of determining the macroscopic characteristics of flow (capacity, length of waiting line) with precision. The microscopic models that have been calibrated and validated remain rare to date [ALG 97]. In general, the calibration of microscopic models consists of reproducing the macroscopic characteristics of the flow, notably the speed—density relationship, by adjusting the parameters related to vehicles or to driver conduct.
Macroscopic models are more appropriate for the development of control laws or the simulation of road flow in a large network. METANET [KOT 99] is a road traffic flow simulator built around a macroscopic model. It is used in Paris and Amsterdam to solve traffic problems in these large metropolitan areas. Moreover, macroscopic models are well adapted to the analysis and reproduction of macroscopic flow characteristics, such as shock waves and waiting lines. Even though some work has been carried out in order to generalize macroscopic models so that they take the different categories of vehicles circulating in a multi-lane infrastructure into account, for the moment this remains very limited and is rarely applicable in practice. In addition, macroscopic representations are not adapted to the study of the microscopic behavior inherent in flow, or to the study of the effects of changes in the geometry of road infrastructure (the removal of a traffic lane, for example) on traffic. Finally, the analytical solution of macroscopic models makes them better suited to tasks such as the assessment of prediction and control of traffic flow.
4.4.3. Movement toward hybrid models
Using this analysis of traffic flow models as a starting point, we would like to be able to represent the flow of traffic with a level of detail adapted to the situation and to the desired objectives. With this in mind, a hybrid approach combining macroscopic and microscopic models seems to be a pertinent solution for the representation of traffic. This modeling approach, adopted more recently by transport researchers, consists of using macroscopic and microscopic models within the same application. However, one of the key points of this approach concerns the study and creation of an interface ensuring communication between the two models. The following section presents the principal hybrid models developed to date and lists their advantages and disadvantages.
4.5. Hybrid models
Work on the hybrid modeling of traffic flow is mainly oriented toward the representation of a dynamic system composed of one continuous part and one discrete part. This modeling approach takes its inspiration from studies conducted on gases. As a material, the behavior of gas can be studied at the microscopic level by representing intermolecular interactions. In this context, the study pertains to local phenomena that govern the behavior of the gas. It can also be studied at the macroscopic level with the help of the description of its overall state (a study of global phenomena) [BOU 03]. Though these descriptions are situated at different levels, the phenomena governing the behavior of a gas are closely linked. In fact, a representation at the local level can be generalized to describe a gas at the macroscopic level. This is possible thanks to the kinetic theory of gases, which is used to create macroscopic behavior equations (Navier-Stokes equations [LET 97]) from the study of interactions between molecules.
This principle of hybrid modeling has been successfully applied in several domains, such as the growth of crystals in solution [MIZ 99] and the spread of a fracture in a piece of silicon [ABR 98], as well as in the field of industrial automation [WIL 96] and liquid flow [FLE 00]. However, in the field of traffic this principle is still at the development stage, and we have not yet got a precise definition of the idea of the hybrid model. Each author brings his or her own definition and vision to this type of model. We will therefore list four types of hybrid models:5
– The first, proposed by Chang [CHA 85], the MacroParticle Simulation Model, is used to calculate the movement of traffic either from flow, or by considering a group of vehicles that adjust their speed to reach the equilibrium speed, depending on the density of traffic on the segment of road being studied.
– The second type of hybrid model was developed by Daganzo [DAG 04]. It is used to study the movement of heavy vehicles (considered to be particles) in the form of trajectories in the flow of traffic assimilated into a liquid.
– The third type of model was developed by Helbing [HEL 00] and Hennecke [HEN 00], in which individual elements of the road network (entry/exit ramps, etc.) are described at the macroscopic level, while the rest of the elements are modeled at the microscopic level. This approach is justified by the fact that it is relatively easy to describe these singular elements using a macroscopic model, since this requires the addition of only one term to the model equation, while the choice of a microscopic model requires the behavior of individual vehicles to be taken into account, which makes the description more difficult.
– The fourth type of model consists of the hybrid modeling of road traffic flow. It is described in more detail later in this section. In this category, Magne, Poshinger and Bourrel [MAG 00, POS 02, BOU 02a] have proposed models intended to solve the problems of scale inherent in the description of traffic according to two perspectives — microscopic and macroscopic. In this context, the principle consists of modeling the elements of the network that do not require too much detail at the macroscopic level, and focusing on the parts that are sensitive to sudden and intermittent changes by studying them at the microscopic level. Thus, we are able to reduce the number of vehicles addressed while ensuring a clear and detailed description of singular elements, such as intersections, filter lanes on and off the highway, etc.
Generally speaking, the main difficulty encountered in the development of this type of hybrid model involves ensuring the transmission of information between two different models of traffic representation. The principle consists of translating microscopic data characterized by vehicle position and speed variables into the terms of macroscopic data determined by parameters such as average speed, density, and outflow of road traffic.
All of these models are distinguished from one another by the choice of models used as well as by the linking procedure. Note that the Bourrel model is considered a generalization of the other models proposed by Helbing, Magne and Poshinger, since it combines the various methods of interfacing while improving them at the same time.
4.5.1. The Magne model (MicMac)
The model proposed by Magne (MicMac) [MAG 00] is based on the joint use of a second-order macroscopic model called SIMERS [MES 90], which is a discretized version of the Payne model and of a microscopic (pursuit) model called SITRA-B [GAB 98].
The work by Magne et al. [MAG 00] emphasizes compatibility between the two models. By observing experimental flow/density data and the fundamental diagram, the authors ascertain that, whatever the model, the following constraints must be respected:
– q = qmax to ρ = ρcr: the flow/density function attains its maximum (capacity) at density ρmax;
– 
the incline of curve q =f(ρ) is zero; dp
– qρ = 0 to ρ = 0: the flow at ρ = 0 is zero;
– 
: the incline of curve q = f(ρ) is equal to the free speed at ρ=0;
– qρ = 0 to ρ = ρmax: the flow at congestion is zero.
The compatibility of the models is ensured while vehicles are stationary since all of the above constraints are respected. Generally speaking, macroscopic models satisfy all of these constraints, while most microscopic models do not.
More precisely, the microscopic model used by Magne cannot satisfy the second and fourth constraints above because for an infinitely large intervehicle distance the maximum speed tends toward infinity.
The principle of communication between the two models is based on the calculation of the parameters characterizing the macroscopic part, and the transmission of conditions to the limits of the transitional zones. These parameters are considered to be constant througout macroscopic step of time. The microscopic model, in contrast, evolves in an independent manner by using the limit conditions provided by the macroscopic model. Note that communication must be ensured:
– In the macro-micro sense: the macroscopic model imposes outflow and speed on the transitional zone. However, vehicles are generated from a distribution of speed (Poisson distribution) and at the times of generation (normal distribution). For the choice of lanes, a binomial distribution based on initial speed and density is used. In order to create an outflow close to that imposed by the macroscopic part, a correction is made to the distribution of generation intervals. However, the absence of space to generate a new vehicle leads to the formation of a vertical waiting line.
– In the micro-macro sense, the process of aggregation of parameters takes into account the average speed of the vehicles leaving the microscopic zone, as well as the outflow and density of the last microscopic cell. This information is then communicated to the macroscopic part. Nevertheless, one of the problems encountered at this level is due to the propagation of the circulation conditions of vehicles that are downstream. In fact, since the law of pursuit is only applicable in the microscopic zone, vehicles are not subject to any constraint and tend to circulate at their desired speed. In order to resolve this problem, Magne proposed the creation of a phantom vehicle representing the first vehicle leaving the transition zone. The parameters of this vehicle are determined by the conditions of circulation in the macroscopic zone (its speed is equal to the speed of flow) and are calculated at each macroscopic time step. In this way, the trajectories of the vehicles leaving the microscopic zone follow that of the phantom vehicle.
The method of validation of the hybrid model proposed by Magne consists of a comparative study with results obtained from a macroscopic model used alone.
Though the model proposed by Magne represents a certain advance in the domain of hybrid modeling, its pairing schema presents several inadequacies and generates some irregularities, mainly in a congested system. In fact, if there is insufficient space, a line of vehicles waiting to be created is formed. The next step of calculation time must then be waited for in order to be able to generate these vehicles, which causes a delay in the spread of congestion.
Magne’s model also has other limitations related to the creation of vehicles in the macro—micro sense. The outflow values imposed on the microscopic part are not necessarily whole numbers. For example, for an outflow of 0.2 veh/s imposed in a macroscopic time step 8 s long, the number of vehicles generated in the microscopic part is 1.6 veh, which requires an adaptation since it is impossible to generate an incomplete vehicle at the microscopic level, which the Magne model does not address. Moreover, the stochastic generation of vehicles and the introduction of a correction at outflow in the Magne model can cause the principle of flow conservation to be broken. Finally, the absence of a procedure to correct downstream outflow values leads to the waiting line being at risk for creating unrealistic density values (greater than the maximum density).
4.5.2. The Poschinger model
The Poschinger model [POS 02] is based on the pairing of the intelligent driver pursuit model and the Payne macroscopic model. This association of two models with totally different behaviors has given rise to a heterogeneous hybrid model.
The transmission of information is done progressively, in the same manner as the Magne model. In fact, this model proposes to insert a transitional zone in order to ensure communication between the two models. In this context, the micro—macro transition is ensured by a stochastic disaggregation algorithm, while the inverse transition is done with the help of an aggregation algorithm.
The two models are synchronized in the same way as the Magne model. To ensure communication between the two worlds, the Euler method has been integrated. This is used to combine the information needed by both models at each instant, taking into consideration the last time step as a predictive piece of information that is necessary for the next time step.
The principle of the micro—macro transition is based on the generation of vehicles while taking into account the following microscopic characteristics:
– distribution of intervehicle distance;
– fluctuation of speed; and
– the law of relaxation toward speed.
Particular attention has been paid to the first characteristic. Intervehicular distance is generated in a deterministic manner by adding a random noise and a distribution of speed. In his books, the author discusses the application of two solutions that have previously been adopted to solve this problem. The first consists of using deterministic speeds by adding a virtual zone within the microscopic part, where the vehicles may eventually self-distribute. The second alternative is based on the integration of interactions (friction) between vehicles. These interactions give rise to accelerations, decelerations, passing, etc. Reaching unconvincing statistical results, the author has chosen to use the Ferrari model [FER 88] to generate the auto-correlated speeds of the cars:
where ei are the values of a normal distribution with zero average and standard deviation σe . α is an average movement parameter. σe and α are estimated using individual vehicle data. The correlated speeds provided by this method are independent of the step-of-time. In Burghout [BUR 04] it has been shown that the level of correlation of speeds is elevated for small periods of time, but diminishes as the time period grows. This effect causes the artificial accelerations and decelerations of vehicles entering the microscopic zone.
The micro—macro interface generates strong oscillations that affect the macroscopic cells situated upstream as well as downstream. These oscillations are due to the discrete character of the outflow at the level of this interface. They have a negative effect, as they do not allow us to validate the model with any ease.
4.5.3. The Bourrel model (HYSTRA)
Bourrel has combined the different pairing schemas cited above and considerably improved them. The Bourrel model (HYSTRA) was developed from the pairing of a first-order LWR model (STRADA) [BUI 96] and a microscopic optimal velocity model [NEW 61] (a pursuit model based on the LWR model). The two models are synchronized, as in the case of the Magne and Poschinger models. This model also includes the ideas of the fictional vehicle and of virtual transitional cells.
In the macro—micro transition, moments of vehicle creation are calculated using the macroscopic cell exit outflow in order to obtain a uniform distribution of vehicles. The distance that separates two successive vehicle generations is therefore equal to the inverse of the outflow. When the generation time of a vehicle is obtained, it is created if, and only if, there is enough space (space > ρmax); if not, its creation is delayed. The number of vehicles is not necessarily whole. However, Bourrel has introduced the idea of a portion of a vehicle in order to ensure the preservation of flow. The problem with this method of vehicle generation is that during congestion vehicles are generated with speeds of zero.
In the same way as in the Magne model, a fictional vehicle is created in the micro—macro transition zone in order to determine the movement characteristics of the vehicles leaving the microscopic part. The exit of vehicles depends on the vacancy in the macroscopic zone. These exits are delayed if the demand is greater than the number of spaces available. In this case, the characteristics of vehicle movement are changed in order to ensure a higher exit time.
The validation of the model consists of verifying the preservation and continuity of flow. To do this, Bourrel [BOU 03] suggested a study of the spread of congestion up- and downstream. The results were then compared to those of the STRADA macroscopic model.
The results are convincing in both the fluid and congested modes, with the exception of several oscillations that appear in the transition cell. These are due to estimates of the idea of portions of vehicles.
This model does present some limitations, however, such as:
– the use of the LWR model is not adapted to address complex phenomena that occur outside the equilibrium state (acceleration and deceleration, the spread of congestion upstream in the case of dense traffic, etc.);
– it does not take into account multilane infrastructures; and
– the homogeneous character of the model does not take into account the different types of vehicles.
Though several extensions based on stochastic considerations (distribution of desired speed, interval between vehicles) have been added to the Bourrel model, their application remains very limited due to the insufficiencies of the first-order model. In fact, since this model is deterministic by nature, the introduction of random distributions affects the compatibility of the two models brought into play in the hybrid structure and breaks the law of outflow conservation.
We will now present models that improve on the Bourrel model; namely those of Mammar, Espié and El Hmam.
4.5.4. The Mammar model
The hybrid model developed by Mammar [MAM 06] emerged from that proposed by Bourrel. It is based on the same pairing schema, but differs mainly in the models used. This hybrid model is based on the joint use of the second-order ARZ model (this acronym is based on the the intials of its developers Aw, Rascale [AW 01] and Zhang [ZHA 02]), and the microscopic optimal velocity model [NEW 61]. These two models were chosen so as to ensure their compatibility. In this context, the ARZ model underwent several improvements and has been used in the form of the LWR model. It is based on the idea of supply and demand.
The model is validated using two scenarios. The first is dedicated to the validation of the model at equilibrium, in order to be able to compare it to other models. The second scenario involves the study of the transmission of data in the macro—micro transition zone. Though the analytical solution to the ARZ model is very difficult to obtain, the results of the validation of the hybrid model proposed are very encouraging in the case of equilibrium and in transitional mode. However, the micro—macro transition still requires several improvements.
4.5.5. The Espié model
An initial feasibility study of the hybrid model was conducted within the INRETS Institute (Institut National de REcherche sur les Transports et leur Securité) by the MSIS (modélisations, simulation et simulateurs de conduite) team in 1995. The resulting model developed jointly uses the ArchiSim microscopic model and the SSMT (simulation semi-macroscopique de traffic) macroscopic model (these two models were developed by INRETS). The transmission of data is ensured by information exchange objects called “interfaces”. This initial approach to hybrid modeling did not, however, take into account the idea of portions of vehicles, or synchronization problems.
Espié [ESP 06] improved this model by proposing a pairing schema ensuring the correct propagation of conditions at the limits, the preservation of flow, and the continuity of information.
Unlike the Bourrel model, the Espié model is heterogeneous; it pairs two models that do not have the same law of behavior. Nevertheless, the pairing procedure is very similar to that adopted by Bourrel. It uses the ideas of transitional zones, portions of vehicles, and the phantom vehicle.
Validation of this proposition is based on the verification of two properties of the hybrid model: preservation of flow and the correct propagation of information. The authors carried out experiments in the case of a single-lane network and a two-lane network. The results show that the Espié model preserves flow and ensures its continuity. Being heterogeneous, no validation (analytical or numerical) of the compatibility of the two models was conducted. Moreover, the same inconveniences observed in the Bourrel hybrid model have been detected here, notably those related to the procedure of discrete generation of vehicles and to the introduction of the idea of portions of vehicles. Though this model addresses multilane sections of road, there is no precision involved at this level with regard to vehicles’ use of a particular lane at the entry point to the microscopic section.
4.5.6. The El Hmam hybrid model
The hybrid model developed by El Hmam is a heterogeneous model pairing a microscopic and a macroscopic model [ELH 05], see Figure 4.2. The microscopic model is based on the agent paradigm [FER 95]. The hybrid model that was developed has the advantage of accepting the pairing of this microscopic model with different macroscopic models taken from the literature.
The choice of representing traffic flow at the microscopic level by agents provides great freedom in calibration and thus ensures compatibility with the macroscopic model with which it will be paired.
Moreover, the two models are synchronized by introducing a linear relationship between the calculation steps of the microscopic and macroscopic models.
Finally, this hybrid model based on the agent paradigm constitutes the basis for a generic hybrid simulator.
4.5.6.1. Principle of microscopic modeling based on the agent paradigm
The quality of a microscopic model is limited by the calculation capacities that can be implemented, even more so if the laws of vehicle behavior present strong nonlinearities. In simulations, a good model is the result of a compromise between the following characteristics defined by Nagel [NAG 95]:
– resolution: the level of detail required by the simulation;
– faithfulness: the degree of realism of the model;
– size of the system;
– speed of execution of the simulation; and
– resources: time and IT material available.
Multi-agent systems allow us to address a large number of entities with complex behaviors, thus eliminating a certain number of difficulties encountered in the development of the aforementioned hybrid models.
4.5.6.2. Architecture of the microscopic model based on the agent paradigm
A network is composed of interconnected sections of road. Each section is made up of lanes on which vehicles drive. When several sections cross each other, they form intersections that can be directed by traffic lights or by right-of-way. To reach a destination, a vehicle moves at a variable speed between nodes (intersections) in the network, leaving from a point of origin and following its route plan toward its destination.
Figure 4.3 presents a simplified schema of a road network composed of two intersections and of sections of multilane road on which vehicles are driving. The entry and exit lanes (on and off ramps) are also represented, as well as various signaling systems intended to control traffic.
This network is itself part of a multi-agent system; the vehicles (car, bus, truck, etc.) are modeled by agents. Each of these agents evolves in an environment marked by road sections. The crossing of several environments gives rise to an intersection. At in an intersection, the management of priorities (right of way) is also ensured by an “intersection” agent. The system making up the network being studied is thus composed of a group of agents, characterized by specific behaviors and objectives.
The following section describes the basic elements of a road network, details the behavior and analyzes the functioning of the various agents.
4.5.6.2.1. Physical elements of the network
a) Sections of road
A road network is composed of a group of interconnected sections of road, characterized according to five types of functions (see Figure 4.4):
– downstream: located upstream of a macroscopic section of road;
– upstream: located downstream of a macroscopic section of road;
– origin: the entry point of a network;
– destination: the exit point of a network; and
– simple: directly connected to a microscopic section of road.
b) Lanes
Lanes are elements forming a section of road, and are consequently the base element from which a road network is constituted. Each lane is characterized by an identifier, a traffic direction, geographic coordinates, and the list of vehicles it contains. Other information, such as the speed authorized in the lane, the position of motion detectors, signal signs, etc., can be added to enrich the characterization of a lane depending on the objective of the simulation. Lanes are characterized by their role in the network (see Figure 4.5). Vehicles adopt relative behaviors, taking into account their environment and their location in these lanes.
c) Nodes
The intersection of two or more sections of road corresponds to a crossroads or a roundabout. A roundabout is made up of a group of lanes leading to circular lanes, while a crossroads is the intersection of a group of rectilinear lanes. The movement of vehicles at intersections is managed by a right-of-way sign or a traffic light.
4.5.6.2.2. Simulation agents
Two types of agents coexist within a road network: “vehicle” agents and “intersection” agents. The lifecycle of these two reactive agents is illustrated in Figure 4.6. Each agent is characterized by a limited lifespan depending on its arrival at its destination if it is a vehicle type of agent, and by the end of the simulation if it is an intersection type of agent.
a) Vehicle agent
A vehicle agent manages the movements of a vehicle while respecting safety regulations (safety distance, speed limit, etc.). Each vehicle agent is characterized by its maximum speed, its maximum acceleration, its itinerary, etc. The vehicle agent is autonomous. It has its own objective and its own understanding of the environment. This understanding, which is supposed to be partial, corresponds to the driver’s field of vision (see Figure 4.7), and allows the agent to evolve within his immediate environment. The vehicle agent is a reactive agent [FER 95]. Its decisions are generated on the basis of its perception of the environment. Its actions are physical movements on a route. Before moving, it evaluates its environment (it detects the vehicles surrounding it and the road infrastructures). It collects all of the information pertaining to it, particularly the positions and speeds of vehicles that are close by within a radius limited by the driver’s field of vision. This field of vision depends on meteorological conditions (rain), traffic (congestion), and the geometry of the road infrastructure.
During its movement, the vehicle agent adjusts its speed to attain the desired speed. It tends to accelerate if its speed is less than the desired value, depending on the space available. In the case of dense traffic, the agent’s first reaction is to verify whether or not it is possible to overtake. At this stage, passing is subject to the regulations dictated by the lane-change model. Faced with the impossibility of passing, it remains on its original route, while adjusting its speed according to the car-following model.
Thus, the behavior of the vehicle agent is subject to two complementary laws; the first concerning the application of a car-following model, and the second based on the lane-change model. The following section describes the principles of these two laws.
b) Laws of pursuit
The model adopted is that developed by Krauss [KRA 98]. It is based on the principle of a safe speed allowing a vehicle to remain a reasonable distance from the vehicle in front of it.
If it is respected, the safety speed enables a collision to be avoided when the downstream vehicle brakes sharply. The safety speed is calculated using the following relationship:
where:
– 
: the speed of the front vehicle at time t;
– gt: the distance separating the two vehicles at time t;
– 
(t): the average of the two speeds v and 
at time t;
– b: maximum deceleration of the vehicle in m = s2; and
– τ. driver’s reaction time in seconds.
In addition, a vehicle’s speed is limited by the speed limit imposed for the lane without passing its desired speed, vdes, and by its maximum acceleration. These limitations are used to calculate an intermediary speed expressed by:
Knowing that a driver cannot maintain a constant speed during a given time span (due to driver imperfection), the model imposes a fluctuation in the speed. This fluctuation is expressed using a random variable η following a normal distribution in the interval [0.1], and the amplitude of the noise, ε, considered equal to the driver’s reaction time. The vehicle’s speed is thus expressed by the following relationship:
The Krauss model is a car-following model that has the advantage of being quick to execute thanks to the limited number of equations that characterize it. It also presents the particularity of being without collisions for a period of time lasting h ≤ 1 second, ε = τ = 1 second, a = 0.2 g, and b = 0.6 g where (g = 9.81 m/s²).
Moreover, this model is capable of perfectly reproducing the macroscopic characteristics of flow [KRA 02, KRA 04].
c) Laws of lane change
These laws constitute a group of rules that ensure the proper execution of overtaking and pulling-back maneuvers. This group of rules ensures safety during a lane change.
If a vehicle wishes to drive at a speed higher than that of the vehicle in front of it, it is prompted to overtake the car in front. This maneuver can only occur if safety conditions allow it.
Before overtaking a vehicle, the agent which models this vehicle must make sure that a certain number of conditions are fulfilled, so that it cannot hit, or be hit by, another vehicle. This certainty is achieved with the help of information collected during the perception of its environment.
d) Intersection agent
An intersection agent manages access to intersections. Depending on the type of intersection (right-of-way or light traffic), it takes on behavior that allows it to ensure the safety of vehicles, notably by estimating the future position of the vehicles after each step of the calculation.
4.5.6.3. Hybrid model validation procedure
A hybrid section of road has been subjected to two tests. The first consists of applying an increased traffic flow, and the second a reduction of this traffic flow at the entry point to the section of road.
The results are compared to those obtained from the same test applied to a section of road that was completely simulated using a macroscopic model (see Figure 4.8). They show the perfect compatibility of the models, which explains the interest in this modeling approach.
4.5.7. Comparison of the hybrid models presented and general remarks
Table 4.2 shows the principal elements used to compare the hybrid models presented above.
The study of hybrid models leads us to eventually draw several conclusions:
– hybrid models are mainly differentiated by:
- the flow models used within the same structure. In this context, when the two models are generated using the same law of behavior, we refer to a homogeneous hybrid model; if not, the model is heterogeneous,
- pairing procedure, and
- the type of network being studied, which can be formed of multilane or single-lane sections of road;
– the compatibility of the two representations within hybrid models is often difficult or even impossible to prove. Compatability is generally obtained by deriving the microscopic model from the macroscopic equation applied solely to the law of pursuit. Given that microscopic representations integrate other behavioral laws besides the law of pursuit (such as the lane-change law, etc.), it is difficult to prove whether the two models are equivalent or not, particularly in the congested mode;
– hybrid models do not truly satisfy the constraints that a hybrid model must satisfy, and they are generally difficult to implement due to their complexity. In fact, the Magne and Poschinger models introduce undesirable effects into the propagation of information, particularly the fluctuations detected at the exit points of sections of road. The Mammar and Espié models have adopted the same pairing procedure as used in the Bourrel model, improving it by the application of second-order macroscopic representations. They retain the same flaws as the second-order model, however, notably the idea of portions of vehicles, which cause fluctuations and can affect the preservation of flow;
– all of the work done on hybrid models emphasizes the aggregation and disaggregation of data and the compatibility of two representations, without focusing on real applications; and
– hybrid models remain underdeveloped, and are rarely used in the development of control laws.
In the following section, we will examine the major principles adopted for the regulation of road traffic.
4.6. Different strategies for controlling road traffic flow systems
4.6.1. Regulation of access: definition and history
Excess demand for access is a major cause of congestion. The regulation of road flow on high-traffic routes is most often carried out by controlling the outflow of vehicles entering a highway using a slip or rapid lane (ramp meter). This outflow control is used to keep demand at a level below the capacity of the highway downstream of the ramp, beyond which there is congestion, and consequently to avoid a capacity drop. Moreover, the regulation of access is also used to control vehicle insertion times, which allows us to reduce conflicts and traffic backlog and consequently to improve the safety of users.
Regulation of access principally addresses the problems related to recurring congestion. It has no effect — and is therefore not implemented — during periods of average or low traffic, particularly at night.
Access can be regulated:
– locally in a fast lane (local control or localized regulation), see Figure 4.9; or
– globally on an axis with several access points (coordinated control or axis regulation).
Regulation of access is not a new traffic management technique. The initial access regulation systems appeared in the United States, with the first experiments occurring in Detroit. The first regulated access point was installed in Chicago in a fast lane in 1963. It was based on the use of a police officer whose role was to allow only a single vehicle at a time to pass, according to a predetermined outflow. This experiment was followed by others, notably in Los Angeles in 1968 and then in Minneapolis in 1970.
The Minneapolis site (Minnesota, United States) is one of the best-known access-regulation systems in the world; it includes nearly 433 regulated access points over 500 km. The access control principle functions with the help of traffic lights, either in an isolated manner using fixed traffic-light plans, or connected to a centralized system that controls the regulation parameters.
This American access regulation has been widely shown to be effective in increasing the average speed of vehicles, increasing outflow, and improving safety, with a significant drop in accident rates at peak times.
In France, the first tests occurred in 1967. They were conducted on the A13 highway. The first five truely regulated access points were installed in 1975, on highway A16. As part of the “Ile-de-France mornings” operation in 1976, a system functioning during the peak hours in the morning and on Sunday evening was installed on the A1, A3 and A6 highways. It was based on regulation via fixed-cycle traffic lights and had 40 access points7.
In the UK in 1986, an access regulation system was installed on the M6 (the Midlands) highway and progressively expanded to six isolated access points in 1989. Currently, only the cities of Birmingham and Southampton are equipped with access regulation8. In Southampton, on the M3 and M27 highways, six two-lane access points are regulated, either via the demand—capacity system or by other control systems (telematics and road security).
Today, numerous other sites have implemented access regulation systems:
– in the Netherlands, on the A10 West highway near Amsterdam;
– in Glasgow, Scotland;
– in Essen, Germany; and
– in Louvain, Belgium;
Many other sites are being studied in Italy, the UK, Switzerland, and Spain. In Canada, such regulatory systems have been installed in Montreal, Ottawa and Toronto.
4.6.2. Access regulation methods (metering systems)
We can find various methods used to implement access regulation in the literature. Among these methods, we can distinguish two levels of regulation: static and dynamic.
4.6.2.1. Static regulation
This type of regulation acts on the number of waiting lines by:
– modifying the access route to reduce its capacity; and
– managing access by channeling vehicles entering in a single line using a ground marker (see Figure 4.10). This access management reduces the ramp outflow and consequently reduces the number of conflicts between cars entering the highway. It also has a very low implementation cost.
4.6.2.2. Dynamic regulation
Dynamic regulation consists of dividing up the number of vehicles entering the highway into smaller groups. It assumes the implementation of traffic lights or barriers on the on-ramps (see the Fourvière tunnel in Lyon shown in Figure 4.11), where the access points are closed for a span of time varying from five to 10 minutes; this closure is communicated to users via changeable message boards. Barrier regulation is effective but sudden and difficult to adjust, and can irritate drivers.
In the following section, we will examine only those access regulation strategies that use traffic lights. In this case, the flow of vehicles authorized to access the fast lane from an on-ramp is converted into a cycle of traffic lights. The access flow is adapted to variations in traffic. These regulation strategies usually use historical measurements, real-time data, and estimations or predictions of demand. Most traffic-light regulation strategies (see Figure 4.12) allow us to ensure the flow of vehicles from ramps in two ways:
– via flow through barriers, which allows “batches of vehicles” to enter during each cycle; and
– via “drop-by-drop” flow, which limits entry to a single vehicle per cycle.
Traffic-light access regulation strategies can be grouped into two main categories: fixed-cycle regulation strategies and adaptive strategies.
4.6.2.3. Fixed-cycle access regulation strategies
Preset operation is the simplest form of access regulation and is used in France and the United States. This approach to regulation was first suggested in [WAT 65], followed by other strategies such as that proposed by Schwartz [SCH 77]. These regulation strategies are also called fixed-cycle strategies; they use fixed-duration green lights, and the number of vehicles admitted per cycle is constant. They are based on simple static models. The reader can find an overview of these control strategies in [PAP 02].
This type of strategy, based on constant historic demands, does not take into account fluctuations of traffic in real time, unlike adaptive strategies that are based on road motion detectors and provide real-time information about the presence or absence of vehicles at given places and times. The calculation of the cycle is always carried out in advance, using historical measurements of ramp and highway demand. Demand is not constant and may vary depending on the day of the week or in the case of events that are not necessarily predictable.
The use of historical data can lead to on-ramp flows that are not suitable to the current situation. The absence of real-time measurements on the state of traffic can cause the highway to be underused or overloaded (congested). The advantage presented by this type of strategy, however, is its very simple material configuration.
4.6.2.4. Adaptive access regulation strategies
Adaptive control strategies offer real-time calculation methods of admissible flow using one or more parameters characteristic of the state of traffic: rates of occupation, outflow, and speed. These are variables measured using sensors placed on the road surface, on the active part of the highway and on the access ramp. The ramp outflow thus calculated is then converted into a traffic-light cycle. These strategies are intended to keep highway traffic conditions close to a desired behavior, based on real-time measurements [PAP 04].
There are local access regulation strategies and coordinated access regulation strategies. The former consist of regulating isolated access points and are based on approximate measurements for each ramp, in order to calculate the corresponding individual access outflows. The latter manage several access points in a portion of highway and use the available measurements for this entire portion.
In the following section, we will examine adaptive strategies.
4.6.3. Adaptive local access regulation strategies (responsive ramp metering control strategy)
These are local regulation techniques that consider access points in a manner isolated from one another. The access outflow in these strategies is based on realtime measurements near the ramp. The goal is to keep traffic conditions on the section of ramp close to a desired behavior. The appropriate access outflow is calculated by analyzing the occupation or flow via sensors on the ramp and the main section.
These systems are costly to install and maintain, but they are able to regulate unusual and unforeseen changes in traffic, and generally give good results. The material configuration is similar to that of fixed-cycle strategies, with the addition of sensors on the main section near the ramp.
The main criticism of these algorithms is that they adjust access outflows after the appearance of congestion. The two regulation strategies, the demand-capacity strategy and the occupancy strategy [MAS 75], are part of this type of access regulation and are popular in the United States.
Another closed-loop access regulation strategy (Asservissement LINéaire d’Entrée Autoroutière, or Linear Highway Entry Control, known as ALINEA) was proposed by [PAP 91]. It is based on classic feedback concepts and is popular in Europe.
Results have shown the effectiveness of ALINEA compared to other regulation strategies, as well as to uncontrolled environments [PAP 98].
According to the INRETS report [HAJ 88], an access point marketing evaluation report, this strategy improves insertion, thus reducing the time spent in congestion by 41%. The method has since been implemented in Paris on the ring road.
Most access regulations pertain to the regulation of access to a ramp toward a highway, but there are also adaptive on-highway access regulation strategies. On-highway control is a local control action that is applicable where the parts of two highways merge.
On-highway control can be considered to be a form of local access regulation. The main difference lies in access outflow. Compared to classic access regulation, the access outflow of on-highway control is larger (depending on the number of lanes in the highway).
In the following, we will explore the best-known adaptive regulation strategies. Since the intention is to keep the state of traffic downstream of the ramp close to a desired behavior, the desired optimum may be a maximum outflow (demand-capacity algorithm) or an occupation rate.
4.6.3.1. The demand-capacity strategy
The demand-capacity strategy [MAS 75] was introduced with the first implementations of local adaptive (reactive) access regulations.
This strategy determines, locally and in real time, the access outflow r(k) (number of vehicles authorized to enter during period k) using entry—exit capacity data, based on the following relationship:
where:
– qcap is the highway capacity measured downstream of the ramp, which is a value predetermined using the fundamental diagram;
– outflow qin(k−1) is the measured flow during the period k−1 in real time by magnetic loops (or other sensors) upstream of the ramp;
– oout(k) is the occupation measured downstream of the ramp, while
– ocr is the critical occupation for which traffic flow is maximal.
This strategy allows us to add a flow r(k) to the entry flow upstream qin(k−1) in order to attain the capacity qcap of the highway downstream of the ramp (see Figure 4.13).
This adaptive strategy requires two measurement stations; one upstream of the access point to calculate outflow qin, and the other downstream to determine the occupation rate oout. Its advantage is that it takes into account real-time variations in traffic, while its disadvantage is that it functions in an open loop.
The INRETS demand—capacity strategy is a variant of the standard demand — capacity strategy wherein another sensor is installed at the access point (the convergence point) in the fast lane in order to take into consideration the state of congestion at this point. It was tested on the Paris ring road in 1988.
4.6.3.2. Occupancy strategy
The second category of local regulation is the occupancy strategy [MAS 75]. This strategy is used in the United States (Chicago, Los Angeles, etc.) and is based on the same principal as the standard demand—capacity strategy defining ramp outflow r(k) as the additional outflow upstream measured in order to obtain the highway capacity downstream. However, in this case, the authorized ramp outflow r(k) is based on the measurement of the occupancy upstream of the ramp.
This strategy is economical and only requires a single sensor upstream of the ramp for the measurement of the occupation rate. It is known for its low installation cost and simplicity of implementation, but since the sensor is located upstream of the access point, this strategy often reacts too late to the appearance of congestion downstream of the access point.
4.6.3.3. Wotton-Jeffreys strategy
Other access regulation strategies have been developed; for example, the British Wotton—Jeffreys (W&J) strategy, which operates in real time. In this strategy, the downstream capacity changes depending on the state of traffic. The operating principle of this strategy is the same as it is for the others: the measurement of the upstream outflow (Qam) is completed by the ramp outflow r(k) in order to obtain the capacity (Qmax) downstream. It is governed by the following rules:
– if Qr + Qam > Qmax then the ramp traffic light changes to red; and
– if Qr + Qam < Qmax then the ramp traffic light changes to green.
This strategy requires three measurement stations placed respectively on the ramp, upstream and downstream of the road section in order to determine capacity. Wotton and Jeffreys developed software to determine this capacity in real time using the measurement of outflow (Qav) and speed (Vav).
This strategy has two advantages: its reactivity, and the optimization of the number of vehicles entering the road. However, it is very difficult to calibrate, and requires the installation of a large number of sensors.
4.6.3.4. Rijkswaterstaat strategy
The Dutch Rijkswaterstaat strategy uses the same principle as the standard demand—capacity strategy. It is used in the Netherlands. Congestion is detected using speed measurements upstream and downstream of the access point (flow is supposed to be in congestion if traffic speed is lower than 35 km/h).
It is used in Amsterdam in the “vehicle-by-vehicle” filtering mode. It has the advantage of easy calibration and only requires two sensors, but it remains an open-loop control.
4.6.3.5. Predictive local access regulation algorithms: the ALINEA strategy
This type of algorithm uses the closed-loop control principle to locally determine the access outflow and to try to anticipate operational problems. ALINEA [PAP 91] is used to determine the appropriate access outflow using real-time traffic measurements downstream of the ramp.
ALINEA is the first local access regulation strategy based on a simple application of the classic feedback control theory. Tested in France, the Netherlands, and the UK, it has given very good results when compared to several other strategies [PAP 98].
The ALINEA algorithm obeys the principles of first-order control systems; its objective is to keep occupation downstream of the ramp on the highway equal to a value od defined in advance, called the set-point or desired value in order to guarantee an absence of congestion. This is preferably — but not necessarily — inferior to the critical value (ocr).
The number of vehicles r(k) (in veh./h) authorized to enter the highway during each period of time k is based on the following relationship:
The ALINEA strategy is activated at each interval of control time, with a length between 10 and 60 seconds. ALINEA is based on the principle of a discrete integral regulator [PAP 03]; entry r(k) is obtained by integrating the control error (od−oout(k)) where:
– r(k) is the number of vehicles (in veh./h) authorized to enter during the interval of time [kT, (k+1)T] and is used to calculate the ramp flow r(k+1), and T is a defined interval of control time, usually between 10 seconds and one minute;
– od is the desired occupation chosen by the user. It is assumed to be known, and can be changed at any time [KOT 04]. It is often equal to critical occupation (which corresponds to a downstream flow close to capacity qcap);
– Oout(k) is the average occupation measured (in Per cent) downstream of the ramp during the preceding interval of control time [(k−1)T, kT]. The measurement oout of occupation used by ALINEA must be taken a few hundred meters downstream of the most probable site of congestion; and
– KR > 0 is the regulator gain determined heuristically at the site (its value is homogeneous to a flow).
Experience shows that the results of regulation using ALINEA are not greatly affected by the choice of parameter KR [HAS 02, KOT 04]. The value KR = 70 veh./h is chosen by most authors.
Bellemans [BEL 06] presents the results of simulations using the ALINEA strategy on the E17 Ghent—Antwerp highway in Belgium. It is noted that fluctuations (in speed, flow, density, and access outflow) are larger and can become more dangerous when the regulator gain KR increases. Therefore, the smaller this gain, the greater the delay in traffic light control with regard to the appearance of congestion; the more important it is; and the more reactive the the control is, causing instabilities.
By definition, vehicle outflow r(k) is often included in the interval [rmin, rmax]. Imposing rmin prevents the total closure of the ramp, which allows us to limit vehicle waiting time in the line on the ramp during regulation during high traffic demand. Imposing rmax limits the outflow r(k), which must not exceed the capacity of the on-ramp.
The flow r(k) calculated by ALINEA is truncated, and is used as r(k−1) in the following period of time, in order to avoid the wind-up effect of the integral regulator.
ALINEA optimizes the occupation rate of the sensor, and not the outflow, as in the capacity—demand strategy, which allows us to determine real traffic conditions. The principal reason for this is that a state of fluid traffic and a state of congested traffic both correspond to a flow value (see the fundamental diagram — Figure 4.1). Moreover, the occupation od that is chosen as is often close to the critical occupation ocr. In addition, in comparison to the capacity qcap used by the demand—capacity strategy, critical occupation is less sensitive to changes in environmental conditions (weather, for example), or to the composition of traffic (such as the presence of trucks) [KEE 86].
ALINEA requires a single sensor on the principal road surface in order to measure the occupancy, oout(k), downstream of the on-ramp. The placement of this sensor must be such that recurring congestion is visible in the measurements. ALINEA operates in a closed loop and reacts to the differences (od − oout(k)), which allows us to avoid congestion and stabilizes traffic flow at a higher level [PAP 02]. However, its parameters are difficult to set, particularly the gain parameter KR. Inadequate setting increases the risk of fluctuations.
Papageorgiou [PAP 97] has presented the main characteristics of ALINEA, as well as the results of the first implementation of the strategy on a single on-ramp in Briançon (France), or in the case of several on-ramps (METALINE: not yet implemented), on the Paris ring road and the A10 West Highway in Amsterdam (the Coentunnel on-ramp).
ALINEA has been compared to other control strategies, including the demand— capacity control strategy, the occupancy strategy, the W&J strategy, and the Rijkswaterstaat strategy. The results show that ALINEA provides the best outcome.
4.6.3.6. Comparison of the ALINEA strategy with the demand—capacity and occupancy strategy strategies
The demand—capacity and occupancy strategy strategies are based on measurements of flow and occupation on the principal section of road upstream of the ramp, while ALINEA is a closed-loop regulator based on occupation measurements downstream of the ramp.
The demand—capacity strategy reacts to heavy occupations oout once a threshold value ocr has been attained, while ALINEA reacts to small differences (od − oout(k)). It can thus avoid the appearance of congestion while stabilizing the traffic flow to a considerable degree.
Recent applications of ALINEA have shown certain problems and needs that are not resolved by ALINEA or by the other access regulation strategies, such as:
– the use of upstream measurements (rather than downstream ones);
– the use of measurements based on flow (rather than occupation);
– a real-time adjustment of the desired values in order to maximize flow in the downstream fast lane; and
– an effective regulation of the waiting line in order to avoid the spread of congestion in the secondary network.
Smaragdis [SMA 04] proposes an extension of ALINEA called the AD-ALINEA (adaptive ALINEA) strategy. It contains an algorithm to estimate critical occupation using real-time measurements of qout(k−1) and oout(k−1) downstream of the ramp. This procedure is useful in the case where there is difficulty estimating critical occupation, or when this occupation changes in real time due to changes in environmental conditions or the composition of traffic (trucks, buses, etc.).
The strategy AD-ALINEA needs downstream real-time measurements to estimate qout and oout. In cases where such measurements are not available, another strategy, UP-ALINEA (which stands for Upstream-ALINEA) is suggested. In this strategy, a method for estimating qout and oout from upstream measurements is proposed [SMA 03].
In [YIB 06], the ALINEA (integral regulator) strategy was modified into a proportional—integral strategy in order to resolve the case of narrowing located downstream of the on-ramp. The results showed that ALINEA was less effective in this case, while its expansion into a proportional—integral regulator brought satisfying improvements.
In a book by Bellemans [BEL 06], the ALINEA strategy is applied to the E-17 Ghent—Antwerp highway in Belgium, and is compared to optimal predictive control using simulation techniques. The results show that ALINEA causes a gain in total time spent and is able to limit waiting lines to values close to the maximum limit. This limit of waiting lines means that fluctuations appear in the access outflow, traffic density and average speed on the section of road to which the ramp is connected. In Bellemans’s work, predictive control of access leads to higher performances (low total time spent) compared to the ALINEA strategy. Moreover, no fluctuations appeared; the access outflows of predictive control were smooth, as were densities, average speed, and flow. Thus, limitations on waiting lines in access lanes were better respected.
All access regulation strategies measure an access outflow to avoid or reduce congestion; this outflow causes a limitation in the number of vehicles accessing the fast lane. This limitation causes an increase in the waiting line in terms of access. The increase in the waiting line can be detected by sensors placed upstream on the ramp.
If the waiting line becomes longer than the ramp’s capacity to hold vehicles, it can stretch to the surface network, so it is important to limit the length of this line. The access regulation strategies mentioned below do not take into consideration the length of the waiting line in their control algorithms. Once the maximum limit of the waiting line has been reached, the action of the regulator can be canceled to allow a larger number of cars to access the fast lane and reduce the number of vehicles waiting. These constraints are valid for any regulation strategy.
The performance of an access system depends largely on the access flow and the control strategy, while the access flow depends on the objective of the regulation system. If the system is planned to eliminate or reduce congestion on the principal section of road, the access flow is based on the demand in the principal section upstream of the ramp, the capacity downstream, and the demand on the ramp. If the combination of flow upstream of the ramp and the flow on the ramp exceed the capacity of the highway, access outflows are activated to reduce ramp flow so that the downstream capacity is not exceeded.
4.6.4. Adaptive strategies for coordinated access regulation (multivariable regulator strategies)
In the case of a fast line with several critical access points, local strategies (demand—capacity strategy, INRETS demand—capacity strategy, W&J, ALINEA, etc.) do not lead to the optimum sought. In this case, access point regulation must be applied globally on all ramps in order to obtain maximum efficiency. This is called coordinated regulation.
Coordinated strategies have the same objectives as local strategies: they attempt to keep traffic conditions close to the desired values. They are intended to minimize the total time spent by the group of vehicles on the highway and at the access points. In contrast, unlike local strategies that optimize the state of traffic locally and only take into account the traffic conditions at the access points, coordinated strategies take into consideration not only all the available measurements taken at all access points, but also those taken between these points [PAP 02]. This allows for an overall optimization of the highway’s capacity.
Additionally, a coordinated strategy is able to act at the moment there is congestion caused by an accident. When congestion forms, it spreads upstream. Access control can only react from the moment when the congestion reaches the measurement station. With its measurement stations between access points, a coordinated strategy detects the occurrence of an incident sooner than this.
Finally, the implementation of coordinated actions allows for the distribution of waiting time over various access points. Without coordination, only the vehicles coming from the access point situated just before the congestion will be able help the traffic to flow more steadily; control of the access point makes them wait so that access points upstream are not affected by the congestion.
The regulation of the coordination of access points can be carried out using synchronized local strategies (pseudo-coordination or synchronization). Local strategies can be fixed-cycle strategies or adaptive strategies. For example, in Amsterdam one coordinated strategy consists of linking the start of the Rijkswaterstaat strategy (with the aid of an operator) to an access point regulating other access points that are downstream of it.
There are also synchronized local ALINEA strategies that have been tested on the A6 highway in France. The synchronization here is different to that of Rijkswaterstaat, as it is automatic. The zones of influence must be determined in order to define which upstream access points need to be activated. The disadvantage here is that coordination only occurs at the access points. There is no taking into account of situations between access points.
Coordinated regulation techniques are more complicated; these seek to optimize a multitude of highway on-ramps. They are based on various control approaches, such as multivariable control strategies [PAP 90, DIA 94] or optimum control strategies [PAP 82, CHE 97, ZHA 99, KOT 02, KOT 04, GOM 06]. The system has more precise information. So, the calculation of the time when vehicles enter onto the highway from the on-ramp is more concise. This is better in order to prevent congestion from forming or to avoid aggravating it in the case of a non-recurring event (such as an accident). However, this assumes a link to an automatic incident detection system.
4.6.4.1. The METALINE strategy
The METALINE multivariable regulator strategy [PAP 90, DOA 94] devised by INRETS is a coordinated feedback control strategy developed as part of the EUROCOR project and based on the linear quadratic optimization theory. It was tested by using the METANET model on the Paris ring road and the A10-West highway in Amsterdam.
For all simulation tests carried out on the A10-West highway, the METANET macroscopic traffic model is used as a modeling and simulation tool. The simulation results are summarized according to evaluation indices: total trajectory time, total waiting time at point of origin, total time spent, total consumption of fuel, and maximum length of waiting lines at the points of origin.
The objective of this strategy is to maximize exit flow (so as to minimize the time spent in the network being studied) in a coordinated manner simultaneously controlling the flows of m on-ramps with respect to the occupation rates measured in real time for a group of n axis stations (where n > m). The principle consists of keeping occupation downstream of controlled ramps in the neighborhood of certain values. This strategy requires a flow traffic model from which a linear quadratic integral control law is obtained. The METALINE strategy can be seen as a generalization and expansion of ALINEA [KOT 04].
4.6.4.2. Comparison of ALINEA and METALINE
A comparison of local and global control strategies leads to the conclusion that local strategies are easier to design and implement [PAP 97].
Simulation tests and results regarding the effectiveness of the ALINEA and METALINE control strategies have led to the following conclusions [PAP 98, PAP 90]:
– faced with recurring congestion, METALINE does no better than the ALINEA strategy when applied locally at each ramp; and
– faced with non-recurring congestion, METALINE gives better results than ALINEA, thanks to the large number of measurements.
This can be explained by the fact that non-recurring congestion (due to a traffic incident, such as an accident) can arise anywhere on the axis before spreading upstream. The ALINEA strategy can only react when congestion reaches the measurement station placed on the ramps. METALINE, on the other hand, takes into account measurements taken between access ramps; it therefore detects the formation of a non-recurring congestion event sooner than ALINEA. Recurring congestion (due to an overly large flow demand) forms on the ramps; therefore, ALINEA and METALINE detect the congestion from the moment of its formation.
4.6.5. Implementation of regulation via traffic lights
Access point control consists of limiting the outflow between the on-ramp and the main route using traffic lights. Most access control strategies do not determine how the lights should change. In fact, from among the access controls that we have presented, only the W&J strategy consists of defining the color of the light according to traffic conditions. The other strategies calculate the outflow of vehicles authorized to leave the ramp. It is thus a matter of converting this outflow into traffic-light cycles. The relationship that exists between the outflow and the traffic-light cycle is shown by the following relationship:
where:
– c: total cycle time;
– g: duration of the green light phase;
– r: ramp outflow calculated by the strategy in vehicles per second; and
– Qsat: outflow of ramp saturation in vehicles per second, corresponding to the ramp’s flow capacity.
Using the outflow calculated by the access control, relationship [4.26] gives only the g/c ratio. It remains to determine the values of g and c. The filtering method is used to obtain them.
The “drop-by-drop” method limits insertion to one vehicle per cycle; the duration of the green light is constant and corresponds exactly to the time of crossing the fast lane by a single vehicle (around four seconds). In this filtering mode, the time of the cycle (and consequently the duration of the red-light phase) varies depending on the flow r calculated by the access control.
In the barrier filtering mode, vehicles are inserted at each cycle. In this case, the duration of the cycle c is fixed (generally at around 40 seconds) and the duration during which the traffic light stays green varies. However, to avoid the complete closure of the access point the duration of the green light cannot drop below a minimum value, gmin.
All access control strategies can function in either the drop-by-drop mode or the barrier mode. The usefulness of the drop-by-drop mode lies in ensuring better insertion of vehicles compared to barrier flow. Its disadvantage is the limitation of the maximum outflow authorized to leave the ramp; it cannot exceed 900 veh./h in the case of an average cycle of four seconds.
4.6.6. Evaluation of access control (effects of access regulation)
Access regulation is a direct traffic management tool that is most frequently used and most effective at controlling and improving highway traffic. Various positive effects have been observed if access point control is properly installed. With regard to the effects on congestion, we have [PAP 04]:
– an increased outflow at the exit of the section of the network being studied;
– a reduction in the total time spent by all vehicles in the network (this time includes the time spent traveling in the network as well as the time spent waiting at access points);
– an increase in average speed; and
– an effective reaction in the number of incidents.
There are also other positive effects, such as the delay in the appearance of congestion and the increased rapidity of the return to fluidity (as shown by experiments on the A10 West highway). Traffic jams last for shorter periods of time when the system is regulated, and they cover a smaller distance in the fast lane. Results of studies conducted at INRETS show that the regulation of access points produces beneficial effects not only on ring road traffic, but also on that of the Boulevards des Maréchaux in Paris with regard to the average circulation speed of vehicles.
In addition, control has a positive impact on safety. This is explained by the fact that access control improves vehicle insertion [MAC 98], as well as by the fact that reduced congestion causes a reduction in the number of accidents.
Access point regulation is implemented in urban areas where surface network problems can appear, so negative effects can also be observed, such as:
– increased waiting time at access points; and
– the transfer of traffic onto a secondary network due to the length of waiting lines.
The regulation of access leads to additional waiting time on access ramps (waiting line management). In these conditions, and when the waiting time becomes too long, users tend to modify their itinerary (traffic transfer). Waiting lines on ramps tend to grow longer and to generate traffic jams in neighboring lanes. This is why access regulation systems also include devices to measure waiting lines. When the waiting line reaches a certain limit, access regulation is temporarily canceled and the system is left to operate freely.
Since surface roads cannot support a large traffic transfer from a saturated highway, the sole objective of fluidity in the fast lane is insufficient. Control strategies must therefore integrate measurements of the state of traffic on roads that are attached as well as information for users regarding traffic predictions.
4.7. Conclusion
Without being exhaustive, this chapter presents as complete an overview as possible of the modeling and control approaches of the most frequently implemented road traffic flow systems. This presentation could be expanded to include the modeling of pedestrian flow, a domain that has obvious connections to the one presented here. Moreover, recent work has shown that more complicated control techniques are being implemented in numerous research laboratories all over the world than those presented here. These have not been addressed; however, the bibliography below will allow the reader to expand his or her knowledge of this subject. Less informed readers have been provided with an overview of the basic elements developed in the past as a foundation on which to build their knowledge.
Many development perspectives exist, particularly in the domain of control algorithm studies. The modeling of traffic flow systems is certainly not perfect, and still requires many improvements. Models are far from perfect and from faithfully representing the real behavior of these systems, which must more accurately integrate aspects related to human behavior. This area of study also has connections to other domains that were not mentioned above, such as the estimation of vehicle travel time, the study of itinerary strategy, vehicle routing, etc.
Most studies are based on simulation techniques. This also assumes the development of more realistic traffic flow simulators, particularly in the case of the studying large road networks integrating both urban and interurban zones. Hybrid models should soon be able to allow for the proper conducting of such studies.
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1 Chapter written by Daniel JOLLY, Boumediene KAMEL and Amar BENASSER.
1 www.senat.fr/europe/textes europeens/e1818.pdf.
2 In France, in 2000 there were 72% more cars than there had been in 1985. Source: CCFA and CSIAM, available at: http://ec.europa.edu/transport/strategies/doc/2001_white_paper_lb_ texte_complet_en.pdf.
3 Here, Δx corresponds not only to a distance traveled by a vehicle, but also to its speed of movement.
4 c0 is a constant that represents the speed of propagation of a disturbance [LEC 02].
5 We can also find hybrid mesoscopic-microscopic models in the literature, such as the Burghout model [BUR 04], intended for the calculation of itinerary plans with the mesoscopic model and for the simulation of flow with the microscopic model.
7 http://lara.inist.fr/bitstream/handle/2332/975/CERTU-97-18.PDF?sequence=2.
8 www.highways.gov.uk/knowledge/documents/Ramp_Metering_Summary_Report.pdf.