GENERALIZED VELOCITY–DENSITY MODEL BASED ON MICROSCOPIC TRAFFIC SIMULATION

. In case of the Intelligent Driver Model (IDM) the actual Velocity–Density law V ( D ) applied by this dynamic system is not defined, only the dynamic behaviour of the vehicles/drivers is determined. Therefore, the logical question is whether the related investigations enhance an existing and known law or reveal a new connection. Specifically, which function class/type is enhanced by the IDM? The publication presents a model analysis, the goal of which was the exploration of a feature of the IDM, which, as yet, ‘remained hidden’. The theoretical model results are useful, this analysis important in the practice in the field of hybrid control as well. The transfer of the IDM groups through large-scale networks has special practical significance. For example, in convoys, groups of special vehicle, safety measures with delegations. In this case, the large-scale network traffic characteristics and the IDM traffic characteristics should be taken into account simultaneously. Important characteristics are the speed–density laws. In case of effective modelling of large networks macroscopic models are used, however the IDMs are microscopic. With careful modelling, we cannot be in contradiction with the application of speed–density law, where there IDM convoy passes. Therefore, in terms of practical applications, it is important to recognize what kind of speed–density law is applied by the IDM convoys in traffic. Therefore, in our case the goal was not the validation of the model, but the exploration of a further feature of the validated model. The separate validation of the model was not necessary, since many validated applications for this model have been demonstrated in practice. In our calculations, also the applied model parameter values remained in the range of the model parameters used in the literature. This paper presents a new approach for Velocity–Density Model (VDM) synthesis. It consists in modelling separately each of the density and the velocity (macroscopic parameter). From this study, safety time headway (microscopic parameter) can be identified from macroscopic data by mean of interpolation method in the developed map of velocity–density. By combining the density and the velocity models, a generalized new VDM is developed. It is shown that from this one, some literature VDMs, as well as their properties, can be derived by fixing some of its parameters.


Introduction
Traffic modelling can be classified into the three following classes.
The first class includes the microscopic models where the traffic is viewed as a system of interacting particles or vehicles.This interaction is modelled by differential equations as done by Bando et al. (1998), Helbing and Tilch (1998), Treiber et al. (2000b), Molina (2005), Ge et al. (2008), Kesting (2008), Rakha and Gao (2010).For interested readers, a general overview of microscopic traffic models is presented by Derbel et al. (2012aDerbel et al. ( , 2012b)).
The second class includes the macroscopic models where the traffic is viewed as one group of particles.Macroscopic parameters are traffic density, traffic flow and velocity, which are used for graphic representation of the fundamental diagram.Several works are based on this approach such as those ones developed by Holden and Risebro (1995), Herty and Klar (2003), Treiber et al. (2000b), Péter (2012), Bede et al. (2013), Dömötörfi et al. (2016), Péter et al. (2015), Bede and Péter (2014).
The third class includes the mesoscopic models, which appears to be an intermediate between the two last classes.Here, the traffic is viewed as clusters of ve-hicles.Several studies have been realized in this field such as those done by Prigogine and Herman (1971), Paveri-Fontana (1975), Mahnke and Kaupužs (1999).
The presented research work deals with microscopic and macroscopic approaches, where the traffic is represented by one cluster.The goal is to predict traffic density and velocity with a minimum of errors.Most of the literature Velocity-Density Models (VDMs) are developed and identified by mean of experimental macroscopic traffic data.Since we do not have these data, our work is based on the simulation data.Here, the idea is to generate macroscopic traffic data from microscopic traffic model simulation.Nonetheless, this microscopic model must faithfully reproduce the traffic behaviours and be validated by the fundamental diagram study.
In this paper, the developed VDM is based on the Intelligent Driver Model (IDM) (Kesting 2008;Treiber et al. 2000a).Our new method proceeds by the identification of the simulated velocity and density data separately.Among the advantages of this method is the ability to identify each microscopic parameter from macroscopic ones by interpolation method using our developed map of macroscopic data related to microscopic parameter.Then, a new VDM is computed with a mathematical development of the velocity and the density models.In addition, it will be shown later that some of existing models in the literature, as well as their properties can be derived from this new VDM by fixing some of its parameters.
In this paper, Section 1 presents a brief state-of-the art of the existing VDMs.Section 2 identifies the velocity and the density functions.Section 3 presents the synthesis method of the new VDM.Section 4 presents the generalized VDM and the last section concludes and gives outlooks.

State-of-the-art of the Velocity-Density Models
Two classes of VDMs can be distinguished: the stochastic models and the deterministic models.This section presents a brief state-of-the-art of deterministic VDMs considering the chronological order.

Greenshields Model
Up to our knowledge, the first deterministic VDM was proposed by Greenshields (1935).This affine model is given by the following expression: where: D max is the maximum density which is the jam density.This model is identified by linear regression method using seven experimental observations.Recently, Wang et al. (2013) have showed that, with more than seven data, the Greenshields model is not enabling the prediction of velocity and density.

Greenberg Model
By the analogy with fluid flow, Greenberg (1959) has developed a logarithmic velocity-density relationship given by: The main drawback of this model is its inability to predict velocities for low densities.Indeed, the velocity tends to infinity when the density tends to zero, which is unrealistic.(1961) has derived an exponential model in order to overcome the drawbacks of the Greenberg and Greenshields models for the free traffic flow condition.This model is given by:

Underwood
The main drawback of the Underwood model is that velocity becomes zero only when density D reaches infinity and not D = D max .Hence, this model cannot be used for predicting velocities at high densities.

Newell Model
The VDM of Newell (1961) is expressed by: where: λ>0 is the slope of inter-distance-velocity curve at V = 0 km/h.Here, when D = D max , then V = 0 km/h; D = 0 then V = V max .Therefore, the limit conditions are verified by the Newell model.et al. (1967) have enhanced the Greenberg model by studying various macroscopic traffic models.By estimating the density from velocity and flow data, they propose the new VDM expressed by: ( )

Drake
According to Ardekani et al. (2011), this model presents a better fitting than the models of Greenshields, Greenberg and Underwood for non-congested conditions.In case of congested conditions, the Drake model presents a poor data fitting.(1967) has generalized the Greenshields model leading to a new velocity-density relationship given by:

Pipes
By varying the values of r and m, a family of models can be developed.For example, Greenshields model is obtained for r = 1 and m = 1.(1968) has proposed another model expressed as follows:

Drew
( ) According to Ardekani et al. (2011), at free-flow phase, the Drew model presents an underestimated velocity, but in the congested phase, the velocity is overestimated.

Del Castillo Model
Del Castillo and Benítez (1995) have developed a VDM, which is given by the following expression: where: C j is the kinematic wave speed given by: According to MacNicholas (2008), the drawback of this model is the large kinematic wave speed range, which makes difficult its estimation.
If D max = 1 and C j = λ, then we have the Newell model.

Van Aerde Model
Van Aerde (1995) developed a new VDM, which is based on a simple car-following model.This last one depends on the free and current velocity and a calibrated constant.The VDM is given by: ( ) where: C 1 , C 2 , C 3 are constants which can be calibrated by nonlinear regression.

MacNicholas Model
MacNicholas (2008) have proposed the following VDM: ( ) where: m, q are real constants.By varying these constants, a family of models can be developed.For example, if D max = 1, m = 0 and 1 2 the Drew model expression is found.

Power Function Model
The power function proposed by Del Castillo ( 2012) is given by: where: θ is a shape parameter; a, b are constants; max r = D D .

Exponential Model
The exponential model is given by Del Castillo (2012) as follows: where: ( )

Negative Power Model
The negative power model is presented by Del Castillo (2012) as: ( ) In this case, the parameter vector e contains 5 parameters (Fig. 1).
The Table 1 demonstrates the favourable and unfavourable parameter domains.The internal domain is located between the two distinct domains, in most of the cases the practical parameter comes from this internal interval.The borders of the intervals are empirical values, in a given case the coordinates of the e = [e 1 , e 2 , e 3 , e 4 , e 5 ] parameter-vector are determined via regression analysis after the velocity-density measurement.
The specialty of the introduced V(r, e) function is that it gives the same results as the linear function of Greenshields, if every parameters' value equals to 1 (Fig. 2).
Del Castillo (2012) (negative power model) Péter and Fazekas (2014) (environmental parameterization model) ( ) This result shows that Greenshields' linear function has parameter values from the mid-range, so that provides really an average V(r) velocity-density function relationship in practice.

Synthesis
Table 2 summarizes the references with the macroscopic traffic models cited in this section.V max is the maximum velocity, D max the jam density, λ the slope of inter-distance-velocity curve at V = 0, C 1 , C 2 , C 3 , m, p, q and r are constants and C j is the kinematic wave speed at jam density.
It was necessary to review the evolution of the velocity-density laws in the literature, since these functions have a decisive role in the fundamental equation in case of traffic models.They shall satisfy two conflicting needs: to be as simple as possible, so that the large complex simulation models provide fast numerical calculations, while satisfying the real practical situations as widely as possible.A common feature of the velocity-density laws is that they result from non-linear regression analysis based on measurements and hypotheses (preconceptions).The common flaw of the classical results is that because they do not refer to other environmental impacts, they only examine the effect of speed on the vehicle density, so essentially any of them can be suitable for a given modelling.Péter and Fazekas (2014) goes much further and examines also the environmental parameters.In case of the IDMs the comprehensive examination of the speed-density functions is an interesting problem, because without preconception an expanded set of functions can be examined from the point of view of which known function types or other new models are considered as valid by this model.Of course, neither the classic IDMs contain the environmental parameters mentioned above, so the studies in this area are far from closed.
The next section is dedicated to generate microscopic and macroscopic simulated traffic data, which are the density and the velocity.In addition, the impact of microscopic parameters to macroscopic ones is investigated.

Macroscopic Traffic Data Generation
Based on microscopic traffic simulation, density-time and velocity-time functions will be computed in this section.
Subsection 2.1 presents the simulation assumptions and the IDM intended to represent the microscopic longitudinal vehicle motion.Subsection 2.2 introduces the used mathematical formula to compute macroscopic parameters (velocity and density).Subsection 2.3 presents the simulation results and discussions and subsection 2.4 studies the impact of microscopic parameter on macroscopic ones.

Microscopic Traffic Model: Intelligent Driver Model
The IDM is an Adaptive Cruise Control (ACC) system intended for adjusting the driver's longitudinal desired velocity and safety time gap.The IDM, developed by Kesting et al. (2008), is expressed by: where: a n is the maximum acceleration of the vehicle n [m/s 2 ]; 0 n v the desired velocity of the vehicle n [m/s]; s n the distance gap [m]: The desired minimum gap of the vehicle n, * n s is given by: Compared to the other Adaptive Cruise Control (ACC) models in the literature, the IDM shows more advantages in terms of easy implementation, calibration and the intuitive and the availability of its parameters.These ones are the desired velocity 0 n v and the safety time gap T n , which are fixed as system inputs

Density Formula
As shown in subsection 1.15 in Table 2, the quotient between the traffic density and the jam density are used in all models in the literature for density normalization.In this paper, the density is already normalized.Here, the traffic road section is limited by the first and the last vehicle.Then, the density formula can be written as: where: l k is the length of the vehicle (k=1, 2, …, N); N the number of vehicle in the platoon; L platoon the length of the platoon.The density d can be normalized in the interval [0,1].Then, we have: where: D max = 1 in this case.For this purpose, in the rest of this paper, the density d will be noted as D.

Velocity Formula
The platoon velocity v is the average velocity of the vehicles in the platoon which is given by: ( ) ( ) where: v n is the velocity of the vehicle n.

The Simulation Model
The following system of non-linear matrix differential equations shows the structure of the IDM under investigation.For a multi vehicle system, the IDM is given by: s s or: ( ) ( ) ( ) This model examines the longitudinal dynamics in the direction of travelling in case of vehicles travelling in a lane and is specifically capable of analysing the evolution of the speed-density law.The movement is performed on a line, which is a given lane.We have dealt with the further generalization of the model in Derbel et al. (2012bDerbel et al. ( , 2013)).

Simulation Assumptions
In our simulations, the following assumptions are taken into account: -the AAC model, which is our microscopic traffic model, is already validated and its parameters are identified based on experimental data (Kesting et al. 2008); -the traffic is homogeneous.It means that the longitudinal motion of all vehicles is controlled by the same ACC model (IDM) with the same parameter values; a heterogeneous traffic includes vehicles with manual and automatic driving modes; -the road has only one lane.
-only the longitudinal vehicle motion is studied; the lane change and the lateral motion are neglected.

The Simulator
The development of traffic simulators has received much attention in recent years.Many companies have invested in such projects and have made simulators integrating various dynamic and kinematic models of vehicles, driver and road architectures available.These simulators are known by their complexities in terms of software development.For example, the simulator ARCHISIM adopts multi-agent approach and techniques from artificial intelligence to simulate complex phenomena.
Our motivations for the development of a mixed traffic simulator (coexistence of automated and manual driving style in the same traffic section) are justified according to the following two aspects: -the high cost of traffic simulators in the market; -the collision management problem.For example, the traffic simulators CarMaker and Microsimulation of Road Traffic Flow (MRTF) do not perfectly manage collisions between vehicles.Our mixed traffic simulator is developed using the C language and the Open Graphics Library (OpenGL) library for graphical interface.Fig. 3 shows the structure of mixed traffic simulator.This simulator has inputs that are used by its different modules to save the output of the simulation in different formats such as video and/or data required for post-processing.
The inputs of the simulator are the following three: -'XML File' in which some parameters can be set: the simulation number, the initial vehicle number, the road length and the minimum and maximum percentage of automated vehicles; -'Screen' -displays the architecture and the scenario; -'Video' -to save the simulation; -'File .txtor .bin'-saves the data needed for postprocessing.Fig. 4 shows the position of the vehicles in a platoon.
The IDM simulation model with the parameters given by Table 3 is applied to simulate the fully automated traffic using our developed microscopic traffic simulator.The presented parameter values in this table are originated from an identification step using experimental data and used to validate the IDM with the fundamental diagram by Kesting et al. (2008).At each simulation step, the simulated microscopic traffic data such as the position, the velocity and the acceleration of each vehicle are computed.Then, the density and the velocity are computed according to Eqs ( 17) and ( 18) respectively.
Simulation parameters are given in Table 4.In this table, the jam distance s n 0 is set to zero in order to have the maximum density equal to 1.To study the sensitivity of macroscopic parameters (velocity and density) according to the microscopic parameters (a n , b n , T n ), traffic simulation is performed made for each of the maximum acceleration a n values (the other microscopic parameters are constants), the desired deceleration b n (the other microscopic parameters are constants) and the safety time headway T n (the other microscopic parameters are constants).

Simulation Results
Fig. 5 shows the density versus the time (first curve) and the velocity versus the time (second curve) during 1000 s of simulation with the microscopic parameters given by Table 3 and the simulation parameters given by Table 4.Here as the density increases, the velocity decreases until the stable state.

From Macroscopic Parameters to Microscopic Parameters
In this section, two goals are fixed: the first one consists in studying the dependency of the shape of the velocity-time and the density-time functions to microscopic parameters.The second goal is to study the ability to identify microscopic parameters from macroscopic parameters.Microscopic parameters are the maximum acceleration a n , the desired deceleration b n and the safety time headway T n .In this step the following assumptions are made: the maximum acceleration a n ∈ [2, 7] m/s 2 the desired deceleration b n ∈ [2, 5] m/s 2 and the safety time headway T n ∈ [1.5, 4] s.The ranges are empirical intervals given by Kesting et al. (2008).In this subsection, it is noted that all vehicles have the same parameters (homogeneous traffic).

Impact of a n , b n and T n on Density
As shown in Fig. 6, the maximum acceleration and the desired deceleration have no impact on traffic density.
Indeed, when the traffic becomes stable: Dv n = 0, ∀n ∈ {1, ..., N} and then The safety time headway T n has an impact on traffic density i.e when T n increases, traffic density decreases.In fact, vehicles tend to increase inter-distance when T n is high, then the traffic density decreases.

Impact of a n , b n and T n on Velocity
As shown in Fig. 7, the maximum acceleration and the desired deceleration have no impact on the velocity.The same reason cited in 3.4 is the cause of this result: when the traffic becomes stable, Dv n = 0 ∀n ∈ {1, ..., N} and then 0 2 . The safety time headway T n has an impact on the velocity i.e. when T n decreases vehicles ve-

New Velocity-Density Model Synthesis Method
The new method consists in studying the variation of each of the two macroscopic parameters (velocity and density given by the Figs 6 and 7 separately along the simulation time.To carry out this study, a family of functions is computed to approximate the density-time and velocity-time functions.For a given γ, this family of candidate functions is expressed by: ( ) ( ) For reasons, which will be later explained, γ is fixed for all candidate functions of the two macroscopic parameters.
The function D ∈ H γ , which is a candidate to approximate the density-time function, is given by: where: D 1 , τ d and D 2 are constants.
The function v ∈ H γ , which is a candidate to approximate the velocity-time function, is given by: where: V 1 < 0, and τ v > 0, V 2 > 0 are constants.γ is fixed for the two candidate functions to have an analytical time independent velocity-density relationship from Eqs ( 20) and ( 21).D and v functions are used to fit separately the density and the velocity data in the least square sense.The fitted curves of the density and the velocity together with the macroscopic data given by IDM simulation.Parameters of these two functions, obtained with setting γ = 0.8, are summarized in Table 5.The mean, the maximum and the minimum absolute errors are given by Table 6.The absolute error between data given by microscopic simulation and the computed functions D and v is small compared to the maximum values of each one.The IDM is written as: ( ) .
The solution for the velocity is given by: When traffic become stable, we obtain 0 =  n v and Then, ( ) ( ) and, as shown in Eq (22), V 2 depends on the interdistance and the safety time headway T n and does not depends on a n and b n .

Velocity-Density Model Synthesis
From the density model defined by Eq. ( 20) and the velocity model defined by Eq. ( 21), the VDM is developed analytically in this section.

Model Synthesis
Eq. ( 20) is given by: The function v ∈ H γ , which is a candidate to approximate the velocity-time function, is given by: From these two equations, we get: Let V, the function defined as: with slight abuse of notation, we have Then, the analytic velocity-density relationship can be written as: To calibrate this model, the limit conditions are applied in the next subsection.

Sub Models
From the model defined by Eq. ( 34), some of existing models in the literature can be derived by fixing some of its parameters: -If D 2 = 0 and e V = 0.
In this case, In this case, ( ) ( ) Eqs ( 20) and ( 21) can be written as: Then: ( ) where: the relationship between velocity and density remains linear.In case of the Greenshields model, V max = V 2 and D max = 1.
In this case, ( ) which is the Drew model given by Eq. ( 7), where D max = 1.
Since τ d and τ v are two time constants, then 0 τ ≥ τ If D 2 → 0, then only lower densities can be predicted.Therefore, velocity can be estimated only in free flow condition.Thus, the model cannot predict lower velocities.This result has been proven by Ardekani et al. (2011), empirically and proved analytically now by our method of velocity-density synthesis and through our generic model.
Taking into account the velocity processes the method is directly connectable with the optimization of environmental pollutions, so further research in this direction may be conducted within the project.It has an important influence in many areas of transportation (Tettamanti et al. 2015;Gumz, Török 2015;Török et al. 2015;Lakatos 2015;Szauter et al. 2014).

Fig. 1 .
Fig. 1.V(r) velocity-density function with e 1 = 2; e 2 = 1; e 3 = 1; e 4 = 1; e 5 = 1 parameters ) where: a n is the desired deceleration of the vehicle n [m/ s 2 ]; 0 n s the jam distance of the vehicle n [m]; T n the safety time gap of the vehicle n [s].

Fig
Fig. 3. Structure of the developed simulator

Fig. 4 .
Fig. 4. Positions of the vehicles in the platoon

Fig. 6 .
Fig. 6.Density for different a n , b n and T n sets

Fig. 8 .Fig. 9 .
Fig. 8. Impact of the time headway on the velocity-density curve 0 0.2 0.4 0.6 0.8 1.0 Density max = 1.One of the advantages of the new model is the finding of the physical meaning of the parameter m.-If D 2 = 0, e V = 0 and τ d = τ v .
then p ≥ −1.In this study, the density is expressed as:

Table 1 .
Demonstration of the e parameter vector

Table 5 .
Parameters of the D and v functions with γ = 0.8