- Original paper
- Open Access

# A macroscopic traffic model for traffic flow harmonization

- Zawar H. Khan
^{1}Email author and - T. Aaron Gulliver
^{2}

**10**:30

https://doi.org/10.1186/s12544-018-0291-y

© The Author(s). 2018

**Received:**12 September 2017**Accepted:**5 April 2018**Published:**20 June 2018

## Abstract

Traffic flow will harmonize to forward conditions. The time and distance required for harmonization can have a significant effect on the traffic density behavior. The flow can evolve into clusters of vehicles or become uniform depending on parameters such as safe time headway and safe distance headway. In this paper, a new model is presented to provide a realistic characterization of traffic behavior during the harmonization period. Results are presented for a discontinuous density distribution on a circular road which shows that this model produces more realistic traffic behavior than other models in the literature.

## Keywords

- Macroscopic traffic model
- Reaction distance
- Reaction time
- Safe distance headway
- Safe time headway
- Traffic flow
- Roe decomposition

## 1 Introduction

This paper considers the behavior of vehicles as they harmonize to forward traffic conditions. The time for traffic harmonization is based on the front traffic stimuli, i.e. the time to react and align (harmonize) to the forward traffic. The time required to react is known as the reaction time, and the time for traffic alignment (harmonization) is known as the transition time [1]^{.} The reaction distance is the distance travelled during the reaction time, while the transition distance is the distance covered during the transition time. The sum of the transition and reaction times is known as the safe time headway. This is the time required for the safe adjustment of velocity. The distance travelled during the safe time is known as the safe distance headway.

Drivers adjust their velocity when a change in traffic flow is observed in an effort to achieve the equilibrium velocity distribution. This distribution depends on the traffic density as well as driver behavior and road characteristics, and will result in a homogeneous traffic flow [2]. This flow will evolve into clusters with a large safe distance headway and small safe time headway. Conversely, a small safe distance headway and large safe time headway will produce a more uniform flow. The goal of this paper is to develop a simple, realistic model to characterize the traffic flow. This will lead to better control of traffic behavior to mitigate congestion, reduce pollution levels, and improve public safety. These models can also be employed for automatic control of traffic flow to reduce travel time.

The main types of traffic models are macroscopic, mesoscopic and microscopic. Macroscopic models consider the aggregate behavior of traffic flow while microscopic models consider the interaction of individual vehicles. These models include parameters such as driver behavior, vehicle locations, distance headways, time headways, and the velocity and acceleration of individual vehicles. Mesoscopic models share the properties of macroscopic and microscopic traffic models. These models characterize the influence of vehicles in close proximity and then approximate the cumulative temporal and spatial traffic behavior [3]. Macroscopic traffic models are typically employed due to their low complexity.

Newel proposed a microscopic traffic model and acknowledged for the first time that the distance headways varies during traffic harmonization [4], but the variable distances between vehicles were not explained [5]. The General Motors model indicates that the distance headway between vehicles increases with velocity. However, this model ignores the variable distances between vehicles at slow speeds [6, 7]. Gipps characterized variable distance headways using a safety rule [8]. According to this model, the distance headways between vehicles should be sufficient to decelerate and harmonize speeds. However, the behavior of the driver population on the distance headway was not considered. Wiedemann proposed a psychophysical model to characterize individual driver behavior based on conscious and unconscious reaction and perception [9]. Drivers unconsciously harmonize small changes in speed for small distance headways, so the reaction of these drivers is slow. Conversely, with large distance headways, drivers consciously harmonize speeds with large reactions and perceptions. The limitations of the Wiedemann model are the constant ranges for perception and reaction which actually vary depending on the speed harmonization [10].

Driver behavior varies with ethnicity, gender, age, psychology and intoxication level. People in the age groups 18–25, 25–55 and 55+ have different cognitive and physical behavior and thus also different driving behavior [11]. Cognitive and physical behavior decline with age which increases the probability of accidents. The crash rate between ages 35–64 is three times less than at 65+ [12]. Older people can have difficulties moving their heads sideways to scan the traffic flow and there can be memory issues. Thus, reaction to changing conditions can be sluggish and is a major cause of accidents at intersections [11, 13]. Young drivers often exhibit limited horizontal road scanning behavior which is similar to the sluggish behavior of intoxicated drivers [14]. Drugs and alcohol change thought processes and typically lead to slow driving behavior. In 1972, teenagers in the US were five times more likely to die in a traffic accident than people in the 35–64 age group. Experienced drivers have a wide horizontal traffic scanning behavior. Thus, they quickly recognize changes in speed and density and so take less time to perceive and react than inexperienced drivers. Existing microscopic models do not characterize differences in driving behavior. To more accurately characterize traffic flow harmonization (alignment), a model is required which captures the effect of driver behavior [5]. This behavior can be characterized for specific groups such as intoxicated drivers, old and young drivers, and considering ethnicity and experience.

One of the most popular macroscopic traffic flow models is the two equation model developed by Payne [15] and Whitham [16] which is known as the Payne-Whitham (PW) model. The first equation is based on the continuity equation for the conservation of vehicles on a road, while the second models the acceleration behavior of traffic based on driver anticipation and relaxation. Driver anticipation results from the presumption of changes in the forward traffic density, while relaxation is the tendency to adjust velocity based on traffic conditions. The relaxation time can be considered the transition time. The PW model is based on the assumption that vehicles on a road have similar behavior. Smooth traffic velocities and density distributions are employed [17], and alignment (harmonization) occurs with a constant velocity [18]. Unfortunately, this results in unrealistic velocity and density behavior [19]. Zhang improved the PW model using the fact that driver anticipation cannot be constant [20], and considered that drivers harmonize their speeds based on the density distribution. However, this model can produce unrealistic results in some traffic flow situations.

Khan and Gulliver [21] improved the PW model using the fact that driver anticipation is based on the velocity of the forward traffic, so that traffic behavior depends on the velocity during transitions. It was shown that this model provides more realistic traffic flow and density than the PW model. The sensitivity of traffic is the rate at which alignment occurs. The limitation of the Khan-Gulliver (KG) model is that this sensitivity depends only on the relaxation time *τ*. For high velocities, *τ* is small, so traffic alignment can occur too quickly, whereas with low velocities, *τ* is large so alignment can be very slow. As a consequence, this model does not have sufficient flexibility to properly characterize traffic behavior.

*d*

_{ s }is the safe distance headway and

*t*

_{ s }is the safe time headway. The traffic density distribution has a greater variance at lower safe velocities [21], and changes in this distribution during alignment depend on the velocity adjustments required to adapt to the equilibrium velocity distribution. In the proposed model, a parameter is introduced to regulate traffic flow behavior so these adjustments are appropriate. With a large regulation value, the flow evolves into a large number of small clusters. Conversely, the traffic flow is more uniform with a small value. This value can be extended to incorporate driving behavior due to factors such as intoxication, experience, ethnicity, and age, to accurately model traffic behavior. It can also be employed for traffic during adverse weather and congestion, and movement in clusters. The effect of this parameter is examined in Section IV.

The remainder of this paper is organized as follows. Section II presents the KG, Zhang and proposed models. In Section III, the well-known Roe decomposition technique is used to implement these models, and performance results are presented for a circular road in Section IV. Finally, some concluding remarks are given in Section V.

## 2 Traffic flow models

*t*and

*x*denote temporal and spatial derivatives, respectively.

*ρ*and

*υ*are the traffic density and average velocity, respectively, so that

*ρυ*is the flow.

*υ*(

*ρ*) is the equilibrium velocity distribution and

*d*

_{ tr }is the transition distance. A large average velocity results in a small relaxation time and thus the alignment (harmonization) can be too fast and produce unrealistic behavior.

*υ*=

*υ*(

*ρ*), and the traffic flow is smooth. The sensitivity of this term is given by

*ζ*

_{1}only depends on the relaxation time

*τ*, which may not be sufficient to produce appropriate traffic behavior.

*υ*(

*ρ*) is greater than the average velocity

*υ*there is acceleration and alignment will occur at a velocity greater than

*υ*. Conversely, if

*υ*(

*ρ*) is smaller than

*υ*there is deceleration, and alignment will occur at a velocity smaller than

*υ*. This can be characterized by the numerator of (7). Further, alignment (harmonization) depends on the physiological and psychological response of the drivers. This behavior can be characterized by modifying the denominator of (7). Thus, the number 2 in (7) is replaced with a flow regulation value

*b*. A small value of

*b*will produce a more uniform flow, while a large value will result in clustered traffic. The new source term is then

*d*

_{ r }and transition distance

*d*

_{ tr }so that

The psychological response of a driver is characterized by the transition distance, and the physiological response by the reaction distance. A sluggish driver responds slowly and takes more time to perceive and process forward traffic conditions. Thus, they will have large reaction and transition distances to align to the traffic. Examples of sluggish drivers are intoxicated or distracted drivers, old and new drivers, and drivers affected by fatigue [22, 23]. The reaction time in a fatigued state has been shown to be 17% longer than in an alert state, and the increase in reaction time is greater for females than males [24]. Further, old drivers have longer reaction times than young drivers. For a sluggish driver, *b* should be large. Conversely, excited or aggressive drivers such as commercial or teenage drivers will have a small reaction time and corresponding small transition distance [18], so *b* should be small.

*C*

_{0}is the velocity constant which characterizes driver response. According to this model, driver response does not depend on the traffic conditions and is a constant. The relaxation term of the PW model is the same as that in the KG model, as shown in Table 1. Several models have been proposed for

*υ*(

*ρ*) [25], but the most commonly employed is the Greenshields model which is given by [26]

*ρ*

_{ m }and

*ρ*are the maximum and average traffic densities, respectively, and

*υ*

_{ m }is the maximum velocity on the road. Therefore, this model is employed here.

Traffic model comparison

Term | KG model | PW model | Proposed model | Zhang model |
---|---|---|---|---|

Anticipation | \( \rho \frac{\partial }{\partial x}\left(\frac{\upsilon^2\left(\rho \right)-{\upsilon}^2}{2{d}_{tr}}\right){\rho}_x \) | \( \rho {C}_0^2{\rho}_x \) | \( \rho \frac{\partial }{\partial x}\left(\frac{\upsilon^2\left(\rho \right)-{\upsilon}^2}{2{d}_{tr}}\right){\rho}_x \) | \( \frac{{\left(\rho \frac{\partial v\left(\rho \right)}{\partial \rho}\right)}^2}{\rho }{\rho}_x \) |

Relaxation | \( \rho \frac{\upsilon \left(\rho \right)-\upsilon }{\tau } \) | \( \rho \frac{\upsilon \left(\rho \right)-\upsilon }{\tau } \) | \( \rho \left(\frac{\upsilon^2\left(\rho \right)-{\upsilon}^2}{b{\upsilon}_s}\right) \) | \( \rho \frac{\upsilon \left(\rho \right)-\upsilon }{\tau } \) |

The relaxation term is the same as the PW and KG models, as shown in Table 1. With this model, the driver response depends on the traffic density. The next section presents Roe’s decomposition technique which is used to implement the KG, PW, Zhang, and proposed models.

## 3 Roe decomposition

*G*is the vector of data variables,

*f*(

*G*) is the vector of functions of these variables, and

*S*(

*G*) is the vector of source terms. The subscripts

*t*and

*x*denote partial derivatives with respect to time and distance, respectively. Equation (14) is then given by

*A*(

*G*) be the Jacobian matrix of the system. Then (15) can be expressed as

The data variables are density *ρ* and flow *ρυ* in the KG, PW, Zhang and proposed models. Roe’s technique is used to linearize the Jacobian matrix *A*(*G*) by decomposing it into eigenvalues and eigenvectors. This is based on the realistic assumption that the data variables, eigenvalues and eigenvectors remain conserved for small changes in time and distance. This technique is widely employed because it is able to capture the effects of abrupt changes in the data variables.

Consider a road divided into *M* equidistant segments and *N* equal duration time steps. The total length is *x*_{
M
} so a road segment has length *δx* = *x*_{
M
}/*M*, and the total time period is *t*_{
N
} so a time step is *δt* = *t*_{
N
}/*N*. The Jacobian matrix is approximated for road segments \( \left({x}_i+\frac{\delta x}{2},{x}_i-\frac{\delta x}{2}\right) \). This matrix is determined for all *M* segments in every time interval (*t*_{n + 1}, *t*_{
n
}), where *t*_{n + 1} − *t*_{
n
} = *δt*.

*G*be a small change in the data variables

*G*and Δ

*f*the corresponding change in the functions of these variables. Further, let

*G*

_{ i }be the average value of the data variables in the

*i*th segment. The change in flux at the boundary between the

*i*th and (

*i*+ 1)th segments is

*λ*

_{1},

*λ*

_{2}, ⋯,

*λ*

_{ p }] of the Jacobian matrix and

*e*is the corresponding eigenvector matrix. From [28], the eigenvalues should be positive so that

*G*= (

*G*

_{i + 1}−

*G*

_{ i }) is used. The flux at the boundary between segments

*i*and

*i*+ 1 at time

*n*is then approximated by

*i*and

*i*+ 1 respectively, at time

*n*. Substituting (19) into (20) gives

This approximates the change in density and flow without considering the source.

*A*(

*G*) is the same for these models and results in the same eigenvalues and eigenvectors, and also average velocity and density.

*A*(

*G*) as well as the corresponding eigenvalues and eigenvectors, average density and velocity for the KG model were derived in [21]. Assuming the right hand side (RHS) of the KG and proposed models are in quasilinear form, traffic flow alignment (harmonization) is not considered so that \( {S}_y(G)=\left(\genfrac{}{}{0pt}{}{0}{0}\right) \). Then (24) takes the form

The LHS of (3) and (11) is approximated using the Jacobian matrix \( (G)=\frac{\partial f}{\partial G} \), which is obtained from (25).

These show that when a transition occurs, the velocity changes according to the equilibrium velocity distribution and the average velocity.

*C*

_{0}is the velocity constant. This shows that traffic velocity alignment is at a constant rate

*C*

_{0}during transitions. The corresponding eigenvectors are

*i*and

*i*+ 1 for the KG, PW and proposed models is

*i*and

*i*+ 1 is given by the geometric mean of the densities in these segments

*ρ*presents the derivative of the equilibrium velocity distribution with respect to density. The corresponding eigenvectors are

*i*and

*i*+ 1 for the Zhang model is given in Appendix B.

- A.
Entropy Fix

*λ*

_{ k }of the Jacobian matrix, and

*e*is the corresponding eigenvector matrix. The Harten and Hayman entropy fix scheme [30] is employed here so that

This ensures that the \( {\widehat{\lambda}}_k \) are not negative and similar at the segment boundaries. The Jacobian matrix *e*|Λ|*e*^{−1} for the proposed, KG and Zhang models are given in Appendix C. The corresponding flux is obtained from (21) using *f*(*G*_{
i
}) and *f*(*G*_{i + 1}) and substituting *e*|Λ|*e*^{−1} for \( A\left({G}_{i+\frac{1}{2}}\right) \), the updated data variables, *ρ* and *ρυ*, are then obtained at time *n* using (24).

## 4 Performance results

*x*

_{ M }= 100 m. A discontinuous density distribution

*ρ*

_{0}at

*t*= 0 with periodic boundary conditions is employed.

*ρ*

_{0}is shown in blue in the figures. The Greenshields equilibrium velocity distribution given in (13) is used with

*υ*

_{ m }= 34 m/s and maximum density

*ρ*

_{ m }= 1. The safe distance headway is 28 m, the safe time headway is

*t*

_{ s }= 1.4 s, and

*d*

_{ tr }is 20 m. For the KG and PW models,

*τ*= 1 s. The total simulation time for the proposed and KG models is 30 s. The total simulation time for the PW model is 3 s and for the Zhang model is 4 s. Based on

*δx*= 1 m, the time step for the proposed, KG and Zhang models is chosen as

*δt*= 0.01 s, and the time step for the PW model is chosen as

*δt*= 0.006 s, to satisfy the CFL condition [31]. The number of time steps and road steps are 3000 and 100, respectively, for both the proposed and KG models, while the number of time steps for the PW model is 500. The number of road steps and time steps for the Zhang model are 100 and 400, respectively. The flow regulation parameters considered for the proposed model are

*b*= 1 and 2. The simulation parameters are summarized in Table 2.

Simulation parameters

Name | Parameter | Value |
---|---|---|

road step |
| 1 m |

equilibrium velocity |
| Greenshields velocity distribution |

maximum velocity |
| 34 m/s |

time step for the KG, Zhang and proposed models |
| 0.01 s |

time step for the PW model |
| 0.006 s |

velocity constant |
| 4.12 s |

safe distance headway |
| 28 m |

transition distance |
| 20 m |

safe time headway |
| 1.4 s |

safe velocity |
| \( \frac{28}{1.4}=20 \) m/s |

normalized maximum density |
| 1 |

total simulation time for the KG and proposed models |
| 30 s |

total simulation time for the PW model |
| 3 s |

total simulation time for the Zhang model |
| 4 s |

flow regulation parameter |
| 1, 2 |

relaxation time constant |
| 1 s |

number of time steps for the PW model |
| 500 |

number of time steps for the KG and proposed models |
| 3000 |

number of time steps for the Zhang model |
| 400 |

number of road steps |
| 100 |

*b*= 1 s at four different time instants. This shows that with a small value of

*b*, the traffic becomes quite smooth over time. At 5 s, the variation in traffic density ranges from 0.07 to 0.17, while at 15 s this variation is 0.1 to 0.14, and at 30 s the range is only 0.12 to 0.13. Figure 3 presents the normalized traffic density behavior with the proposed model and

*b*= 2 s at four different time instants. There are larger variations in the density than with

*b*= 1, but smaller than with the KG model. The traffic evolves into two clusters with a smooth density between them. The variation in density is between 0.09 and 0.16 at 15 s, and between 0.1 and 0.15 at 30 s.

*b*= 1. This corresponds to the density shown in Fig. 2. At 5 s, the variations in velocity are the greatest, ranging from 29 to 31 m/s. At 15 s, this variation is 29.5 to 31.5 m/s, while at 30 s, the difference is less than 1 m/s. These variations are smaller than with the KG model. Figure 6 presents the velocity behavior at four different time instants for the proposed model with

*b*= 2. This corresponds to the density shown in Fig. 3. The fluctuations in velocity are greatest at 5 s, with a range of 28 to 31 m/s, At 15 s, the range is 29 to 30.5 m/s, while at 30 s it is only 29 to 30.2 m/s. Thus, the velocity fluctuations are larger than with

*b*= 1, but smaller than with the KG model.

*b*= 1. At 5 s, the flow varies from 2.5 to 4.5 veh/s. The maximum and minimum flows occur at 50 m and 40 m, respectively. At 15 s, the flow varies from 3.2 to 4.2 veh/s, which is less than at 5 s. The maximum and minimum flows now occur at 70 m and close to 60 m, respectively. At 30 s, the flow is only in the range 3.5 to 4.0 veh/s, and the maximum flow occurs at 40 m. Figure 9 presents the corresponding traffic flow behavior with the proposed model and

*b*= 2. The behavior is more oscillatory at 5 s, and the flow varies from 2.5 to 6.0 veh/s, which is greater than with

*b*= 1. At 15 s, the flow varies from 3.0 to 4.5 veh/s, and it is almost the same at 30 s. However, the locations of the maximum and minimum traffic flows are different.

*b*in the proposed model can be used to adjust traffic oscillations and cluster behavior. For smaller values of

*b*, traffic becomes more uniform. Thus, unrealistic oscillations can be eliminated with this parameter, and traffic behavior can be properly characterized.

## 5 Conclusion

In this paper, a new model was proposed to characterize the physiological and psychological response of drivers to changes in the traffic flow. For a slow response, the traffic becomes clustered, while for a fast response the traffic flow is more uniform. A regulation parameter was introduced to characterize driver response to forward conditions. This allows for more realistic traffic characterization than with other models in the literature. With smaller values of the regulation parameter in the proposed model, changes in velocity are reduced and the traffic flow becomes smooth. Pollution emissions increase with changes in velocity, in particular carbon monoxide, nitric oxide and hydrocarbon emissions are greater at higher velocities. However, velocities between 16.7 m/s and 22.2 m/s result in reduced fuel consumption and pollution [32]. Variations in velocity can be reduced with the regulation parameter *b* in autonomous vehicles to reduce fuel consumption and pollution. The proposed model can help in analyzing the behavior of autonomous vehicles. This will lead to more accurate results which can be employed to reduce fuel consumption and pollution.

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