As the number of automobiles continues to increase year after year, the associated problem of traffic congestion has become a serious societal issue. Initiatives to mitigate this problem have considered methods for optimizing traffic volumes in wide-area road networks, and traffic-flow simulation has become a focus of interest as a technique for advance characterization of such strategies. Classes of models commonly used for traffic-flow simulations include microscopic models based on discrete vehicle representations, macroscopic models that describe entire traffic-flow systems in terms of average vehicle densities and velocities, and mesoscopic models and hybrid (or multiscale) models incorporating both microscopic and macroscopic features. Because traffic-flow simulations are designed to model traffic systems under a variety of conditions, their underlying models must be capable of rapidly capturing the consequences of minor variations in operating environments. In other words, the computation speed of macroscopic models and the precise representation of microscopic models are needed simultaneously. Thus, in this study we propose a multiscale model that combines a microscopic model—for detailed analysis of subregions containing traffic congestion bottlenecks or other localized phenomena of interest—with a macroscopic model enabling simulation of wide target areas at a modest computational cost. In addition, to ensure analytical stability with robustness in the presence of discontinuities, we discretize our macroscopic model using a discontinuous Galerkin finite element method (DGFEM), while to conjoin microscopic and macroscopic models, we use a generating/absorbing sponge layer, a technique widely used for numerical analysis of long-wavelength phenomena in shallow water, to enable traffic-flow simulations with stable input and output regions.

As the number of owned automobiles continues to increase year after year, the associated problem of traffic congestion has become a serious societal issue, responsible not only for lost time but also for increased emission of greenhouse gases and other harmful environmental effects. To date, proposals for reducing traffic congestion have called for the construction of bypasses and similar remedies, but the problem will not be solved by simply increasing road space alone. The primary reason for this is that, when the transportation capacity of a given thoroughfare is increased, it receives new users who had previously been using other roads, and this release of latent demand ensures that the congestion state remains unchanged. Moreover—as shown, for example, by the paradox discussed in [

Initiatives to mitigate these problems have considered not only simply expanding traffic capacity, but also optimizing traffic volumes in wide-area road networks, and traffic-flow simulation has become a focus of interest as a technique for advance characterization of such strategies. Indeed, through effective use of traffic-flow simulation one can reconstruct traffic systems, under a variety of conditions, at a tiny fraction of the cost of experiments using actual roads. Moreover, solving traffic problems requires an understanding of the causes, timing, and location of traffic congestion, as well as insight into how congestion propagates through road networks, and traffic-flow simulations also offer an efficient way to obtain results from this perspective. This situation has spurred the development of a variety of traffic-flow simulators, including ADVENTURE_Mates (Multi-Agent based Traffic and Environment Simulator) [

In this study, we design a multiscale traffic-flow model in which traffic-flow information is mutually exchanged between microscopic and macroscopic models. The proposition of this model is the first step in future simulations of large-scale road networks considering localized effects such as lane regulation and street parking, which are difficult to express in macroscopic models. The specific models we combine to yield our multiscale model are the generalized force model [

This paper is organized as follows. In this Introduction, we reviewed the fundamentals of traffic-flow models and their societal significance, surveyed prior research, and outlined the research presented here and the objectives of our study. In Section 2, we conducted a survey of conventional researches. In Section 3, we present an outline of our traffic-flow model. In Section 4, we discuss a method for discretizing our macroscopic traffic-flow model. In Section 5, we will describe the methodology of connecting the two models described up to this point, In Section 6, present an example of a simulation using the traffic-flow model of this paper. Finally, In Section 7, we present our conclusions and discuss avenues for future work.

Traffic-flow models may be broadly divided into two categories based on granularity: microscopic models, which separately track the motion of all vehicles to yield detailed descriptions of the behaviour of each individual vehicle in a model, and macroscopic models, which consider the vehicle densities in a traffic flow and apply methods of fluid mechanics. Because microscopic traffic-flow models consider the behaviour of individual vehicles, they are capable of producing highly detailed results, but the very nature of these models makes their simulations computationally costly, and enormous computation time is required to recreate traffic flows over wide geographical areas. In contrast, macroscopic traffic-flow models do not attempt to resolve the behaviour of individual vehicles, but rather represent traffic flows in terms of macroscopic quantities—such as traffic volumes, vehicle densities, and space mean velocities—associated with specific points or sections in time and space. These models do not attempt to capture the state of individual vehicles, so their simulation requires relatively little computation time, even for models spanning wide geographical areas.

Because traffic-flow simulations are designed to model traffic systems under a variety of conditions, they must produce detailed results that reflect minor variations in the operating environment. Moreover, it is impossible to measure parameters of individual cars and drivers accurately in actual traffic systems, and thus traffic-flow simulation must be constructed by components incorporating randomness in order to represent parameter distributions. This requires that simulations be performed multiple times, even with identical parameter settings, with different random seeds; consequently, each individual simulation must require only a short period of time to complete if one is to have any hope of testing multiple sets of operating conditions.

In actual traffic flows, certain road segments—such as segments with merge points, sags, or traffic lights—act as bottlenecks from which traffic congestion frequently arises. The challenge of simulating such phenomena has motivated the introduction of multiscale models, which combine a microscopic model for specific subregions with a macroscopic model for all other subregions. Using a microscopic model enables detailed local analysis of segments where traffic congestions or accidents often occur, and combining macroscopic models allows wide peripheral geographical areas to be calculated at a low computational cost.

To date, studies of multiscale traffic-flow models have employed mathematical techniques to connect microscopic and macroscopic models and convert between the two domains. Reference [

The scope of both of these studies was restricted to presenting transformation formulas and mathematical proofs for transferring computational results between microscopic and macroscopic traffic-flow models, and to date the models have not been verified in the way of traffic engineering or applied to practical situation. Moreover, the model of Garavello et al. used an equilibrium traffic model for the macroscopic domain; such models, in which the velocity is uniquely determined by the density, cannot reproduce realistic traffic flows. The model of Lattanzio et al. used a non-equilibrium traffic model, but improvements to the stability of this model were left as a topic for future work. In a related research field, recent efforts have been devoted to the derivation of macroscopic models from kinetic equations (mesoscopic models) [

In this study, we combine two traffic-flow models, one microscopic and one macroscopic, with the goal of achieving high-precision traffic-flow simulation at low computational cost. In this section, we discuss the key features of each model; our procedure for combining the models is discussed in the following section.

The microscopic model used in this study is the generalized force model (GFM) of Helbing et al. [

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

_{n}: Minimal vehicles distance |
1.38 [m] |

_{n}: Acceleration time |
2.45 [s] |

_{n}: Deceleration time |
0.77 [s] |

_{n}: Safe time headway |
0.74 [s] |

_{n}: Acceleration interaction range |
5.59 [m] |

_{n}: Deceleration interaction range |
98.78 [m] |

A traffic flow comprising the motion of many vehicles may be viewed macroscopically as a continuous fluid. Continuous macroscopic traffic-flow models are models that adopt this point of view, describing vehicle traffic as the motion of a continuous fluid and interpreting coarse-grained fluctuations in traffic density as the propagation of waves through this fluid. These models boast the advantage of modest computational cost even for simulations encompassing large numbers of vehicles but suffer from the drawback that they are essentially incapable of capturing the behavior of individual vehicles. Although there do exist a small number of macroscopic traffic-flow models that distinguish vehicles of different types—such as passenger vehicles, trucks, and buses—even in such cases, all vehicles of a given type are treated as identical indistinguishable elements. Descriptions of this sort are clearly limited in their ability to represent actual real-world traffic flows, which involve vehicles of many different types, operated by drivers of many different driving styles, thereby giving rise to unpredictable traffic flows. Moreover, the data-acquisition capabilities of continuous macroscopic models are limited to two macroscopic quantities, the density and average velocity of vehicles within a finite segment of a road, and are insufficient to yield a complete re-creation of actual realistic traffic flows. The continuum equation, which means the total number of vehicles is conserved, is written as follows:

In this context, the term “equilibrium” refers to the fact that, in ETMs, all traffic phenomena arise in accordance with a single equilibrium curve (_{m}]; NETMs relax this restriction by adding a new partial differential equation describing the rate of change of the velocity, thus enabling the emergence of non-equilibrium states.

To compensate for this limitation of ETMs—that, by imposing a velocity-density relation, they constrain traffic flows to lie within the equilibrium state curve—higher-order equations approximating the fluid-mechanical law of momentum conservation have been proposed; for example, for the velocity-density relation one might introduce a partial differential equation describing the rate of change of the velocity. In this study, we adopt the PW model, an NETM described by the following equations:

In this study, for the equilibrium velocity of the macroscopic traffic-flow model, we use the equation derived by Greenshields et al. [

We adopt an generating/absorbing layer model [

where ^{in}_{1} and ^{in}_{2} are additional terms for inflow boundary condition, and ^{out}_{1} and ^{out}_{2} are ones for outflow boundary condition defined as follows:

Here, ^{in} and ^{in} are the imposed density and flow rate at the inflow boundary, and ^{out} and ^{out} are the imposed density and flow rate at the outflow boundary. _{L} and _{R} are positions of inflow and outflow boundaries. ^{in} and ^{out} are lengths of layers at inflow and outflow boundaries. ^{in} and ^{out} are parameters to adjust the effectiveness of layers. A schematic view of these layers is shown in

The discontinuous Galerkin finite element method (DGFEM) [

The divergence form of

_{i} (_{i} (

where

as shape functions as follows:

Letting ^{e} ∈ (^{e}_{L}, ^{e}_{R}) be the area of element

where

to determine the numerical flux in this study.

Substituting

where

Applying the explicit Euler method to

In this section, we use a working example to illustrate our procedure for connecting the PW model to the GFM and converting between the two models. We begin by converting computational results obtained for the PW model into data that may be processed by the GFM. This involves computing the number of vehicles passing through the PW model flow outlet on each timestep. After using DGFEM to analyze the PW model, the number of vehicles passing through the flow outlet may be computed ^{n} of outflowing vehicles on the

However, in this approach, it is difficult to determine uniquely the positions and velocities of outflowing vehicles. The alternative option is to place the connecting region at a point ^{con} lying upstream from the sponge layer. The concept of the approach is shown in ^{con} at that point then determines ^{n}. The “con” superscript labels quantities associated with the connecting region.

Using this method, the density ^{con} at the location of the connecting region may be used to determine the velocity ^{con} at that point according to the following:

Next, consider the number of vehicles arising on the GFM side. For the GFM, we use a smaller timestep than for the PW model; denoting the ratio of timesteps by ^{NE} = ^{AG}, where (^{NE} and (^{AG} are respectively the timesteps used for the PW model and for the GFM.

For the PW model, we denote by

For an arbitrary time

As the PW model proceeds from step

In this study, we use the parameter settings listed in

Microscopic traffic-flow model | Macroscopic traffic-flow model | |
---|---|---|

Model | Car-following model (GFM) | Continuum model (PW model) |

1 time step ( |
0.12 [s] | 0.36 [s] (= 0.0001 [h]) |

Road length | 500 [m] | 20000 [m] (= 20 [km]) |

Number of road segments | 20 (25 [m]/road segment) | 50 (400 [m]/road segment) |

Maximum velocity | 25 [m/s] (= 90 [km/h]) | |

Maximum density | 0.18 [veh/m] (= 180 [veh/km]) |

We also generate graphical visualizations to illustrate results of the microscopic model simulation; an example is shown in

More precisely, whenever the number of outflowing vehicles—given by the product of the macroscopic-model traffic volume and the length of a single timestep—attains or exceeds an integer value, that number is input by the microscopic model as the number of inflow vehicles. Thus, the relationship between outflow and inflow counts may be stated more accurately in the form

where 0

We next compare the density distributions of the macroscopic and microscopic models.

The section from 0 to 4 km is the sponge layer flowing into the macroscopic model, which in this case we have set to an inflow volume of 34 vehicles/km. On the other hand, the section from 16 to 20 km is the sponge layer for flowing out from the macroscopic model. To determine density values in the microscopic-model region, we obtain the positions of all vehicles on each segment, then evaluate a weighted average, with higher weights assigned to vehicles lying closer to segment centers, to obtain and plot quasi-density values for each segment.

To compare computation times for various simulation models, we prepared three types of simulation and measured the computation time required to complete each:

Local (500 m) | Global (20000 m) | Total | |
---|---|---|---|

Microscpic model (offline simulation) | (Micro) 15.94 [s] | (Macro) 636.40 [s] | 652.34 [s] |

Multiscale model (micro + macro) | (macro) 20.80 s. [s] | 36.54 [s] | |

Microscpic model | 2023.38 [s] |

In this table, “Microscopic model (offline simulation)” refers to the total time required to complete two separate simulations of our microscopic model for roads of lengths 500 and 20000 m, whereas “Multiscale (micro + macro) model” refers to the time required to simulate the multiscale model proposed in this study, and “Microscopic model” refers to the computation time required to complete a single simulation of our microscopic model for a road of length 20500 m. For cases

CPU | Intel Core i7-2600 |
---|---|

Memory | 15.4 GiB |

OS | Ubuntu 16.04.2 LTS |

We first analyze our simulation results to assess the extent to which the values of physical quantities are preserved by our methods for converting computational results between microscopic and macroscopic models. First considering vehicle counts from a microscopic point of view,

Possible explanations for the fluctuations on the microscopic side include

In contrast, in the microscopic model, the number of vehicles is obtained by multiplying the traffic volume by the timestamp

Finally, with regard to computation times we note that replacing the microscopic model with the macroscopic model for simulating the greater part of the target area yields a computational cost reduction of 615.80 s—that is, nearly 10 min and 16 s—for a computation spanning 10 h of simulation time. That we failed to achieve even greater cost reduction in this case is probably due to the simplicity of our simulations, which involved only a single road and omitted complications such as traffic signals; we expect even larger computational savings for simulations of more realistic traffic-flow scenarios involving inter-vehicle interactions.

In this study, for the purpose of realizing high-precision traffic-flow simulations at low computational cost, we combined a microscopic traffic-flow model (the GFM, a car-following model) with a macroscopic traffic-flow model (the PW model, a continuous macroscopic model).

To discretize the PW model, we used a DGFEM, which not only offers robustness in the presence of discontinuities but also boasts outstanding stability properties. We also used a sponge layer to enhance stability in outflow/inflow regions. Then, we used our new model to simulate vehicle outflow from the macroscopic to the microscopic model and validated the accuracy of the results in two separate ways. First, measurements of the number of vehicles flowing out of the macroscopic model and flowing into the microscopic model confirmed that these quantities were in agreement at each timestep. Next, a comparison of density values obtained in the macroscopic model to post-conversion density values for the microscopic model confirmed that the requirement of density-value conservation was largely satisfied, indicating proper function of our multiscale model. Since there are countless combinations of DGFEM and multi-agent models, the proposed method may be applied to other combinations of macroscopic and microscopic models as well as the PW model and the GFM. The question of why macroscopic and microscopic density values were not in perfect agreement is a topic for future work, in which we hope to improve our procedure by using identical volume–density relations for the macroscopic and microscopic models.

A comparison of computational times confirmed that, thanks to the high speed of computations using the macroscopic model, our multiscale traffic-flow model yields high-quality results from simulations that complete in relatively little time.

In future work, we hope to study a reversed version of the multiscale model presented in this paper, in which quantities from a microscopic model are converted to a macroscopic model. To ensure vehicle outflow from a microscopic to a macroscopic traffic-flow model, we are considering methods for computing the mean velocity of vehicles arriving at the endpoints on each timestep, as well as techniques for converting vehicle counts to vehicle densities. In the macroscopic traffic-flow model considered here, one would like to use a sponge layer to yield fixed values for inflow and outflow, but this gives rise to fluctuations in values that had been held fixed. Thorough comparisons of the accuracy and applicability of advanced macroscopic models such as the Aw-Rascle-Zhang model [

We expect that successfully attaining these milestones would pave the way towards traffic-flow simulations that focus on subregions of wide-area systems.