Development of Dynamic Micro- and Macroscopic Hybrid Model for Efficient Highway Traffic Simulation

Helbing [7] proposed a hybrid model using the Intelligent Driver Model (IDM) [29] as a micro model and a macro model derived from IDM. However, the conservation of traffic volume at model boundaries was not fully verified. Mammar et al. [15] proposed a hybrid model using their own defined micro model and the Aw-Rascle-Zhang (ARZ) model [2, 31] as a macro model. Transition cells are placed at the macro-micro boundaries to maintain consistency between models. El Hmam et al. [5] used the Lighthill-Whitham-Richards (LWR) model [14, 23] and Payne model [21] as macro models and constructed a hybrid model combining them with an agent-based micro model. Storani et al. [26] used the Cell Transmission Model (CTM) [3, 4] and Nagel-Schreckenberg model [17] of microscopic cellular automata (CA).

To achieve traffic simulation with accuracy and computational speed using a hybrid model, high-resolution micro models should be applied only to the spatiotemporal domain, where individual vehicle behaviors significantly affect the occurrence and elimination of traffic phenomena. When focusing on traffic jams, these are the areas where queues are formed or dissolved. For example, Nishi et al. [18, 19], Taniguchi et al. [28], and He et al. [6] argue that traffic jam absorption driving can suppress the propagation of jam waves. This means that microscopic vehicle behavior can alter a kind of macroscopic phenomenon.

The problem with the hybrid models listed above, which are called static hybrid models, is that the application domains of the macro and micro models are fixed for the entire simulation duration. It may cause inaccuracy or computational inefficiency.

Several hybrid models that dynamically change model application domains during simulation have also been proposed. These models can adequately reproduce propagating jam waves while keeping computational costs low. Joueiai et al. [8] combined LWR model and IDM. They classified the consistency at the boundaries between the two models into “local consistency” and “global consistency”, and demonstrated their model satisfied each consistency. They also introduced dynamic boundaries. Sewall et al. [24] proposed a dynamic hybrid model in which the region of interest is simulated by IDM and the rest by the ARZ model. Their simulator has 3D traffic flow visualization and interactive operation capabilities, and the dynamic hybrid model improved the operability by ensuring sufficient computation speed.

Unlike these dynamic hybrid models, Takahashi et al. [27] focused on constructing a model switching criterion that appropriately captures the tails of traffic jams. Through verification and validation, it was confirmed that the application domain of the micro model changes with traffic jam conditions, and results similar to those of the micro model can be obtained with low computational cost.

Instead of spatially dividing the application domain of micro- and macro-models, Laval and Daganzo [11] proposed a method in which only lane-changing vehicles are modeled microscopically as particles and other vehicles are represented macroscopically as kinematic wave (KW) stream. They defined the lane-changing vehicles to behave like obstacles to the KW stream and successfully reproduced density differences between lanes. Leclercq et al. [12] used a similar concept to introduce a microscopic merging mechanism into a macroscopic model.

The model proposed in this study is based on Takahashi’s model [27]. This model employes IDM and the minimizing overall braking induced by lane change (MOBIL) model [9] as a micro model and CTM as a macro model. The road is divided into cells, as in CTM, and the calculation model is determined for each cell according to the criteria mentioned later. Figure 1 is a flowchart showing the procedure of each simulation time step. Here, \({\Delta } t_\textrm{M}\) and \({\Delta } t_\textrm{m}\) are the time intervals for the macro and micro models, respectively. In addition, \({\Delta } t_\textrm{sw}\) is used as the time interval to decide the model applied to a cell. The cells where the macro and micro models are applied are hereinafter referred to as the macro model cells and micro model cells, respectively.

To reduce computational cost, the micro model should be applied only to the cells where the difference in vehicle interactions significantly affects the formation and elimination of traffic jams, and the macro model should be applied to the rest. Therefore, the micro model is applied only to a non-stationary state of congested flow, and the macro model is applied to a free flow state and a stationary state of congested flow. A free flow state is the state in which each vehicle can drive without much influence from other vehicles, while a congested flow state is the state in which each vehicle drives at low speeds under the influence of other vehicles. In addition, congestion states are classified as non-stationary states and stationary states. A Non-stationary state is the state with high variabilities, such as the heads and tails of traffic jams, and a stationary state is the state inside a queue. In a stationary congestion state, vehicles are strongly influenced by other vehicles, still, the variability of the interaction among vehicles is not significant enough to be reproduced in the micro model.

Takahashi et al. successfully detected the tails of standing queues using cell density and its coefficient of variant and applied the micro model to it. However, due to the detection rule based on the values of two adjacent cells, some vehicle platoons that could not be called traffic jams were detected as moving jams, and the micro model was applied to the heads and tails of the platoons [27]. This is not good for improving the computational efficiency of the dynamic hybrid model. In this study, the authors propose a new method to detect the heads and tails of traffic jams by setting a new density threshold, as described later. In addition, merging behavior, which has not been treated in Takahashi’s model, is added to improve the model’s generality. A comparison of the Takahashi’s model and this study is shown in Table 1.

In CTM [3, 4], a target road is divided into cells as shown in Fig. 2. According to the Modified CTM by Muñoz et al. [16], \(k_i(t)\), the density of cell i at time t, is updated at each step Eq. 1.

\(L_i\) is the length of cell i and \({\Delta } t\) is the time interval. The flow rate \(q_i\) of cell i is calculated based on the maximum flow \(S_{i-1}(t)\) that cell \(i-1\) can supply and the maximum flow \(R_i(t)\) that cell i can receive, using the following equations:where u is the forward wave speed, w is the backward wave speed, \(q^\textrm{max}_i(t)\) is the maximum flow, and \(k^\textrm{jam}_i(t)\) is the jam density.

Equation 2 can be extended to handle merging. In the sections shown in Fig. 3, \(S_{i-1,1}\) and \(S_{i-1,2}\) are the maximum flows that the upstream cells \((i-1,1)\) and \((i-1,2)\) can supply, respectively, and \(q_{i,1}\) and \(q_{i,2}\) are the flow rates entering cell i.

When the road is not crowded (\(S_{i-1,1}+S_{i-1,2}<R_i\)), the vehicles attempting to flow out of both upstream cells can flow into the downstream cell, so the flow rate can be calculated as follows:During congestion (\(S_{i-1,1}+S_{i-1,2}\ge R_i\)), not all vehicles attempting to flow out of the upstream cells can flow into the downstream cell. The bottleneck that determines the total flow rate is \(R_i\).Herein, in the case that \(S_{i-1,1}\) and \(S_{i-1,2}\) are sufficiently large, the constants \(p_{i,1}\) and \(p_{i,2}\) \((p_{i,1}+p_{i,2}=1)\) representing priority are used to calculate the flow rate as follows:On the other hand, when \(S_{i-1,1}<p_{i,1}R_i\) or \(S_{i-1,2}<p_{i,2}R_i\) holds, \(q_{i,2}=p_{i,2}R_i\) or \(q_{i,1}=p_{i,1}R_i\) cannot be satisfied. In this case, the flow rate is determined as follows:

IDM [29] is one of the car-following models in which a vehicle determines its acceleration under the influence of the preceding vehicle. The acceleration of vehicle j is calculated as follows:where \(j-1\) represents the index of the preceding vehicle, \(x_j\), \(l_j\), and \(v_j\) are the position, the body length, and the speed of vehicle j, respectively. \(v_0\) is the desired speed, \(s_0\) is the minimum distance, and T is the safe time headway. a is the maximum acceleration, and b is the comfortable deceleration. \(\delta \) is the acceleration exponent.

A modified implementation of Al-Obaedi & Yousif’s merging model [1], which reproduces vehicle merging behavior relatively simply, is employed in the proposed model. After evaluating the gap using the gap acceptance model, decisions on merging, acceleration, or deceleration are made at each time step.

Gap Acceptance Model Figure 4 shows the lead gap (the gap between the preceding vehicle J1 on the mainline and the merging vehicle C) and the lag gap (the gap between the merging vehicle C and the following vehicle J2 on the mainline). Note that vehicles drive on the left side of the road in Japan. The gap acceptance model judges whether the lead and lag gaps exceed their minimum values \(g^\textrm{lead}_\textrm{th}\) and \(g^\textrm{lag}_\textrm{th}\) defined in the following equations:where \(T_\textrm{r}\) is the driver’s reaction time, and v and \(b'\) represent the speed and maximum deceleration, respectively. The meanings of indices C, J1, and J2 are the same as those in Fig. 4. The value of the minimum gap varies depending on the situation. The parameter \(\alpha \) is set to 0.3 when considering the gap with J1 and 0.5 when considering the gap with J2. It is also set to 0.2 when C approaches the head of the acceleration lane and when the mainline vehicles take cooperative actions. Furthermore, when J1 is faster than C, the minimum gap with J1 is set to \({1.0\,\textrm{m}}\). When C is faster than J2, the minimum gap with J2 is also set to \({1.0\,\textrm{m}}\).

Decisions of Merging, Acceleration, or Deceleration The merging rules shown in Table 2 are applied depending on the size of the gap. A checkmark in the table indicates that the gap is acceptable; that is, the gap is equal to or greater than the threshold.

For simplicity, it takes a fixed value of \({1.4\,\mathrm{m/s^2}}\) for acceleration and \({-3\,\mathrm{m/s^2}}\) for deceleration. When a following vehicle on a mainline engages in cooperative behavior, it attempts to change lanes into the overtaking lane. If it cannot change lanes due to insufficient space, it decelerates by \({-4.9\,\mathrm{m/s^2}}\).

A simulation was conducted to reproduce traffic jams around the merge section of a real highway interchange A (hereafter IC-A) to validate the implemented merging model and the model switching rules.

Figure 8 shows the time-space mean speed obtained from the probe data observed around IC-A on November 22, 2020. The horizontal axis in the chart is a relative coordinate to IC-A, with negative and positive values meaning upstream and downstream, respectively. There is a toll gate 13.0 km upstream from IC-A. The spatiotemporal domain in Fig. 8 is discretized in the time direction in units of 60 s and the space direction in units of 100 m. Each subdomain in the figure is colored according to the harmonic mean of the measured speed data aggregated in it. Since the penetration ratio of an onboard device for data acquisition is lower than 100%, traffic volume cannot be obtained from the measured data alone; however, we can understand an approximate condition of the traffic congestion from the time-space mean speeds. For example, a phenomenon that a low-speed region propagates upstream is represented by a red- or orange-colored area extending toward the lower left of the figure.

Figure 8 shows that the traffic jam at speeds of 20 km/h or less maximally extends to around the toll gate, starting at IC-A. Since the purpose of this simulation is to reproduce the traffic jam caused by the merging at IC-A, the target area is 26.0 km in total, from 13.0 km upstream of IC-A to 13.0 km downstream, and the target time is 17 hours and 50 minutes from 2:40 to 20:30. The target spatiotemporal domain is indicated by the red dotted line in Fig. 8. Although there are actually several other interchanges in the simulation target area beside the toll gate and IC-A, for this study, the inflow was limited to the toll gate as the upstream end and IC-A, and the outflow was limited to the downstream end. These conditions were set because the impacts of merging and diverging at other ICs on the formation of jams around IC-A are expected to be relatively small and because the available data is limited. In addition, only jams formed upstream of IC-A were considered for evaluation.

Figure 9(a) is a conceptual diagram of the simulation area, with three main lanes and two merging lanes. Merging lanes are extended approximately 1 km upstream from the acceleration lanes. Figure 9(b) shows the cell representation. In this simulation, the time intervals for the micro model, the macro model, and model switching are set to 0.1 s, 10 s, and 60 s, respectively (\({\Delta } t_\textrm{m}=0.1\textrm{s}, {\Delta } t_\textrm{M}=10\textrm{s}, {\Delta } t_\textrm{sw}=60\textrm{s}\)). Cell length is set to \(u{\Delta } t_\textrm{M}\approx 236.1 \textrm{m}\) to satisfy the Courant-Friedrichs-Lewy (CFL) condition of CTM, and the free-flow speed u was set to 85 km/h. As an exception, only the length of the cell corresponding to the acceleration lane was set to 270 m to match the actual length at IC-A.

The inflow traffic volume was estimated based on the measured data. The inflow and through traffic volumes at IC-A can be directly obtained from the available data. In the simulation, not only the traffic volume at IC-A but also that at the upstream end of the simulation area is required. The method proposed by Okano et al. using the tracking time sum [20] was used to estimate the traffic volume at the upstream end.

The parameters of CTM were set as shown in Table 3 based on the Q-K diagram obtained from the observed traffic counter data. IDM parameters were set as shown in Table 4 based on Kesting et al. [10], consistent with CTM. The ratio of trucks was determined from the observed traffic counter data.

As in Spiliopoulou et al. [25] and Wang et al. [30], different parameters are used in CTM only for the merging section. The entire area was simulated using the micro model as a preliminary analysis, and the results were obtained with a traffic capacity of 3611veh./h at the merging section and a merging ratio of \(p_1(\textrm{main}):p_2(\textrm{merging})=0.68:0.32\), which are applied to CTM. Although there is a method to determine the merging rate dynamically, it was not used in this simulation. This is because the micro model is applied when dynamic merging rates are needed during congestion.

The thresholds for model switching are \(k_\textrm{th}=18.0\) and \(CV_\textrm{th}=0.05\). To apply the micro model from just upstream of the tails of traffic jams, \(k_\textrm{th}\) needs to be slightly less than the critical density in CTM \((1700/85.0=20.0)\). On the other hand, \(k'_\textrm{th}\) is tested at three values, 18.0, 23.0, and 27.0. When \(k'_\textrm{th}=18.0\), i.e., \(k'_\textrm{th}=k_\textrm{th}\), the detection rule improved in this study can not work and is identical to that in the conventional Tkahashi’s model.

Using the road conditions, inflow volumes, and model parameters described in the previous subsection, a simulation was first conducted by applying the micro model to the entire area. The time-space mean speed is shown in Fig. 10. The meanings of the colors are the same as in Fig. 8. The black dotted lines in the figure indicate the start/end times of a traffic jam obtained from the measured data. Only the cell where merging behavior occurs continues to be a low-speed state unrealistically, and correcting this is future work; however, a jam length and start/end times close to the observed values were obtained. As mentioned earlier, this study only reproduces the traffic jam caused by the merging at IC-A and thus does not evaluate the low-speed region due to bottlenecks downstream of IC-A.

Subsequent simulations with the proposed dynamic hybrid simulation confirmed that the jam length and start/end times were reproduced appropriately. The time-space mean speed obtained by the simulation with \(k'_\textrm{th}=27.0\) is shown in Fig. 10 (bottom). Since the micro model is applied at the head of a traffic jam described in Section 3.5.1, the unrealistic nature of the continuation of a low-speed state in the cell of the merging section is similar to that of the simulation in which the micro model is applied to the entire area.

Figure 11 shows the model application domains, with the micro model cells in red and the macro model cells in white. The micro model was dynamically applied at the head and tail of a traffic jam. In the case that the conventional detection rule by Takahashi et al. was applied, the micro model was always applied to the merging section regardless of whether or not a traffic jam actually occurred. In addition, some micro model cells were found downstream of the bottleneck as well. This might be because small platoons at the merging section and downstream of the bottleneck are mistakenly detected as traffic jams. As mentioned in Section 3.1, from a computational cost perspective, it is best to avoid applying the micro model to cells where no traffic jams occur. Using \(k'_\textrm{th}\) higher than \(k_\textrm{th}\) was confirmed to suppress the unnecessary application of the micro model downstream of a traffic jam.

Figure 12 compares the computation time and the RMSE of the measured congestion length at each time in the case where the micro model was applied to the entire area, the case where the macro model was applied to the entire area (a preliminary analysis mentioned before) and the cases where different values of \(k'_\textrm{th}\) were used in the proposed method. Since a jam length cannot be obtained from the probe data, the results of the micro model are used as the ground truth. Compared to the macro model, the proposed method was consistent with the micro model. The jam length in the macro model was always shorter by one or two cells than in the micro model. This means there is a discrepancy between the macro and micro models of merging during congestion. The heterogeneity that the macro model could not represent might determine the jam length. In addition, the computation time of the proposed method was about 7% of simulating the entire area by the micro model.

Among the dynamic hybrid models, the case with \(k'_\textrm{th}=27.0\) has the fewest micro model cells; therefore, its computation time was 13.5% and 11% shorter than those with \(k'_\textrm{th}=18.0\) (conventional rule) and \(k'_\textrm{th}=23.0\), respectively. Although the quantitative improvement over the conventional rule in this experiment was slight, the number of micro model cells was steadily reduced. If the proposed model were to be applied to areas where the proportion of the spatiotemporal domain of traffic jams is relatively small, the difference in computational cost would be significant.