Systematic Evaluation of a Multi-Lane Green-Driving Algorithm in a Mixed Connected Environment

Abstract

Vehicles traveling under oscillated traffic have low energy efficiency and high air pollutant emissions. Green driving with the help of connected vehicles (CVs) attracts a lot of research effort to improve vehicle energy efficiency. However, it is very challenging to perform green driving on multi-lane freeways under a mixed connected environment. In the researchers’ previous work, one innovative green-driving algorithm was proposed to solve the multi-lane problem with only one CV. In this study, a systematic analysis of the algorithm is conducted to understand its benefits and limitations on smoothing traffic oscillations. The effect of the steady states and the number of lanes is also analyzed. In addition, the algorithm is extended to a more general scenario with multiple CVs. The extended system coordinates multiple CVs to form multiple moving bottlenecks to mitigate traffic oscillation more efficiently as well as providing more realistic instructions to CVs. The evaluation of the extended system concludes with the most effective strategies to control CVs to smooth oscillations.

With the increasing demands of transportation services, the current transportation sector consumes more fuel and produces more vehicle emissions. In U.S., more than 28% of fuel was used by the transportation sector in 2016 ( 1 ). The situation becomes even worse in the developing countries with the rapid increase of motor vehicles. Improving vehicle energy efficiency and reducing air pollutant emissions is an urgent issue for transportation engineers and policy makers. In the literature, the causes of high fuel consumption levels and air pollutant emissions in transportation systems have been widely investigated. Frequent accelerations associated with stop-and-go waves, excessive speed (over 60 mph), slow movements on congested roads, and extra idling time can all dramatically increase fuel consumption and emission levels ( 2 – 4 ). Consequently, it is clear that smoothing speed and movement fluctuations and reducing idling time are two critical ways to reduce fuel usage.

Recently, the development of connected vehicles (CVs) enables the real-time individual vehicle control to minimize vehicle fuel usages. Eco- or green-driving with CVs is one viable and cost effective solution ( 5 ). The CV technology can help vehicles to gather real-time road traffic status, signal phasing and timing (SPaT) information, and individual vehicle dynamics, which can be applied to estimate the real-time suggestions for the vehicles, such as advisory speed limits, acceleration and deceleration rates, speed alerts, and so forth, to minimize fuel usages. It has been proven that on average green-driving algorithms can reduce the fuel consumption and emissions by up to 10% ( 6 ).

A vehicle green-driving system can be applied on both arterial corridors and freeways. On arterial roads, the system focuses on optimizing traffic signal timings using traffic volumes and vehicular queue lengths ( 7 , 8 ). With the help of CVs, many green-driving strategies, such as eco-approach and departure (EAD) applications, speed optimization, trajectory optimization, Green Light Optimized Speed Advisory (GLOSA), eCoMove, eco-cooperative adaptive cruise control, and so forth, have been developed to optimize individual vehicle behaviors ( 9 – 17 ). All the research focused on estimating fuel-optimized trajectories with the consideration of SPaT information. On freeways, vehicles are rarely affected by traffic signs or signals, and most green-driving algorithms on freeways apply CVs to understand road traffic conditions and compute advisory speed profiles or trajectories for vehicles to follow so as to minimize fuel consumption. Barth and Boriboonsomsin proposed a dynamic eco-driving system to increase energy efficiency of individual vehicles ( 18 ). The system utilized the real-time traffic sensing and telematics to monitor road speed, density, and flow, and estimated advisory speeds based on the average speed and its standard deviation for eco-driving vehicles to drive smoothly on freeways so as to reduce fuel consumption and vehicle emissions. However, this system was designed without considering the effect of surrounding traffic, which made it less reliable on multi-lane freeways. Park et al. developed a predictive eco-cruise control system, and Ahn et al. invented an eco-drive application ( 19 – 21 ). Both of them were designed based on a model predictive control to search for optimal speed profiles for individual vehicles to drive on freeways with grades. However, they only considered optimizing the behaviors of individual vehicles with regard to the changes of road grades, and the road traffic conditions were only incorporated in the system development. Yang and Jin developed a green-driving system with CVs to smooth traffic oscillations on freeways ( 22 ). The system applied a distributed feedback control system to estimate advisory speed limits for individual vehicles to move at constant speed without experiencing any disturbances. However, the system was only designed to work for single-lane roads. Wang et al. developed a CV-enabled variable speed limit and acceleration control system to smooth traffic oscillations ( 23 ). Similarly, a speed harmonization system was developed, based on connected autonomous vehicles, to smooth individual vehicle trajectories and to mitigate traffic congestion at bottlenecks ( 24 – 26 ). All the aforementioned green-driving systems were only proposed for single-lane roads or multi-lane roads with high market penetration rates (MPRs) of CVs. The studies ignored the effect of the proposed systems on the surrounding traffic. In that sense, they were not quite appropriate to be deployed in realistic transportation systems with multi-lane roads and low MPRs, especially in the early-stage development of CVs.

In the literature, there are also some studies working on analyzing the effect of individual vehicle control on surrounding traffic. The moving bottleneck model is one effective model to describe such effects ( 27 – 31 ). Generally, a moving bottleneck is caused by one or multiple slow vehicles on a road, which generates upstream traffic congestion and restricts the overtaking rates to downstream. With the moving bottleneck model, it is possible to control the speed of the slow vehicle to regulate the surrounding vehicles to improve traffic performance. In Čičić and Johansson, a cell transmission model was extended to describe traffic dynamics with moving bottlenecks, so as to estimate control strategies for individual vehicles to resolve traffic jams ( 32 ). In Piacentini et al., the moving bottleneck concept was applied to control traffic flow with the goal of reducing fuel consumption of vehicles passing fixed road bottlenecks ( 33 ). Both studies explored the possibility of manipulating traffic streams by introducing moving bottlenecks.

In Yang and Oguchi, an innovative multi-lane green-driving (MGD) algorithm was developed to smooth traffic oscillations with only one CV ( 34 ). The algorithm applied the moving bottleneck mechanism to analyze the effect of a slow vehicle on the dynamics of surrounding traffic stream on a multi-lane freeway, and it estimated an advisory speed limit for the CV to manipulate the overtaking rate so as to mitigate traffic oscillations ahead. The algorithm was proven to be effective on smoothing traffic oscillations with both microscopic and macroscopic simulations. However, it lacked a systematic analysis of its reliability and limitations. In this paper, the proposed MGD algorithm is further evaluated to understand the effect of the steady state on roads and the number of lanes on the estimation of advisory speed limits for CVs. Its limitations on extremely congested traffic are further evaluated to understand its feasibility at different conditions. Moreover, the algorithm is extended to coordinate multiple CVs on roads so as to work for a more realistic mixed connected environment.

The rest of the paper is organized as follows. The next section reviews the development of the MGD algorithm. The section after that makes a sensitivity analysis of steady states and the number of lanes. The penultimate section extends the MGD algorithm to multiple CVs and evaluates its benefits on advising CVs to smooth oscillations. The final section concludes the findings and suggests some recommendations for future work.

Review of Multi-Lane Green-Driving Algorithm

In Yang and Oguchi, a MGD algorithm was proposed to mitigate freeway oscillations ( 34 ). The algorithm applies a moving bottleneck model to smooth traffic oscillations by controlling one CV on multi-lane freeways ( 28 ). In this section, the algorithm is briefly reviewed to introduce the mechanism of multi-lane vehicle control.

First, the moving bottleneck model, which is critical to the algorithm development, is reviewed. As shown in Figure 1a, if there are two vehicles moving side-by-side at a lower speed than their surrounding traffic on a four-lane road, they will partially block the freeway and form a moving bottleneck. Note that, this example can be extended to $N$ -lane roads ( $N \geq 2$ ) with $n$ lanes blocked ( $0 < n \leq N$ ).

Figure 1.

Freeway fundamental diagram (flow-density relationship): (a) freeway with slow vehicles, (b) stationary coordinates, and (c) moving coordinates.

The fundamental flow-density relationships at the open and blocked areas are defined as $q = Q_{o} (ρ)$ , and $q = Q_{b} (ρ)$ , respectively. Figure 1b shows the fundamental diagrams of the two scenarios under steady coordinates, where ${Q_{o}^{c}, ρ_{o}^{c}, ρ_{o}^{j}}$ and ${Q_{b}^{c}, ρ_{b}^{c}, ρ_{b}^{j}}$ represent the road capacities, the densities at capacity, and the jam densities of the open and blocked areas, respectively. Assume that the controlled vehicles are moving at the speed $v_{0}$ , then the flow that vehicles overtaking the slow vehicles can be defined as

q_{0} = q - ρ v_{0} .

(1)

Therefore, under the moving coordinates with the speed $v_{0}$ , as shown in Figure 1c, the fundamental diagram in Figure 1b is modified as

q_{0} = Q_{o} (ρ) - ρ v_{0} = Q_{o}^{*} (ρ),

(2a)

q_{0} = Q_{b} (ρ) - ρ v_{0} = Q_{b}^{*} (ρ),

(2b)

where $Q_{0}^{*} (ρ)$ and $Q_{b}^{*} (ρ)$ represent the flow-density relationship under the moving cooridinate.

As vehicles can only overtake the slow ones from the unblocked lanes, only the positive part of the dash line in Figure 1c represents the overtaking rates under different densities. If the upstream traffic is extremely light (state $S_{1}$ ) or very heavy (state $S_{2}$ ), which has an overtaking rate lower than the capacity at the blocked area, it can all be discharged, and there is not any state change at the upstream and downstream of the moving bottleneck. While, at state $S_{3}$ which demands an overtaking rate higher than the capacity of the blocked area, the bottleneck can only discharge the flow with its capacity. In that sense, it will generate upstream congestion $S_{3, u}$ and downstream free flow $S_{3, d}$ . And, the line connecting $S_{3, d}$ and $S_{3, u}$ is tangent to the dash fundamental diagram and crosses its highest point (see Figure 1c).

The major idea of the MGD algorithm is estimating an advisory speed limit for a CV to travel at a slower speed than its surrounding traffic so as to generate a moving bottleneck. Similar to the state $S_{3}$ , the moving bottleneck creates a downstream freeway flow with a forward shock wave to smooth the traffic oscillations ahead of the CV. In that sense, all the traffic upstream of the CV will not experience any oscillations.

To estimate the advisory speed limit, the density distribution along the road with traffic oscillation shall be detected as the input for the MGD algorithm. Assume that there are some probe vehicles traveling ahead of the CV; they have communciation devices, but they are not controlled by the MGD algorithm. The probe vehicles can share their trajectories and speed profiles with the CV. Then, the trajectory and speed profiles can be used to estimate traffic density distribution on a freeway, that is, $ρ (x, t)$ at the location $x$ and the time $t$ , and to capture traffic oscillations on the road ( 35 ). It is also known that traffic oscillations on freeways are generally periodic under stop-and-go traffic, that is, the density distribution along the freeway is also periodic ( 36 , 37 ). Assume that the length of one period is $L_{p}$ , and the density distribution on the segment $x \in [x_{0}, x_{0} + L_{p}]$ is $ρ (x, t)$ at location $x$ and time $t$ , where $x_{0}$ is the starting point of the period. Then, the average density of one period is

ρ_{D} = \frac{1}{L_{p}} \int_{x_{0}}^{x_{0} + L_{p}} ρ (x, 0) dx,

(3)

which is also the density at the steady state, $D$ on the road.

The MGD algorithm is proposed to smooth the oscillation in the period and change the road to the steady state. If the steady state is $D$ , as shown in Figure 2, a line can be drawn from the point $D$ and tangent to the fundamental diagram of the blocked road. The tangent point is $E$ , and the point crossing $Q_{o} (ρ)$ is $C$ . Assume that the slope of the line is $v_{CD}$ . Then, the MGD algorithm sets the advisory speed limit of the CV $k$ as $v_{0} = v_{CD}$ . Therefore, the line $CD$ becomes horizontal in the moving coordinates. Then, the upstream vehicles can overtake the CV with the maximum rate, and the upstream state becomes $D$ . At the same time, the downstream state of the CV is $C$ , and a forward shock wave is then generated to mitigate oscillations downstream of the CV. As the state of the upstream is steady, the forward shock wave can completely mitigate the oscillations downstream, and the CV will not meet any disturbances, that is, the road traffic is smoothed.

Figure 2.

Advisory speed limit estimation with the multi-lane green-driving (MGD) algorithm.

Figure 3 shows one example of the MGD algorithm with a triangular fundamental diagram. Figure 3a shows the base case with traffic oscillations, where states $A$ and $B$ represent the less and more dense traffic, respectively. Here, the density at state $A$ must be higher than the critical density $ρ_{o}^{c}$ ; otherwise, the oscillation can be mitigated by itself even without control. Vehicles traveling on the road will experience stop-and-go traffic (higher speed at $A$ and lower speed at $B$ , and $v_{AB}$ is the speed of the waves). After introducing the MGD algorithm, a CV is controlled to move at the speed $v_{0}$ , and its trajectory is shown as the red dash-dot line in Figure 3b. The forward shock wave generated by the free-flow traffic $C$ , as shown in Equation 2, smooths all the oscillations ahead of the CV, and the vehicle will meet the end of the oscillation just when it is smoothed. And, all the traffic behind the CV will become the steady state, that is, all other vehicles traveling in that region will not experience any oscillations.

Figure 3.

Traffic waves along the freeway: (a) base case without control, and (b) control with multi-lane green-driving (MGD) algorithm.

Note that, the MGD algorithm only provides advisory speed limits, $v_{0}$ , to CVs. The instructions can be taken either by human drivers or by advanced vehicle control system, such as ADAS and ACC, to implement the MGD system. The CVs do not need to follow the suggested speed exactly. Their speeds are only controlled to not exceed the speed limits. Therefore, the algorithm can be easily implemented for CVs to ensure high compliance rate.

System Evaluation of the MGD Algorithm

In this section, a systematic analysis of the proposed MGD algorithm is proposed to understand its reliability and the limitations. The effect of the steady states and the numbers of blocked lanes are analyzed. To simplify the analysis, the straight road segment in Figure 1a is modeled (the number of lanes can be different), and a triangular fundamental diagram, Figure 4, is applied to describe the traffic conditions on the road and to estimate the advisory speed limit for the MGD algorithm.

Figure 4.

Triangular fundamental diagram for the multi-lane green-driving (MGD) algorithm.

Sensitivity of Steady States

In the subsection, the relationship between the steady states and the advisory speed limits of CVs are analyzed to understand the effect of steady states on the reliability of the MGD algorithm as well as to explore its limitation.

Assume that there are only two lanes on the road in Figure 1a, and one CV is controlled to smooth the oscillations. As the proposed MGD algorithm requires the steady state to fall in the congested region, that is, $ρ_{D} > ρ_{o}^{c}$ , the tangent point from the steady state $D$ to the blocked triangular fundamental diagram is always the capacity at the critical density $ρ_{b}^{c}$ . Therefore, a line can always be drawn from $D$ to $C$ to find out the advisory speed limit. For the two-lane road, assume that the free-flow speed, the road capacity, and the jam density for the open and blocked areas are ${108 km / hr, 4320 vph, 400 veh / km}$ and ${108 km / hr, 2160 vph, 200 veh / km}$ , respectively.

Figure 5 shows the advisory speed limits estimated by the MGD algorithm under different steady-state vehicle densities on the two-lane road. It indicates that with higher vehicle densities, that is, the steady states are under more congested region, the CV shall be advised to travel at lower speeds to smooth oscillations, and the lower speed limits may reduce the compliance rate of the MGD algorithm. In addition, when the steady state is under an extremely congested condition (in this example, more than $220$ veh/km), the MGD algorithm estimates a negative advisory speed limit to the CV, that is, the algorithm fails to smooth oscillations, as the vehicle cannot move backward on the road. This analysis reveals the limitations of the algorithm on severe traffic congestion. Therefore, it is not realistic to apply the MGD algorithm in that situation, since it is almost impossible to make CVs move at very low speeds or even stop on roads. In that sense, it requires multiple CVs to be coordinated at multiple locations or on multiple lanes to gain higher advisory speed limits.

Figure 5.

Sensitivity analysis of steady states on a two-lane freeway.

Sensitivity of Blocked Lanes

In this subsection, a sensitivity analysis of the number of lanes blocked by the CVs is conducted to further evaluate the reliability of the proposed MGD algorithm on multi-lane freeways.

Assume the number of lanes on the road in Figure 1a is 5, and there are multiple CVs traveling side-by-side on the road to generate a moving bottleneck by blocking multiple lanes (from 1 to 4). As before, a triangular fundamental diagram is applied to describe the traffic on the road, and free-flow speed, the road capacity, and the jam density for the open and blocked areas are all set as ${108 km / hr, 4320 vph / lane, 400 veh / km / lane}$ . Figure 6 shows the advisory speed limits with respect to different steady states under different numbers of blocked lanes. Similar to the two-lane analysis, with more congested steady states the advisory speed limits are lower, and the MGD algorithm will fail when the vehicle density at the steady state is much higher. In addition, with more lanes blocked by the CVs to form a moving bottleneck, the advisory speed limits are higher, and the MGD algorithm can still work successfully at higher steady-state densities. The reason is that, with more lanes blocked, there are fewer lanes left for the rest vehicles to overtake the CVs. Therefore, it is easier for the algorithm to adjust the advisory speed limit to generate a traffic stream with a lower density for the state $C$ in Figure 4. Then, it can create a forward shock wave with a high speed to mitigate the downstream oscillations. The analysis proves that with more lanes blocked, the algorithm is more efficient at providing higher advisory speed limits so as to increase the compliance rate.

Figure 6.

Sensitivity analysis of blocked lanes on a five-lane freeway.

Extension of the MGD Algorithm to Multiple CVs

Earlier in this paper, the MGD algorithm is proposed to control only one CV or multiple CVs driving side-by-side at one location on the road. It was illustrated that, to make the MGD algorithm work for more lanes, it is better to block multiple lanes simultaneously. However, in reality, managing multiple CVs driving side-by-side is not easy, as it might be difficult to find multiple CVs traveling close to each other. Therefore, the algorithm should be improved to coordinate and control multiple CVs at different locations to achieve the goal of mitigating traffic oscillations. Instead of forming one single bottleneck from the previous MGD algorithm, the extended one will generate multiple bottlenecks, and each one can only block one lane.

Assume that the oscillations are periodic on a freeway segment, and the length of one oscillation is $L_{p}$ . Given the vehicle density distribution on the oscillation, $ρ (x, t), x \in [x_{0}, x_{0} + L_{p}]$ at the location $x$ and time $t$ , where $x_{0}$ is set as the starting point of the oscillation, the steady-state density of the oscillation is $ρ_{D} = \frac{1}{L_{p}} \int_{x_{0}}^{x_{0} + L_{p}} ρ (x, t) dx$ . Also, the length of the segment with the density higher than $ρ_{D}$ was set as $L_{c}$ . Suppose that there are $K$ CVs distributed in the oscillation, and the extended MGD algorithm will coordinate them to smooth the oscillation. For the CV $k, k = 1, 2, \dots, K$ (ordered from downstream to upstream), it is responsible for smoothing a portion, $p (k), 0 < p (k) < 1$ , of the over-congested region, where $\sum_{k = 1}^{K} p (k) = 1$ . Without loss of generality, it is assumed that the road is less congested at $[x_{0}, x_{0} + L_{p} - L_{c}]$ , and the rest are over-congested. For the CV $k$ , it will smooth the region $[x_{0}, x_{0} + L_{p} - L_{c} + \sum_{i = 1}^{k} p (i) \cdot L_{c}]$ to reach its steady state $D_{k}$ , where the steady-state density is

ρ_{D}^{k} = \frac{1}{L_{p} - L_{c} + \sum_{i = 1}^{k} p (i) \cdot L_{c}} \int_{x_{0}}^{x_{0} + L_{p} - L_{c} + \sum_{i = 1}^{k} p (i) \cdot L_{c}} ρ (x, t) dx .

(4)

In the deployment of the extended MGD algorithm, the value of $p (k)$ is searched to minimize the duration of smoothing one oscillation as well as avoiding negative advisory speed limits. In general, the CVs at the downstream must have higher values to satisfy the two requirements.

To evaluate the performance of the extended MGD algorithm, it is deployed in the road segment (see Figure 1a) with five lanes. The triangular fundamental diagram is also applied to describe traffic dynamics on the road, and the free-flow speed, the road capacity, and the jam density are set as ${108 km / hr, 4320 vph / lane, 400 veh / km / lane}$ . Assume that there exist periodical oscillations on the road, and the initial vehicle density distribution is defined as

\begin{matrix} ρ (x, 0) = {\begin{matrix} 500 & 400^{*} i \leq x < 400^{*} i + 200 \\ 125 & 400^{*} i + 200 \leq x < 400^{*} (i + 1) \end{matrix} \\ i = 0, 1, 2, \dots \end{matrix},

(5)

With this setting, the average speed of the traffic along the road can be estimated based on the triangular fundamental diagram as $26.4$ km/h. In addition, four CVs are introduced to smooth the oscillation in a coordinated way, and they are initially uniformly distributed in the region $[0, 200]$ meters. Figure 7 shows the advisory speed limits assigned to four CVs from downstream to the upstream. The optimal values of $p (k)$ are set as ${0.4, 0.3, 0.2, 0.1}$ . Figure 7 indicates that the CVs at the downstream have higher advisory speed limits. For the upstream vehicles, the speed limits are close to zero, that is, they are almost stationary on the road to smooth the oscillations. Moreover, the scenarios with one, two, and three CVs are also tested. However, they all fail to smooth the oscillations with the setting in Equation 5. Four is the minimum number of vehicles to make it a success. This illustrates the advantages of the extended MGD algorithm. While, on the other hand, Figure 7 also shows that the advisory speed limits are all much lower than the average speed, which may result in low compliance rates of the extended MGD algorithm.

Figure 7.

Advisory speed limit estimation with one lane blocked from the extended multi-lane green-driving (MGD) algorithm.

To further improve the extended algorithm, multiple CVs blocking multiple lanes at multiple locations are introduced, to smooth oscillations. Figure 8 shows the results with two, three, and four lanes blocked. In Figure 8, each vehicle ID represents one block location, and the number of CVs at the locations is the same as the number of lanes blocked in the multi-lane blocked scenarios. Similarly, the downstream CVs (lower IDs) have higher advisory speed limits. And, with more CVs coordinated, the limits are also higher. The control at only one location always has the lowest limits. Furthermore, with more lanes blocked, the limits can also be higher. Figure 8c indicates that with four lanes blocked and the control at four locations, the advisory speed limits can be close to or even higher than the average speed.

Figure 8.

Advisory speed limit estimation from the extended multi-lane green-driving (MGD) algorithm: (a) Two lanes blocked, (b) Three lanes blocked, and (c) Four lanes blocked.

Finally, combining the results in Figures 7 and 8, it can be concluded that if the CVs are close to each other, it is better to assist them driving side-by-side to block more lanes, which can give them higher advisory speed limits to increase the compliance rate of the MGD algorithm. If the CVs are far away from each other, then coordinating multiple CVs with the extended algorithm can also improve their performance on smoothing oscillations.

Conclusion

In this paper, a systematic analysis of the multi-lane green-driving algorithm proposed in Yang and Oguchi was conducted to understand its reliability and the limitation on smoothing traffic oscillations on freeways with multiple lanes ( 34 ). The effect of steady states and numbers of lanes were evaluated to understand the estimation of advisory speeds for CVs. The analysis indicated that with more congested steady states, the advisory speed limits would be lower; and at certain congestion levels, the MGD algorithm would fail to estimate the speed limits necessary to smooth oscillations. The sensitivity analysis of the number of lanes proved that the MGD algorithm estimated higher advisory speed limits for CVs when more lanes were blocked simultaneously, and the higher values could increase the compliance rates of the algorithm. Besides the sensitivity analysis, the MGD algorithm was also extended in this paper to work for multiple CVs traveling at different locations. The algorithm coordinated them to smooth traffic oscillations with higher advisory speed limits. Generally, with more CVs controlled, the limits assigned to them would be higher, and the vehicles at downstream had higher limits. The systematic analysis showed that controlling CVs to move side-by-side was more efficient for smoothing traffic oscillations. While, if it was not possible, coordinating CVs at multiple locations with the extended algorithm could also benefit oscillation mitigation.

Based on the analysis in this paper, the authors have some recommendations for future work. First, the proposed system is currently evaluated numerically with macroscopic models. The microscopic simulations will be applied in the future to understand the detailed control strategies to each CV and its effects on the surrounding traffic. Second, the analysis showed that controlling CVs moving side-by-side is more effective. In the future, an advanced vehicle driving assistant system shall be developed to assist CVs moving together to block multiple lanes. Third, the current algorithm still has limitations on very congested steady states and freeways with many lanes. It shall be further investigated to solve these limitations with small number of CVs. Fourth, the MGD algrithm is developed based on a triangular fundmental diagram and a simplified moving bottleneck model. The algorithm will be further improved with more realistic moving bottleneck models with general fundamental diagrams ( 38 , 39 ). Finally, the algorithm is only proposed for a homogeneous straight freeway segment. In the future, it will be extended to traffic oscillations on non-homogeneous roads with different road geometries, such as lane drop, merging, diverging, and so forth.

Footnotes

Acknowledgements

The authors confirm contribution to the paper as follows: study conception and design: H. Yang, K. Oguchi; data collection: H. Yang, K. Oguchi; analysis and interpretation of results: H. Yang, K. Oguchi; draft manuscript preparation: H. Yang, K. Oguchi All authors reviewed the results and approved the final version of the manuscript.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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