Variable Speed Limit Strategies Based on the Macro Hierarchical Control Traffic Flow Model

Department of Traffic Management Engineering, Shandong Police College, Jinan 250014, China Department of Traffic Engineering, Shandong Jianzhu University, Jinan 250101, China Department of Automation and Electronic Engineering, Qingdao University of Science and Technology, Qingdao 266061, China MOE Key Laboratory for Urban Transportation Complex Systems /eory and Technology, Beijing Jiaotong University, Beijing 100044, China Shandong High-Speed Information Group Co., Ltd, Jinan 250013, China Department of Rail Transportation, Shandong Jiaotong University, Jinan 250357, China


Introduction
e expressway is a modern transportation infrastructure which has its specific characteristics, such as high flow and high speed. Building expressway becomes a major project to relieve traffic congestion and traffic delay. is project could improve regional development and promote industrial restructuring and social progress. However, it also attracts more traffic demand for its high efficiency.
en, the problem of traffic congestion and traffic safety occurred inevitably. According to the survey results of Baidu Map in 2020, the degree of highway congestion in China has increased significantly, and the number of traffic jams has also increased.
is phenomenon is especially obvious during holidays and peak travel periods. erefore, it is urgent to take effective control measures to address these problems.
At present, the static speed limit sign is widely used in the speed limit control of expressways, but the determination of the static speed limit value is often based on ideal traffic conditions, and the influence of variable factors such as road traffic flow and weather conditions on the safe driving speed of vehicles is seldom considered. In order to overcome the deficiency of static speed limit control, the dynamic traffic control method represented by variable speed limit (VSL) has gradually become a control strategy considered by scholars and traffic control departments. A large number of studies have found that a smooth traffic flow can effectively improve the safety of vehicles and the efficiency of road sections. Variable speed limit can consider the influence of the change of traffic environment on safe speed more comprehensively and carry out real-time control by adjusting the speed limit value, which is of great significance to improve the traffic safety and operation efficiency of expressways. e dynamic traffic control is an effective method to relieve traffic congestion, which is mainly divided into two types: global control strategy and local control strategy. Both of them play essential roles in relieving the congestion of expressway and ensuring the travel safety [1][2][3]. Compared with local control, the global control measures have many advantages from the network level, such as global logistics network optimization [4] and global state space-time network model [5,6], take the solution of local traffic congestion as an example for analysis. e local control mitigates the congestion of a specific location on the traffic network. However, this good result comes at the cost of worse congestion at other links in the traffic network, while the control strategy on the network level could avoid this deficiency and reduce the adverse impact of local control on the traffic states of other links. Local controllers only use part of the traffic information, while in the controller on the traffic network level, the real origin-destination (OD) is crucial. e predictions of traffic flow will be easily applicable to control on the network level when the dynamic OD is available.
erefore, how to implement variable speed limit control strategy in expressway from the perspective of global control becomes the focus of this paper. In addition, in order to adapt the control strategy to the future traffic situation, this paper studies the combination of traffic volume prediction and control of expressway trunk line with variable speed restriction. e paper is divided into eight sections. Section 1 introduces the background and research significance of the problem. Section 2 reviews previous research. Section 3 states the traffic demand estimation and prediction. Section 4 gives the proposed control model and the improved macroscopic traffic flow model. Section 5 presents the simulation method and algorithm. Section 6 shows the numerical examples.Section 7 compares the research results of this model with other models. Section 8 draws the conclusion.

Literature Review
In variable speed limits research, scholars have done a lot of excellent research. Lin et al. [7] proposed VSL algorithms which were combined with ramp control (RM) with the aim of improving traffic efficiency. Hegyi et al. [8] used the variable speed limit strategy to study the traffic wave and it can compress the wave. Bertini et al. [9] studied the variable speed limit and traveler information system provided by overhead dynamic information signs in order to improve the understanding of how these systems affect driver behavior and bottleneck formation and location. Abdel et al. [10] analyzed the relationship between the implementation of variable speed limit and the possibility of vehicle collision under different traffic conditions and gave some suggestions on the implementation of variable speed limit. Yao proposed a trajectory smoothing method based on individual variable speed limits with location optimization (IVSL-LC) and verified that the method can improve traffic efficiency and fuel consumption [11]. Ellen proposed a variable speed limit system based on connected vehicles and verified that, in most cases, the VSL system can improve traffic efficiency through microsimulation software [12]. In addition, some other applications and studies have also demonstrated the important role of variable speed limit in the safety and efficiency of expressways [13][14][15][16].
Besides the direct application of VSL on traffic system, it is also introduced into micro-and mesotraffic flow model [17,18] and other optimization models [19][20][21][22]. Abdel et al. used microscopic simulation models to analyze the effects of different speed limit strategies on alleviating and improving road safety [17]. Inanjko established a microscopic simulation framework to analyze the layout and application scenarios of the VSL controller using travel time and vehicle emissions as indicators [18]. Fang proposed a VSL control algorithm based on the model predictive control (MPC) framework.
e improved model reduces the speed prediction accuracy error and improves the performance of the system [19]. Alasiri used microscopic simulation software to combine the cell transmission model (CTM) and variable speed limit strategy to verify the effectiveness of the robust VSL controller [20]. Qu used the proposed single-lane cellular automaton model to simulate the traffic flow characteristics under VSL control. e simulation results are consistent with the actual situation, which verifies the validity of the model [21]. Li et al. combined the variable speed limit strategy with the cellular conveyor model to optimize the traffic flow of expressways [22]. e good performance of VSL combined with traffic flow model is demonstrated in their work [23][24][25][26], but there are still some unresolved problems need to be addressed. e gaps need to be filled are listed as follows: (1) the research area is small, and a large number of studies mainly focus on the ramp area of expressways, lacking the research on long-distance trunk lines; (2) the simulation data are mainly based on historical traffic volume, and lack of prediction of future traffic volume changes, resulting in low reference of simulation results. At present, the research on traffic flow modeling and prediction has also made great progress [27,28], but it is rarely applied to VSL; (3) a large number of studies start from the microperspective, but there is lack of analysis from the macro perspective. In view of the abovementioned problems, this paper selects the expressway trunk line as the main research area from a macro point of view and introduces the traffic flow prediction into the control strategy before the implementation, so as to predict the future traffic changes. erefore, this paper studies the combination of traffic volume prediction and control of expressway trunk line with VSL strategy. e contributions of this paper are as follows: (1) this paper simulates the overall situation of vehicles on the road based on the macroscopic traffic flow model and gives the control strategy from the macro level, which fills up the gaps in related research and (2) generates real-time dynamic demand that can be loaded on the transportation network through traffic estimation and forecasting technology. e dynamic road network traffic demand can more realistically verify the effect of the control strategy. (3) An optimal speed control strategy for highway arterials is proposed, based on optimization models and design algorithms. (4) e feasibility of the proposed model is verified by traffic simulation technology.

Traffic Demand Estimation and Prediction
Based on the prediction of traffic demand, the real-time traffic demand is obtained. Also, the traffic demand can be obtained by performing OD prediction function. e decision about the boundary of the queue length should be made in advance. us, the estimated or predicted traffic demand will be loaded on traffic network, and then the traffic state can be acquired to realize active management for dynamic traffic.
e key to real-time dynamic traffic demand forecasting is to obtain the OD amount in the future time period based on the historical time-varying OD matrix. Okutani [29] first established a state space model for dynamic OD estimation and verified the effect of the model through Kalman filtering. Kachroo [30] studied the applicability of the Kalman filter method in estimating the origin-destination of travelers within the network from the link traffic. On this basis, Ashok [31] made further corrections and analyzed the deviation between the predicted OD value and the historical value. In this paper, the state space model of Ashok is used as the estimation and prediction of real-time OD matrix. Using deviation to define state variable, h . e deviation shows an autoregressive process: As an autoregressive coefficient matrix, f p h+1 is the impact of δx p on δx h+1 ; qis the autoregressive process times; w h+1 is the white Gaussian noise. Satisfying In the formula, is an assignment dynamic OD matrix, which is the contribution of δx p to δy p . e fraction of rin OD departs from the original place during the interval of p. en, during the interval h, it crosses the counting point. p ′ + 1stands for the maximum time intervals in travelling between the OD pairs of the traffic network. v h is the error measurement, E(v h ) � 0. To achieve the estimation and prediction of the dynamic OD, equations (1) and (2) are employed for the construction of the state space model. Also, the Kalman filter [32] is helpful for solving this problem.

e Improved Traffic Flow Model Integrated with VSL.
Generally, there are three types of traffic flow models: macro-, micro-and mesoscopic models. e macroscopic model focuses on the collect behaviour of vehicles, the speed and density are the research objects. e microscopic model aims to describe the behaviour of individual vehicle in detail. e description range of mesotraffic model is between micro and macro, which is suitable for medium-sized traffic network. In this paper, the macro hierarchical control model [33] will be explored and improved based on the characteristics of highway. At present, the research on variable speed control strategy is not thorough. Besides, the control models employed in these studies are only based on few control strategies. A comprehensive and effective variable speed decision system is not established yet. To develop an effective decision system, the VSL control strategy is introduced in the macro hierarchical traffic flow control model. e speed-density relation will change if the speed limits are added to the macro hierarchical control model. e definitions of indexes and variables are shown in Table 1. e improved macromodel-integrated VSL is proposed as follows: At the entrance and exit of the boundary ramp, namely, i � 1and i � N, e queue length of the on-ramp is as follows: In addition to the state equation (6), r(i, k)must meet the following constraints: . . , is the variable speed limits strategy. e relationship between density and speed on a segment can be described by equation (4) (see Figure 1).
In addition to the presented model, the dynamic characteristics of the vehicles are described. In this paper, the control objective is to minimize the total delay and the total travel time of all vehicles. is is a global optimal control algorithm on the basis of the mentioned state equations of the traffic flow model.

4.2.
e Optimization Control Model. e multiobjective optimization control model, in this paper, aims to minimize Journal of Advanced Transportation the total delay and the total travel time of all vehicles of all simulation intervals through the network. In other words, the objective is to ensure all vehicles reach destinations in the shortest possible time. One of the best variable speed limit strategies can be selected by means of simulation method. e following notations are used to provide a clear description of the problem. e traffic network consists of nodes and connecting lines.
e lines are divided into segments, which are represented by j. e total simulation time is divided into several intervals, which are represented by h, Q(j, h)is the average traffic flow in hinterval on jsegment, V(j, h)is the average traffic speed in hinterval on jsegment, and p(j, h)is the average traffic density in hinterval on jsegment. Equation (8), formula of the objective function, reads where C min is the minimum traffic capacity and C max is the maximum capacity. e constraints are up threshold and nonnegative.

The Simulation Method
e feasibility of online macroscopic traffic simulation and optimal traffic control is demonstrated in this paper. is approach can adapt to the changeable traffic conditions. A situation is proposed in which the VSL can be used in upstream main roads. e following is how the approach performs its function. e VSL can prevent traffic congestion at the bottleneck by creating a free flow zone at the upstream section. e existence of VSL changes the fundamental diagram shown in Figure 1and reduces the output capability of this segment. e following section shows the process of executing the proper simulation with VSL. A closed cycle scheme plays an important part in solving the problem (see Figure 2). VSL is integrated into the mesoscopic traffic simulator to estimate network operation state. e optimal objective is the optimal control speed limits strategy by minimizing the total delay. In order to improve the estimation accuracy of macroscopic model parameters (such as α, β, k jam etc. ), it has a process of parameter correction. A point deserves noting that the parameters usually become constant when the parameter calibration horizon passes. e procedure reduces the number of variables and enhances the stability of the traffic system. e parameters calibration process is chosen with the purpose of correcting system parameters. In this way, the network description and traffic dynamics characteristics can be obtained more accurately. In the following, a new optimization's performing functions are triggered to find the best strategy. In this regard, traffic estimation and prediction are of great help to this. A more comprehensive network state results from their interaction. Known as rolling horizon, this scheme enables updating the state from measurement and it can even update the model in every iteration step.

e Simulation Environment.
As shown in the simulation structure diagram in Figure 3, the highway network is divided into 9 sections. Each section is a one-way four-lane connecting line. On this highway, there are 4 entry ramps and 4 exit ramps. e section of each road is 3 km. e proposed model is used in the modeling and simulation. In order to simulate the real traffic situation and reveal its essence, the parameters of the traffic flow model are set according to the specific characteristics of the road section (length, width, level, capacity, and traffic flow), where α � 1.7, β � 2, r max � 900 veh/h, r min � 180 veh/h, v f � 100 km/h, p jam � 130 (veh/km)/Lane, and the    Table 3. After completing the traffic demand forecast, we first distribute the forecast results to the road network in the study area to obtain the traffic flow status and traffic flow parameters of each road section. en, on this basis, the changes of traffic flow parameters before and after the implementation of the VSL strategy under the   overall network are compared, and the effect of the model in this paper has been verified. Table 4shows the traffic density of each road section in the network before and after the implementation of the VSL strategy. What is worth noting here is that the entire network is taken into consideration in obtaining the simulation results. Obviously, before the VSL comes into play, the overall average density of the network is 0.00778. After it is performed, the value is 0.00792, a slight increase in density indicates that a decrease in speed results in an increase in the number of vehicles stranded on the network. e average traffic and speed of road sections in the network before the implementation of VSL were 387.4769 and 20.7170. However, after the implementation of VSL, the average network traffic was 387.2130 and the average speed was 20.5129. Due to the implementation of the variable speed limit strategy, the speed of the vehicle in the network has become lower, so the driving time of the vehicle in the network has increased. e following two-dimensional and three-dimensional distribution map, respectively, depicts the average density and speed of each segment in each time period. Figure 4is the three-dimensional distribution of the average density on each segment in each interval without VSL. Figure 5is the three-dimensional distribution of the average density on each segment in each interval with VSL. Figures 6and 7are their two-dimensional representations.
According to Figures 4-7, the average density of segments which are implemented with the VSL strategy increases significantly. But overall the other segments' average density becomes smoother. Because of the proposed method with VSL strategy, the simulation results present a better queue formation than those without a non-VSL. Owing to the VSL, it is convenient for vehicles to keep the earlier speed, because they can adjust speed according to the network state. In this way, the queue can be transferred from the bottleneck to upstream timely. However, the queue without the help of VSL cannot achieve that goal. Although after using VSL, the mean speed, especially during the rush hour, of most sections is lower than that in the non-VSL case, there is no negative impact on travel time. is highlights the advantages of the VSL in enabling the network to operate at a more stable speed. A comparison of the average speed of each interval from the eighth to twelfth interval implies   used. e network of traffic flow becomes more stable than that in the previous time. e traffic flow of some segments becomes less congested. eoretically, the more stable the traffic flow is, the better the operation efficiency of the traffic system will be.
In the same way, Figures 8-15show the average flow and speed of each segment in each interval in three-dimensional and two-dimensional forms.
In the four locations, the values of variable speed limit are set 100 km/h, 80 km/h, 80 km/h, and 90 km/h, respectively. Different values should be taken to be in line with limiting strategies. Because traffic congestion does not occur at those segments, variable speed limit strategies are acceptable in relieving traffic density while the overall traffic flow remains stable. When critical density appears on traffic network, the VSL plays an important role in avoiding potential danger. Both traffic density and traffic flow on the sections are far away from variable speed limit sections which undergo no changes. It can be concluded that without reducing road capacity, the variable speed limit strategies contribute to traffic safety.
Undoubtedly, if the vehicle speed is stable, it benefits to traffic flow operation. is avoids the risk of traffic accidents. To sum up, the variable speed limit strategies can effectively prevent the potential risk factors and improve the operation efficiency of the system in some cases. Although it does not show obvious effect in relieving traffic congestion, it is the best application in traffic management.
On the basis of the abovementioned research, in order to further verify the effectiveness of the method proposed in this article, we adopted a method of randomly perturbing the test data, that is, randomly changing each OD pair with a range of 30%. Since the traffic demand of the expressway network is time-varying, adding random disturbances can better describe the dynamic changes of the traffic volume on the road network and then verifies the effectiveness of the method in this paper more realistically. e simulation results show that when the demand increases by 30% on individual road sections with high OD demand, the variable speed limit has little effect on the traffic flow of the road section.
is is because the density of the road section    Journal of Advanced Transportation roughly makes the speed of the road section much lower than the dynamic speed limit value. When the demand is reduced by 30%, the speed of the vehicles on the road section is greater than the speed limit value. At this time, the variable speed limit value can act on most vehicles on the road section, thus producing a more obvious improvement and regulation effect. In general, it can be concluded through simulation: on the one hand, the variable speed limit method proposed in this paper can increase the traffic in some road sections, and the density is evenly distributed in the overall road network. On the other hand, the model has a more obvious improvement effect before the occurrence of congestion, so the implementation of the strategy needs to cooperate with accurate traffic state prediction methods. In addition, if the OD is adjusted again, the control measures of closed ramp can be adopted, combined with the variable speed limit of this article to adjust the main line traffic flow, and better control effect is desirable.

Comparing with Other Research Results
e proposed method in this paper is suitable for specific sections of the expressway. We used the assumed theoretical data to verify it. In this section, we compared the results of this paper with three other traffic flow models integrated with VSL, namely, the mesotraffic flow model, CTM (the cell transport model), and LTM (the link transport model) [14,34,35]. CTM is suitable for studying the micromotion of urban road vehicles and is combined with VSL to control the congestion at the intersection at the same time ensuring traffic safety. LTM is suitable for studying the traffic conditions at the section level which can nip the traffic congestion in the bud according to the speed limit strategy of different road sections. e mesoscopic traffic flow model integrated with VSL can control regional urban traffic. At the same traffic environment, we use VSL-integrated CTM and the VSL-integrated mesoscopic traffic flow model to study the traffic flow parameters such as density, and the results are shown in Figures 16and 17.
From the abovementioned results, there is little difference among the two types of models. e density calculated by the LTM-based model is a little higher than any one of others. Although the optimization effect of the traffic flow on the road is not very different, it is worth noting that the research scope of the expressway in this paper is very large, which shows that the calculation speed of the model in this paper is faster than other methods, so it is suitable for the macrotraffic flow based on VSL.

Research Conclusions and Prospects
is paper explores the VSL strategy on expressway by means of traffic simulation method, and the real-time traffic demands are estimated and predicted. With the help of motion equation, a cooperative systems extension is developed. en, a new speed limits strategy is proposed by using the proposed method. e recommendations of speed are provided in advance for the vehicles at predefined points. Additionally, according to the actual speed of the vehicles, the location of releasing speed limits strategy is determined.
According to the simulation results, the proposed method produces some different queue formations under the VSL strategy. ese vehicles adapt their earlier speed to guiding vehicle speed, which continuously produces a queue that moves upstream from the bottleneck. In the non-VSL case, however, the queue is stationary and the front of the queue grows upstream from the bottleneck, which reduces the capacity of bottleneck. e new control method can also smooth the density oscillation of traffic flow . en, the traffic flow is in a stable state. In the cooperative VSL case, the mean speed of most road segments during the rush hour is lower than that in the non-VSL case. But the lower mean speed does not affect the total travel time. is provides a new way to control the traffic flow with a stable speed pattern. It can be concluded that the proposed VSL strategy could harmonize traffic flow, promote traffic efficiency, and then improve the regional traffic conditions. Data Availability e data of this study come from mesoscopic simulation software.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.