# Design and Control Method of Switchable On- or Off-Ramp for Urban Highway

Authors: Yugang Liu; Hongbo Yi; Jiali Li; Bin Chen; Hongtai Yang
Journal: Journal of Advanced Transportation
 (2020)

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## Abstract

High volume of traffic is loaded on urban highway during peak hours as well as the ramp connecting main road and auxiliary road of highway. Once the demand exceeds the capacity of the ramp, congestion may happen on both main road and the auxiliary road. To deal with this problem, an new urban highway ramp, switchable ramp (SR) that can switch between on and off condition, is proposed in this paper. A macroscopic time-space discrete model is built in this paper to model this process by dividing the studied time period into short time periods and the studied road segment into small sections. The impact of the type of ramp on upstream and downstream traffic flow is analyzed. A segment of urban highway in Chengdu, China, is used as a case study. The traffic flow data in peak hours is collected and used. Mathematical analysis proves that the SR system proposed in this paper can reduce the total travel time (TTT) in the studied area. Results show that the SR system is more useful when the traffic volume is high.


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## Body

## 1. Introduction

The urban highway is an important part of urban road network. It is usually composed of a main road where the vehicle can run at high speed and ramps connecting the main road and auxiliary road. The main road and the auxiliary road of the urban highway are always on the same level and parallel to each other. Vehicles get on and off the main road of highway through on-ramp and off-ramp.As shown in Figure1, when the traffic volume of on-ramp and off-ramp exceeds the queue storage space, which usually happens during peak hours, there will be vehicles queueing in front of the ramp. This results in lower speed of traffic in the exit or entrance lance [1]. There will be bottlenecks in these lanes. In order to avoid the occurrence of traffic congestion, studying ramp metering and new type of ramp is very important.Figure 1
Queue of vehicles caused by traffic volume exceeding the queue storage space of ramp.Related studies have proved that there are some measures to increase the capacity of ramps. Regarding on-ramp, the vehicles that enter the main road through the on-ramp will interweave with vehicles that are already on the main road. So, controlling the traffic entering main road through the on-ramp can improve the capacity of merging area [2, 3]. Similarly, controlling the traffic entering auxiliary road through the off-ramp can improve the capacity of merging area on auxiliary road. However, the interaction between adjacent ramps along the expressway is inevitable. If the coordinated control of multiple ramps is considered, the capacity of ramps and highway could be further improved [4–11]. Considering the traffic demand on high fluctuates over time, if the traffic flow information is obtained and analyzed in real time, the best control method can also be applied in a real-time manner. Some researchers also find that real-time coordinated ramp metering strategy could better deal with the fluctuation of traffic flow [12–14]. It has also been found that the control method of variable speed limits (VSLs) along the main road has similar effect and ramp metering combined with VSL could achieve better results [1, 15, 16]. A control model for reducing congestion and emissions by combining ramp metering and route guidance was proposed by Pasquale et al. [17].Ramp metering could reduce the highway congestion by limiting the number of vehicles passing through the ramp, which could also reduce the turbulence of platooned on-ramp vehicles and thus improve traffic safety. However, although ramp metering and other control methods can improve the traffic condition of main roads, these methods only consider the condition of the main road. When the demand to get on or to get off the highway varies, traditional ramp cannot meet varying demand well to balance the traffic pressure on the main road or auxiliary road.Usually, existing ramps cannot meet the high traffic demand during peak hours. If we use fixed ramps, a pair of ramps (an on-ramp and an off-ramp) are needed. Besides, new fixed ramps may cause unnecessary weaving of vehicles during non-peak hours according to the shortcomings of the fixed ramp mentioned above. Therefore, it is necessary to design a type of switchable ramp (SR), which can switch condition according to the actual traffic condition. It can also be closed when it is not needed to avoid attracting extra demand. In terms of implementation, it is easy to let road users know the condition of the SR in real time using various methods such as variable message system (VMS) set upstream and map APP. The using of SR can meet the excessive traffic demand either on the main road or auxiliary road and reduce the possibility of platooned vehicles passing through the ramp as well. In addition, SR can also be used for other purposes, such as open during emergency for fire trucks or emergency vehicles.This paper focuses on the design of SR and its optimal control method and discusses the security and operability issues of SR in the implementation process. The structure of this paper is as follows. Section2 addresses the geometric design, operation, and implementation of SR. A macroscopic time-space discrete control model for SR is built in Section 3. A real case study is discussed in Section 4. Sections 5 and 6 present the conclusion and acknowledgments of this paper.


## 2. Switchable Ramp on Urban Highway

### 2.1. SR System

The SR can switch condition based on real-time traffic conditions. By using the cloud server, the optimal solution of the SR form can be determined based on real-time traffic data and a preset algorithm. We take two questions into consideration as follows when it comes to the implementation of SR.First of all, we must inform the drivers of the real-time status of the SR. We recommend two ways of notification in this paper. One is VMS and the other is map APP. As long as drivers know the real-time condition of the SR, drivers could make better decision.Secondly, the safety issues of SR in implementation are also considered. If the geometry of the SR is similar with that of a traditional fixed ramp, the SR will have similar safety performance. So, the safety issue should not be a concern.The schematic diagram of the SR system is shown in Figure2.Figure 2
The geometric design of switchable ramp and two of its forms.


### 2.2. Geometry Design of SR

According to the reasons mentioned above, a type of switchable ramp is designed in this paper, which can switch its condition by simple physical transformation or traffic light. The form of SR could be one of these three: on-ramp, off-ramp, and closed. The choice of the condition depends on the real time traffic condition. The method to determine the condition of SR is explained in the Section3. Generally speaking, it often turns into on- or off-ramp during peak hours and into closed condition when traffic demand is low. When the SR is in use, it can alleviate traffic congestion. When it is closed, it can reduce weaving traffic by reducing unnecessary lane changing. The design scheme of SR is introduced as follows.As is shown in Figure3, the arrow in each subgraph indicates the direction of the traffic flow. The upper side is the main road of urban highway, and the lower side is the auxiliary road of the highway. The shaded area represents the central isolation belt between main road of highway and auxiliary road. It often uses flower bed with vegetation of appropriate height. This would be the basis of realization of SR. The geometric design is described below.Figure 3
The geometric design of switchable ramp and two of its forms.
(a)(b)(c)The whole SR system are embedded into central isolation belt (the shaded area between auxiliary road and main road). The SR system is divided into three segments: area (a), area (b1), and area (b2). Every segment is different with area (a) approximating a triangle and area (b1) and area (b2) approximating two trapezoids. It can be seen that area (b1) and area (b2) are axisymmetric. Area (a), area (b1), and area (b2) should be designed with curved edges to accommodate the vehicles' running streamline.Each area is equipped with a board that can be raised and lowered. The upper limit of the rise of the board is an appropriate height at which vehicles cannot pass through. The lower limit at which the board can be lowered is at the same level as the surrounding road surface. If all boards rise to the highest point, the SR becomes closed and the vehicle cannot pass through. If the area (a) is lowered and then one of the areas (b1) and (b2) is lowered, an open ramp is formed. For example, when area (b1) is open and (b2) is blocked, it is an on-ramp.In the following sections, we will introduce a control method for SR and verify its validity.


## 2.1. SR System

The SR can switch condition based on real-time traffic conditions. By using the cloud server, the optimal solution of the SR form can be determined based on real-time traffic data and a preset algorithm. We take two questions into consideration as follows when it comes to the implementation of SR.First of all, we must inform the drivers of the real-time status of the SR. We recommend two ways of notification in this paper. One is VMS and the other is map APP. As long as drivers know the real-time condition of the SR, drivers could make better decision.Secondly, the safety issues of SR in implementation are also considered. If the geometry of the SR is similar with that of a traditional fixed ramp, the SR will have similar safety performance. So, the safety issue should not be a concern.The schematic diagram of the SR system is shown in Figure2.Figure 2
The geometric design of switchable ramp and two of its forms.


## 2.2. Geometry Design of SR

According to the reasons mentioned above, a type of switchable ramp is designed in this paper, which can switch its condition by simple physical transformation or traffic light. The form of SR could be one of these three: on-ramp, off-ramp, and closed. The choice of the condition depends on the real time traffic condition. The method to determine the condition of SR is explained in the Section3. Generally speaking, it often turns into on- or off-ramp during peak hours and into closed condition when traffic demand is low. When the SR is in use, it can alleviate traffic congestion. When it is closed, it can reduce weaving traffic by reducing unnecessary lane changing. The design scheme of SR is introduced as follows.As is shown in Figure3, the arrow in each subgraph indicates the direction of the traffic flow. The upper side is the main road of urban highway, and the lower side is the auxiliary road of the highway. The shaded area represents the central isolation belt between main road of highway and auxiliary road. It often uses flower bed with vegetation of appropriate height. This would be the basis of realization of SR. The geometric design is described below.Figure 3
The geometric design of switchable ramp and two of its forms.
(a)(b)(c)The whole SR system are embedded into central isolation belt (the shaded area between auxiliary road and main road). The SR system is divided into three segments: area (a), area (b1), and area (b2). Every segment is different with area (a) approximating a triangle and area (b1) and area (b2) approximating two trapezoids. It can be seen that area (b1) and area (b2) are axisymmetric. Area (a), area (b1), and area (b2) should be designed with curved edges to accommodate the vehicles' running streamline.Each area is equipped with a board that can be raised and lowered. The upper limit of the rise of the board is an appropriate height at which vehicles cannot pass through. The lower limit at which the board can be lowered is at the same level as the surrounding road surface. If all boards rise to the highest point, the SR becomes closed and the vehicle cannot pass through. If the area (a) is lowered and then one of the areas (b1) and (b2) is lowered, an open ramp is formed. For example, when area (b1) is open and (b2) is blocked, it is an on-ramp.In the following sections, we will introduce a control method for SR and verify its validity.


## 3. Control Method

### 3.1. Traffic Conditions on Main Roads under Normal Condition

According to the previous studies [18], the interaction of the three parameters to describe traffic state can be described as follows:(1)v=Vρ,q=ρ⋅Vρ,where Vρ indicates that the parameter of velocity v is a function determined by the density ρ. The velocity v decreases with the increase of density ρ and equals to zero when the density ρ reaches the congestion density.A macroscopic dispersed traffic flow model is used in model building in this paper, by describing the road network of the study area as a directed graph containing nodes and links. The traffic condition of a road segment is regarded as homogeneous. The road segments could be divided in a way to make sure that there are no ramps, big curves, or changes in the number of lanes in any segments. The location of the node should be the point where the ramp and the curves are located, and the point where the number of lanes changes. Similarly, we divide the entire study time into several time periods of lengthΔT. Therefore, the traffic state of the road segment i of length Li within ΔT can be described as follows when Li and ΔT are small [1]:(2)vi,ΔT=Vρi,ΔT,Qi,ΔT=ρi,ΔT⋅Vρi,ΔT⋅ΔT,where vi,ΔT represents the average vehicle speed of the road segment i within time ΔT and Qi,ΔT represents the traffic volume of the road segment within the time ΔT at segment i, which is expressed as the number of vehicles moving away from the downstream boundary of the road segment i within ΔT. Density ρi,ΔT is affected by road geometry, number of vehicles in the road segment, and lane change behavior of the vehicle [19, 20]. The way that lane change behavior influences density ρi,ΔT will be introduced in Section 3.2.


### 3.2. Influence of Lane Change Behavior on Traffic Condition of Main Roads

When a vehicle changes lane, it will travel from the current lane to the target lane. During this period, the vehicle occupies those two lanes. It can be considered as having two cars traveling in line. As shown in Figure4, the dotted rectangle in Figure 4(a) represents the actual lane change behavior, and the dotted rectangle in Figure 4(b) represents the equivalent case.Figure 4
Lane change behavior and its equivalent case.
(a)(b)According to the analysis above, the equivalent densityρ¯ on the segment i in the time ΔT can be described as follows [21]:(3)ρ¯=ρi,ΔT+Nl_cTl_cLiΔT,where ρLi,ΔT represents the traffic flow density when there is no lane change behavior in the studied area, ΔT and Li represent the studied time period and space, respectively, Nl_c represents the number of lane changes in ΔT and Li, and Tl_c represents the time needed for lane change behavior.If we denote the traffic volume of the on-ramp asq′ and the traffic volume of the main road as q, then the downstream traffic volume is q+q′. The upstream or downstream boundaries of the studied area is set to satisfy the requirement that this area can contain all the lane change behavior caused by the ramp. The traffic volume of each lane in the upstream and downstream is q/n and q+q′/n, respectively, where n is the number of lanes. In this condition, represented by Figure 5(a), the equivalent density ρ¯ on the main road at downstream of the on-ramp is expressed as(4)ρ¯=ρi,ΔT+q′Tl_cn−12Li,Nl_c=q′n−12⋅ΔT.For off-ramp, the lane change behavior happens on the main road on the upstream of the ramp, as shown in Figure5(b). If we denote the traffic flow on the ramp as q′, the time for lane change is the same as case of on-ramp. The equivalent density ρ¯ on the main road at upstream of the off-ramp is(5)ρ¯=ρi,ΔT+q′Tl_cn−12LiΔT.Figure 5
Analysis of traffic flow of on-ramp and off-ramp.
(a)(b)The lane change behavior increases the equivalent density of the segment. As mentioned earlier, increase in the number of lane changes reduces the traffic flow speed of the vehicle. The expression of functionVi,ΔT is(6)Vi,ΔT=minVlim,Vfi1−ρ¯i,ΔT/ρconi,where ρconi represents the congestion density, Vfi represents the free flow velocity, and Vlim represents the speed limit of the road.It can be seen from the function above that change of the equivalent density results in change of the average speed on the road segment, which leads to change of traveling time of all the vehicles. It is assumed that the traffic condition on the road segmenti is homogeneous during time ΔT when ΔT and Li are small enough. The number of vehicles in the section within ΔT can be recorded as Ni,ΔT. The expected total traveling time (TTT) of those vehicles in the road segment i during ΔT is(7)TTTi,ΔT=Ni,ΔT⋅Li/Vi,ΔT,TTTΔT=∑iTTTi,ΔT.where TTTi,ΔT represents the expected TTT of vehicles in road segment i, and TTTΔT represents the overall expected TTT of vehicles of the studied area.


### 3.3. Nodes Model between Road Segments

In this paper, the road to be studied is divided into several segments and nodes connecting these segments for modeling purposes. It should be noted that the node acts as a virtual point connecting the upstream and downstream road segments and does not have spatial attributes. So that nodes cannot store vehicles like road segments. Therefore, for the nodeNi, at any time,(8)qNiin=∑i∈INiqiout,qNiout=∑i∈ONiqiin,where the parameters qNiin and qNiout represent the inflow and outflow of the node Ni, respectively, INi represents a set of road segments on the upstream side of node Ni, and similarly, ONi represents the set of road segments on the downstream side of node Ni. The relationship between the segments and the node is shown in Figure 6.Figure 6
Relationship between segments and nodes.The following three principles should be followed when selecting the location of the node:(i)
The traffic conditions on the upstream and downstream side of the ramp are often different. So, the point where the ramp is located should be selected as the node.(ii)
When the distance between two adjacent ramps is too long, in order to ensure that the length of each segment is small enough, appropriate number of points should be selected along the road as nodes.(iii)
When road geometry changes, these points should be selected as nodes.A node can only be one of the three types: normal node (without ramp), on-ramp, or off-ramp. The type of nodeNi can be denoted as SNi, with values of 1, 0, and −1 indicating on-ramp, normal, and off-ramp nodes.


### 3.4. Model of Time Dimension

The traffic condition of each road segments is changing constantly over time. In this section, we discuss the relationship between traffic flow in thek+1thΔT and traffic flow in the previous time period kth ΔT. The equivalent density recursion function between two time periods is as follows:(9)ρ¯i,ΔTk+1=ρi,ΔTk+1+ρi,ΔT′k+1,(10)ρi,ΔT′k+1=Nl_ck+1Tl_cLiΔT,(11)ρi,ΔTk+1=ρi,ΔTk+ΔTqi,ΔTink−qi,ΔToutkLi,(12)qNioutk+1=min∑j∈ONiQjc−Qjk∑j∈ONiqjink.where Qjc represents the maximum number of vehicles that can be accommodated by road segment j and Qjk represents the vehicle that is already on the road segment j in the kthΔT time. According to the equations above, the traffic condition at k+1thΔT can be calculated by the following equations:(13)Vi,ΔTk+1=minVlimi,Vfi1−ρ¯i,,ΔTk+1ρci,(14)qi,ΔTink+1qi,ΔTink∑i∈ONiqi,ΔTink=qNi−1,ΔToutk+1,(15)∑i∈INiqi,ΔToutk+1=qNiink+1=qNioutk+1,(16)qi,ΔToutk+1=qi,ΔToutk∑i∈INiqi,ΔToutk∑i∈INiqi,ΔToutk+1.Equations (9) to (16) can be summarized as follows:(17)xk+1=fxk,sNik+1,where xk represents the state of traffic flow during the time period kthΔT and sNik+1 represents the type of nodes during the time period k+1thΔT.


### 3.5. Optimization Objective

During peak hours or in an emergency, the capacity of fixed ramps may not be able to meet the high traffic demand. In some time periods, the traffic demand from the main road to the auxiliary road is greater, and sometimes it is the contrary. In these two cases, the condition of SR could be adjusted according to the actual traffic state. When the traffic demand is not high, fixed ramps can meet all the requirements. Thus, the SR should be turned into the closed condition to reduce unnecessary weaving of vehicles.In this paper, we set minimizing TTT as the optimization objective. The form of SR ink+1thΔT is determined by the traffic state of the time period kthΔT:(18)minTTTk+1=∑iTTTi,ΔTk+1,s.t.xk+1=fxk,sNik+1,SNik+1=0or1,k=1,2,…,n,When the traffic demand is high and the capacity of existing ramp cannot meet the demand, there will be a large number of vehicles accumulating in the upstream segment of the ramp. According to equation (6), when the traffic flow density of a road segment reaches congestion density, the traffic flow speed of the road segment will be equal to zero. In this case, opening the SR could allow vehicles to pass through to reduce congestion. The probability that the density of weaving segments reaches congestion density is also reduced. In the Section 4, a case study is performed to prove this point.


## 3.1. Traffic Conditions on Main Roads under Normal Condition

According to the previous studies [18], the interaction of the three parameters to describe traffic state can be described as follows:(1)v=Vρ,q=ρ⋅Vρ,where Vρ indicates that the parameter of velocity v is a function determined by the density ρ. The velocity v decreases with the increase of density ρ and equals to zero when the density ρ reaches the congestion density.A macroscopic dispersed traffic flow model is used in model building in this paper, by describing the road network of the study area as a directed graph containing nodes and links. The traffic condition of a road segment is regarded as homogeneous. The road segments could be divided in a way to make sure that there are no ramps, big curves, or changes in the number of lanes in any segments. The location of the node should be the point where the ramp and the curves are located, and the point where the number of lanes changes. Similarly, we divide the entire study time into several time periods of lengthΔT. Therefore, the traffic state of the road segment i of length Li within ΔT can be described as follows when Li and ΔT are small [1]:(2)vi,ΔT=Vρi,ΔT,Qi,ΔT=ρi,ΔT⋅Vρi,ΔT⋅ΔT,where vi,ΔT represents the average vehicle speed of the road segment i within time ΔT and Qi,ΔT represents the traffic volume of the road segment within the time ΔT at segment i, which is expressed as the number of vehicles moving away from the downstream boundary of the road segment i within ΔT. Density ρi,ΔT is affected by road geometry, number of vehicles in the road segment, and lane change behavior of the vehicle [19, 20]. The way that lane change behavior influences density ρi,ΔT will be introduced in Section 3.2.


## 3.2. Influence of Lane Change Behavior on Traffic Condition of Main Roads

When a vehicle changes lane, it will travel from the current lane to the target lane. During this period, the vehicle occupies those two lanes. It can be considered as having two cars traveling in line. As shown in Figure4, the dotted rectangle in Figure 4(a) represents the actual lane change behavior, and the dotted rectangle in Figure 4(b) represents the equivalent case.Figure 4
Lane change behavior and its equivalent case.
(a)(b)According to the analysis above, the equivalent densityρ¯ on the segment i in the time ΔT can be described as follows [21]:(3)ρ¯=ρi,ΔT+Nl_cTl_cLiΔT,where ρLi,ΔT represents the traffic flow density when there is no lane change behavior in the studied area, ΔT and Li represent the studied time period and space, respectively, Nl_c represents the number of lane changes in ΔT and Li, and Tl_c represents the time needed for lane change behavior.If we denote the traffic volume of the on-ramp asq′ and the traffic volume of the main road as q, then the downstream traffic volume is q+q′. The upstream or downstream boundaries of the studied area is set to satisfy the requirement that this area can contain all the lane change behavior caused by the ramp. The traffic volume of each lane in the upstream and downstream is q/n and q+q′/n, respectively, where n is the number of lanes. In this condition, represented by Figure 5(a), the equivalent density ρ¯ on the main road at downstream of the on-ramp is expressed as(4)ρ¯=ρi,ΔT+q′Tl_cn−12Li,Nl_c=q′n−12⋅ΔT.For off-ramp, the lane change behavior happens on the main road on the upstream of the ramp, as shown in Figure5(b). If we denote the traffic flow on the ramp as q′, the time for lane change is the same as case of on-ramp. The equivalent density ρ¯ on the main road at upstream of the off-ramp is(5)ρ¯=ρi,ΔT+q′Tl_cn−12LiΔT.Figure 5
Analysis of traffic flow of on-ramp and off-ramp.
(a)(b)The lane change behavior increases the equivalent density of the segment. As mentioned earlier, increase in the number of lane changes reduces the traffic flow speed of the vehicle. The expression of functionVi,ΔT is(6)Vi,ΔT=minVlim,Vfi1−ρ¯i,ΔT/ρconi,where ρconi represents the congestion density, Vfi represents the free flow velocity, and Vlim represents the speed limit of the road.It can be seen from the function above that change of the equivalent density results in change of the average speed on the road segment, which leads to change of traveling time of all the vehicles. It is assumed that the traffic condition on the road segmenti is homogeneous during time ΔT when ΔT and Li are small enough. The number of vehicles in the section within ΔT can be recorded as Ni,ΔT. The expected total traveling time (TTT) of those vehicles in the road segment i during ΔT is(7)TTTi,ΔT=Ni,ΔT⋅Li/Vi,ΔT,TTTΔT=∑iTTTi,ΔT.where TTTi,ΔT represents the expected TTT of vehicles in road segment i, and TTTΔT represents the overall expected TTT of vehicles of the studied area.


## 3.3. Nodes Model between Road Segments

In this paper, the road to be studied is divided into several segments and nodes connecting these segments for modeling purposes. It should be noted that the node acts as a virtual point connecting the upstream and downstream road segments and does not have spatial attributes. So that nodes cannot store vehicles like road segments. Therefore, for the nodeNi, at any time,(8)qNiin=∑i∈INiqiout,qNiout=∑i∈ONiqiin,where the parameters qNiin and qNiout represent the inflow and outflow of the node Ni, respectively, INi represents a set of road segments on the upstream side of node Ni, and similarly, ONi represents the set of road segments on the downstream side of node Ni. The relationship between the segments and the node is shown in Figure 6.Figure 6
Relationship between segments and nodes.The following three principles should be followed when selecting the location of the node:(i)
The traffic conditions on the upstream and downstream side of the ramp are often different. So, the point where the ramp is located should be selected as the node.(ii)
When the distance between two adjacent ramps is too long, in order to ensure that the length of each segment is small enough, appropriate number of points should be selected along the road as nodes.(iii)
When road geometry changes, these points should be selected as nodes.A node can only be one of the three types: normal node (without ramp), on-ramp, or off-ramp. The type of nodeNi can be denoted as SNi, with values of 1, 0, and −1 indicating on-ramp, normal, and off-ramp nodes.


## 3.4. Model of Time Dimension

The traffic condition of each road segments is changing constantly over time. In this section, we discuss the relationship between traffic flow in thek+1thΔT and traffic flow in the previous time period kth ΔT. The equivalent density recursion function between two time periods is as follows:(9)ρ¯i,ΔTk+1=ρi,ΔTk+1+ρi,ΔT′k+1,(10)ρi,ΔT′k+1=Nl_ck+1Tl_cLiΔT,(11)ρi,ΔTk+1=ρi,ΔTk+ΔTqi,ΔTink−qi,ΔToutkLi,(12)qNioutk+1=min∑j∈ONiQjc−Qjk∑j∈ONiqjink.where Qjc represents the maximum number of vehicles that can be accommodated by road segment j and Qjk represents the vehicle that is already on the road segment j in the kthΔT time. According to the equations above, the traffic condition at k+1thΔT can be calculated by the following equations:(13)Vi,ΔTk+1=minVlimi,Vfi1−ρ¯i,,ΔTk+1ρci,(14)qi,ΔTink+1qi,ΔTink∑i∈ONiqi,ΔTink=qNi−1,ΔToutk+1,(15)∑i∈INiqi,ΔToutk+1=qNiink+1=qNioutk+1,(16)qi,ΔToutk+1=qi,ΔToutk∑i∈INiqi,ΔToutk∑i∈INiqi,ΔToutk+1.Equations (9) to (16) can be summarized as follows:(17)xk+1=fxk,sNik+1,where xk represents the state of traffic flow during the time period kthΔT and sNik+1 represents the type of nodes during the time period k+1thΔT.


## 3.5. Optimization Objective

During peak hours or in an emergency, the capacity of fixed ramps may not be able to meet the high traffic demand. In some time periods, the traffic demand from the main road to the auxiliary road is greater, and sometimes it is the contrary. In these two cases, the condition of SR could be adjusted according to the actual traffic state. When the traffic demand is not high, fixed ramps can meet all the requirements. Thus, the SR should be turned into the closed condition to reduce unnecessary weaving of vehicles.In this paper, we set minimizing TTT as the optimization objective. The form of SR ink+1thΔT is determined by the traffic state of the time period kthΔT:(18)minTTTk+1=∑iTTTi,ΔTk+1,s.t.xk+1=fxk,sNik+1,SNik+1=0or1,k=1,2,…,n,When the traffic demand is high and the capacity of existing ramp cannot meet the demand, there will be a large number of vehicles accumulating in the upstream segment of the ramp. According to equation (6), when the traffic flow density of a road segment reaches congestion density, the traffic flow speed of the road segment will be equal to zero. In this case, opening the SR could allow vehicles to pass through to reduce congestion. The probability that the density of weaving segments reaches congestion density is also reduced. In the Section 4, a case study is performed to prove this point.


## 4. Case Study

### 4.1. Data of the Case

A four-lane main road and two-lane auxiliary road of about 2.03 kilometers, located in Chengdu, China, is used for our case study. It is presented in Figure7. The road segment is between Supo-flyover (103.9906E, 30.6763N), on the upstream of the section, and Yangxi-flyover (104.0047E, 30.7037N), on the downstream of the section. We divided the studied area into 20 small segments, with the length of each segment being about 100 meters. There are only one fixed on-ramp (node N5) and one fixed off-ramp (node N13) along the road. Distance between these two ramps is about 800 meters. Traffic data of Monday, July 8, 2019, were used, from 5:00 PM to 5:30 PM. Traffic volume on the main road and auxiliary road is shown as Tables 1 and 2. A hypothetical SR (node N9) is set at the midpoint between the on-ramp and off-ramp.Figure 7
Case study area.Table 1
Traffic volume of main road.
Time indexkFlow rate (pcu/h/lane)1 to 101548126085590082812781800144091863011 to 20756585720810846630900145836036021 to 3036036036036036036036036036036031 to 40360360360558360630738990675118841 to 501980229520702187198022681818540720144051 to 6014409001683108014401800117011881350108061 to 70234021331845189018361800153016381809247571 to 8012781782118821241926171099011162340217881 to 902043165620791710170121242205144063054091 to 10014401350109881010808101170141310801260101 to 11011701755630270152113141530855360648111 to 120828127812961008153063069312961080450121 to 1301026990154815308101530127863011251260131 to 140810162015481440153010809009009361350141 to 150153016831368153011887201098918990360151 to 160144072063072023589006305586301566161 to 170125172012157207383782703603816180Table 2
Traffic volume of auxiliary road.
Time indexkFlow rate (pcu/h/lane)1 to 1015663603603603609001080720108036011 to 2036036036036036036036054036036021 to 3036036036036036036036036036036031 to 4036036036036036036036036036072041 to 50900252018001170165672097254036036051 to 6036072012601062198010801260900126090061 to 7019801692187214401080162010809001440144071 to 80990900720108012607205408461260126081 to 9014409001116126090016561080108054072091 to 1001530108099036011163601296540810990101 to 110720900720450360576360576720360111 to 120576360360360360360360720720360121 to 130360576360360360360540360360756131 to 140540360540720360720360360540360141 to 150360360360360540540360360540360151 to 16057654072054072054054018008101116161 to 17054036075672090054054011521152180The average velocity of traffic in initial condition at starting point on the upstream of the main road is 43.808 kilometers per hour and that of the auxiliary road is 36.768 kilometers per hour. The average density of traffic at starting point of main road and auxiliary road is 19.325% and 32.263%, respectively. According to the study of Greenshields [22]; the free flow velocities of the main and auxiliary roads are around 67 kilometers per hour and 49 kilometers per hour, respectively. The speed limit of the main road is 80 kilometers per hour and that of the auxiliary road is 60 kilometers per hour. The time needed for a vehicle to change lanes is set as 10.5 seconds [20].


### 4.2. Optimization Effect When Adding an SR at NodeN9

In this section, we analyze the change in traffic condition after adding an SR under two types of traffic demands. The results are as follows.Figure8(a) indicates the traffic condition of the studied area without SR. The x axis represents the road segment i (from 1 to 20), and the y axis represents the time k (from 1 to 170). Step size ΔT is selected as 10s. Different colors represent different TTTs of the segment i in kthΔT, i.e., TTTi,ΔTk. Figure 8(b)1) represents the curve of overall TTT TTTΔTk over time in the no SR case. When the traffic demand in the studied area increases by 5%, the traffic condition without SR is shown in Figure 8(a)). The coordinate system in Figure 8(a)) is the same as the coordinate system in Figure 8(a)). Figure 8(b)) shows the overall TTT curve over time when there is no SR.The results of both two cases above when SR is used are shown in other subfigures. When the traffic demand is set as the current traffic demand, results are shown in Figures8(c) and 8(d). When the traffic demand increases by 5%, results are shown in Figures 8(c) and 8(d). Moreover, Figure 8(e) shows the curve of sN9 (N9 is the node with SR in this case). As defined before, 1, 0, and −1 represent on-ramp, normal, and off-ramp, respectively.The mathematical analysis proves that the SR system proposed in this paper can reduce TTT contrast to no SR case. Under the current traffic flow condition, the TTT can be reduced from 148.107 hours to 142.024 hours, with a reduction of 4.107%. When traffic demand increases by 5%, the TTT of highway system can be reduced from 157.289 hours to 150.262 hours, with a reduction of 4.468%.Figure 8
Results of case study.
(a)(b)(c)(d)(e)The reduction mentioned above is the reduction in the TTT over the study period. But SR is not open all the time. If we only compare the reduction of TTT during the opening period of SR, the reduction rate of TTT under the surveyed traffic flow is 6.416%, and the reduction rate of TTT when traffic flow increases by 5% is 9.526%. The increase in traffic flow leads to more significant effects.


### 4.3. Comparison of Effects of SR and Traditional Ramp

The analysis above demonstrates that the SR proposed in this paper is effective in reducing the overall TTT at the surveyed traffic flow condition and when the volume increases by 5%. In order to verify that adding a SR has better effect than adding a traditional on-ramp or off-ramp, we analyze TTT under different traffic flow conditions. The results are shown in Table3. Four scenarios are considered:Scenario 1: no changeScenario 2: add an SR atN9Scenario 3: add an on-ramp atN9Scenario 4: add an off-ramp atN9Table 3
TTT (hours) under different traffic flows.
Multiple of original flow0.900.930.9350.940.951.001.05Scenario 1131.082136.694137.546138.398140.828148.107157.289Scenario 2131.082136.694137.281137.724140.105142.024150.262Scenario 3131.853136.868137.414138.208140.514146.837155.382Scenario 4132.144136.970137.335138.061140.467145.066153.776The first row in Table3 indicates the input traffic flow, represented by the number multiplying the original traffic flow.The gray cells in each column indicate the lowest TTT in the column, which are the optimal cases. We found that the TTT of Scenario 1 is the shortest when the input flow is less than or equal to 0.93 times the surveyed flow. Adding a ramp in these conditions will increase TTT. Thus, when the traffic flow is less than or equal to 0.93 times, the SR should be in the closed form, which is equivalent to no ramp. When the input traffic flow is greater than or equal to 0.94 times the surveyed traffic flow, no matter which type of ramp is added, the TTT will decrease. When SR is added, the TTT is the lowest.In order to find out the critical point where SR can improve the traffic condition, we studied in more detail about the traffic flow between 0.93 and 0.94 times the surveyed flow and found that the critical point is 0.935 times the surveyed traffic flow in this case.


## 4.1. Data of the Case

A four-lane main road and two-lane auxiliary road of about 2.03 kilometers, located in Chengdu, China, is used for our case study. It is presented in Figure7. The road segment is between Supo-flyover (103.9906E, 30.6763N), on the upstream of the section, and Yangxi-flyover (104.0047E, 30.7037N), on the downstream of the section. We divided the studied area into 20 small segments, with the length of each segment being about 100 meters. There are only one fixed on-ramp (node N5) and one fixed off-ramp (node N13) along the road. Distance between these two ramps is about 800 meters. Traffic data of Monday, July 8, 2019, were used, from 5:00 PM to 5:30 PM. Traffic volume on the main road and auxiliary road is shown as Tables 1 and 2. A hypothetical SR (node N9) is set at the midpoint between the on-ramp and off-ramp.Figure 7
Case study area.Table 1
Traffic volume of main road.
Time indexkFlow rate (pcu/h/lane)1 to 101548126085590082812781800144091863011 to 20756585720810846630900145836036021 to 3036036036036036036036036036036031 to 40360360360558360630738990675118841 to 501980229520702187198022681818540720144051 to 6014409001683108014401800117011881350108061 to 70234021331845189018361800153016381809247571 to 8012781782118821241926171099011162340217881 to 902043165620791710170121242205144063054091 to 10014401350109881010808101170141310801260101 to 11011701755630270152113141530855360648111 to 120828127812961008153063069312961080450121 to 1301026990154815308101530127863011251260131 to 140810162015481440153010809009009361350141 to 150153016831368153011887201098918990360151 to 160144072063072023589006305586301566161 to 170125172012157207383782703603816180Table 2
Traffic volume of auxiliary road.
Time indexkFlow rate (pcu/h/lane)1 to 1015663603603603609001080720108036011 to 2036036036036036036036054036036021 to 3036036036036036036036036036036031 to 4036036036036036036036036036072041 to 50900252018001170165672097254036036051 to 6036072012601062198010801260900126090061 to 7019801692187214401080162010809001440144071 to 80990900720108012607205408461260126081 to 9014409001116126090016561080108054072091 to 1001530108099036011163601296540810990101 to 110720900720450360576360576720360111 to 120576360360360360360360720720360121 to 130360576360360360360540360360756131 to 140540360540720360720360360540360141 to 150360360360360540540360360540360151 to 16057654072054072054054018008101116161 to 17054036075672090054054011521152180The average velocity of traffic in initial condition at starting point on the upstream of the main road is 43.808 kilometers per hour and that of the auxiliary road is 36.768 kilometers per hour. The average density of traffic at starting point of main road and auxiliary road is 19.325% and 32.263%, respectively. According to the study of Greenshields [22]; the free flow velocities of the main and auxiliary roads are around 67 kilometers per hour and 49 kilometers per hour, respectively. The speed limit of the main road is 80 kilometers per hour and that of the auxiliary road is 60 kilometers per hour. The time needed for a vehicle to change lanes is set as 10.5 seconds [20].


## 4.2. Optimization Effect When Adding an SR at NodeN9

In this section, we analyze the change in traffic condition after adding an SR under two types of traffic demands. The results are as follows.Figure8(a) indicates the traffic condition of the studied area without SR. The x axis represents the road segment i (from 1 to 20), and the y axis represents the time k (from 1 to 170). Step size ΔT is selected as 10s. Different colors represent different TTTs of the segment i in kthΔT, i.e., TTTi,ΔTk. Figure 8(b)1) represents the curve of overall TTT TTTΔTk over time in the no SR case. When the traffic demand in the studied area increases by 5%, the traffic condition without SR is shown in Figure 8(a)). The coordinate system in Figure 8(a)) is the same as the coordinate system in Figure 8(a)). Figure 8(b)) shows the overall TTT curve over time when there is no SR.The results of both two cases above when SR is used are shown in other subfigures. When the traffic demand is set as the current traffic demand, results are shown in Figures8(c) and 8(d). When the traffic demand increases by 5%, results are shown in Figures 8(c) and 8(d). Moreover, Figure 8(e) shows the curve of sN9 (N9 is the node with SR in this case). As defined before, 1, 0, and −1 represent on-ramp, normal, and off-ramp, respectively.The mathematical analysis proves that the SR system proposed in this paper can reduce TTT contrast to no SR case. Under the current traffic flow condition, the TTT can be reduced from 148.107 hours to 142.024 hours, with a reduction of 4.107%. When traffic demand increases by 5%, the TTT of highway system can be reduced from 157.289 hours to 150.262 hours, with a reduction of 4.468%.Figure 8
Results of case study.
(a)(b)(c)(d)(e)The reduction mentioned above is the reduction in the TTT over the study period. But SR is not open all the time. If we only compare the reduction of TTT during the opening period of SR, the reduction rate of TTT under the surveyed traffic flow is 6.416%, and the reduction rate of TTT when traffic flow increases by 5% is 9.526%. The increase in traffic flow leads to more significant effects.


## 4.3. Comparison of Effects of SR and Traditional Ramp

The analysis above demonstrates that the SR proposed in this paper is effective in reducing the overall TTT at the surveyed traffic flow condition and when the volume increases by 5%. In order to verify that adding a SR has better effect than adding a traditional on-ramp or off-ramp, we analyze TTT under different traffic flow conditions. The results are shown in Table3. Four scenarios are considered:Scenario 1: no changeScenario 2: add an SR atN9Scenario 3: add an on-ramp atN9Scenario 4: add an off-ramp atN9Table 3
TTT (hours) under different traffic flows.
Multiple of original flow0.900.930.9350.940.951.001.05Scenario 1131.082136.694137.546138.398140.828148.107157.289Scenario 2131.082136.694137.281137.724140.105142.024150.262Scenario 3131.853136.868137.414138.208140.514146.837155.382Scenario 4132.144136.970137.335138.061140.467145.066153.776The first row in Table3 indicates the input traffic flow, represented by the number multiplying the original traffic flow.The gray cells in each column indicate the lowest TTT in the column, which are the optimal cases. We found that the TTT of Scenario 1 is the shortest when the input flow is less than or equal to 0.93 times the surveyed flow. Adding a ramp in these conditions will increase TTT. Thus, when the traffic flow is less than or equal to 0.93 times, the SR should be in the closed form, which is equivalent to no ramp. When the input traffic flow is greater than or equal to 0.94 times the surveyed traffic flow, no matter which type of ramp is added, the TTT will decrease. When SR is added, the TTT is the lowest.In order to find out the critical point where SR can improve the traffic condition, we studied in more detail about the traffic flow between 0.93 and 0.94 times the surveyed flow and found that the critical point is 0.935 times the surveyed traffic flow in this case.


## 5. Conclusion

This paper proposes a type of SR for urban highway that can be easily implemented. It can be used in the case that the main road and auxiliary road of urban highway are on the same level. A spatial-temporal discrete model is proposed to minimize the TTT of all the vehicles in the studied area by adjusting the condition of SR. The idea of the model is to divide the studied road segment into several short sections and the studied time period into small intervals to calculate the travel time of the vehicles in each section and time interval. The impact of the condition of SR on the traffic flow of adjacent main road and auxiliary road is analyzed.At the end of the paper, an expressway segment of about 2 kilometers in Chengdu, China is used to demonstrate the control method. By applying the mathematical analysis, the adoption of SR could reduce the TTT of vehicles traveling in the segment. When the traffic demand is higher, the percent of reduction of TTT is also higher, indicating the SR is more effective. The critical threshold value is also determined. When the traffic demand is higher than the critical threshold value, the SR should be turned on.There are also several limitations of this paper. First of all, this paper only considers the impact of SR on traffic condition of surrounding area when a new SR is added. In the following research, we will study the coordinated control of multiple SRs in the overall urban road network. Another limitation of this study is that the decision of turning the SR to on, off, or closed condition depends largely on the traffic volume on the main and auxiliary road. The method to determine the condition of SR is somehow complicated. Thus, there is no simple guideline to show when to turn the SR condition to on, off, or closed. Thirdly, the decision of adding a ramp or switching ramp between on and off condition depends on a lot of factors such as human factors. Those factors are not considered in this paper. Future studies could focus on how to consider these factors when designing SR. However, the setup of SR depends on existing technologies and could be regarded as a combination of on and off-ramp with metering. This combination is not considered before and a control method is proposed for this type of random. This type of ramp may not be implemented immediately because a lot of factors such as travellers’ reaction are not considered. But, we believe it is a valuable exploration and this type of ramp may be implemented in the future.



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*Source: 7375416-2020-09-01.xml*