# Distributed Cooperative Backpressure-Based Traffic Light Control Method

Authors: Shenxue Hao; Licai Yang; Li Ding; Yajuan Guo
Journal: Journal of Advanced Transportation
 (2019)

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## Abstract

On the foundation of the original backpressure-based traffic light control algorithm, a distributed cooperative backpressure-based traffic light control method is proposed in this paper. The urban traffic network is modeled as a smart agent-controlled queuing network, in which the intersection agents exchange the queue length information and the selected activating light phase information of neighboring intersections through communications and determine the activating light phase at each time slot according to local traffic information. The improved phase pressure computation method considers the phase state of downstream intersections instead of only the queue length of the local intersections. Light phase switching coordination among adjacent intersections is achieved using the consensus-based bundle algorithm, in which the cooperative light phase switching problem is viewed as a task assignment issue among adjacent intersections. Simulation results illustrated that the proposed cooperative backpressure-based traffic light control method obtained better performance than the original backpressure-based and fixed-time traffic control methods.


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## Body

## 1. Introduction

With the increase of vehicles travelling on roads in cities, congestion has become a major problem for urban traffic systems. Optimal traffic control strategy can improve traffic efficiency and balance the traffic load, which plays an important role in urban traffic systems. There are three types of conventional traffic control methods: fixed-time traffic control method, actuated traffic control, and adaptive traffic control. In the 1960s, TRANSYT [1] and MAXBAND [2] were proposed based on offline historical traffic data which selected optimal timing plans at different times of the day. However, these fixed-time traffic control methods cannot deal with the fluctuation of traffic demand due to predefined traffic parameters. Next, an actuated traffic control method was proposed to extend the light phase duration according to the detected traffic flow data in real time, using detectors installed at upstream segments. However, the detection of sparse traffic has a considerable influence on delay time [3] and it was applied mainly at isolated intersections [4]. Adaptive traffic control methods (SCOOT [5] and SCATS [6]) were proposed to adjust signal timing plans based on online traffic information for responding to real-time traffic demand. These methods can respond to traffic fluctuation using detector input, historical trends, and predictive models [7]. Although adaptive traffic control systems have been implied in real urban traffic networks, the cooperation of multiple intersections requires a control center, which has obstructed further development. In addition, the centralized signal timing scheme is always calculated according to the current traffic conditions, but is implied in the next cycle when the traffic situation may have already changed [4].To fix these deficiencies, there is an agreement among many researchers that a distributed traffic control strategy is an ideal alternative. In a distributed traffic control system, signalized intersections are viewed as smart agents, and each agent determines the traffic control parameters according to current local information. For real-time dynamic traffic conditions, this kind of traffic control method can achieve better performance and adaptation [8, 9]. Many agent-based traffic control approaches have been studied in the past decades, and different theories and methods have been utilized to optimize traffic control among intersections according to different traffic parameters. Researchers have utilized intelligent computation methods to obtain the cycle length and splits by minimizing traffic parameters, such as total travel time, etc. These types of methods avoid huge computations, such as the colony optimization approach [10] and reinforcement learning algorithm [11]. However, it is still difficult for these types of traffic control methods to achieve online traffic timing decision making, since the traffic signal timing problem usually is NP hard, it may take long time to find an optimal solution for simple transportation systems [12, 13]. Therefore, to simplify the traffic control problem, a better way of achieving distributed traffic system control with lower computation is needed. In 2012, Wongpiromsarnet al. first introduced a backpressure algorithm to solve traffic control problems and develop a traffic signal control strategy by viewing the traffic network as a queuing network [14]. The backpressure algorithm is usually used in wireless multihop networks as an optimal strategy for resource allocation. It has many features, including throughput optimality, achievable adaptive resource allocation, and simplicity [15].Since the backpressure-based traffic signal control algorithm can maximize the throughput of the traffic network in a completely distributed manner, researchers have paid more attention to this algorithm and have obtained many achievements. In 2013, Varaiya presented a maximum pressure traffic control strategy based on the backpressure algorithm. At each intersection, the active phase is selected depending on the local queue length, mean turn ratios, and saturation rates [16]. Considering the influence of the routing rate of the queued vehicle on links, Gregoireet al. presented a backpressure-based traffic signal control algorithm with unknown routing rates and an estimated aggregated queue length. The vehicle routing information can be detected using detectors on dedicated lanes [17]. In their later research, a routing model of the traffic network was established with partial controllable vehicles to be used for pressure computation [18]. With the development of communication technology, more information could be obtained from a vehicular network and a multicommodity backpressure algorithm for traffic light control was proposed, where it is assumed that all vehicles’ routes are known [19]. To further improve the efficiency of the traffic network, Taaleet al. integrated route guidance technology with traffic signal control based on the backpressure algorithm [20], and Leet al. proposed a cyclic phase backpressure control policy with online estimation of turning fraction and measurement of queue size [21]. However, these backpressure-based traffic light control methods only determine the activating light phase according to the phase pressure computed using local information and neglect the possible coordination with adjacent intersections for phase pressure computation.In this paper, we propose a cooperative backpressure-based traffic control method, in which the phase pressure is computed by considering the phase state of downstream intersections. Actually, the phase pressure is influenced by the queue length on the downstream segment. When the vehicles queued in front of the downstream intersections obtain the right of way, the corresponding phase of the downstream intersection is activated, and the current phase pressure will increase due to the decrease of queued vehicles on downstream segment. These phase pressure changes may affect the choice of the activating light phase. Therefore, we propose a modified phase pressure computation method, which considers the phase state of downstream intersections to achieve cooperative light phase switching among intersections. In addition, for the cooperative light phase switching decision among intersections, the traffic light switching problem is viewed as a task assignment issue. Moreover, the consensus-based bundle algorithm (CBBA), which is usually utilized for decentralized task selection for a multiagent system [22], is introduced to achieve coordination among intersections.To summarize, there are two main innovations in this paper:(1) A modified phase pressure computation method is proposed considering the phase state of downstream intersections; this method is appropriate for cooperative light phase switching among intersections. (2) CBBA is introduced to solve the conflicts in cooperative light phase switching decision by viewing the cooperative traffic light switching as a cooperative task assignment problem.The remainder of this paper is organized as follows: In Section2, the urban traffic network is modeled as an agent-controlled queuing network. The phase pressure computation method is presented in Section 3. In Section 4, the CBBA based traffic light cooperative control algorithm is described. In Section 5, the stability of the proposed algorithm is analyzed. Simulations are carried out, and the results are discussed in Section 6. This paper is concluded in Section 7.


## 2. Traffic Network Modeling

The urban traffic network is modeled as an agent-controlled queuing network. Road segments are considered link nodes. There are 3 types of link nodes: ingress node, internal node, and exit node. Both ingress nodes and internal nodes have a downstream intersection. Vehicle flows entering a node generate vehicle queues when the corresponding light phase is red. The exit nodes, from which vehicles are leaving the traffic network, have no downstream intersection, and therefore do not generate queued vehicles. It assumes that vehicle flows entering internal nodes are coming completely from an upstream intersection.The intersections controlled by Smart Traffic Light Control Agents (STLCA) are viewed as junctions that connect link nodes. Although it is unnecessary to explain the light phase of each intersection under backpressure-based traffic control method used in this paper, to describe the topology of vehicle flows in traffic network clearly, it assumes that there are 4 light phases at each intersection: north-south straight phase, north-south left-turn phase, west-east straight phase, and west-east left-turn phase, as shown in Figure1. There are 3 dedicated lanes on each road segment, i.e., a left-turn lane, a straight lane, and a right-turn lane. It assumes that the vehicles entering a road segment will drive into the dedicated lane immediately and the right-turn vehicle flows are free up to the traffic light. Vehicle flow on the straight lane or left-turn lane is controlled by the corresponding light phases. For example, as shown in Figure 2, there are 12 vehicle flows passing through intersectioni. The right-turn flows {fr1,fr2,fr3,fr4} are not controlled by the traffic light of STLCAi, the straight flows {fs1,fs3} and {fs2,fs4} are controlled by west-east straight phase and north-south straight phase of intersectioni, and the left-turn flows {fl1,fl3}, {fl2,fl4} are controlled by west-east left-turn phase and north-south left-turn phase, respectively.Figure 1
The light phases of one intersection.Figure 2
The topology of queuing traffic network.Time is slotted for the queuing network control. LetAa(t) denote the exogenous arrivals from upstream intersection,Dab(t) represent the number of vehicles moving from segmenta tob, andTab(t) is the turn ratio of exogenous arrivals, that is, there areTab(t)Aa(t) vehicles add to queued vehiclesqab(t) during time slott. Then, queue lengthqab ona waiting for moving tob can be computed using (1). Shown as Figure 2, the exogenous arrivalAa(t) ofa comes from the three vehicle flows {fl4,fs1,fr2} of intersectioni-1.(1)qabt+1=qabt+TabAat-DabtIn the agent-controlled queuing network, STLCAi selects a light phase to activate with a given green time or extends the current activating light phase with a short green time according to the backpressure algorithm at each time slott. In this paper, we only consider the cooperative light phase switching strategy. The activating phase extension strategy will be researched in the future work.


## 3. Pressure Computation Method of Light Phase

In the traffic network, vehicle flows on segmenta, waiting to pass through the intersection, will generate traffic pressure to the downstream intersection. According to the original backpressure algorithm, the pressure of each light phasepressurep is defined as the sum of the pressure associated with all vehicle flows controlled by light phasep, computed using (2) and (3). The pressure associated with vehicle flowfab∈p is defined as the current flow rate, weighted bywab(t) (the difference betweenqa andqb), wherefab∈pindicates thatfab obtains the right of way whenp is activated [14, 23]:(2)wabt=qat-qbt(3)pressurep=∑fab∈pμabpwabtwhereqa(t) is the total number of vehicles waiting on segmenta at the beginning of slott andμab(p) is the number of vehicles passing through intersection per unit time whenp is activated. Ifp is not activated, thenμab(p) = 0.Only these vehicles queued on segmenta, waiting for moving to nodeb, generate pressure to the light phasep, instead of all the vehicles on segmenta. Therefore, in Varaiya’s paper [17], the weight wab(t) was modified as (4)wabt=qabt-∑crbcqbctwhererbc denotes the proportion of vehicle flow leavingb and enteringc and ∑crbcqbc(t) indicates the average queue length ofb.According to the backpressure-based traffic control algorithm [15], STLCAi chooses a phase with maximum pressure to activate in the next time slot using (5) to achieve maximum throughput of a single intersection.(5)p∗=argmaxp∈Pi⁡pressureptwherePi is the light phase set of intersectioni and p∗ is the selected phase to activate in the next time slot.However, when the vehicle queue on the downstream segments of an intersection is longer than the vehicle queue on an upstream segment, or when the two queue lengths are close (as shown in Figure3), according to the backpressure algorithm, phase p′ will be activated instead of phasepin the next time slot. In this situation, the two pairs of queue lengths (qa1,qb1) and (qa2,qb2) both have a small difference, and the two pairs of (qc1,qd1) and (qc2,qd2) have a larger difference. This results in the pressure of phasep being smaller than phase p′, and phase p′ with a smaller volume is to be activated in the next time slot. It is obvious that in this circumstance the backpressure-based phase switching strategy described above cannot achieve an ideal effect.Figure 3
The case with similar queue length for downstream and upstream links.Considering this situation, the cooperative traffic control strategy may be a good choice for some subareas of the traffic network. Based on this view, in this paper the phase pressure is calculated considering the phase state of downstream intersections to achieve a cooperative backpressure-based phase switching algorithm. Shown in Figure4(a), for the straight phase pij of intersectioni, there is fab∈pij. For the downstream intersectioni+1, there are fbc′∈pki+1 and fbc′′∈pk+1i+1. pij denotes thejth light phase of intersectioni. dbc′ and dbc′′ denote the number of vehicles departing fromb to c′ or c′′, respectively. If pjk or pk+1j is activated, then the queued vehicles onb will decrease, and the pressure of pij will increase. Then, we can conclude that the pressure of pij of intersectioni is affected by the phase state of the downstream intersectioni+1. A similar situation of the left-turn light phase of intersectioni is shown in Figure 4(b). According to the above-stated analysis, there are two cases for the phase switching coordination of adjacent intersections. Case 1 occurs when intersectioni is determining the activating phase of the next time slot, the corresponding downstream phase pki+1 or pk+1i+1 is still activating. Case 2 occurs when the two intersectionsi andi+1 are determining the activating phase at the same time. For these two cases, the phase pressure computation method of pij should be modified as (6)-(9). (6)wabt=qabt-∑c∈c′,c′′rbcqbct-dbct(7)pressurep=∑fab∈pμabpqabt-∑c∈c′,c′′rbcqbct-dbctwheredbc(t) is the number of vehicles departing fromb toc defined as (8), (9).(8)dbct=spktqbctforfbc∈pktifqbct<fmaxspktfmaxforfbc∈pktifqbct>=fmaxwhere(9)spkt=1ifpktisactivated0ifpktisnotactivatedwhere fmax is the maximum number of vehicles passing through the intersection when the downstream light phasepk is activated.Figure 4
Pressure of straight phase and left-turn phase of intersectioni affected by downstream intersectioni+1. (a) Straight phase of intersectioni affected by downstream light phase; (b) left-turn phase of intersectioni affected by downstream light phase.
(a) (b)Using the proposed method of phase pressure computation by considering the phase state of downstream intersections, the light phase is to be switched in cooperation mode. The light phase switching of adjacent intersections should be able to make a cooperative decision without conflicts. However, it is difficult to obtain the global optimal traffic control strategy among multiple intersections with a complex traffic state. CBBA, which is usually used for task assignment problems [22], is introduced to solve this problem in the next section.


## 4. Cooperative Light Phase Switching Decision Using CBBA

According to the idea of the backpressure traffic light control algorithm, the phase with maximum pressure will be selected to activate in the next time slot. However, the phase pressure computed using the proposed method is affected by the phase state of the downstream intersection. The selected phasepij of intersectioni may be in conflict with the phase pki+1 of downstream intersectioni+1. This is because the pressure of pij may be computed by assuming that phase pk+1i+1 is activated. To solve the conflicts of phase switching among intersections, the CBBA is introduced in this section.CBBA is a market-based distributed task assignment algorithm and has been shown to produce a conflict free solution [24, 25]. There are two stages in CBBA [22], i.e., the auction stage and the consensus stage. In the auction stage, each agent bids on a task and calculates a score based on the local information to decide whether it assigns a new task to the task bundle or not. The task bundle of agenti is defined as the possible task set with task score. If agenti does not win any tasks in the auction stage, the task bundle of agenti is empty. In the consensus stage, agents exchange the task information through communication. If an agent is outbid for a task, then it is released from the task bundle. For a task, the agent with the highest score is the winner. Based on the idea of CBBA, in the next section, the cooperative phase switching problem in urban traffic network is modeled as a task assignment problem.


### 4.1. Cooperative Light Phase Switching Modeled as Task Assignment

There are four light phases for the light controller of each intersection, which are managed by a smart agent, STLCA. The task of STLCA is to determine which phase is to be activated in the next time slot. For each intersection, there are two vehicle flows with opposite directions, controlled by each light phase. For each vehicle flow, there is one downstream intersection at most. If there is one downstream intersection for one vehicle flow, then there are three cooperative phase switching possibilities. Shown in Figure5, vehicle flowsf1 andf2 are controlled by phase pij. When pij is activated, the vehicles waiting on segmentsa1 anda2 obtain the right of way to pass through intersectioni. The pressure of pij is consists of two parts: the pressure fromf1 and the pressure fromf2. The pressure generated from vehicle flowf1 is associated with the cooperative phasescp11 andcp12. Ifcp11 orcp12 are activated, vehicles on segmentb1 can drive into segmentsc11 orc12. In this case, the number of vehicles departing from segmentb1 should be considered. Ifcp11 andcp12 are not to be activated or theDownstreamIntersection1 does not exist, the number of vehicles departing from segmentb1 is zero. Similarly, the computation of pressure generated from vehicle flowf2 should consider the downstream phase statescp21 orcp22. Therefore, when considering the downstream phase state, there are 9 possible cooperative phase switching strategies for phase pij, as shown in Figure 6. For each intersection, there are at most 36 possible cooperative phase switching.Figure 5
Coordination relationships of light phases among adjacent intersections.Figure 6
Nine cooperative phase switching possibilities of one light phase.In order to solve the cooperative phase switching problem using CBBA, these possible cooperative phase switching strategies are viewed as the task bundle of STLCAi. Each possible phase switching strategy is viewed as a possible task of intersectioni. The task score is represented by the phase pressure calculated using (6)-(9) according to the downstream phase state.


### 4.2. Cooperative Backpressure-Based Light Phase Switching Algorithm

In this section, the CBBA is utilized to solve the cooperative phase switching problem among intersections. According to the CCBA, the cooperative backpressure-based light phase switching algorithm consists of three steps as follows.(a) Task Bundle Construction. STLCAi constructs the task bundle according to its topology in the traffic network. The possible switching strategies of intersectioni are considered as the possible tasks of STLCAi. The amount of possible switching strategies for a particular intersection is fixed, since the topology of an intersection in urban traffic network is fixed. The task bundle of each intersection is constructed before phase switching decision making can improve the algorithm performance. Shown in Table 1, there are 9 possible phase switching strategies for one phase of intersectioni. Therefore, there are 36 possible switching strategies in one intersection at most.Table 1
The possible cooperative phase switching strategies ofpij.
Strategy Number Light phase DownstreamIntersection 1 CooperativePhase 1 DownstreamIntersection 2 CooperativePhase 2 1 p i j 0 × 0 × 2 p i j 1 cp 11 0 × 3 p i j 1 cp 12 0 × 4 p i j 0 × 1 cp 21 5 p i j 0 × 1 cp 22 6 p i j 1 cp 11 1 cp 21 7 p i j 1 cp 11 1 cp 22 8 p i j 1 cp 12 1 cp 21 9 p i j 1 cp 12 1 cp 22 Note: × indicates that the cooperative phase is not to be activated. 0 indicates that the downstream intersection does not exist or does not coordinate withpij. cp11, cp12, cp21, and cp22 are the cooperative phases of downstream intersection1 and downstream intersection2.(b) Phase Pressure Computation. STLCAi collects the queue length information of each segment and broadcasts the information to the adjacent intersections. The pressure of each strategy is calculated using (6)-(9) according to the local queue length information and queue length information from adjacent intersections. The possible strategies of intersectioni are ordered by the phase pressure, and all the possible strategies are set to be available at beginning. After the pressure computation, for each intersection, the strategy with maximum pressure is selected to be the candidate strategy, and the responding phase to be the candidate light phase for activating in the next time slot.(c) Activating Phase Decision and Conflicts Resolution. In this step, STLCAi determines the candidate phase as the activating phase of intersectioni at timet+1 directly if the candidate strategy need not coordinate with the neighboring downstream intersections. These intersections whose light phase has been determined are called activating-phase-determined intersections. If STLCAi has determined the activating phase, STLCAi broadcasts this information to its adjacent intersections. Then, the adjacent STLCAs of intersectioni release the strategies conflict with intersectioni. For example, if intersectioni has determined the activating light phase to be west-east straight phase, the strategies of neighboring intersections (if the neighboring intersections are not activating-phase-determined intersections) which are conflict with the activating light phase of intersectioni, will be released, that is, the invalid flag of these strategies are set to be true. For these activating-phase-undermined intersections, STLCAs elect a strategy with maximum pressure in distributed mode by communication. The elected intersection determines the activating phase and broadcasts it to adjacent intersections. Step (a) and (b) are repeated until all STLCAs have determined the activating phase. A detailed description of the algorithm is given in Algorithm 1.Algorithm 1:The algorithm of intersection i for activating phase decision and conflicts resolution.
IfSelectedStrategy.DownstreamIntersection1 = 0 And SelectedStrategy.DownstreamIntersection2 = 0 Then Intersectioni.DeterminedFlag = True Intersectioni.SelectedPhase = SelectedStrategy.phase Let the selected phase be the activating phase in the next time slot. Broadcast the decision ofIntersectioni to the adjacent intersections. End If IfIntersectioni.DeterminedFlag = False Then If STLCAi receives the decisions from adjacent intersections Then For eachAdjacentIntersection of Intersectioni IfAdjacentIntersectionj.DetermineFlag= True Then Release the strategies ofIntersectioni conflicting with AdjacentIntersectionj.SelectedPhase End If Next Reselect theIntersectioni.SelectedStrategy with maximum pressure from the available strategies BroadcastIntersectioni.SelectedStrategy to the adjacent intersections. End If Elect the maximum pressure strategy among the available strategies of all intersections using distributed election algorithm. IfIntersectioni.SelectedStrategy is the elected strategy with maximum pressure Then Intersectioni. DeterminedFlag = True Intersectioni. SelectedPhase = SelectedStrategy.phase Let the selected phase be the activating phase in the next time slot. Broadcast the decision ofIntersectioni to the adjacent intersections. End If End IfBased on the three steps, the cooperative backpressure-based traffic control method (simplified as CBP method) can obtain a conflict free phase switching strategy with maximum phase pressure. The backpressure-based traffic light control method (simplified as BP method) only considers the queue length of the current intersection and neglects the decrease of queued vehicles on the downstream segment when the corresponding downstream phase is activated. The CBP method fixes this deficiency and considers the phase state of downstream intersections to achieve coordination among intersections. Furthermore, all the switching possibilities based on BP method (that is, the strategies withDownstreamIntersection1 = 0 andDownstreamIntersection1 = 0) are contained in the task bundle of each intersection in CBP method. In other words, the CBP method considering downstream phase state can obtain equal or greater traffic performance compared to the BP method. To illustrate the feasibility of the proposed CBP method, the stability is discussed in the next section.


## 4.1. Cooperative Light Phase Switching Modeled as Task Assignment

There are four light phases for the light controller of each intersection, which are managed by a smart agent, STLCA. The task of STLCA is to determine which phase is to be activated in the next time slot. For each intersection, there are two vehicle flows with opposite directions, controlled by each light phase. For each vehicle flow, there is one downstream intersection at most. If there is one downstream intersection for one vehicle flow, then there are three cooperative phase switching possibilities. Shown in Figure5, vehicle flowsf1 andf2 are controlled by phase pij. When pij is activated, the vehicles waiting on segmentsa1 anda2 obtain the right of way to pass through intersectioni. The pressure of pij is consists of two parts: the pressure fromf1 and the pressure fromf2. The pressure generated from vehicle flowf1 is associated with the cooperative phasescp11 andcp12. Ifcp11 orcp12 are activated, vehicles on segmentb1 can drive into segmentsc11 orc12. In this case, the number of vehicles departing from segmentb1 should be considered. Ifcp11 andcp12 are not to be activated or theDownstreamIntersection1 does not exist, the number of vehicles departing from segmentb1 is zero. Similarly, the computation of pressure generated from vehicle flowf2 should consider the downstream phase statescp21 orcp22. Therefore, when considering the downstream phase state, there are 9 possible cooperative phase switching strategies for phase pij, as shown in Figure 6. For each intersection, there are at most 36 possible cooperative phase switching.Figure 5
Coordination relationships of light phases among adjacent intersections.Figure 6
Nine cooperative phase switching possibilities of one light phase.In order to solve the cooperative phase switching problem using CBBA, these possible cooperative phase switching strategies are viewed as the task bundle of STLCAi. Each possible phase switching strategy is viewed as a possible task of intersectioni. The task score is represented by the phase pressure calculated using (6)-(9) according to the downstream phase state.


## 4.2. Cooperative Backpressure-Based Light Phase Switching Algorithm

In this section, the CBBA is utilized to solve the cooperative phase switching problem among intersections. According to the CCBA, the cooperative backpressure-based light phase switching algorithm consists of three steps as follows.(a) Task Bundle Construction. STLCAi constructs the task bundle according to its topology in the traffic network. The possible switching strategies of intersectioni are considered as the possible tasks of STLCAi. The amount of possible switching strategies for a particular intersection is fixed, since the topology of an intersection in urban traffic network is fixed. The task bundle of each intersection is constructed before phase switching decision making can improve the algorithm performance. Shown in Table 1, there are 9 possible phase switching strategies for one phase of intersectioni. Therefore, there are 36 possible switching strategies in one intersection at most.Table 1
The possible cooperative phase switching strategies ofpij.
Strategy Number Light phase DownstreamIntersection 1 CooperativePhase 1 DownstreamIntersection 2 CooperativePhase 2 1 p i j 0 × 0 × 2 p i j 1 cp 11 0 × 3 p i j 1 cp 12 0 × 4 p i j 0 × 1 cp 21 5 p i j 0 × 1 cp 22 6 p i j 1 cp 11 1 cp 21 7 p i j 1 cp 11 1 cp 22 8 p i j 1 cp 12 1 cp 21 9 p i j 1 cp 12 1 cp 22 Note: × indicates that the cooperative phase is not to be activated. 0 indicates that the downstream intersection does not exist or does not coordinate withpij. cp11, cp12, cp21, and cp22 are the cooperative phases of downstream intersection1 and downstream intersection2.(b) Phase Pressure Computation. STLCAi collects the queue length information of each segment and broadcasts the information to the adjacent intersections. The pressure of each strategy is calculated using (6)-(9) according to the local queue length information and queue length information from adjacent intersections. The possible strategies of intersectioni are ordered by the phase pressure, and all the possible strategies are set to be available at beginning. After the pressure computation, for each intersection, the strategy with maximum pressure is selected to be the candidate strategy, and the responding phase to be the candidate light phase for activating in the next time slot.(c) Activating Phase Decision and Conflicts Resolution. In this step, STLCAi determines the candidate phase as the activating phase of intersectioni at timet+1 directly if the candidate strategy need not coordinate with the neighboring downstream intersections. These intersections whose light phase has been determined are called activating-phase-determined intersections. If STLCAi has determined the activating phase, STLCAi broadcasts this information to its adjacent intersections. Then, the adjacent STLCAs of intersectioni release the strategies conflict with intersectioni. For example, if intersectioni has determined the activating light phase to be west-east straight phase, the strategies of neighboring intersections (if the neighboring intersections are not activating-phase-determined intersections) which are conflict with the activating light phase of intersectioni, will be released, that is, the invalid flag of these strategies are set to be true. For these activating-phase-undermined intersections, STLCAs elect a strategy with maximum pressure in distributed mode by communication. The elected intersection determines the activating phase and broadcasts it to adjacent intersections. Step (a) and (b) are repeated until all STLCAs have determined the activating phase. A detailed description of the algorithm is given in Algorithm 1.Algorithm 1:The algorithm of intersection i for activating phase decision and conflicts resolution.
IfSelectedStrategy.DownstreamIntersection1 = 0 And SelectedStrategy.DownstreamIntersection2 = 0 Then Intersectioni.DeterminedFlag = True Intersectioni.SelectedPhase = SelectedStrategy.phase Let the selected phase be the activating phase in the next time slot. Broadcast the decision ofIntersectioni to the adjacent intersections. End If IfIntersectioni.DeterminedFlag = False Then If STLCAi receives the decisions from adjacent intersections Then For eachAdjacentIntersection of Intersectioni IfAdjacentIntersectionj.DetermineFlag= True Then Release the strategies ofIntersectioni conflicting with AdjacentIntersectionj.SelectedPhase End If Next Reselect theIntersectioni.SelectedStrategy with maximum pressure from the available strategies BroadcastIntersectioni.SelectedStrategy to the adjacent intersections. End If Elect the maximum pressure strategy among the available strategies of all intersections using distributed election algorithm. IfIntersectioni.SelectedStrategy is the elected strategy with maximum pressure Then Intersectioni. DeterminedFlag = True Intersectioni. SelectedPhase = SelectedStrategy.phase Let the selected phase be the activating phase in the next time slot. Broadcast the decision ofIntersectioni to the adjacent intersections. End If End IfBased on the three steps, the cooperative backpressure-based traffic control method (simplified as CBP method) can obtain a conflict free phase switching strategy with maximum phase pressure. The backpressure-based traffic light control method (simplified as BP method) only considers the queue length of the current intersection and neglects the decrease of queued vehicles on the downstream segment when the corresponding downstream phase is activated. The CBP method fixes this deficiency and considers the phase state of downstream intersections to achieve coordination among intersections. Furthermore, all the switching possibilities based on BP method (that is, the strategies withDownstreamIntersection1 = 0 andDownstreamIntersection1 = 0) are contained in the task bundle of each intersection in CBP method. In other words, the CBP method considering downstream phase state can obtain equal or greater traffic performance compared to the BP method. To illustrate the feasibility of the proposed CBP method, the stability is discussed in the next section.


## 5. Stability Analysis

In this section, the stability of the proposed CBP method is analyzed. As described in references [26] and [14], for a network with queue vector U={U1,U2,…,UN}, a sufficient condition for stability can be provided using Lyapunov drift, which is given as below.Lemma 1.
SupposeE{Ui(t)}<∞ for all i∈{1,2,…,N}, and there exist constantsB >0 andε>0 which satisfies(10)ELUt+1-LUt∣Ut≤B-ε∑i=1NUitthen the network is stable, where the Lyapunov function is defined as(11)LU=∑i=1NUi2To describe simplicity, define the functionVabin(p(t)) and Vabout(p(t)) and denote the vehicles enteringqab(t) and vehicles departing fromqab(t) under the current light phase switching strategyP(t) during slott.(12)VaboutPt=∑fab∈pitμabpitwherepi(t) is the activated light phase of intersectioni, and vehicle flowfab is controlled bypi(t);b is the downstream node ofa.(13)VabinPt=∑fca∈pi-1tμcapi-1twherepi-1(t) is the activated light phase of intersectioni-1 and vehicle flowfca is controlled bypi-1(t);c is the upstream node ofa.Then, (1) is rewritten as(14)qabt+1=qabt+VabinPt-VaboutPtAccording to the Lyapunov function and (14), Vabin(P(t)) and Vabout(P(t)) are simplified as Vabin and Vabout, we obtain(15)LUt+1-LUt=∑a,bVabin2+Vabout2-2∑a,bqabtVabout-Vabin-2∑a,bVaboutVabin=∑a,bVabin2+Vabout2-2∑a,bqabtVabout-Vabin+∑a,bVaboutVabinFor queue network in this paper, it assumes that the arrival vehicles of ingress node are all come from the upstream nodes. For example, shown in Figure 7, there are two vehicle queues {qbc1,qbc2} on nodeb; the arrival vehicle Vbc1in is computed using Vbc1in=rbc1Vabout, whererbc1 is the proportion of vehicles enteringqbc1 fromqab when the corresponding light phase is activated. Similarly, there is Vbc2in=rbc2Vabout. Furthermore, for the nodeb, under a given light phase strategyP(t), one of Vbc1out and Vbc2out is zero at least.Figure 7
The relationship of arrival and departing vehicle flow in traffic network.Based on these properties of the traffic network, the right term of (15) is expanded as(16)∑a,bqabtVabout-Vabin+∑a,bVaboutVabin=qabtVabout-qabtVabin+qbc1tVbc1out-qbc1tVbc1in+qbc2tVbc2out-qbc21tVbc2in+…+VabinVabout+Vbc1inVbc1out+Vbc2inVbc2out+…=qabtVabout-qbc1trbc1Vabout-qbc2trbc2Vabout-qabtVabin+…+VabinVabout+rbc1VaboutVbc1out+rbc2VaboutVbc2out+…=qabtVabout-qbc1trbc1Vabout-qbc2trbc2Vabout+rbc1VaboutVbc1out+rbc2VaboutVbc2out+-qabtVabin+VabinVabout_+…For the term underline-marked in the above equation, -qab(t)Vabin+VabinVabout,a is an ingress node. It assumes that nodex is a virtual node withqx=0, Vxout is the vehicle input on nodea, andrab is the proportion of Vxout for vehicle enteringqab during time slott. Then, this term can be rewritten as follows:(17)-qabtVabin+VabinVabout=qxtVxout-qabtrabVxout+rabVxoutVaboutSimilarly, the queues in the exit nodes also can be expressed as the similar forms. Therefore, the right term of (15) can be rewritten as(18)∑a,bqabtVabout-Vabin+∑a,bVaboutVabin=∑a,bqabtVabout-qbc1trbc1Vabout-qbc2trbc2Vabout+rbc1VaboutVbc1out+rbc2VaboutVbc2out=∑a,bqabt-qbc1trbc1-qbc2trbc2+rbc1Vbc1out+rbc2Vbc2outVabout=∑a,bqabt-qbc1trbc1+qbc2trbc2-rbc1Vbc1out-rbc2Vbc2outVabout=∑a,bqabt-∑cqbctrbc-rbcVbcoutVabout=∑a,bqabt-∑crbcqbct-VbcoutVaboutThen,L(U(t+1))-L(U(t)) can be expressed as(19)LUt+1-LUt=∑a,bVabin2+Vabout2-2∑a,bqabt-∑crbcqbct-VbcoutVaboutLet B=∑a,bsup⁡(Vabin)2+sup⁡(Vabout)2, then there is(20)LUt+1-LUt≤B-2∑a,bqabt-∑crbcqbct-VbcoutVaboutIn the above equation, Vbcout is the departing vehicles fromqbc, and if the corresponding downstream light phase of nodea is not activated, Vbcout will be zero. Here Vbcout is same todbc in (6). Inject (6), (12) into (17); we obtain(21)LUt+1-LUt≤B-2∑a,b∑fab∈ptμabptwabtSince the queuing network modeled in this paper is similar to the model in reference [14, 16], there are similar properties:(22)∑a,bqabtVaboutPt-VabinPt=∑a,b∑fab∈ptμabptqabt-∑crbcqbctFrom (22), the following equation can be deduced:(23)∑a,bqabtVaboutPt-VabinPt=∑a,b∑fab∈ptμabptqabt-∑crbcqbct≤∑a,b∑fab∈ptμabptqabt-∑crbcqbct-dbct≤∑a,b∑fab∈ptμabptwabtIt means that, under the same given light phase switching strategyP(t), if the downstream phase is coordinated with the phase of current intersection, it achieves equal or better throughput. Ifdbc(t) is zero, the CBP method is degenerated to the BP method. When the downstream light phase is coordinated with the current light phase, it achieves better throughput.According to (21) and (23), we obtain(24)LUt+1-LUt≤B-2∑a,b∑fab∈ptμabptwabt≤B-2∑a,bqabtVaboutp-VabinpIn additional, in [14], it has been proved that, for the admissible arrival rate λ in capacity region Λ, and ε >0, there is(25)EVaboutPt-VabinPt∣Ut=λab+εThen, we obtain(26)ELUt+1-LUt∣Ut≤B-2ε∑abqabtAccording toLemma1, the network controlled by the proposed CBP method is stable under the admissible arrival rate λ in capacity region Λ. To illustrate the effectiveness of the proposed CBP method, simulations are conducted, and a description is given in the following section.


## 6. Simulation and Discussion

Three traffic light control methods are implemented, including fixed-time control, BP control, and CBP control. The BP method implemented in simulations is proposed by Varaiya [16]. Although there are several improved versions of backpressure-based traffic control methods, these methods do not consider the coordination of light phase switching among neighboring intersections. The CBP control method considering the downstream light phase state is proposed on the foundation of basic backpressure-based traffic control method proposed by Varaiya. Therefore, in this section we compare the three traffic control methods. In the further research work, based on these improved versions the coordination, should be considered.The simulation traffic network consists of 15 intersections and 76 links constructed in Vissim. The network includes 16 ingress links and 16 exit links, as shown in Figure8(a). The vehicle input of ingress links is 800veh/h. There are 15 signal controllers and 4 light phases for each controller, i.e., north-south straight, north-south left-turn, west-east straight, and west-east left-turn. There are 3 lanes on each link. The left-turn vehicle flow drives on Lane 3 and the straight vehicle flow drives on Lane 2. The two vehicle flows are controlled by a traffic light. The right-turn vehicle flow drives on Lane 1 and is free of the traffic light. To obtain the traffic parameters during simulation, 60 queue counters, 60 travel time sections, and 60 routing decisions are laid on the simulation traffic network, as shown in Figures 8(b), 8(c), and 8(d). During the simulation, the green light time of each light phase is assumed to be 21s, which is the pedestrian clearance time (the road width is about 21m, and the pedestrians speed is about 1m/s) [4]. The yellow light time is assumed to be 3s.Figure 8
Topology and settings of simulation traffic network. (a) The topology of simulation traffic network; (b) route decisions setting; (c) travel time sections setting; (d) queue Counters setting.
(a) (b) (c) (d)The three traffic light control methods are implemented in Visual Studio 2010. The simulation programs communicate with Vissim through the Vissim COM programming interface to obtain the traffic parameters and decide the traffic light signal at each time slot. The simulation runs for 7,200s, and the traffic network performance is evaluated from 1,000s to 7,200s using Vissim. This is because in the first 1,000s of simulation time there are not enough vehicles entering the simulation traffic network.The simulation results of traffic network performance are given in Table2. The results show that under similar traffic volume condition, the fixed-time control algorithm results in a higher delay time and travel time due to the lower average speed. Vehicles stopping in front of intersections have to wait for the next cycle to get the right of way to pass through the intersection, which results in an increase in the vehicle delay time. Although the BP method can switch light signal according to the pressure of each light phase, it cannot deal well with the situation described in Figure 3. This kind of situation results in an increase of vehicle delay, because the vehicles ona1 anda2 cannot be released in time. The CBP method solves this problem well. From Table 2, we can conclude that the CBP control can obtain better traffic network performance than the other two methods.Table 2
Traffic network performance comparison.
Parameters Fixed-Time Method BP Method CBP Method Total travel time (h) 1494.11 1440.434 1332.647 Total delay time (h) 824.652 761.403 648.577 Number of Stops 120850 70359 68554 Total stopped delay (h) 593.744 587.938 482.059 Average delay time (s) 133.379 123.199 105.175 Average number of stops 5.43 3.162 3.088 Average stopped delay (s) 96.032 95.131 78.172 Average speed (km/h) 23.549 24.761 26.965 Number of vehicles in network 938 880 806 Number of vehicles left network 21320 21369 21394 Total Number of vehicles 22258 22249 22200To further illustrate the effectiveness of the CBP method, the average delay, average stop delay, average queue length, and the maximum queue length during the simulation are extracted from the Vissim evaluation files and listed in Figures9, 10, 11, and 12.Figure 9
Average vehicle delay during simulation.Figure 10
Average stop delay during simulation.Figure 11
Average queue length during simulation.Figure 12
Maximum queue length during simulation.Figure9 displays the average delay of the three methods during simulation time. The delay time of fixed-time method is larger than the other two methods since each light phase is activated in same time interval in the fixed-time control, and the segments with longer queue length cannot get the right of way in priority. The BP and CBP methods have a close average vehicle delay, because these two methods have similar light phase switching strategy by selecting the light phase with the maximum pressure for activation. The CBP method considers the phase state of downstream intersections and achieved coordination among intersections. It can effectively speed up the release of queued vehicles. Therefore, the CBP method obtains a smaller average delay during simulation.For the average stop delay, shown in Figure10, the CBP method shows a smaller average stop delay during simulation than the other two methods. By considering the phase state of downstream intersections, the CBP method achieves coordination among intersections in a distributed pattern, which releases more vehicles due to the cooperative light phase switching among adjacent intersections.During the simulation, there is an average of three queue lengths increase in the first 1,000s with the increase of simulation time, as shown in Figure11. Under the same vehicle input, the average queue length of the CBP method generates a smaller average queue length in the traffic network. Figure 11 illustrates that the cooperative backpressure-based traffic light control method can speed up the queued vehicle releasing.From Figure12, according to the curve of the fixed-time traffic light control method, queue lengths of some segments in the traffic network reaches the maximum value at about 2,600s. The queue length of the BP method reaches the maximum value at about 6,600s. The queue length of the CBP method does not reach the maximum value in the entire simulation time. This simulation result indicates that the fixed-time control method is more likely to lead to queue spillover in the segments whose queue length reaches the maximum value. The queue spillover of some segments may result in traffic congestion propagation or a traffic collapse. This is a seriously negative effect to urban traffic. Although the time of the queue length for reaching the maximum value of the BP method is later than the fixed-time control method, it still reaches the maximum queue length after all. As shown in Figure 12, the queue lengths of all the segments do not reach the maximum value during the CBP method simulation. This result indicates that the CBP method can obtain better performance in the traffic network.Based on the above discussion, the proposed traffic control method performs better than the other two methods in the simulations. However, the CBP method requires communication among intersections due to the distributed phase switching decision. This results in an increase of the communication cost. With the development of an intelligent transport system, the communication problem in the proposed method can be solved completely. Therefore, this proposed cooperative traffic light control method may be an useful attempt for improving the efficiency of the urban traffic system.


## 7. Conclusion

In this paper, a cooperative backpressure-based traffic light control method is proposed to improve the efficiency of urban traffic network. The traffic network is modeled as a queuing network, in which the light phases of each intersection are controlled by STLCA. The light phase pressure is computed using the proposed light phase pressure computation method considering the phase state of downstream intersections, which is the basis for cooperative light phase switching decision among intersections. In the cooperative light phase switching decision, the CBBA is introduced to solve the conflicts in the switching strategies of adjacent intersections in a distributed mode. Simulations show that the proposed method obtains better performance than the original backpressure-based traffic light control method and fixed-time traffic light control method.However, there are further works to be done in the future. Firstly, the pressure computation should be improved to consider other factors such as the capacity of the downstream segment, the priority of queued vehicles, and the number of passengers in queued vehicles. These factors may influence the light phase switching decision in different scenarios. Secondly, for simplicity, the cooperative backpressure-based traffic light control method only considers situations of synchronous traffic light switching. In reality, traffic lights do not switch at the same time, and there are also different green times at different intersections. Actually, some group-based traffic control methods have considered this problem, in these methods the current light phase can be decided to extend the green time or finish it according to the max-pressure signal control policy [27, 28]. In the future work, on the foundation of these methods the cooperative light phase switching method can be further improved for better traffic network performance. Thirdly, the physical traffic model should be designed in the future research work to achieve the actual application in the urban traffic network.



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*Source: 7481489-2019-03-05.xml*