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Urban road networks may benefit from left turn prohibition at signalized intersections regarding capacity, for particular traffic demand patterns. The objective of this paper is to propose a method for minimizing the total travel time by prohibiting left turns at intersections. With the flows obtained from the stochastic user equilibrium model, we were able to derive the stage generation, stage sequence, cycle length, and the green durations using a stage-based method which can handle the case that stages are sharing movements. The final output is a list of the prohibited left turns in the network and a new signal timing plan for every intersection. The optimal list of prohibited left turns was found using a genetic algorithm, and a combination of several algorithms was employed for the signal timing plan. The results show that left turn prohibition may lead to travel time reduction. Therefore, when designing a signal timing plan, left turn prohibition should be considered on a par with other left turn treatment options.

Left turns at signalized intersections may cause efficiency problems, because they have comparatively high potential for conflicts with other movements. When permitted left turns are used, the delay of vehicles is determined both by the traffic signal and by the opposing vehicles. As a result, vehicles turning left would usually wait longer than other movements. Protected left turns can be applied at the expense of increasing the intergreen time and decreasing the effective green time, resulting in efficiency problems. Therefore, to improve efficiency, left turn prohibition (LTP) is investigated in urban networks as a remedy for urban congestion problems.

Previous left turn treatment guidelines provide the treatment of permitted/protected left turns at signalized intersections [

The methods for signal timing plan design are the stage-based method [

An example of two stages sharing movements [

This paper presents a method for selecting left turns with a genetic algorithm for prohibition and designing a fixed signal timing plan including LTP for a time in a day, by minimizing total travel times. After prohibiting left turns, the proposed method adjusts signal settings using the stage-based method. Once the selected left turns are prohibited, all stages are regenerated and then the stage sequence is optimized. Besides, cycle length and green duration are calculated. The intersections in networks are coordinated by the common cycle length, which is the largest cycle length of all intersections. Stochastic user equilibrium (SUE) is integrated for forecasting redistributed traffic flows, and it is used for the evaluation of the LTP. To test the proposed method, one should set the total travel time without LTP as a base case and compare it with the total travel time with LTP.

The following abbreviations are used in this paper: LTP represents left turn prohibition; SUE means stochastic user equilibrium; OD indicates Origin-Destination; TSP stands for traveling salesman problem.

The methodology consists of three parts: selecting the prohibited left turns, signal timing plan design including permitted/protected/prohibited left turns for each intersection, and an SUE used for a realistic assignment of the demand.

The flowchart in Figure

Flowchart of signal timing plan design including left turn prohibition.

The following sections explain each processing box and subprocessing step of the flowchart in detail.

Each network includes many left turns. Each left turn has two possible states: it can be either allowed or prohibited. An LTP combination is defined as the decision regarding the state of each left turn in the network. Considering the number of left turns in the network, the total number of LTP combinations can be very large. Therefore, using the algorithm detailed in Section

The overall objective is to minimize the total travel time in the network as described in (

For each selected prohibited left turn, we assumed that the lane would be used for through movement. We also made sure that each selected left turn can be prohibited. Specifically, a check was performed that the prohibition of this turn would still allow network connectivity. Thus each OD path set includes at least one path (see (

A traffic assignment model describes how traffic flows are distributed in the network. In SUE, one of the traffic assignment models, drivers choose routes with minimal perceived travel time. The relevant link flows change according to the result of path choice and influence the travel time. Drivers iteratively react to updated travel time until any change of routes makes the perceived travel time longer.

Link flow calculation and travel time estimation are critical steps in SUE. Link flow is calculated by the product of OD demand and probabilities of paths being chosen. The probabilities are calculated with a simple logit model in which the utility function is path travel time. In this research, we applied the simple logit model with the same configuration as Tang and Friedrich [

Link travel times are initialized by applying free flow travel time, and then travel time is differently estimated before and after signal timing plan design. Before signal timing plan is designed, as no signal timing information is available, the link costs are updated by BPR function (see (

If link

This section first introduces the method of conflict matrix determination according to left turn phasing types. With the conflict matrix, the stages can be generated and their sequences can be optimized. To estimate the delays, one should also calculate the lane flow and adjust the saturation flow. Finally the signal timing is calculated, especially for the stages sharing movements.

Conflict matrix records conflicts between movements which cannot be in the same signal stage for safety reasons. If left turns are protected, they will not be on the same stage as opposing through movements; if they are permitted, they can be on the same stage as opposing through movements. Thus, types of left turn phasing affect conflict matrix and saturation flow of left turns. As the permitted left turn vehicles are interrupted by opposing through movements, the saturation flows decrease compared with the saturation flows of protected left turns.

To determine whether left turns need to be protected, one can apply recommendations from different guidelines. However, most guidelines focus on safety aspects [

if

if

if

Here

Please note that when a left turn is permitted according these conditions, the left turn could be either permitted or protected. In the conflict matrix, if two movements do not conflict, they still could be in the different stages. For the same reason, the permitted left turn and the opposing through movement could be in the different stage, so that the left turn is protected in this case, and when calculating the saturation flow of the left turn, one should regard the left turn as protected left turn. The final left turn phasing type is determined according to the generated stages.

When generating the stages, one of the conventions is to include all nonconflict movements in the same stage (compatibility). In this regard, a feasible set would be one which contains different stages with nonconflict movements. It is equivalent to the problem of finding all subgraphs in graph theory. The adjacent matrix indicating the compatibility among movements is generated from the conflict matrix. If the value of the conflict matrix is 1, two movements conflict; if the value is 0, the movements are not in conflict. Respectively, if the value in the adjacent matrix is 1, meaning that both movements are compatible, they can be in the same stage; if this value is 0, the two movements are incompatible and therefore cannot be in the same stage, as shown in Figure

An example of stage generation: (a) lane configuration; (b) obtaining conflict matrix; (c) generation of adjacent matrix; (d) generation of feasible set by finding subgraphs; (e) selecting stages by minimizing the number of stages.

Once we have all possible stages by finding all subgraphs (see Figure

When the stages are generated, one movement may be shared by multiple stages. For example, in the numerical example of Memoli et al. [

Generally the most favorable stage sequence is determined by the total necessary intergreen times, which leads to shortest cycle time [

An example of stage sequence optimization: (a) obtaining adjacent matrix; (b) building intergreen time matrix; (c) obtaining generated stages; (d) generating sum of intergreen time matrix; (e) determining stage sequence by solving traveling salesman problem.

If one movement appears in multiple sequential stages, the signal of the movement can keep green in the intergreen time between stages; i.e., the intergreen time is 0. The movement thereby has extra green time and capacity. However, if these stages are not sequential, the extra green time cannot be gained. Therefore, when determining the stage sequence, one should consider the potential gain of extra green time. In this paper, the stage sequence optimization problem is represented as the traveling salesman problem (TSP), where the sequence of stages is equivalent to the visited cities and the minimum total intergreen time is equivalent to the minimum distance (see Figure

Denote

Set formally, for intersection

For example, in Figure

As one movement may occupy multiple lanes, drivers have to select one lane for turning preparation. To well adjust signal timing, it is necessary to calculate the number of vehicles turning a direction on each lane, i.e., the assigned flow. Lane flow is the sum of assigned flows of each movement.

Denotations

If a movement

If movement

By solving linear equations in (

Consequently, the lane flow can be calculated via summing the assigned flow of each movement on the lane:

In Figure

The saturation flow of permitted left turns per lane is adjusted by estimating the filtered saturation flow based on gap acceptance theory and adjusting the saturation flow of an exclusive lane [

The saturation flow of movements on the shared lane is weighted according to the flow of each movement [

The formulas of signal timing calculation are originally deduced by Webster [

To calculate cycle length, the flow ratio, which is the flow to the saturation flow, for each stage must be determined. As green times are determined by the lane with maximum flow in a stage, for all lanes in the same stage, the maximum flow ratio of lanes is the flow ratio of the stage (see (

For all

Cycle length must be greater than or equal to the minimum cycle length and not larger than the maximum cycle length. Thus, if

Due to the constraints of minimum green times, if the green time of stage

Cycle lengths are firstly calculated for each intersection, and the largest cycle length is decisive as the common cycle length for the network. Green times are then recalculated with the common cycle length by (

As mentioned, to calculate signal timing, the flows of movements/lanes in each stage should be known. However, if one movement is in two stages, it is hard to separate the movement flows into two stages beforehand and one cannot calculate the flow ratio. This case is neglected by HBS [

The rule to deal with the stages sharing movements is that green times are determined by the lane with the maximum flow. When calculating the green time of stages sharing movements, two cases need to be considered: either the required green time for the movement shared in multiple stages is bigger than the required green time for the other movements included in these stages or this green time is smaller. The final green time will be the larger one of these two cases. For example, in Figure

Pohlmann [

Examples of three stages sharing movements.

In “prohibiting left turn” of Figure

SUE is solved with the Method of Successive Average [

For signal-relevant steps in Figure

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The test network is an artificial network with six signalized intersections, ten origins, and ten destinations (Figure

Demands between origins and destinations (unit: veh/h).

Origin/Destination | A | B | C | D | E | F | G | H | I | J | Total |
---|---|---|---|---|---|---|---|---|---|---|---|

A | 0 | 70 | 90 | 160 | 100 | 60 | 150 | 50 | 180 | 200 | 1060 |

B | 200 | 0 | 130 | 60 | 140 | 70 | 90 | 130 | 80 | 110 | 1010 |

C | 180 | 90 | 0 | 180 | 90 | 160 | 70 | 80 | 120 | 70 | 1040 |

D | 40 | 70 | 140 | 0 | 80 | 120 | 60 | 50 | 150 | 80 | 790 |

E | 160 | 130 | 70 | 140 | 0 | 90 | 140 | 150 | 50 | 50 | 980 |

F | 90 | 80 | 110 | 80 | 170 | 0 | 50 | 100 | 120 | 50 | 850 |

G | 60 | 70 | 200 | 150 | 80 | 80 | 0 | 90 | 140 | 100 | 970 |

H | 90 | 80 | 40 | 110 | 200 | 70 | 160 | 0 | 80 | 120 | 950 |

I | 60 | 80 | 100 | 120 | 50 | 100 | 90 | 200 | 0 | 90 | 890 |

J | 90 | 180 | 40 | 70 | 0 | 120 | 60 | 90 | 100 | 0 | 750 |

Total | 970 | 850 | 920 | 1070 | 910 | 870 | 870 | 940 | 1020 | 870 | 9290 |

Values of parameters.

Parameters | Notations | Values |
---|---|---|

Parameter in BPR function | | 0.15 |

Parameter in BPR function | | 4 |

Number of exit lanes | | 3 |

Saturation flow of through movements | | 1900 veh/h |

Saturation flow of right turns | | 1615 veh/h |

Saturation flow of protected left turns | | 1805 veh/h |

Saturation flow of permitted left turns | | by Eq. ( |

Maximum cycle length | | 60 s |

Minimum cycle length | | 100 s |

Minimum green duration | | 6 s |

Intergreen time | | 4 s |

Observation time | | 0.25 h |

Layout of “toy” network with six intersections.

With the OD matrix and test network, 3 of 24 left turns are prohibited: the left turn in the northern arm at Intersection 2, the left turn in the southern arm at Intersection 3, and the left turn in the western arm at Intersection 6. Before LTP, the total travel time is 564.9 h whereas, after LTP, the minimal total travel time is 435.5 h with reduction of 22.9%. Prohibiting left turns reduces the total travel time. The average degrees of saturation for each intersection before and after LTP are also compared (see Figure

Average degree of saturation for each intersection before and after LTP.

Signal settings are also adjusted before and after LTP. The common cycle lengths before LTP and after LTP are 100 s. The stage generation results and their sequences before and after LTP are shown in Figure

Green durations of stages before and after LTP (s).

Green durations of movements before LTP

Intersection | Stage 1 | Stage 2 | Stage 3 | Stage 4 |
---|---|---|---|---|

1 | 38 | 27 | 23 | |

2 | 34 | 22 | 14 | 14 |

3 | 33 | 11 | 24 | 16 |

4 | 27 | 18 | 12 | 27 |

5 | 23 | 16 | 26 | 19 |

6 | 22 | 17 | 22 | 23 |

Green durations of movements after LTP

Intersection | Stage 1 | Stage 2 | Stage 3 | Stage 4 |
---|---|---|---|---|

1 | 35 | 28 | 25 | |

2 | 22 | 18 | 22 | 22 |

3 | 15 | 19 | 17 | 33 |

4 | 26 | 21 | 11 | 26 |

5 | 27 | 17 | 20 | 20 |

6 | 30 | 27 | 31 |

(a) Generated stages of test network and their sequence before LTP; (b) generated stages of test network and their sequence after LTP.

The algorithms of the proposed model are evaluated with increasing scale of networks. Table

Network scale information for algorithm evaluation.

Network | Layout | Number of intersections | Number of nodes | Number of links |
---|---|---|---|---|

1 | | 6 | 68 | 96 |

2 | | 7 | 76 | 122 |

3 | | 8 | 84 | 138 |

4 | | 9 | 88 | 140 |

5 | | 10 | 96 | 156 |

6 | | 11 | 104 | 172 |

7 | | 12 | 108 | 174 |

Computing time with increasing scale of networks.

The flows of prohibited left turn and their types are analyzed. LTP is related to left turn flows according to previous research. Hajbabaie et al. [

The flows and types of left turns without LTP and the LTP results from running the proposed model are collected via testing eight OD matrices. The OD matrices are generated by the demands in Table

The results are displayed in Figure

(a) Flow comparison between permitted left turns and prohibited/permitted left turns; (b) flow comparison between protected left turns and prohibited protected left turns.

The goal of this paper is to design signal timing plane including prohibiting left turns. The numerical example shows that prohibiting left turns reduces the total travel time. This study could potentially lead to useful insights regarding congestion management.

The main reason why prohibiting left turns can reduce the total travel time is that LTP reduces the number of conflict points at intersections. Due to the reduction of the conflict points, the number of stages decreases so that the intergreen time between two stages decreases (e.g., Intersection 6). If the intergreen time between two stages is longer, e.g., 6 s, the total travel time reduction is expected to be larger because longer intergreen time indicates a high potential of more effective green time. Moreover, some movements are in multiple stages which also lengthens effective green times for those movements (e.g., the right turn in the northern arm of Intersection 3). Further, the lanes of prohibited left turns are assigned for through movements which also increases the capacity of the through movements (e.g., the through movement in the northern arm of Intersection 2). Therefore, the delays at the intersection go down and the total travel time decreases.

The prohibited left turns may be related to types of left turn phasing. The protected left turns with minor flows should be prohibited, but as the influence of opposing flows is not clear, we cannot conclude that the permitted left turns with minor flows should also be prohibited. Meanwhile, these findings are specific for the network. More generalized findings would require an additional analysis.

The signal timing plan with the proposed method does not aim at reaching global optimum. The design of signal timing consists of “stage generation”, “stage sequence optimization”, and “signal timing determination”. Although “stage generation” and “stage sequence optimization” are conducted with optimization methods, “signal timing determination” is based on the formulas from the relevant manuals. The accumulation of the reasonable steps does not ensure the global optimization. Meanwhile, the interaction between signal timing and traffic flow is conducted in a straightforward way: with given traffic flows obtained from an SUE with a cost based on BRP function, a signal timing plan is designed for each intersection, and then the traffic flows adopt to the signal timing plan. The multiple interactions are not considered as the case in the dynamical scenarios.

We propose a method of designing signal timing plan including LTP by minimizing total travel time. The total travel time reduces after LTP. Types of left turn phasing and relevant left turn flows may be related to LTP. This paper provides an idea of congestion management in urban road networks. Prohibiting left turns should actually be considered among other left turn treatments in signal timing plan design. Planned future research includes analyzing the other factors influencing LTP.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that they have no conflicts of interest.

This research is supported by the German Research Foundation (DFG) through the Research Training Group SocialCars (GRK 1931). The focus of the SocialCars Research Training Group is on significantly improving the city’s future road traffic through cooperative approaches. This work is also carried out within the project OptimUM (Optimization of Urban Traffic Management towards Environment Friendly and Safe Mobility) which is financially supported by the Helmholtz CAS Joint Research Groups (HCJRG) of the Helmholtz Association of German Research Centers. The support is gratefully acknowledged.