Bandwidth is defined as the maximum amount of green time for a designated movement as it passes through an arterial. In most previous studies, bandwidth has been referred to arterial bandwidth. In practice, a balance between link bandwidth and arterial bandwidth has proven to be important in optimizing coordinated signal timing plans, because not all drivers need to pass through all the intersections on an arterial. This study proposes an algorithm on how to obtain an optimal coordinated signal timing plan with both optimal link bandwidth and optimal arterial bandwidth considering practical vehicles’ speed. The weighted link bandwidth attainability is introduced as an additional measure of effectiveness for assessing the optimization results. The link bandwidth optimization is built based on the improvement of Messer’s algorithm about bandwidth optimization. The arterial bandwidth optimization algorithm takes into consideration the weighted link bandwidth attainability while selecting phase sequences. The proposed algorithm is demonstrated in a case study, and many improvements are archived when a balanced consideration is given to both link bandwidth and arterial bandwidth. Fine-tuning of initial signal timing plan is done using practical travel speed. The evaluation results show a rather significant improvement which is achieved.
The objective of a signalized arterial progression is to provide continuous movement of vehicles and/or minimize the delay along an arterial. Bandwidth is defined as the maximum amount of green time for a designated movement as it passes through an arterial. It is an outcome of the signal timing plan that is determined by the offsets between intersections and the allotted green time for the coordinated phase at each intersection. Bandwidth (measured in seconds) can be defined in terms of two consecutive intersections (referred to as link bandwidth) or in terms of an entire arterial (referred to as arterial bandwidth). Bandwidth and with its associated measures of efficiency and attainability are measures that are often used to assess the effectiveness of a coordinated signal timing plan [
A larger progression bandwidth implies that more traffic on an arterial can progress through the signals without stops [
From the late 1960s to the early 1980s, many researchers, including Little, Messer, and Brooks, made a significant progress and developed a series of bandwidth optimization algorithms [
In these studies mentioned above, bandwidth is always defined in terms of an entire arterial. Arterial bandwidth is commonly used to describe capacity or maximized vehicle throughput. While arterial bandwidth is a good indicator for progression opportunities, it may not fully capture the overall arterial operation. For example, on an arterial with 10 signalized intersections, an arterial bandwidth solution allows vehicles to travel through the entire system. In reality, one must consider how many vehicles actually travel through all the intersections. In some cases, link bandwidth is more important than arterial bandwidth because of high volumes (especially in the case of high left-turn volumes) between some links. Therefore both link and arterial bandwidth should be considered in optimizing coordinated signal timing plans. In 2002, Gartner and Stamatiadis provided mathematical programming models for the development of optimal arterial-based progression schemes considering an individually weighted band that can be adapted to the prevailing traffic flows on that link [
In an ideal coordinated system with optimal bandwidth, platoons with desired travel speed from an upstream intersection at the start of green should arrive at downstream intersection near start of the green indication. In reality, vehicle’s speed might be lower or higher than the desired speed. Speed limit is generally used when the actual speed data are not available. Recently, transportation professionals have new technologies and instruments to collect traffic data. The use of Global Positioning Systems (GPS) has become a springboard for many transportation related projects. GPS tools can capture, store, and transfer real-time traffic data on the field. Vehicle counts, speed, time, and delay are the key components of information to develop proper signal coordination timing plan. The expanding Geographic Information System (GIS) and GPS technologies have made it easy to collect spatial data (latitude, longitude) with attributes of time and velocity [
This study proposes an arterial progression optimization approach using link-based bandwidth optimization algorithm and a GPS tool on an arterial. The rest of paper is organized as follows. Section
Relative offset of
The bandwidth optimization algorithm developed by Brooks and Little establishes the primary principles of bandwidth optimization. The algorithm was originally developed for two-phase signals. Messer et al. [
Figure
Illustration of bandwidth optimization algorithm.
The bandwidth optimization principle reveals that the location of
After calculating upper/lower interferences, there are four additional cases to adjust these upper/lower interferences and obtain the valid interferences, as shown in Table
Calculation of upper/lower interference for link bandwidth.
Common situation | Upper interference | Lower interference |
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Special situation | Upper/lower interfence | Interference adjustment |
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The calculation of upper interference for situation 4.
In Figure
Phase sequence combinations of two consecutive intersections.
Phase sequence combination | No. |
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IBL | OBL |
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IBL | OBL |
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1 | Lead | Lead |
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Lead | Lead |
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2 | Lead | Lag |
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3 | Lag | Lead |
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4 | Lag | Lag |
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5 | Lead | Lag |
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Lead | Lead |
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6 | Lead | Lag |
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7 | Lag | Lead |
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8 | Lag | Lag |
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Lead | Lead |
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10 (Figure |
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11 | Lag | Lead |
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12 | Lag | Lag |
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13 | Lag | Lag |
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Lead | Lead |
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14 | Lead | Lag |
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15 | Lag | Lead |
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16 | Lag | Lag |
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All upper/lower interferences to be valid must be less than the minimum inbound through movement green time and greater than the slack time between
Assume that the referencing phase is the start of green time of outbound through movement, there are six cases to calculate the relative offset of two consecutive intersections.
Consider
Then
The calculation of relative offset.
Case
Case
Consider
Then
Consider
Then
Consider
Then
Consider
Then
Consider
Then
The signal timing plans of two intersections are shown below. The distance between them is 2015 ft, and the speed limit is 40 mph; then travel time is 34 s and cycle length is 130 s:
Upper/lower interferences of two consecutive intersections.
Phase sequence | ||||||||
Intersection |
1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
Intersection |
1 | 2 | 3 | 4 | 1 | 2 | 3 | 4 |
Interference of intersection |
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Upper interference | 55 | Common: |
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53 |
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53 |
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Lower interference | −74 | Common: |
−54 | −72 | −54 | −72 | −34 | −52 |
Link bandwidth | — |
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— |
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Offset of intersection |
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— |
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Phase sequence | ||||||||
Intersection |
3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 |
Intersection |
1 | 2 | 3 | 4 | 1 | 2 | 3 | 4 |
Interference | ||||||||
Upper interference | 84 | 102 | 64 | 82 | 64 | 82 |
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62 |
Lower interference |
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−83 |
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−83 |
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−63 | −81 |
Link bandwidth |
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Offset of intersection |
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According to the rule of valid interference in Section
Bandwidth efficiency and attainability are two measures to describe the quality of a coordinated signal timing plan. Both are computed from a time-space diagram that represents the signal timing plan. The two parameters are first defined by Messer et al. [
Besides the arterial bandwidth efficiency and attainability, maximum link bandwidth, possible link bandwidth, and link bandwidth attainability are first proposed as additional measures of effectiveness in our research. Maximum link bandwidth is the maximum value of a link bandwidth obtained using link bandwidth optimization algorithm. Possible link bandwidth is the value of a link bandwidth when the signal system has the optimal arterial progression bandwidth. In this case, the link bandwidth may not be the best solution to get the arterial bandwidth. In Figure
Example of maximum and possible link bandwidth and arterial bandwidth.
Maximum link bandwidth
Possible link and arterial bandwidth
After the outbound direction is determined, link bandwidth
Optimized results for
Timing parameter | Valid solutions | ||||||||
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1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
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2 | 3 | 1 | 4 | 2 | 4 | 3 | 1 | 2 |
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3 | 2 | 3 | 3 | 4 | 2 | 1 | 2 | 1 |
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72 | 66 | 54 | 52 | 52 | 48 | 46 | 46 | 43 |
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1.0 | 0.92 | 0.75 | 0.72 | 0.72 | 0.67 | 0.64 | 0.64 | 0.60 |
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83 | 96 | 83 | 83 | 83 | 96 | 96 | 96 | 83 |
Similarly, the link bandwidth
Optimized results for
Timing parameter | Valid solutions | ||||||||||||||
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1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | |
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1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 3 | 1 | 4 | 3 | 3 |
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1 | 3 | 4 | 1 | 2 | 4 | 3 | 1 | 4 | 3 | 3 | 2 | 2 | 1 | 4 |
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81 | 81 | 81 | 81 | 81 | 81 | 81 | 73 | 72 | 71 | 70 | 64 | 55 | 53 | 52 |
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1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 0.90 | 0.89 | 0.88 | 0.86 | 0.79 | 0.68 | 0.65 | 0.64 |
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56 | 56 | 56 | 37 | 56 | 37 | 56 | 56 | 56 | 37 | 56 | 56 | 56 | 56 | 56 |
Using the same approach, the link bandwidth
Optimized results for
Timing parameter | Valid solutions | |||||||||||
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1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
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2 | 4 | 1 | 2 | 2 | 3 | 4 | 1 | 4 | 1 | 2 | 3 |
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3 | 3 | 3 | 1 | 4 | 3 | 1 | 1 | 4 | 4 | 2 | 2 |
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98 | 83 | 82 | 80 | 78 | 65 | 63 | 62 | 61 | 60 | 58 | 54 |
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1.0 | 0.85 | 0.84 | 0.82 | 0.80 | 0.63 | 0.64 | 0.63 | 0.62 | 0.61 | 0.59 | 0.55 |
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81 | 81 | 81 | 81 | 81 | 81 | 81 | 81 | 81 | 81 | 81 | 86 |
Using the conditions
Selected solutions of different phase sequences on Kietzke Lane (2nd Street → Plumb Lane).
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1 | 2 | 1 | 3 | 0.92 | 3 | 0.92 | 3 | 0.92 | 1 | 0.75 |
2 | 3 | 0.86 | 2 | 1 | 2 | 1 | 2 | 1 | 3 | 0.86 |
3 | 3 | 0.63 | 2 | 1 | 4 | 0.85 | 1 | 0.84 | 3 | 0.63 |
4 | 3 | — | 3 | — | 3 | — | 3 | — | 3 | — |
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— | 0.8249 | — |
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— | 0.9247 | — | 0.9214 | — | 0.7485 |
The inbound through movement green time determines which intersection is the reference intersection for one link and how the relative offset of each intersection is calculated. According to the value of inbound through movement green time, there are four cases to calculate the offsets of
Consider
Consider
Consider
Consider
(1) Consider
(2) Consider
A Windows program bandwidth optimization and time space diagram (BOTSD) was developed based on the proposed algorithm. Time-space diagrams can be produced by BOTSD with additional information to show the start and end of each phase. This information is useful for field implementation of signal timing plans, providing easiness of verifying that the current plan is running.
Kietzke Lane, an arterial in Reno, NV, USA, is selected as a case study to check the performance of the proposed bandwidth optimization algorithm. There are eight intersections (E 2nd Street no. 1, Mill Street no. 2, Vassar Street no. 3, Plumb Lane no. 4, Grove Street no. 5, Gentry Way no. 6, Moana Lane no. 7, and Peckham Lane no. 8) on the Kietzke Lane selected in this study. The coordinated cycle length of this arterial is 130 seconds (16:00 pm–18:00 pm) and speed limit is 40 mph. Table
Phase splits in the initial signal timing plan of Kietzke Lane (2nd Street → Peckham Lane).
INTID | SBL | NBT | WBL | EBT | NBL | SBT | EBL | WBT | Distance (ft) |
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1 |
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20 | 41 |
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30 | 31 | |
2 |
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25 | 40 |
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25 | 40 | 2015 |
3 |
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23 | 35 |
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18 | 40 | 3294 |
4 |
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18 | 42 |
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22 | 38 | 2600 |
5 |
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— | 35 |
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— | 35 | 1841 |
6 |
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— | 35 |
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— | — | 2180 |
7 |
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20 | 48 |
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19 | 49 | 724 |
8 |
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21 | 32 |
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19 | 34 | 2168 |
Traffic volumes of every intersection in the Kietzke Lane from field data.
INTID | NBL | NBT | NBR | SBL | SBT | SBR | EBL | EBT | EBR | WBL | WBT | WBR |
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1 |
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820 | 169 |
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430 | 54 | 248 | 585 | 143 | 160 | 418 | 63 |
2 |
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903 | 259 |
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492 | 121 | 166 | 684 | 134 | 318 | 742 | 146 |
3 |
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971 | 120 |
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625 | 146 | 149 | 238 | 100 | 200 | 408 | 256 |
4 |
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780 | 313 |
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483 | 143 | 230 | 870 | 244 | 282 | 757 | 168 |
5 |
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1193 | 72 |
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800 | 148 | 113 | 71 | 70 | 128 | 98 | 30 |
6 |
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1143 | 51 |
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883 | 79 | 170 | 56 | 170 | 61 | 66 | 46 |
7 |
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897 | 168 |
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738 | 140 | 250 | 631 | 111 | 257 | 681 | 176 |
8 |
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845 | 249 |
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633 | 168 | 81 | 419 | 19 | — | 514 | 300 |
Total |
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7552 | 1401 |
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5084 | 999 | 1407 | 3554 | 991 | 1406 | 3684 | 1185 |
Figure
TSD of signal timing plan with Messer’s algorithm.
Southbound is the outbound direction. Figure
TSD of signal timing plan with proposed algorithm.
Using the same method of fine-tuning, we have got time-space diagrams with other travel speeds, which are less than the speed limit and same for every link. In this case study, the speed limit is 40 mph; we can use 38 mph, 35 mph, and 32 mph to design the signal timing plans with the proposed bandwidth optimization algorithm and get the TSD with maximum progression of arterial bandwidth and link bandwidth for comparison.
We have collected more than ten study runs of GPS data at the PM peak hours, using GPS receiver and GPS2LT [
Average practical travel speeds for links.
Southbound
Northbound
Using the GPS and TS tool and the collected GPS data, we have drawn the GPS trajectory on the TSD for three types of study runs for PM peak hours (16:00 pm–18:00 pm). In Figure
GPS trajectory on initial TSD on the speed of 40 mph.
According to the results of fine-tuning of initial signal timing plan link by link, we have got the best solution of TSD for the arterial. The arterial progression bandwidth of southbound is 28 seconds, and 39 seconds in northbound. The bandwidth efficiency is 25.8%. Attainability of every link is more than 85% and three of them are equal to 100%, as shown in Figure
TSD of signal timing plan with practical travel speed.
Table
Attainability of link bandwidth on Kietzke Lane with two algorithms.
Link | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Arterial MOE | |||
Max link bandwidth (s) | 72 | 81 | 98 | 98 | 134 | 84 | 84 | ||||
Weighted factor of link | 0.1155 | 0.1345 | 0.1280 | 0.1391 | 0.1827 | 0.1656 | 0.1347 | ||||
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Link bandwidth attainability | |||||||||||
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Algorithm | Outbound |
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Proposed | SB | 0.89 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 0.99 | 0.9855 | 0.2231 | 0.7532 |
NB | 0.89 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | 0.99 | 0.9856 | |||
Messer | SB | 0.81 | 0.94 | 0.96 | 0.94 | 0.84 | 0.99 | 0.77 | 0.8903 | 0.1923 | 0.6494 |
NB | 0.89 | 0.99 | 0.85 | 0.97 | 0.83 | 0.99 | 0.73 | 0.8915 | 0.2038 | 0.6883 |
According to arterial bandwidth efficiency and attainability in Figures
In the other way, simulation results from SYNCHRO show that the proposed algorithm is better than Messer’s algorithm. Total intersection control delay of 8 intersections with optimized signal timing plan with proposed algorithm is 266.2 s, less than that of Messer’s algorithm 288 s. Total Link delay in arterial direction with proposed algorithm is 401.3 s, less than that of Messer’s algorithm 445.2 s.
Additionally, when the number of signals in a system increases, it becomes more difficult to obtain a good bandwidth solution using Messer’s algorithm. In some cases there is no valid solution with arterial bandwidth. However, the new proposed algorithm can be used to obtain an optimal solution of coordinated signal timing plan with both optimal link bandwidth and optimal arterial bandwidth.
The simulation model of this arterial is built up in VISSIM 5.30. Individual parameters in VISSIM 5.30 were adjusted and tuned with model calibration and validation. To reduce stochastic variation, ten random-seeded runs were conducted in VISSIM 5.30 for the case study. At the same time, the signal timing plan has been input into SYNCHRO 7.0 to get the control delay for the intersections and links on the arterial and check the traffic flow lines on the TSD. Average speed and delay per vehicle for the whole network, average delay and queue length for the nodes, and average delay for the links are the main simulation results, shown in Table
Performance index of signal timing plans with different speeds.
Type | Index | 32 | Practical speed | 35 | 38 | 40 (speed limit) |
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Network performance | Average speed (mph) | 18.01 |
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17.95 | 17.9 | 17.69 |
Average delay (s) | 57.25 |
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57.91 | 58.29 | 61.83 | |
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Nodes | Total average delay (s) | 210.7 |
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209.1 | 213.2 | 221.4 |
Total average queue (ft) | 519.8 |
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527.6 | 532.8 | 548.2 | |
Total intersection control delay | 257.2 |
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260.6 | 267.2 | 266.2 | |
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Links | Delay from VISSIM | 378.7 |
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370.3 | 382.1 | 392.7 |
Delay from SYNCHRO | 378.3 |
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382.6 | 397.6 | 401.3 |
The results show that a rather significant improvement is achieved after the fine-tuning of signal timing plan with practical travel speed data is done.
Bandwidth can be defined in terms of two consecutive intersections (link bandwidth) or in terms of an entire arterial (arterial bandwidth). In most previous studies, bandwidth is always referred to as arterial bandwidth. In practice, a balance between link bandwidth and arterial bandwidth has proven to be important in optimizing coordinated signal timing plans, because not all drivers need to pass through all the intersections on an arterial. This paper proposes an algorithm on how to obtain an optimal coordinated signal timing plan with both optimized link and arterial bandwidth. The proposed algorithm has two parts: link bandwidth optimization and arterial bandwidth optimization. Link bandwidth attainability is defined as an additional measure of effectiveness for assessing the optimized results. In the link bandwidth optimization, there are eight cases to calculate upper/lower interference and six cases to calculate offset between two consecutive intersections, based on the improvement of Messer’s research. The arterial bandwidth optimization has 3-fold processes, such as calculation of link bandwidth, selection of phase sequence, and calculation of offset, generally link by link. Weighted means of link bandwidth attainability is defined as another measure of effectiveness for selecting the best solution of different phase sequences. The results in the case study show that the proposed algorithm can be used to get a much better signal timing plan than that from Messer’s algorithm. Additionally when the number of signals in a system increases, it becomes more difficult to obtain a good bandwidth solution with Messer’s algorithm. There is no valid solution of arterial bandwidth using Messer’s algorithm when the number of signals is very high. However, the proposed algorithm of this paper can be used to obtain an optimal coordinated signal timing plan with both optimal link bandwidth and arterial bandwidth. Fine-tuning of initial signal timing plan is done to improve link and arterial progression bandwidth using practical travel speed compared to speed limit. The evaluation results show that a rather significant improvement is achieved. In future, we will continue our research on optimizing coordinated signal timing plans under the consideration of the traffic volumes of left-turns and earlier release of left-turn phase.
There is no known conflict of interests associated with this paper and there has been no significant financial support for this work that could have influenced its outcome. The authors do not have any possible conflict of interests.
This study is supported by Project “The Fundamental Research Funds for the Central Universities (no. 2013JBM049)” and “National Basic Research Program of China (no. 2012CB725403).”