Shortest-Way: An Improved Empirical Transition Method for ... › journals › jat › 2017 › 7670521.pdfSignal Coordination MingmingZheng,1 HongfengXu,2 KunZhang,2 andRonghanYao2

Research ArticleShortest-Way: An Improved Empirical Transition Method forSignal Coordination

Mingming Zheng,1 Hongfeng Xu,2 Kun Zhang,2 and Ronghan Yao2

1School of Traffic and Transportation Engineering, Dalian Jiaotong University, Dalian, Liaoning 116028, China2School of Transportation and Logistics, Dalian University of Technology, Dalian, Liaoning 116024, China

Correspondence should be addressed to Hongfeng Xu; [email protected]

Received 18 October 2016; Accepted 7 February 2017; Published 20 March 2017

Academic Editor: Yue Liu

Copyright © 2017 Mingming Zheng et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

Four fundamental insights into transition methods are provided from the perspective of traffic engineers. An improved empiricaltransition method (i.e., Shortest-way) is developed with the goal of reducing the time spent on offset correction and the offsetdeviations of the coordinated phases during the transition period. Shortest-way operates stepwise and can be activated to correctoffset at the scheduled time to switch plans. The maximum amount of adjustment that can be made to a transition cycle length iscalculated based on the timing parameters of active phases in the old and new plans. The problem of cycle length distribution isformulated as a nonlinear integer programming problem, aiming at minimizing the sum of the squares of the intersection offsetdeviations of all the transition cycles. The portion of the cycle length that can be allocated to each phase in a transition cycle iscalculated based on its splits in the old and new plans and its potential contribution to the maximum amount of adjustment tothe cycle length. The numerical experimental results proved the potential advantage of Shortest-way over CORSIM Shortway andjustified the necessity for managing the time to switch plans at the intersection level.

1. Introduction

Signal coordination requires that cycle length, splits, andoffsets be defined to generate signal plans for intersectionsalong an arterial. Changes in demand patterns dictate theneed for switching signal plans from one another. If the oldand new plans have different cycle lengths or offsets, thesignal controller at an intersection will spend a period of timeon correcting offsets by using a transition method before thenew plan can be started.

Plan transition is an important engineering issue of signalcoordination exercises. The existing transition methods canbe categorized as empirical ones and theoretical ones. Theprevailing empirical methods (i.e., Dwell, Max Dwell, Add,Subtract, and Shortway) have been widely used by signalcontroller vendors and practitioners [1–6]. Beginning at theend of the cycle where the scheduled time to switch plans ispresent, offset correction will be completed in a maximum ofthree to five cycles. A limited amount of adjustment ismade to

the cycle length and splits for all the phases in a stepwise anddeterministic fashion. The key advantage of these methods istheir low cost and simplicity in generating transition plans;that is, transition plans can be generated based on no moreinformation than the timing parameters in old and newplans.There is a large body of evidence showing that Shortway (alsosometimes called Bestway, Fastway, or Smooth) performs thebest under most of the scenarios. The theoretical methodshave been proposed by researchers for some years but werenot found to be used in practice [7–10]. Of greatest concernto these methods is to optimize intersection performance(e.g., vehicle delay and vehicle emission) when transition-ing plans. Sophisticated optimization models with real-timetraffic volumes as independent variables are established tojointly determine the number of cycles, the cycle lengths, andthe splits for all the phases during the transition period. Alarge amount of funding and effort for collecting traffic countdata may be the primary reason that hinders the theoreticalmethods from being implemented in the field. Regardless of

HindawiJournal of Advanced TransportationVolume 2017, Article ID 7670521, 12 pageshttps://doi.org/10.1155/2017/7670521

https://doi.org/10.1155/2017/7670521

2 Journal of Advanced Transportation

the complexity of a transition method, there is a potential forplan transition to lead to serious disruptions of traffic flowand long-lasting effects [4, 11].

Transition methods play a supporting role in exercisingsignal coordination and have been a subject of relatively littleresearch within the scope of coordinated signal timing. Inrecent years, some research efforts on the theoreticalmethodshave merited special attention [8–10]. From the perspectiveof traffic engineers, there are four fundamental insights intotransition methods. First, changes to signal plans that aretoo frequent can be a detriment because the arterial neverachieves signal coordination long enough tomeet the desiredoperational objective. It is a common practice to remain ina signal plan for at least 30 minutes [12]. It is the case thatthe cycle length will be lengthened or shortened during thetransition period. Typically, to ensure sufficient capacity foran intersection, a high-demand plan is implemented beforethe onset of the high-demand period, and a low-demand planis implemented after the onset of the low-demand period.Quality signal plans and appropriate times to switch plansfar outweigh a transition method in affecting the systemwideperformance of an arterial. Second, transitionmethod is inde-pendently executed by each signal controller without regardfor the state of adjacent signal controllers. Offset deviationand the resulting poor quality of progression are commonproblems facing transition plans. The goal of reducing thecycle-by-cycle change in the offset may be unattainable due tobig difference between the offsets in the old and new plans orbe attainable at the expense of an excessive amount of timespent on offset correction. Third, the duration of transitionperiod varies from a case to another from a few seconds toa few minutes. But performance measures are required to beevaluated over time periods of a minimum of 15 minutes [12–16]. It is very difficult to distinguish and capture the transientimpact of transition method on the performance measures.In a general sense, the time spent on offset correction and theoffset deviations of the coordinated phases during the transitionperiod should be the first focus when assessing the effectivenessof transition methods. Fourth, there is a long way to go beforethe costs and benefits of using the theoretical methods canbe fully understood outside laboratory environment. Theempirical methods, however, have been proven to be cost-effective in the long-term engineering practice. Improvementsin the empirical methods are required for practical purposesand will be more likely to gain acceptance and support fromsignal controller vendors and practitioners.

In this study, an improved empirical transition method(i.e., Shortest-way) is developed for pretimed or actuatedcoordinated signal systems. Technically, Shortest-way differsfrom the prevailing empirical methods in three aspects. First,the signal controller can shift the sync reference points andstart correcting the offset at the scheduled time to switchplans. Second, the cycles during the transition period areunequally lengthened or shortened to optimize intersectionoffset deviations. Third, the amount of time added to orsubtracted from each phase in a transition cycle is basedon its potential contribution to the maximum amount ofadjustment to the cycle length.

The remainder of this paper is organized as follows. Thekey techniques of Shortest-way are presented in Section 2.The numerical experiments are conducted in Section 3.Conclusions and future studies are provided in Section 4.

2. Key Techniques

Sync reference points are the points on the master clockto which each signal controller is referenced in order toestablish an offset between coordinated phases. An offset isthe time from the sync reference point to the start of greenof a specific coordinated phase. An intersection offset is thetime that elapses between the sync reference point and thestart of green of the first phase in the sequence. From anarterial perspective, it is convenient to develop transitionmethods by using the concept of intersection offset. Figure 1illustrates the relationship between the phase-specific offsetsand intersection offset. A signal plan or a cycle can beregarded as being started as soon as the intersection offset isachieved.

Shortest-way can be activated at the scheduled times toswitch plans to correct the intersection offset. There are fivemain steps that present the working procedure of Shortest-way.

Step 1. Shift the sync reference points and locate the candidatestart points of the new plan.

Step 2. Calculate the minimum and maximum cycle lengthsduring the transition period.

Step 3. Determine the actual start point of the new plan, theduration of transition period, and the number of transitioncycles.

Step 4. Calculate the cycle length for each transition cycle.

Step 5. Calculate the portion of the cycle length that can beallocated to each phase in a transition cycle.

The key techniques of Shortest-way are presented fora typical four-leg intersection, as shown in Figure 2. Left-turnmovements are protected and right-turnmovements arepermitted on all the approaches. There are a total of eightvehicle phases and four pedestrian phases. The right of wayis assigned to the major street phases, followed by the minorstreet phases. Lead-lead left-turn sequence is applied on themajor and minor streets. Pedestrian phases run concurrentlywith their adjacent through vehicle phases. For simplicityof presentation, assume that the minimum green times ofthe through vehicle phases are sufficient to accommodatethe pedestrian timing requirements. None of the pedestrianintervals are discussed in the following.

2.1. Sync Reference Points. Figure 3 illustrates the sync refer-ence points of a coordinated signal system. The master clockis referenced to a real-time clock and the variables in theboxes will be assigned with the times of day. The last syncreference point of the old plan (SR𝑚) is next to the left ofthe scheduled time to switch plans (STW), followed by the

Journal of Advanced Transportation 3

Sync reference pointsO�set of phase A

Intersectiono�set

Intersectiono�set

O�set of phase B

Cycle length

Phase A

Phase BCycle length

Splits for coordinated phasesSplits for noncoordinated phases

(a) Intersection offset < phase-specific offsets

Sync reference pointsO�set of

Intersectiono�set

Intersectiono�set

Phase A

Phase BCycle length

Cycle length

phase A

O�set ofphase B

Splits for coordinated phasesSplits for noncoordinated phases

(b) Intersection offset = phase-specific offsets

Figure 1: Relationship between the phase-specific offsets and intersection offset.

K8K3

F3 F3

K4 K7

K1K5

F2

F2

F4

F4K2

K6

F1F1

N

S

EW

Pedestrian signal

Minor street

Minor street

Maj

or st

reet

Maj

or st

reet

Vehicular signal

(a) Intersection layout and phase numbering

Green

K2K1 K4K3

K6K5 K8K7

Minor streetMajor street

Barrier Barrier

Ring

1Ri

ng 2

YellowRed

(b) Ring-and-barrier diagram showing the lead-lead left-turnsequence

Figure 2: Intersection for presenting the key techniques of Shortest-way.

first sync reference point of the new plan (SR𝑚+1). The timeinterval between two adjacent sync reference points is as longas the cycle length of the old or new plan (𝐶old or 𝐶new).

The cycle where STW is present serves as the firsttransition cycle. The start point of transition period (STP) isinfluenced by the location of STW.The value of STP is equalto the last point where the intersection offset of the old plan(𝑂old) is achieved, given bySTP

= {{{SR𝑚−1 + 𝑂old, if SR𝑚 ≤ STW < SR𝑚 + 𝑂oldSR𝑚 + 𝑂old, if SR𝑚 + 𝑂old ≤ STW < SR𝑚+1.

(1)

As shown in Figure 3, based on the sync reference pointsof the new plan (i.e., SR𝑚+1, SR𝑚+2, . . .), the intersection offsetof the new plan (𝑂new) can be achieved at multiple points.The earlier the new plan is started, the shorter the duration oftransition period is. The 𝑛th candidate start point of the newplan (CSNP𝑛) is given by

CSNP𝑛 = SR𝑚+𝑛 + 𝑂new, (2)where SR𝑚+𝑛 is the 𝑛th sync reference point of the new plan.2.2. Minimum and Maximum Transition Cycle Lengths. Inthe first transition cycle, the signal controller will remainin the old plan until there exists an opportunity to adjust


Real-timeclock

SRm SRm−1 SRm+1 SRm+2 SRm+3 SRm+4

STP

STWOold Oold

Cold Cold

Onew

Cnew

Onew

Cnew

Onew

Cnew

CSNP1 CSNP2 CSNP3

(a) Scenario 1: SR𝑚 ≤ STW < SR𝑚 + 𝑂old

Real-timeclock

SRmSRm−1 SRm+1 SRm+2 SRm+3 SRm+4

OoldOold

Cold Cold

Onew

Cnew

Onew

Cnew

Onew

Cnew

STW

STP CSNP1 CSNP2 CSNP3

(b) Scenario 2: SR𝑚 + 𝑂old ≤ STW < SR𝑚+1

Figure 3: Sync reference points illustrated.

Table 1: Time window definition illustrated.

Time window Start point End point Active phases

TW1 STP STP +min{{{{{𝐺minK1old𝐺minK5old

K1, K2, K3, K4, K5, K6, K7, and K8

TW2 STP +min{{{{{𝐺minK1old𝐺minK5old

STP +max{{{{{𝑆K1old + 𝐺minK2old𝑆K5old + 𝐺minK6old

K2, K3, K4, K6, K7, and K8

TW3 STP +max{{{{{𝑆K1old + 𝐺minK2old𝑆K5old + 𝐺minK6old

STP + 𝐶maold +min{{{{{𝐺minK3old𝐺minK7old

K3, K4, K7, and K8

TW4 STP + 𝐶maold +min{{{{{𝐺minK3old𝐺minK7old

STP + 𝐶maold +max{{{{{𝑆K3old + 𝐺minK4old𝑆K7old + 𝐺minK8old

K4 and K8

TW5 STP + 𝐶maold +max{{{{{𝑆K3old + 𝐺minK4old𝑆K7old + 𝐺minK8old

STP + 𝐶old None𝐺minK𝑖old is the minimum green time of phaseK𝑖 in the old plan (𝑖 = 1, 2, . . . , 8); 𝑆

K𝑖old is the split for phaseK𝑖 in the old plan; and𝐶

maold is the portion of the cycle

length allocated to the major street phases in the old plan.

the splits for some phases. The time interval between STPand (STP + 𝐶old) can be divided into five time windows(i.e., TW1,TW2, . . . ,TW5), as shown in Table 1. The keyinformation for defining the timewindows includes the phasesequence used and the timing parameters of the old plan.Theactive phases that are capable of adjusting their splits in thefirst transition cycle can be identified according to the timewindow where STW is present. The adjustment of the splitsfor the active phases results in a lengthening or shorteningcycle length.Themore the number of active phases, the largerthe amount of adjustment that can be made to the cyclelength.

The second and subsequent transition cycles, if they exist,are obtained by lengthening or shortening the cycle lengthof the new plan within a specified range. All the phases inthe sequence are the active phases being involved in the cyclelength adjustment.

Specifically, the minimum green time that is defined foreach phase in the old and new plans limits the maximum

amount of time subtracted from the split for the phase.Under the constraint of pedestrian timing requirements, aminimum green time that is relatively large can be assignedto a coordinated phase or an uncoordinated phase withinsufficient capacity, while a minimum green time that isrelatively small can be assigned to an uncoordinated phasewith sufficient capacity.

The minimum and maximum cycle lengths of the 𝑥thtransition cycle [𝐶min(𝑥)𝑡 and 𝐶max(𝑥)𝑡] are given by

𝐶min (𝑥)𝑡= max{{{

𝑆min (𝑥)K1𝑡 + 𝑆min (𝑥)K2𝑡𝑆min (𝑥)K5𝑡 + 𝑆min (𝑥)K6𝑡

+max{{{𝑆min (𝑥)K3𝑡 + 𝑆min (𝑥)K4𝑡𝑆min (𝑥)K7𝑡 + 𝑆min (𝑥)K8𝑡 ,


𝐶 max (𝑥)𝑡

= min{{{𝑆 max (𝑥)K1𝑡 + 𝑆 max (𝑥)K2𝑡𝑆 max (𝑥)K5𝑡 + 𝑆 max (𝑥)K6𝑡

+min{{{𝑆 max (𝑥)K3𝑡 + 𝑆 max (𝑥)K4𝑡𝑆 max (𝑥)K7𝑡 + 𝑆 max (𝑥)K8𝑡 ,

𝑆min (𝑥)K𝑖𝑡

={{{{{{{{{{{

𝐺minK𝑖old + 𝑌K𝑖 + 𝑅K𝑖, if (𝑥 = 1) , (𝐴K𝑖𝑡 = 1)𝑆K𝑖old, if (𝑥 = 1) , (𝐴K𝑖𝑡 = 0)𝐺minK𝑖new + 𝑌K𝑖 + 𝑅K𝑖, if 𝑥 > 1,

𝑆max (𝑥)K𝑖𝑡

={{{{{{{{{{{

⌊𝑆K𝑖old ∗ (1 + 𝑓+)⌉ , if (𝑥 = 1) , (𝐴K𝑖t = 1)𝑆K𝑖old, if (𝑥 = 1) , (𝐴K𝑖t = 0)⌊𝑆K𝑖new ∗ (1 + 𝑓+)⌉ , if 𝑥 > 1,

(3)

where 𝑆min (𝑥)K𝑖𝑡

and 𝑆max (𝑥)K𝑖𝑡

are the minimum andmaximum splits for phase K𝑖 in the 𝑥th transition cycle (𝑖 =1, 2, . . . , 8); 𝐺minK𝑖new is the minimum green time of phase K𝑖in the new plan; 𝑌K𝑖 and 𝑅K𝑖 are the yellow change intervaland red clearance interval of phase K𝑖; 𝐴K𝑖

𝑡= 1 if phase K𝑖

is the active phase in the first transition cycle and 𝐴K𝑖𝑡

= 0otherwise; 𝑓+ is the maximum percentage that the split fora phase in the old or new plan can be lengthened during thetransition period; 𝑆K𝑖new is the split for phaseK𝑖 in the newplan;and ⌊ ⌉ is the round-to-the-nearest-integer operator.2.3. Duration of Transition Period. Assuming that there is atotal of 𝑋 transition cycles (𝑋 = 1, 2, . . .), the minimum andmaximum durations of transition period (𝑇min𝑡 and 𝑇max𝑡)can be expressed by

𝑇min𝑡 = 𝑋∑𝑥=1

𝐶min (𝑥)𝑡 ,

𝑇max𝑡 = 𝑋∑𝑥=1

𝐶max (𝑥)𝑡 .(4)

As shown in Figure 4, the actual start point of the newplan (SNP), the duration of transition period (𝑇𝑡), and thenumber of transition cycles (𝑋) can be identified by analyzingthe relationship among CSNP𝑛, 𝑇min𝑡, and 𝑇max𝑡. It shouldbe noted that there is no constraint placed on the maximum

Start

n = 1 x = 1

Locate CSNPn and Tmaxt

n = n + 1

x = x + 1

Yes

NoYes

No

CSNPn > STP + Tmaxt

CSNPn < STP + Tmint

SNP = CSNPn

Tt = SNP − STP

X = x

End

Calculate Tmint

Figure 4: SNP, 𝑇𝑡, and𝑋 determination process.

number of transition cycles because this may override theminimum and maximum transition cycle lengths for theworst case scenarios.

2.4. Cycle Length Distribution. One of the consequencesof adjusting the cycle length and splits during the transi-tion period is the potential for the coordinated phases tobegin earlier or later than expected. The signal controllersat adjacent intersections execute the transition method inisolation, resulting in early or late return to the coordinatedphases which is difficult to manage along an arterial. For thescenarios in which a large amount of adjustment needs bemade to the cycle length, there is no immediate and effectivealternative to prevent poor quality of progression withoutincreasing the duration of transition period. One techniquethat can be used to allow for the coordinated relationshipbetween intersections is to minimize the intersection offsetdeviations during the transition period.

Knowing STP and 𝑋, the actual start point of the 𝑥thtransition cycle [ASTC(𝑥)𝑡] is given by

ASTC (𝑥)𝑡 ={{{{{{{{{

STP, if 𝑥 = 1STP + 𝑋∑

𝑥=2

𝐶 (𝑥 − 1)𝑡, if 𝑥 > 1,(5)

where 𝐶(𝑥)𝑡 is the cycle length of the 𝑥th transition cycle.Beginning from the sync reference point next to the left

of STP, the optimal start point of the 𝑥th transition cycle[OSTC(𝑥)𝑡] is the point where the intersection offset of theold or new plan (𝑂old or 𝑂new) is achieved, given by


OSTC (𝑥)𝑡 ={{{{{{{{{{{{{{{{{{{

SR𝑚+𝑥−2 + 𝑂old, if (SR𝑚 ≤ STW < SR𝑚 + 𝑂old) , (𝑥 = 1 or 2)SR𝑚+𝑥−2 + 𝑂new, if (SR𝑚 ≤ STW < SR𝑚 + 𝑂old) , (𝑥 > 2)SR𝑚+𝑥−1 + 𝑂old, if (SR𝑚 + 𝑂old ≤ STW < SR𝑚+1) , (𝑥 = 1)SR𝑚+𝑥−1 + 𝑂new, if (SR𝑚 + 𝑂old ≤ STW < SR𝑚+1) , (𝑥 > 1) .

(6)

The intersection offset deviation of the 𝑥th transitioncycle [OD(𝑥)𝑡] can be expressed by

OD (𝑥)𝑡 = ASTC (𝑥)𝑡 −OSTC (𝑥)𝑡. (7)The value of OD(𝑥)𝑡 may be positive or negative and

should be squared when being used to define an objectivefunction. In order to distribute 𝑇𝑡 to 𝑋 transition cycles,a nonlinear integer programming model is established tominimize the sum of the squares of the intersection offsetdeviations of all the transition cycles, given by

min𝑋∑𝑥=1

[OD (𝑥)𝑡]2s.t. 𝐶min (𝑥)𝑡 ≤ 𝐶 (𝑥)𝑡 ≤ 𝐶max (𝑥)𝑡

𝑋∑𝑥=1

𝐶 (𝑥)𝑡 = 𝑇𝑡𝐶 (𝑥)𝑡, 𝑥, 𝑋 are integers.

(8)

2.5. Split Distribution. The amount of adjustment to the cyclelength during the transition period is distributed to themajor street andminor street phases, based on their potentialcontributions to the maximum amount of adjustment tothe cycle length. The portions of the cycle length that canbe allocated to the major street and minor street phasesin the 𝑥th transition cycle [𝐶(𝑥)ma

𝑡and 𝐶(𝑥)mi

𝑡] are given

by

𝐶 (1)ma𝑡 ={{{{{{{{{{{{{{{

𝐶maold + ⌊[𝐶 (1)𝑡 − 𝐶old] ∗ 𝐶max (1)ma𝑡

− 𝐶maold𝐶max (1)𝑡 − 𝐶old ⌉ , if (𝐴ma𝑡 = 1) , [𝐶 (1)𝑡 ≥ 𝐶old]𝐶maold − ⌊[𝐶old − 𝐶 (1)𝑡] ∗ 𝐶

maold − 𝐶min (1)ma𝑡𝐶old − 𝐶min (1)𝑡 ⌉ , if (𝐴ma𝑡 = 1) , [𝐶 (1)𝑡 < 𝐶old]

𝐶maold, if 𝐴ma𝑡 = 0

𝐶 (𝑥)ma𝑡 ={{{{{{{{{

𝐶manew + ⌊[𝐶 (𝑥)𝑡 − 𝐶new] ∗ 𝐶max (𝑥)ma𝑡

− 𝐶manew𝐶max (𝑥)𝑡 − 𝐶new ⌉ , if (𝑥 > 1) , [𝐶 (𝑥)𝑡 ≥ 𝐶new]𝐶manew − ⌊[𝐶new − 𝐶 (𝑥)𝑡] ∗ 𝐶

manew − 𝐶min (𝑥)ma𝑡𝐶new − 𝐶min (𝑥)𝑡 ⌉ , if (𝑥 > 1) , [𝐶 (𝑥)𝑡 < 𝐶new]

𝐶 (𝑥)mi𝑡 = 𝐶 (𝑥)𝑡 − 𝐶 (𝑥)ma𝑡𝐶min (𝑥)ma𝑡 = max{𝑆min (𝑥)

K1𝑡+ 𝑆min (𝑥)K2𝑡

𝑆min (𝑥)K5𝑡 + 𝑆min (𝑥)K6𝑡𝐶max (𝑥)ma𝑡 = min{𝑆max (𝑥)

K1𝑡+ 𝑆max (𝑥)K2𝑡

𝑆max (𝑥)K5𝑡+ 𝑆max (𝑥)K6

𝑡,

(9)

where 𝐶min(𝑥)ma𝑡

and 𝐶max(𝑥)ma𝑡

are the minimum andmaximumportions of the cycle length that can be allocated tothe major street phases in the 𝑥th transition cycle; 𝐶manew is theportion of the cycle length allocated to themajor street phasesin the newplan;𝐴ma

𝑡= 1 if there aremajor street active phases

in the first transition cycle and 𝐴ma𝑡

= 0 otherwise; and ⌊ ⌉ isthe round-to-the-nearest-integer operator.

The amount of adjustment to the portion of the cyclelength allocated to the major street or minor street phasesduring the transition period is further distributed to all the


phases on the street, based on their potential contributionsto the maximum amount of adjustment to the portion of the

cycle length. The split for phase K𝑖 in the 𝑥th transition cycle[𝑆(𝑥)K𝑖𝑡] is given by

𝑆 (1)K1𝑡 ={{{{{{{{{{{{{{{{{

𝑆K1old + ⌊[𝐶 (1)ma𝑡 − 𝐶maold] ∗ 𝑆max (1)K1𝑡− 𝑆K1old∑2

𝑖=1[𝑆max (1)K𝑖𝑡 − 𝑆K𝑖old]⌉ , if (𝐴

K1𝑡= 1) , [𝐶 (1)𝑡 ≥ 𝐶old]

𝑆K1old − ⌊[𝐶maold − 𝐶 (1)ma𝑡 ] ∗ 𝑆K1old − 𝑆min (1)K1𝑡∑2𝑖=1

[𝑆K𝑖old − 𝑆min (1)K𝑖𝑡 ]⌉ , if (𝐴K1𝑡= 1) , [𝐶 (1)𝑡 < 𝐶old]

𝑆K1old, if 𝐴K1𝑡 = 0

𝑆 (𝑥)K1𝑡 ={{{{{{{{{{{

𝑆K1new + ⌊[𝐶 (𝑥)ma𝑡 − 𝐶manew] ∗ 𝑆max (𝑥)K1𝑡− 𝑆K1new∑2

𝑖=1[𝑆max (𝑥)K𝑖𝑡 − 𝑆K𝑖new]⌉ , if (𝑥 > 1) , [𝐶 (𝑥)𝑡 ≥ 𝐶new]

𝑆K1new − ⌊[𝐶manew − 𝐶 (𝑥)ma𝑡 ] ∗ 𝑆K1new − 𝑆min (𝑥)K1𝑡∑2𝑖=1

[𝑆K𝑖new − 𝑆min (𝑥)K𝑖𝑡 ]⌉ , if (𝑥 > 1) , [𝐶 (𝑥)𝑡 < 𝐶new]

𝑆 (𝑥)K2𝑡 = 𝐶 (𝑥)ma𝑡 − 𝑆 (𝑥)K1𝑡

𝑆 (1)K3𝑡 ={{{{{{{{{{{{{{{{{

𝑆K3old + ⌊[𝐶 (1)mi𝑡 − 𝐶miold] ∗ 𝑆max (1)K3𝑡− 𝑆K3old∑4


K3𝑡= 1) , [𝐶 (1)𝑡 ≥ 𝐶old]

𝑆K3old − ⌊[𝐶miold − 𝐶 (1)mi𝑡 ] ∗ 𝑆K3old − 𝑆min (1)K3𝑡∑4𝑖=3



𝑆 (𝑥)K3𝑡 ={{{{{{{{{{{

𝑆K3new + ⌊[𝐶 (𝑥)mi𝑡 − 𝐶minew] ∗ 𝑆max (𝑥)K3𝑡− 𝑆K3new∑4


𝑆K3new − ⌊[𝐶minew − 𝐶 (𝑥)mi𝑡 ] ∗ 𝑆K3new − 𝑆min (𝑥)K3𝑡∑4𝑖=3


𝑆 (𝑥)K4𝑡 = 𝐶 (𝑥)mi𝑡 − 𝑆 (𝑥)K3𝑡

𝑆 (1)K5𝑡 ={{{{{{{{{{{{{{{{{

𝑆K5old + ⌊[𝐶 (1)ma𝑡 − 𝐶maold] ∗ 𝑆max (1)K5𝑡− 𝑆K5old∑6


K5𝑡= 1) , [𝐶 (1)𝑡 ≥ 𝐶old]

𝑆K5old − ⌊[𝐶maold − 𝐶 (1)ma𝑡 ] ∗ 𝑆K5old − 𝑆min (1)K5𝑡∑6𝑖=5



𝑆 (𝑥)K5𝑡 ={{{{{{{{{{{

𝑆K5new + ⌊[𝐶 (𝑥)ma𝑡 − 𝐶manew] ∗ 𝑆max (𝑥)K5𝑡− 𝑆K5new∑6


𝑆K5new − ⌊[𝐶manew − 𝐶 (𝑥)ma𝑡 ] ∗ 𝑆K5new − 𝑆min (𝑥)K5𝑡∑6𝑖=5

[𝑆K𝑖new − 𝑆min (𝑥)K𝑖𝑡 ]⌉ , if (𝑥 > 1) , [𝐶 (𝑥)𝑡 < 𝐶new]𝑆 (𝑥)K6𝑡 = 𝐶 (𝑥)ma𝑡 − 𝑆 (𝑥)K5𝑡

𝑆 (1)K7𝑡 ={{{{{{{{{{{{{{{

𝑆K7old + ⌊[𝐶 (1)mi𝑡 − 𝐶miold] ∗ 𝑆max (1)K7𝑡− 𝑆K7old∑8


K7𝑡= 1) , [𝐶 (1)𝑡 ≥ 𝐶old]

𝑆K7old − ⌊[𝐶miold − 𝐶 (1)mi𝑡 ] ∗ 𝑆K7old − 𝑆min (1)K7𝑡∑8𝑖=7




𝑆 (𝑥)K7𝑡 ={{{{{{{{{{{{{

𝑆K7new + ⌊[𝐶 (𝑥)mi𝑡 − 𝐶minew] ∗ 𝑆max (𝑥)K7𝑡− 𝑆K7new∑8


𝑆K7new − ⌊[𝐶minew − 𝐶 (𝑥)mi𝑡 ] ∗ 𝑆K7new − 𝑆min (𝑥)K7𝑡∑8𝑖=7


𝑆 (𝑥)K8𝑡 = 𝐶 (𝑥)mi𝑡 − 𝑆 (𝑥)K7𝑡 ,(10)

where 𝐶miold and 𝐶minew are the portions of the cycle lengthallocated to the minor street phases in the old and new plans;and ⌊ ⌉ is the round-to-the-nearest-integer operator.3. Numerical Experiments

In view of the complex nature of the operating environmentwhere transition methods are used, there are two typesof in-house experiments (i.e., numerical experiments andsimulation experiments) that can be conducted to assess theeffectiveness of transition methods. Numerical experimentsmeasure the static performance measures that are irrele-vant to the operating environment. Simulation experimentsmeasure the dynamic performance measures that vary witha variety of factors, such as simulation tool, simulationmodeling, test bed arterial, traffic demands, signal plans,scheduled times to switch plans, and transition method.

In this section, a total of six numerical experiments wereconducted at the typical intersection (see Figure 2(a)) tomakea comparison of the static performance Shortest-way and oneof the most representative and successful empirical methods,CORSIM Shortway [17].

3.1. Old and New Plans. Table 2 shows the old and newplans for the experiments. Phases K2 and K6 were thecoordinated phases.There were three phase sequence optionsavailable on the major street: lead-lead left turn, lead-lag leftturn, and lag-lag left turn. Lead-lead left-turn sequence wasapplied on the minor street. The signal controller switchedfrom the pre-peak-period plans to the peak-period plans inexperiments 1 to 3 and switched from the peak-period plansto the post-peak-period plans in experiments 4 to 6. For eachexperiment, two scheduled times to switch plans (STW1 andSTW2) were randomly selected between SR𝑚 and SR𝑚+1.

For Shortest-way, 𝑓+ was set to 20%. For CORSIMShortway, the splits for all the phases could be lengthened orshortened proportionally to their splits in the new plan withthe maximum percentage of +20% or −17%.3.2. Results. Figure 5 shows the cycle lengths between SR𝑚−3and SR𝑚+4. The plotted values varied from the cycle lengthof the old plan, via the ones during the transition period,to the one of the new plan. For the transition periodscovering multiple cycles, there was a noticeable variationin the transition cycle lengths yielded by Shortest-way. Thiswas saying that the nonlinear integer programming model

had played an effective role in optimizing the intersectionoffset deviations when distributing the duration of transi-tion period to each transition cycle. Shortest-way methodcompleted offset correction in a maximum of four cycles(see experiment 2). The computational time for the signalcontroller to solve the nonlinear integer programmingmodelcould be negligible. By contrast, CORSIM Shortway, asexpected, equally lengthened or shortened the cycle lengthof the new plan to obtain the transition cycle lengths.

The time spent on offset correction was calculated asfollows: SNP − STW. As shown in Figure 6, Shortest-way completed the offset correction no later than CORSIMShortway, which meant that Shortest-way had a potentialadvantage over CORSIM Shortway in reducing the timespent on offset correction. Most of the transition periodslasted for a few minutes regardless of the transition methodused. There seemed to be no better way to maintain theoperational objective of an arterial than to avoid switchingplans during the conditions when the intersections alongthe arterial needed to operate at maximum efficiency. It wasnoted that either STW1 or STW2 could lead to less time spenton offset correction. In order to mitigate the interruptionof transition methods on the signal coordination along anarterial, it was viable to manage the time to switch plans atthe intersection level.

The intersection offset deviation could be converted intothe offset deviation of a coordinated phase according to thephase sequence used and the splits for the phases beforethe coordinated phase. Figure 7 shows the offset deviationsof the coordinated phases (i.e., phases K2 and K6) relativeto SR𝑚−3, SR𝑚−2, . . ., and SR𝑚+3, respectively. It was quiteclear that the offset deviations yielded by Shortest-way weretypically smaller than those yielded by CORSIM Shortway;that is, Shortest-way had a potential advantage over CORSIMShortway in reducing the offset deviations of the coordinatedphases during the transition period. This was the conse-quence of optimizing the intersection offset deviations anddistributing the splits more fairly and equitably. Nevertheless,there still existed large offset deviations in some instancesregardless of the transition method used. There seemed tobe no better way to maintain the coordination relationshipsbetween intersections than to start the new plan as quickly aspossible. Again, the phenomenon that either STW1 or STW2could lead to smaller offset deviations further strengthenedthe necessity for managing the time to switch plans at theintersection level.


88 88 8897

115 115 115 115

88 88 8897

115 115 115 115

88 88 88106 106

115 115 115

88 88 88 88

124115 115 115

020406080

100120140

m m

Shortest-wayCORSIM Shortway

Cycle

leng

th (s

)

m−3

m−3

m−2

m−2

m−1

m−1

m+1

m+1

m+2

m+2

m+3

m+3

m+4

m+4

STW2 = 07:20:47STW1 = 07:20:04

(a) Exp. 1 (SR𝑚 = 07:20:00; SR𝑚+1 = 07:21:28)


83 83 8091 91

109 111 111

83 83 8375

102111 111 111

83 83 8396 96 96

111 111

83 83 83 83

103 102111 111

020406080

100120

Cycle

leng

th (s

)

m m

m−3

m−3

m−2

m−2

m−1

m−1

m+1

m+1

m+2

m+2

m+3

m+3

m+4

m+4

STW2 = 07:21:05STW1 = 07:20:13

(b) Exp. 2 (SR𝑚 = 07:20:00; SR𝑚+1 = 07:21:23)


95 95 95114

130118 118 118

95 95 95104

140

118 118 118

95 95 95 95

134 133118 118

95 95 95 95

134 133118 118

020406080

100120140160

Cycle

leng

th (s

)

m m

m−3

m−3

m−2

m−2

m−1

m−1

m+1

m+1

m+2

m+2

m+3

m+3

m+4

m+4

STW2 = 07:21:10STW1 = 07:20:29

(c) Exp. 3 (SR𝑚 = 07:20:00; SR𝑚+1 = 07:21:35)


109 109 109101

86 86 86 86

109 109 109101

86 86 86 86

109 109 109101

86 86 86 86

109 109 109 109

7886 86 86

020406080

100120

Cycle

leng

th (s

)

m m

m−3

m−3

m−2

m−2

m−1

m−1

m+1

m+1

m+2

m+2

m+3

m+3

m+4

m+4

STW1 = 18:51:13STW1 = 18:50:06

(d) Exp. 4 (SR𝑚 = 18:50:00; SR𝑚+1 = 18:51:49)


120 120 120104

92 92 92 92

120 120 120104

92 92 92 92

120 120 120104

92 92 92 92

120 120 120 120

84 8492 92

020406080

100120140

Cycle

leng

th (s

)

m m

m−3

m−3

m−2

m−2

m−1

m−1

m+1

m+1

m+2

m+2

m+3

m+3

m+4

m+4

STW1 = 18:50:56STW1 = 18:50:21

(e) Exp. 5 (SR𝑚 = 18:50:00; SR𝑚+1 = 18:52:00)


116 116 116

89 89 89 89 89

116 116 116 109

76 8289 89

116 116 116

89 89 89 89 89

116 116 116 116

80 80 8089

020406080

100120140

Cycle

leng

th (s

)

m m

m−3

m−3

m−2

m−2

m−1

m−1

m+1

m+1

m+2

m+2

m+3

m+3

m+4

m+4

STW1 = 18:51:30STW1 = 18:50:16

(f) Exp. 6 (SR𝑚 = 18:50:00; SR𝑚+1 = 18:51:56)

Figure 5: Cycle lengths between SR𝑚−3 and SR𝑚+4.

4. Conclusions and Future Studies

Transition method is an indispensable component of pre-timed or actuated coordinated signal systems.This research ismotivated by the need for developing an improved empiricalmethod that is more rapid in correcting offset and lessdetrimental to quality of progression.

The four fundamental insights into transition methodspave the way for signal controller vendors, practitioners,and researchers to address engineering issues related to plantransition. Shortest-way inherits the stepwise working proce-dure of prevailing empirical methods but is unique in somekey techniques. The point within the cycle where a limited

amount of adjustment is started to be made to the cyclelength and splits enables Shortest-way to reduce the timespent on offset correction.Theway the transition cycle lengthand the transition splits are calculated enables Shortest-wayto reduce the offset deviations of the coordinated phasesduring the transition period. The numerical experimentalresults proved the potential advantage of Shortest-way overCORSIM Shortway. It was important to note that morebenefits of using Shortest-way could be anticipated if the timeto switch plans was carefully managed at the intersectionlevel. Nevertheless, the fact that the negative impact ofShortest-way on the quality of progression could not be com-pletely eliminated supports the judgement that quality signal


050

100150200250300350400

STW

1

STW

2

STW

1

STW

2

STW

1

STW

2

STW

1

STW

2

STW

1

STW

2

STW

1

STW

2

Experiment 1 Experiment 2 Experiment 3 Experiment 4 Experiment 5 Experiment 6


Tim

e spe

nt o

n o�

set c

orre

ctio

n (s

)

Figure 6: Time spent on offset correction.

02468

Phase K2 Phase K6 Phase K2 Phase K6

−8−6−4−2

m


m−3

m−2

m−1

m+1

m+2

m+3 m

m−3

m−2

m−1

m+1

m+2

m+3 m

m−3

m−2

m−1

m+1

m+2

m+3 m

m−3

m−2

m−1

m+1

m+2

m+3

STW2 = 07:20:47STW1 = 07:20:04

O�s

et d

evia

tions

of t

he

coor

dina

ted

phas

es (s

)

(a) Exp. 1 (SR𝑚 = 07:20:00; SR𝑚+1 = 07:21:28)

05

101520253035

Phase K2 Phase K6 Phase K2 Phase K6m

m−3

m−2

m−1

m+1

m+2

m+3 m

m−3

m−2

m−1

m+1

m+2

m+3 m

m−3

m−2

m−1

m+1

m+2

m+3 m

m−3

m−2

m−1

m+1

m+2

m+3


STW2 = 07:21:05STW1 = 07:20:13

O�s

et d

evia

tions

of t

he

coor

dina

ted

phas

es (s

)−5

(b) Exp. 2 (SR𝑚 = 07:20:00; SR𝑚+1 = 07:21:23)


m

m−3

m−2

m−1

m+1

m+2

m+3 m

m−3

m−2

m−1

m+1

m+2

m+3 m

m−3

m−2

m−1

m+1

m+2

m+3 m

m−3

m−2

m−1

m+1

m+2

m+3


STW2 = 07:21:10STW1 = 07:20:29

O�s

et d

evia

tions

of t

he

coor

dina

ted

phas

es (s

)

−35

−30

−25

−20

−15

−10

−5

0

(c) Exp. 3 (SR𝑚 = 07:20:00; SR𝑚+1 = 07:21:35)

02468


m−3

m−2

m−1 m m m m

m+1

m+2

m+3

m−3

m−2

m−1

m+1

m+2

m+3

m−3

m−2

m−1

m+1

m+2

m+3

m−3

m−2

m−1

m+1

m+2

m+3


STW2 = 18:51:13STW1 = 18:50:06

O�s

et d

evia

tions

of t

he

coor

dina

ted

phas

es (s

)

−8

−6

−4

−2

(d) Exp. 4 (SR𝑚 = 18:50:00; SR𝑚+1 = 18:51:49)

05

101520


m

m−3

m−2

m−1

m+1

m+2

m+3 m

m−3

m−2

m−1

m+1

m+2

m+3

m+3m

m−3

m−2

m−1

m+1

m+2 m

m−3

m−2

m−1

m+1

m+2

m+3


STW2 = 18:50:56STW1 = 18:50:21

O�s

et d

evia

tions

of t

he

coor

dina

ted

phas

es (s

)

−10

−5

(e) Exp. 5 (SR𝑚 = 18:50:00; SR𝑚+1 = 18:52:00)

05

1015202530

m−3

m−2

m−1 m m m m

m+1

m+2

m+3

m−3

m−2

m−1

m+1

m+2

m+3

m−3

m−2

m−1

m+1

m+2

m+3

m−3

m−2

m−1

m+1

m+2

m+3


Phase K2 Phase K6 Phase K2 Phase K6STW2 = 18:51:30STW1 = 18:50:16

O�s

et d

evia

tions

of t

he

coor

dina

ted

phas

es (s

)

(f) Exp. 6 (SR𝑚 = 18:50:00; SR𝑚+1 = 18:51:56)

Figure 7: Offset deviations of the coordinated phases.


Table 2: Old and new plans for experiments 1 to 6.

Number Old plan New plan

1

K1 K2 K3 K4K8K7K6K5

22/12 24/16 17/10 25/14

23/1419/1027/1619/1222/12 Programmed split/minimum green

YellowRed

Green

Cold = 88 s Oold = 21 s 43 s 40 s

Lead-lead sequence and pre-peak-period plan

OK2

old= O

K6

old=

25/15 28/2024/15 38/24

K2K1 K4K3K6K5 K8

28/15

K7

Programmed split/minimum green

YellowRed

Green

28/15 34/24 22/15 31/20Cnew = 115 s Onew = 30 s 58 s 54 s

Lead-lead sequence and peak-period plan

OK2

new = OK6

new =

223/1620/10 16/10 24/14

K1K5K2

K6K3K7

K4K8

25/16 Programmed split/minimum green

YellowRed

Green

Cold = 83 s Oold = 34 s 34 s 54 s

25/16 18/10 18/10 22/14

Lead-lag sequence and pre-peak-period plan

OK2

old= O

K6

old=

31/20

28/2024/12

21/1236/24

26/1533/24

23/15K5

K2

33/24

K1K6

K3K7

K4K8


YellowRed

Green

Cnew = 111 s Onew = 17 s 17 s 40 s

Lead-lag sequence and peak-period plan

OK2

new = OK6

new =

3

20/10

18/1024/1230/20

30/20K2

27/20K6

21/12K1

K5

24/16

26/16

K3K7

K4K8


YellowRed

Green

Cold = 95 s Oold = 14 s 14 s 14 s

Lag-lag sequence and pre-peak-period plan

OK2

old= O

K6

old=

25/15

25/1528/1538/26

38/26K2

34/26K6

24/15K1

K5

31/20

31/20

K3K7

K4K8


YellowRed

Green

Cnew = 118 s Onew = 45 s 45 s 45 s

Lag-lag sequence and peak-period plan

OK2

new = OK6

new =

4

24/15

27/15

K1K5

24/1530/22

33/22K2K6

23/15

25/15K3K7

27/18

29/18

K4K8


YellowRed

Green

Oold = 18 s 42 s 45 sCold = 109 s

Lead-lead sequence and peak-period plan

OK2

old= O

K6

old=

20/10

18/10

K1K5

20/1028/18

26/18K2K6

18/10 22/14

16/10 24/14

K3K7

K4K8


YellowRed

Green

Onew = 10 s 30 s 28 sCnew = 86 s

Lead-lead sequence and post-peak-period plan

OK2

new = OK6

new =

5

38/30

24/16K5

27/16

K2

41/30K6

27/16K1

29/22

32/2223/16

26/16

K3K7

K4K8


YellowRed

Green

Oold = 42 s 69 s 42 sCold = 120 s

Lead-lag sequence and peak-period plan

OK2

old= O

K6

old=

27/20

19/12K5

21/12

K221/12K1

29/20K6

18/10

20/10

K3K7

24/16

26/16K4K8


YellowRed

Green

Onew = 26 s 47 s 26 sCnew = 92 s

Lead-lag sequence and post-peak-period plan

OK2

new = OK6

new =


Table 2: Continued.

Number Old plan New plan

6

22/15

25/1523/1535/28

35/28K2

39/28K6

27/15K1K5

32/22

29/22

K3K7

K4K8


YellowRed

Green

Oold = 35 s 35 s 35 sCold = 116 s

Lag-lag sequence and peak-period plan

OK2

old= O

K6

old=

17/10

19/1019/1225/18

25/18K2

28/18K6

22/12K1K5

25/16

23/16

K3K7

K4K8


YellowRed

Green

Onew = 8 s 8 s 8 sCnew = 89 s

Lag-lag sequence and post-peak-period plan

OK2

new = OK6

new =

𝑂K𝑖old and𝑂

K𝑖new were the offset deviations of phase K𝑖 in the old and new plans (𝑖 = 1, 2, . . . , 8).

plans and appropriate times to switch plans far outweigh atransition method in affecting the systemwide performanceof an arterial.

In future studies, a screening method for optimal timeto switch plans will be developed for Shortest-way. Also,extensive simulation experiments will be conducted to exam-ine the dynamic performance of Shortest-way and provideengineering guidance on implementing Shortest-way.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This research is a part of the project “Development andApplication of an Event-Driven Bus Rapid Transit SignalPriority at Arterials,” which is sponsored by the NationalNatural Science Foundation of China (no. 61374193).

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Shortest-Way: An Improved Empirical Transition Method for ... › journals › jat › 2017 › 7670521.pdfSignal Coordination MingmingZheng,1 HongfengXu,2 KunZhang,2 andRonghanYao2

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