Distributed Coordinated Signal Timing Optimization in Connected Transportation Networks

Transcription

1 Disribued Coordinaed Signal Timing Opimizaion in Conneced Transporaion Neworks Link o he paper: hps://urldefense.proofpoin.com/v2/url?u=hps-3a auhors.elsevier.com_a_1v3f6-2cm0mr5bao&d=dwifaq&c=c3yme8gmkxg_ihjnxs06zywk4ejm8ldrrvxqb- Je7sw&r=nAfV9kl2rgoyTkdl0ccp8anEYaGBCUH7z2SvoVRnIYQ&m=xedJ-Vlm0GeRH- TIiklIacGEpiw4YeO0eRYi2PKVEM&s=kCQUy0LWMCTCui0x1xQP1rLxq-hzjHeO22cwm7-62Go&e= S M A Bin Al Islam Graduae Research Assisan, Civil and Environmenal Engineering Deparmen Washingon Sae Universiy smabinal.islam@wsu.edu Ali Hajbabaie, Ph.D., Corresponding Auhor Assisan Professor, Civil and Environmenal Engineering Deparmen Washingon Sae Universiy PO Box , Pullman, WA , USA Phone: (509) , Fax: (509) ali.hajbabaie@wsu.edu Absrac This paper presens a Disribued-Coordinaed mehodology for signal iming opimizaion in conneced urban sree neworks. The underlying assumpion is ha all vehicles and inersecions are conneced and inersecions can share informaion wih each oher. The novely of he work arises from reformulaing he signal iming opimizaion problem from a cenral archiecure, where all signal iming parameers are opimized in one mahemaical program, o a decenralized approach, where a mahemaical program conrols he iming of only a single inersecion. As a resul of his disribuion, he complexiy of he problem is significanly reduced hus, he proposed approach is real-ime and scalable. Furhermore, disribued mahemaical programs coninuously coordinae wih each oher o avoid finding locally opimal soluions and o move owards global opimaliy. We proposed a real-ime and scalable soluion echnique o solve he problem and applied i o several case sudy neworks under various demand paerns. Thealgorihm conrolled queue lengh and maximized inersecion hroughpu (beween 1% o 5% increase compared o he acuaed coordinaed signals opimized in VISTRO) and reduced ravel ime (beween 17% o 48% decrease compared o acuaed coordinaed signals) in all cases. KEY-WORDS Disribued-Coordinaed Approach; Conneced Vehicles; Real-ime and Scalable; Signal Timing Opimizaion; Oversauraed Flow Condiions 1

2 1. Inroducion Majoriy of signal iming opimizaion algorihms uilize a cenralized formulaion and archiecure. As a resul, hey opimize various signal iming parameers (i.e., phase plan, cycle lengh, green imes, and offses) of all inersecions a he same ime in one mahemaical program. However, nework signal iming opimizaion is an NP-Complee problem (1, 2) and a cenral opimizaion echnique will no be scalable and applicable o large ransporaion neworks. Hierarchal approaches decompose he signal iming opimizaion problem o several inerconneced sub-problems wih a cenral conrol uni. The underlying concep of mos hierarchal approaches is o handle slow-varying and wide-area-level decisions a upper levels and perform real-ime and small-area compuaions in lower levels. As a resul, he upper and lower levels may have objecive funcions ha compee wih each oher. Therefore, designing signal conrol mehods wih a reasonable balance beween hese levels is challenging (3). There are also a number of sudies ha focused on decenralized signal conrol in urban seings. As a resul of he decenralizaion, hese approaches are scalable and can be real-ime; however, raher han global opimizaion, hey mosly locally conrol he signals and may find sub-opimal soluions. This paper develops a Disribued-Coordinaed (DC) approach for nework-level signal conrol in a conneced urban sree environmen. We propose a new disribued formulaion for signal iming opimizaion problem, where a mahemaical program finds he opimal iming of only a single inersecion wih he objecive of maximizing is hroughpu while penalizing for long queue lengh on differen approaches. However, each mahemaical program is aware of oher inersecions decisions and raffic flow condiions. As a resul, he decisions a neighboring inersecions are coordinaed o ensure smooh raffic flow hroughou he nework and avoiding locally opimal soluions. This sudy also proposes an innovaive soluion echnique o solve he DC problem. The proposed rolling horizon soluion echnique gahers demand and queue lengh daa a each inersecion a each ime sep and opimizes he iming of signalized inersecions for several ime seps in he fuure (a.k.a. predicion period); however, only implemens he decisions on erminaing or exending he signals for he nex ime sep (i.e., could be as shor as one second). As a resul, in addiion o coordinaion beween differen inersecions, he approach finds soluions ha are expeced o be near-opimal for he enire sudy period. The proposed DC approach is real-ime and scalable o neworks of any size as long as enough compuaional power is available and he righ duraion of ime sep and predicion periods are seleced. The underlying assumpion is ha he enire nework (all vehicles and inersecions) is conneced. I also addresses he oversauraed condiions wih selecion of an appropriae objecive funcion and inclusion of explici consrains on maximum queue lengh. The proposed approach is dynamic and can reac o abrup changes in demand level and roadway capaciy (e.g., lane closure due o a raffic acciden). The remainder of his paper is organized as follows. Nex secion presens a review of relevan lieraure on raffic signal iming mehods and highlighs he research gaps. Then, problem formulaion and he proposed soluion echnique are inroduced. Informaion on he sudy sie and implemening he algorihm is deailed nex. The mehod is applied o wo case sudy neworks under four differen demand paerns and he resuls are discussed and compared wih differen raffic signal conrol mehods. Finally, concluding remarks and rends for furher research are presened. 2. Lieraure Review Much research is devoed o raffic signal conrol. This secion provides a review of relevan sudies and presens hem in hree differen groups: cenral, hierarchal, and decenralized signal iming approaches. A he end of his secion, he conribuions of his paper will be highlighed. Cenral Signal Timing Approaches Longley (4), Abu-Lebdeh and Benekohal (5), and Chang and Sun (6) developed cenral mehods o conrol raffic signals dynamically in oversauraed neworks. Hajbabaie e al. (7) proposed Geneic Algorihms and Approximae Dynamic Programming approaches for raffic signal coordinaion and queue managemen in oversauraed neworks. These mehods were compared (8) and used o evaluae he effecs of prohibiing 2

3 or allowing lef urns a signalized inersecions (9), using a common or variable cycle lengh in signal coordinaion (10), and meering raffic enering a congesed nework (11) (12) on raffic operaions. Puha e al. (13) proposed a raffic signal coordinaion mehod under oversauraion flow condiions using an colony opimizaion and compared i o simple geneic algorihms. Canarella e al. (14) developed an ieraive procedure for equilibrium nework raffic signal seing and developed heurisics for opimizing signal seing and lane layou (15). Ceylan and Bell (16) and Teklu e al. (17) proposed mehodologies for simulaneous signal iming opimizaion and raffic assignmen using geneic algorihm. Beard and Ziliaskopoulos (18) simulaneously opimized raffic signal iming and sysem opimal raffic assignmen problems in a mixed ineger linear programming model. Sun e al. (19) combined signal conrol in oversauraed neworks wih sochasic roue choice in a bi-level mahemaical programming model and used a heurisic soluion echnique o solve i. Karoonsoonawong and Waller (20) proposed a robus opimizaion formulaion for nework capaciy expansion, raffic signal opimizaion, and dynamic raffic assignmen. Ukkusuri e al. (21) proposed a bi-level formulaion o opimize he signal seing o minimize he sysem-level ravel ime a he upper level, and opimize ravelers users roues o minimize heir ravel cos a he lower level. He e al. (22) developed a mixed-ineger linear program o find fuure opimal signal saus considering ransi and passenger car modes in a conneced vehicle environmen. The compuaional complexiy of heir approach increased wih raffic demand; hus, a real-ime soluion was no possible. He e al. (23) proposed muli-modal prioriy conrol wih he consideraion of signal coordinaion for conneced vehicles. The sudy includes emergency vehicles, ransi buses, commercial rucks, and pedesrians. Their approach minimizes he acual ravel ime along he roue o he desinaion. Hajbabaie and Benekohal (24) developed a bi-level program for simulaneous signal iming opimizaion and raffic assignmen for oversauraed ransporaion nework. This sudy inroduced a new objecive funcion, maximizaion of he weighed number of compleed rips, and explici consrains on queue lengh. Feng e al (25) applied dynamic programming o opimize minimum and maximum green duraions in he upper level of a bi-level opimizaion program. In he lower level, hey opimized phase sequence and acual green duraion simulaneously. Spli, Cycle, and Offse Opimizaion Technique (SCOOT) operaes a group of adjacen juncions. A cenral compuer deermines he common cycle, green duraions, and offses over ime. SCOOT adjuss he imings o reduce oal delay and sops based on he predicion of oal delay and sops caused by he signal iming parameers (26). All sudies ha were highlighed above used a cenral opimizaion archiecure and were aimed a finding opimal signal iming parameers; however, were no scalable. As a resul, hey canno find opimal signal iming parameers when he size of he nework increases. Hierarchal Signal Timing Approaches Hierarchical approaches decompose he nework opimizaion problem ino a muli-level conrol problem wih disinc objecives a each level. The underlying concep of mos hierarchal approaches is o make nework level decisions a he upper (or cenral) level and he real-ime, small-area compuaions in he lower (or inersecion) level. Sydney Coordinaed Area Traffic Sysem (SCATS) uses wo levels of conrol: sraegic and acical. In he sraegic conrol level, a regional compuer deermines suiable signal imings and offses for all inersecions in sub-areas (up o en inersecions) based on average prevailing raffic condiions. A he acical level, each of he local conrollers has he flexibiliy o opimize green ime for meeing he cyclic demand variaion a each inersecion (27). Opimized Policies for Adapive Conrol (OPAC) has hree layers: synchronizaion layer, coordinaion layer, and local conrol layer. The synchronizaion layer opimizes he nework-wide cycle ime by a cenralized compuer. In coordinaion layer, each signal conroller opimizes offses once per cycle. In addiion, o adap wih he changing raffic condiions, he cycle lengh and offses are updaed over ime. The local conrol layer coninuously calculaes opimal swiching of signal phases considering he oupu of upper layers (28). 3

4 Urban Traffic Opimizaion by Inegraed Auomaion (UTOPIA) consiss of wo levels: an area level and a local level. The area level consiss of a cenral sysem responsible for medium and long erm forecasing and conrol over he whole area of concern. A his level, he signal plans ha are needed for he adapive coordinaion are opimized dynamically. A he local level, conrollers in each inersecion deermine he sequence and opimum lengh of signal phases in real ime using he coordinaion crieria esablished by he area level, local raffic measuremens, and informaion received from he conrollers of adjacen inersecions. Boh area and local levels use rolling horizon echnique for opimizaion (29). Real-ime Hierarchical Opimizing Disribued Effecive Sysem (RHODES) has hree levels of conrol: dynamic nework loading, nework flow conrol, and inersecion conrol. The dynamic nework loading level forecass he general ravel demand over an exended ime. Nework flow conrol level uses he prediced ravel demand and opimizes green ime for each demand paern. The lowes level deermines he signal phase and duraion decisions based on observed vehicle flows, coordinaion consrains, and flow predicions on a second-by-second basis (30). Decenralized Signal Timing Approaches In decenralized approaches, he nework is decomposed ino some regions wih varying number of inersecions. As a resul of decenralizaion, hese approaches are scalable and can be real-ime; however, raher han global opimizaion, hey mosly locally conrol signals and may find a sub-opimal soluion. A review of some decenralized signal iming approaches follows. Porche and Laforune (31) developed Adapive Limied Look-Ahead Opimizaion of Nework Signals-Decenralized (ALLONS-D) archiecure. They decomposed he nework o inersecion level and used a dynamic programming based ree searching algorihm o selec he phase receiving green signal a each ime inerval. The inersecions did no coordinae heir decisions; however, higher weighs were assigned o major srees wih higher volumes. ALLONS-D is compuaionally expensive and canno solve he problem in real-ime. Priemer and Friedrich (32) divided he nework ino individual inersecions and proposed a dynamic programming algorihm o minimize oal queue lengh a each inersecion. There is no exchange of informaion beween he signal conrollers as such decisions are no coordinaed. The algorihm decides he phase sequence of each inersecion separaely in every five-second inerval over a 20-second predicion horizon. Weighing facors are used o assign higher prioriy o major roads. Lee and Park (33) proposed Cumulaive Travel-Time Responsive (CTR) inersecion conrol algorihm. They claimed he CTR algorihm can handle he signal coordinaion among inersecions by adoping weighing facors o vehicles on major srees; however, his was no esed. The algorihm allocaes green signal o he movemen wih longer cumulaive ravel ime. Goodall e al. (34) proposed a rolling horizon based predicive microscopic simulaion algorihm (PMSA) based on cumulaive vehicle delay. PMSA divides he nework o individual inersecions and conrols hem separaely in a conneced vehicle environmen. I does no consider coordinaion among adjacen inersecions. Therefore, vehicles need o sop more frequenly compared o coordinaed-acuaed conrol especially a high demands and low peneraion raes. This algorihm uilized a microscopic simulaion model o predic fuure raffic condiions and as a resuls, may no be real-ime. Adacher e al. (35) proposed raffic signal synchronizaion sraegy o ensure a beer flow of raffic. They decomposed he whole area o fixed regions using disribued consensus algorihm. Each region consiss several inersecions. They used a disribued K-means algorihm o find he posiion of he conroller in each region. This conroller uses surrogae measures for finding signal iming and offses for all inersecions wihin he region. Each inersecion shares informaion abou is raffic condiion wih he region conroller. The regions and posiions of he conrollers are fixed in his mehod. Wongpiromsarn e al. (36) divided he nework o individual inersecions. They opimized phase ime of each inersecion wihou considering explici coordinaion among inersecions. Each inersecion is opimized individually using backpressure rouing. They compue pressures a every inersecion of he nework based on queue lenghs, where a longer queue yields higher pressure, and allow flows wih a high 4

5 upsream pressure and a low downsream pressure o be served. Gregoire e al. (37) showed he inabiliy of his mehod o avoid congesion propagaion under finie queue capaciies. Conribuion of he Paper As shown previously, signal iming opimizaion in an arerial sree is an NP-Complee problem (1) and a cenral approach will no be able o find he opimal soluion in a reasonable amoun of ime. The hierarchal approaches, while can find soluions faser, require significan invesmen in infrasrucure o provide communicaions beween a cenral uni and each local opimizer. Finally, he exising decenralized approaches are more prone o finding sub-opimal signal iming parameers. The mehodology proposed in his paper can efficienly find near-opimal soluions wihou a need o a cenral uni. Specific conribuions of his paper are as follows: 1- Reformulaing he signal iming opimizaion problem from cenral archiecure o disribuedcoordinaed archiecure. 2- Developing a mahemaical program ha can find opimal signal iming parameers for an inersecion efficienly 3- Developmen of a novel soluion echnique o solve he problem in real-ime. 3. Problem Formulaion This secion explains he problem formulaion. The decision variables, objecive funcion, and key consrains are inroduced. This approach relies on consrucing a se of local conrollers, each of which is associaed wih each inersecion. Every signal conroller is able o independenly opimize local inersecion and has he abiliy o pass he raffic flow condiion o adjacen conrollers. This message passing among conrollers ensures he coordinaion of heir acions. This sudy does no follow any sric phase sequence or convenional cycle mehod. The lis and definiion of all ses, decision variables, and parameers used in he Mixed Ineger Linear Program (MILP) are summarized in TABLE 1 and FIGURE 1. The definiion of cells is very similar o cell ransmission model (38); however, we define wo differen ypes of cells: source and movemen cells. Source cells are used o keep rack of he number of vehicles ha are released from upsream inersecions and ravel owards he downsream inersecion. One movemen cell is allocaed o each lane group. In each approach, all movemen cells are fed by he corresponding upsream source cell and will feed differen source cells a downsream inersecions, see FIGURE 1. The ravel ime beween movemen cells and source cells is shown by. 5

6 TABLE 1 Definiion of Ses, Decision Variables, and Parameers Ses: T se of all ime seps I se of all cells in an inersecion M se of all movemen cells (i.e., serving a movemen lef urn, hrough, or righ urn a an inersecion) S se of all source cells a an inersecion J (i) se of all cells wih movemens ha conflic wih he movemen of cell i I R se of all receiving cells S (k) se of all cells receiving vehicles from cell k R a a neighboring inersecion C (k) se of all movemen cells sending vehicles o receiving cell k R D (i) se of all cells ha are immediaely downsream of cell i I U (i) se of all cells ha are immediaely upsream of cell i I Decision Variables: x i number of vehicles in he source cell i S a ime sep T y ij number of vehicles flowing from he source cell i S o cell j D(i) a ime sep T q i number of vehicles in cell i M a ime sep T Z i oal available capaciy a he cell receiving vehicles from cell i M a ime sep T n i number of vehicles leaving cell i M a ime sep T g i sae of signal serving cell i M a ime sep T ; 0 if red, and 1 if green Parameers: D i demand a source cell i S a ime sep T n k number of vehicles sen ou of receiving link k R a ime sep T; is values are deermined by neighboring inersecions Q i sauraion flow rae for cell i I L i number of lanes in cell i M N i maximum number of vehicles ha cell i M can have h s sauraion headway in seconds w i weigh facor for cell i M ime sep in seconds LT sar-up los ime in seconds f fracional reducion of sauraion flow rae o accoun for he sar-up los ime ravel ime from upsream inersecion o he conneced vehicle range of downsream inersecion FIGURE 1 shows schemaics of a hypoheical nework and how i will be disribued o inersecion level. We have assumed ha he disance beween inersecions is less han he communicaion range of Road Side Unis (RSU) in FIGURE 1. As a resul, vehicles will arrive o he source cell of downsream inersecions righ afer leaving upsream inersecions. 6

7 FIGURE 1 Discreizaion of he nework ino individual inersecions wih corresponding informaion exchange (for simpliciy, only hrough movemens are shown) The objecive funcion of he problem is o maximize he inersecion hroughpu while penalizing for queue lengh. Inersecion hroughpu is found by summaion of he number of vehicles n i processed over all movemen cells i M over all ime seps T. We define w i as a weigh facor for a movemen cell i M o penalize for he queue lengh. I is calculaed by he cubic of acual queue lengh in ha cell. As such, w i is fixed in each mahemaical program or in oher words, in each predicion period. As a resul of implemening his weigh facor, he movemen ha has longer queue lengh ges more weigh and is more likely o receive more green during he predicion period. We define he penaly erm as he summaion of he produc of w i and number of vehicles q i over all movemen cells i M over all ime seps T. The objecive funcion of he problem is shown in (1). max (n i ) T i I M (w i q i ) T i I M (1) s.. 2 g i 4 i I M T (2) g i + g j 1 i M, j J (i), T (3) q i = q 1 i n 1 1 i + y ji x i = x i 1 j D (i) 1 y ij + D i 1 i M, j U (i), T (4) i S, T (5) 7

8 Z k = Z 1 k + n k 1 k R, T (6) j C (k) n j 1 y ij = min(q i, Q j, x i, N j q j ) i S, j D (i), T (7) n i L i ( Δ hs ) g i i M, T (8) n i n i (L i ) ( Δ hs ) (f)(l i ) (Δ hs ) (g i g i 1 ) i M, T (9) q i i M, T (10) n i i C (k) Z k k R, T (11) q i N i i M, T (12) x i, y ij, n i, q i, Z k 0 i M, k R j D (i), T g i {0,1} i M, T (14) Consrains (2) and (3) make sure ha no more han neiher four movemens (e.g., easbound lef, hrough, and righ plus souhbound righ) will receive he green signal a he same ime nor conflicing movemens receive he green ime a he same ime, respecively. Consrains (2) ensures ha he summaion of he binary variable g i over all movemen cells i M a ime sep T is greaer han or equal o wo and less han or equal o four. Consrain (3) ensures ha he summaion of green imes for movemen cell i M for all conflicing movemen j J (i) is less han or equal o one hus, only one of hem will receive green signal a a ime sep. This formulaion does no follow any ring-barrier conrol sraegy. Consrain (4) updaes he number of vehicles in each movemen cell over ime based on he following concep: he number of vehicles q i in movemen i M a a ime sep, is equal o he number of vehicles ha he cell had a he previous ime sep q 1 i, minus he number of vehicles n 1 i ha lef he cell a he previous ime sep, plus he number of vehicles y 1 ji ha enered he cell from upsream cell j U (i) in he previous ime sep. Consrain (5) represens he same concep for he source cells. Consrain (6) updaes he available capaciy of each receiving cell a each ime sep. The available capaciy Z k of receiving cell k R a ime sep T is equal o is available capaciy a he previous ime sep, minus he oal number of vehicles 1 j S k n j ha lef each sending cell j C k a previous ime sep, plus he number of vehicles n k 1 ha lef he receiving cell k R a he previous ime sep. Noe ha n k 1 is an inpu parameer o his mahemaical program and is found by opimizing he iming of neighboring inersecions. More deails are available in he soluion echnique secion. Consrains (7) limis he number of vehicles y ij ha flow from source i S o movemen cell j D (i) a ime sep T o four erms: a) sauraion flow rae of source cell i S, b) sauraion flow rae of movemen cell j D (i), c) he number of vehicles available a source cell i S, and d) he available capaciy on movemen cell j D (i) a ime sep T. The available capaciy is found by subracing he number of vehicles q j on cell j D (i) a ime sep T from is iniial capaciy N j. Consrain (8) limis he number of vehicles n i processed on movemen cell i M a ime sep T o he maximum number of vehicles ha could be processed in each ime sep T if he signal is green (g i ). The maximum number of vehicles ha can be processed during a ime sep wih duraion of Δ second wih a sauraion headway of h s is equal o Δ hs per lane. Assuming link i M has L i lanes, and having defined g i as a binary variable, he maximum number of vehicles ha can be processed can be (13) 8

9 found as shown in he Consrain. Consrain (9) accouns for he sar-up los ime in he even of changing he signal from red o green. Consrain (10) ensures ha he number of vehicles n i processed in movemen cell i M is less han or equal o he number of vehicles ha exis on ha cell a each ime sep T. In addiion, Consrain (11) limis he oal number of vehicles i C (k) n i released from all inersecion cells i C (k) ha are feeding receiving cell k R o is available capaciy Z k a each ime sep T. Finally, he number of vehicles q i in movemen cell i M should be less han or equal o is capaciy N i in each ime sep T. Consrain (13) is he non-negaiviy consrain and Consrain (14) ensures g i is a binary variable. In consrucing he proposed formulaion and he soluion echnique, we have assumed ha all vehicles are conneced, inersecions are equipped wih RSUs and neighboring inersecions communicae wih each oher, and he communicaion ime beween inersecions is negligible. We have also assumed ha when he disance beween wo inersecions is greaer han RSU communicaion range, addiional RSUs will be used o cover he corridor and communicae he arrival ime of vehicles o he downsream inersecion (see equaion 16). 4. Soluion Technique A cusomized soluion echnique is developed o find near-opimal signal iming parameers in he nework over he enire sudy period. To accoun for unforeseen changes in raffic demand and capaciy, and reducing complexiy of he problem a rolling horizon soluion echnique is developed. The proposed approach collecs daa a a ime sep, predics raffic sae for a cerain number of ime seps (a.k.a. predicion period), and finds opimal signal iming parameers ha yield highes objecive funcion values a each inersecion over he predicion period. The decision on erminaion or coninuaion of exising green phases will be implemened for he nex ime sep (or a number of hem). The proposed DC approach decomposes he nework ino several individual inersecions. As such, he proposed MILP needs o be solved for each inersecion. However, Consrain (7) is no linear and makes he mahemaical program hard o solve. To avoid inroducion of new binary variables, we conver i o Consrains (7.1) (7.4). They limi he number of vehicles flowing from he source cell i S o downsream cell j D (i) o cerain values; however, he opimal soluion may no be binding any of hem. In such case flow may be unnecessarily held in he source cells. This problem is called flow holding back problem (39). y ij Q i i S, T (7.1) y ij Q j i I, j D (i), T (7.2) y ij y ij x i N j q j i I, j D (i), T (7.3) i I, j D (i), T (7.4) To avoid he flow holding back problem, y ij needs o be maximized. As such he objecive funcion is updaed o reflec his change. The updaed mahemaical program is as follows: max (n i ) T i I M (w i q i ) T i I M + y ij s.. (2)-(6), (7.1)-(7.4), and (8)-(12). The demand level a each source cell will be passed from he neighboring inersecions as will be discussed laer in his secion. The soluion o he updaed MILP model can be found in real-ime for each inersecion wih selecing an appropriae combinaion of predicion period and ime sep duraions and T i I j D (i) (15) 9

10 availabiliy of compuer resources. In fac, allocaing one compuer o each inersecion, seing he duraion of ime sep o wo seconds, and he predicion period o hiry ime seps guaranees real-ime soluions. Assuming ha each inersecion is equipped wih RSUs and all vehicles are conneced, he number of vehicles in each inersecion approach and lane wihin he RSU range (1,000 f) is colleced a each ime sep. In addiion, inersecions will share he number of vehicles n k hey will be sending o oher inersecions from differen lanes and heir available capaciy Z k o receive vehicles in each cell k C (k) over all ime seps wihin he predicion period T. + The demand level D s a cell s S on an inersecion a ime + is equal o he oal number of vehicles i C (s) n i processed from differen cells i C (s) a upsream inersecion a ime, where is he associaed ravel ime, see Equaion (16). + D s = s S (16) i C (s) ni Finally, o updae he available capaciy of a cell a an approach, he number of vehicles n k ha are leaving he receiving cell k R a ime sep T is needed. This number is equal o he number of vehicles processed from he same link in he downsream inersecion. Since he opimizaions a differen inersecions happen simulaneously, he number of vehicles being sen and he available capaciy from he previous opimizaion over he predicion period will be used for he curren opimizaion run. FIGURE 2 shows he layou of a single inersecion wih corresponding parameers as described in TABLE 1. Three cells show hree movemens of an approach: lef, righ and hrough movemen. We implemened he proposed approach in a simulaed environmen creaed by VISSIM (40). The proposed approach collecs daa, opimizes he signals and implemens he soluions in VISSIM. We used VISSIM s Componen Objec Model (COM) for his purpose and CPLEX (41) as he opimizer. FIGURE 2 Layou of a single inersecion 10

11 5. Case Sudy We esed he proposed DC approach on wo case sudy neworks. The firs nework consiss of wo inersecions, 14 wo-lane links, and six origin-desinaion pairs, as shown in FIGURE 3.a. All roadways are wo-way and cells 1 o 6 are source cells of he nework. 1 2 (a) Case sudy 1 nework of wo inersecions (b) Case sudy 2 nework of nine inersecions FIGURE 3 Schemaics of he case sudy neworks The second case sudy nework has nine inersecions, 48 wo-lane links and welve origin-desinaion pairs, as shown in FIGURE 3.b. All he roadways are wo-way and no lef urn is considered. Cells 1 o 12 are nework source cells. Furher deails abou he wo case sudies are presened in TABLE 2. TABLE 2 Case Sudy Informaion Parameer Case Sudy 1 Case Sudy 2 Lengh of links (m) Free flow speed (mph) Sauraion flow rae of links (vphpl) Sauraion headway (sec) 2 2 Time sep duraion (sec) 2 2 Predicion period (ime sep) Reducion of he sauraion flow, f (%) Maximum queue lengh, N i (vehicle/link) Travel ime from upsream inersecion o range of downsream inersecion, (sec) 0 0 Analysis period (sec) To es he abiliy of he proposed approach o respond o differen demand condiions, he following four demand paerns are used, covering boh symmeric and asymmeric paerns wih boh undersauraed and oversauraed flow condiions as follows: 1. Symmeric undersauraed demand paern (DP1): 400 vehicles per hour per lane (vphpl) in each enry link. 11

12 2. Symmeric oversauraed demand paern (DP2): 900 vphpl in each enry link. 3. Asymmeric undersauraed demand paern (DP3): 1,000 vphpl in eas wes srees and 400 vphpl in norh souh srees. 4. Asymmeric parially oversauraed demand paern (DP4): a. Case sudy 1: 900 vphpl in corridors 1-2, 5-6 and 6-5, 400 vphpl in corridors 2-1, 3-4 and 4-3. b. Case sudy 2: 900 vphpl in corridors 3-4 and 9-10, 700 vphpl in corridors 7-8, 11-12, and 600 vphpl in corridors 1-2, and 400 vphpl in corridors 5-6. The formulaion was run on a compuer wih a 3.67 GHz quad core CPU and 32 GB of memory. For sixy seconds of predicion period wih wo-second ime seps, he average running ime of he MILP is seconds. 6. Resuls In his secion, he resuls of he applicaion of he proposed DC approach o he wo case sudies will be discussed. In addiion, he soluions are compared o wo ses of signal iming parameers: a) a coordinaed fixed-ime and b) and an acuaed coordinaed signal seing. VISTRO (42) was used o find opimal signal iming parameers for boh cases. Afer finding he soluion in VISTRO, hey were implemened in VISSIM for he comparison and five nework-level measures of effeciveness (i.e., number of compleed rips, average ravel ime, average delay, oal number of sops, and average speed) were used o compare he performances of he hree approaches. We acknowledge ha VISTRO do no uilize he addiional daa ha conneced vehicle echnology provides, and as a resul, is soluions may no be as efficien as he soluions of he proposed DC approach. However, in he absence of signal iming mehods for conneced environmen, VISTRO provides benchmark soluions on nework performance. TABLE 3 summarizes average ravel ime, number of compleed rips, average delay oal number of sops, and average speed for all demand paerns in case sudy 1. The resuls of he DC approach, fixed coordinaed, and acuaed coordinaed approach are presened as well. Demand Paern Symmeric Undersauraed Symmeric Oversauraed Asymmeric Undersauraed Asymmeric Parially Oversauraed TABLE 3 Nework Performance of Case Sudy 1 for Differen Signal Timing Sraegies Signal Sysem Average Travel Time (sec/veh) Performance measures Number of Average Compleed Delay Trips (veh) (sec/veh) Number of sops Average speed (mph) DC Approach Fixed-Time Coordinaed Acuaed Coordinaed DC Approach Fixed-Time Coordinaed Acuaed Coordinaed DC Approach Fixed-Time Coordinaed Acuaed Coordinaed DC Approach Fixed-Time Coordinaed Acuaed Coordinaed The DC approach significanly improved nework level raffic operaions in all demand paerns compared o benchmark soluions. For insance, in symmeric oversauraed demand paern, i reduced average ravel ime by 48% compared o he fixed ime and acuaed coordinaed approaches. A he same ime, i increased he number of compleed rips by 5.8%. Similar rends were observed in oher demand 12

13 Queue Lengh (Veh) Queue Lengh (Veh) Queue Lengh (Veh) Queue Lengh (Veh) Islam and Hajbabaie paerns, of course, o differen exens. The DC approach also found fewer number of sops and higher average speeds in he nework indicaing a more efficien signal coordinaion. FIGURE 4 shows he queue lengh and signal saus on easbound approach of inersecion one of he firs case sudy from ime sep 100 o 200. Solid line shows ha he signal serving he easbound approach is green, while he doed line shows queue lengh on he same approach over ime. The proposed DC approach conrolled he queue lengh in all demand paerns. The maximum queue lengh observed in case sudy one was 14 vehicles (in boh lanes) in symmeric oversauraed demand paern. As expeced, queue lengh in undersauraed flow condiions was shorer han ha in oversauraed flow condiions. In addiion, he proposed approach effecively allocaed he green ime o he easbound approach o limi he queue lengh. The DC algorihm provides almos equal green imes o each approach (51% and 49% of green ime for eas-wes direcion in DP1 and DP2, respecively) in boh symmeric demand paerns. For he asymmeric demand paerns, he signal conroller give prioriy o he srees wih higher demand. As a resul, i allocaes longer green imes o phases serving eas-wes direcions (71% and 68% green ime for eas-wes direcion in DP3 and DP4-(a), respecively) Time Sep (2 sec) (a) Time Sep (2 sec) (c) Time Sep (2 sec) (b) Time Sep (2 sec) (d) FIGURE 4 Queue lengh and green ime for inersecion 1 of case sudy-1 in differen demand paerns. (a) Symmeric unsauraed, (b) Symmeric oversauraed, (c) Asymmeric undersauraed, and (d) Asymmeric parially oversauraed. 13

14 FIGURE 4 also shows ha queue sars o grow or remain he same righ afer he signal urns red. If he demand is high, queue may coninue growing even when he signal urns green; however, afer a few seconds, i sars o decrease. TABLE 4 summarizes average ravel ime, number of compleed rips, average delay, oal number of sops, and average speed in he nework for all demand paerns in case sudy 2. The resuls of he DC, fixed-ime coordinaed, and acuaed coordinaed approaches are presened as well. Demand Paern Symmeric Undersauraed Symmeric Oversauraed Asymmeric Undersauraed Asymmeric Parially Oversauraed TABLE 4 Nework Performance of Case Sudy 2 for Differen Signal Timing Sraegies Signal Sysem Average Travel Time (sec/veh) Performance measures Average Delay (sec/veh) Number of Compleed Trips (veh) Number of sops Average speed (kmph) DC Approach Fixed-Time Coordinaed Acuaed Coordinaed DC Approach Fixed-Time Coordinaed Acuaed Coordinaed DC Approach Fixed-Time Coordinaed Acuaed Coordinaed DC Approach Fixed-Time Coordinaed Acuaed Coordinaed The DC approach significanly improved nework operaions in all demand paerns. For insance, in asymmeric parially oversauraed demand paern, i reduced average ravel ime by 27% compared o boh fixed-ime and acuaed coordinaed approaches. As he same ime, i increased he number of compleed rips by 5.45% and 5.53%, respecively. Similar rends were observed in oher demand paerns wih differen exens. FIGURE 5 shows he queue lengh and signal saus on easbound approach of inersecion five of he second case sudy from ime sep 100 o 200. The DC approach allocaed he green imes almos equally o boh direcions in all demand paerns excep for he asymmeric undersauraed demand paern. In his demand paern, signal conrollers give eas-wes direcion higher prioriy (74%) because of is high demand level (1000 vphpl and 400 vphpl demands in eas-wes and norh-souh direcions, respecively). Noe ha demand level on norh-souh and eas-wes direcions of inersecion five is 900 vphpl in asymmeric parially oversauraed demand paern. 14

15 Queue Lengh (Veh) Queue Lengh (Veh) Queue Lengh (Veh) Queue Lengh (Veh) Islam and Hajbabaie Time Sep (2 sec) (a) Time Sep (2 sec) (c) Time Sep (2 sec) (b) Time Sep (2 sec) (d) FIGURE 5 Queue lengh and green ime for case sudy 2 in differen demand paerns. (a) Symmeric unsauraed, (b) Symmeric oversauraed, (c) Asymmeric undersauraed, and (d) Asymmeric parially oversauraed. FIGURE 6 shows vehicle rajecories in a ime-space diagram on easbound direcion of corridor 3 4 in case sudy 2 for all differen demand paerns. The rajecories are shown for he ime seps 100 o 300 (200 seconds) o make sure ha he graphs are legible. Noe ha for he signal plans shown in FIGURE 6, a green period for he easbound direcion is shown as a solid green line. The green imes a each inersecion appear o be coordinaed o reduce he number of sops for plaoons raveling on he corridor. Similar resuls were found for oher approaches as well. Noe ha our proposed formulaion did no include any explici consrain on offses for signal coordinaion. The DC approach yielded signal coordinaion by he exchange of informaion in neighboring inersecions aiming a maximizing nework hroughpu. As eviden, mos vehicles go hrough he corridor wih a mos one sop while a significan number of hem do no sop a all. Furhermore, no vehicle needed o sop a all hree inersecions. In all he demand paerns excep asymmeric undersauraed flow condiion, volume from boh norh-souh and eas-wes direcions are he same. However, in asymmeric undersauraed demand paern, eas-wes direcion has more demand level (1000 vphpl) han he oher direcion (400 vphpl). Therefore, he DC approach allocaes more green ime o eas-wes direcion. 15

16 FIGURE 6 Trajecories of vehicles wih signal sae for eas direcion. (a) Symmeric unsauraed, (b) Symmeric oversauraed, (c) Asymmeric undersauraed, and (d) Asymmeric parially oversauraed. FIGURE 7(a) shows he oal number of sops on each link for differen demand paerns. The DC approach reduced he number of sops in all cases excep for he symmeric oversauraed condiions compared o VISTRO soluions. In addiion, DC approach found soluions wih significanly lower delays compared o oher wo approaches. This indicaes ha while DC soluions in symmeric oversauraed condiions sopped vehicles more ofen, he duraion of sops were shorer. In addiion, he DC soluion yielded higher average speeds in each nework links compared o VISTRO soluions, see FIGURE 7(b). These findings are ineresing in ha wih no explici consrain for signal coordinaion, he DC approach has found soluions ha coordinae he signals o some degree. 16

17 (a) (b) FIGURE 7 Link Level Performance measures. (a) number of sops and (b) average speed in mph 7. Conclusion This sudy presens a novel disribued-coordinaed formulaion and soluion echnique for signal iming opimizaion on conneced urban sree neworks. Raher han making nework-level decision wihin a cenral archiecure, decisions on erminaion or coninuaion of green imes are made a he inersecion level. As a resul of his disribuion, he complexiy of he problem is significanly reduced, signal iming parameers are found in real-ime, and he approach is scalable. Inersecions coordinae heir decisions o avoid finding locally opimal soluions and insead move owards global opimaliy. The proposed rolling horizon soluion echnique collecs daa on queue lengh and he number of incoming vehicles a each ime sep, finds signal iming parameers ha yield opimal hroughpu values over a predicion period, and makes decisions on green signal coninuaion or erminaion for he nex ime sep. The proposed soluion echnique was used o conrol signals in wo case sudies under four differen demand paerns and he resuls were compared o fixed-ime coordinaed and acuaed coordinaed signal iming parameers ha were obained by VISTRO. The proposed disribued coordinaed approach yielded significanly more efficien nework-wide raffic performance in all cases, conrolled queue lengh and prevened queue spillbacks, coordinaed he signals wihou having any explici offse variables and consrains, and can respond o unforeseen and abrup changes in demand level or capaciy. This sudy was aimed a paving he way for disribued-coordinaed signal iming opimizaion and assumed insananeous and disrupion free communicaion, 100% conneced vehicles, full coverage of corridors wih addiional RSUs, and communicaions beween inersecion conrollers. I is noed ha accouning for differen conneced vehicle peneraion raes will solidify he conribuions of he proposed algorihm, however, such exensive experimen and analysis are ou of he scope of his work. Sudying he impacs of disrupions in communicaions, communicaions delays, lower conneced vehicle peneraion raes, and lack of communicaion beween inersecions should be considered in fuure sudies. 17

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20 40. PTV Planung Transpor Verkehr, H.-u.-N.-S., Karlsruhe Germany., Pv Vissim 7 User Manual, , I.C., Ibm Ilog Cplex Opimizaion Sudio Cplex User s Manual Version 12 Release America, P., Pv Visro User Manual,