Studies of Emergency Evacuation Strategies based on Kinematic Wave Models of Network Vehicular Traffic

Transcription

1 Sudes o Emergency Evacuaon Sraeges based on Knemac Wave Models o Nework Vehcular Trac KAI-FU QIU Deparmen o Auomaon, Unversy o Scence and Technology o Chna Hee, Chna 2327 prc@mal.usc.edu.cn Absrac- How o ecenly conrol rac durng emergency evacuaon s an mporan research ssue. An emergency evacuaon sraegy, one o he man conrol sraeges, ams o deny he bes roung sraegy so as o ully ulze he avalable capacy o a ransporaon nework. In hs sudy, we model he evacuaon rac wh a knemac wave model o nework vehcular rac. We presen wo evacuaon roue gudance sraeges: one s o maxmze he oal number o vehcles evacuaed rom he orgn zone durng a perod o me, and he oher s a myopc sraegy based on local rac supples o downsream lnks a an nersecon. The rs sraegy s an olne sraegy and can be solved by a genec algorhm, whle he second one can be solved onlne. The perormances o he proposed mehods are esed wh a smple road nework. Keywords- Emergency evacuaon; Roue gudance; Knemac wave model; Genec algorhm; Trac supply Ι. INTRODUCTION Naural and soceal dsasers may cause wdespread and severe damages or congeson o avalable ransporaon neworks. For some dsasers, s also necessary o evacuae people ou o a hazard zone n order o releve urher negave mpacs. Thereore, s mporan o develop eecve emergency evacuaon sraeges, whch am o deny he bes roung sraegy so as o ully ulze he avalable capacy o a ransporaon nework. In leraure here have been many sudes on emergency evacuaon. Many sudes adoped rac assgnmen models o deal wh he evacuaon roue gudance problem. MASSVAC [1] adoped olne rac assgnmen algorhms o deermne evacuaon roues or evacuees deparng rom her orgns. Oher sudes adoped dynamc rac assgnmens (DTA) o solve he evacuaon roue gudance problem, e.g. n he wake o a nuclear power plan alure [2]. In addon o rac assgnmen mehod, myopc roue gudance sraeges have also been proposed o deermne roue gudance by rac condons o each oubound lnks n each smulaon nerval. NETVAC1 [3] allows dynamc roue gudance n each me nerval, and he selecon s based on wo acors: he rs s he speed on each o he alernave oubound lnks, and he second he pror knowledge, such as he number o lanes o each lnk. CEMPS evacuaon model [4] akes accoun o mmedae congeson o he avalable roads. WEN-LONG JIN Deparmen o Cvl and Envronmenal Engneerng Unversy o Calorna Irvne, CA wn@uc.edu, Correspondng auhor Dependng on he obecves o emergency evacuaon, many oher models were used n sudyng he emergency evacuaon. Cova e al. [5] presened a lane based evacuaon sraegy wh wo roung prncples. The rs s o roue vehcles o her closes evacuaon zone, and he second s o mnmze he number o nersecon conlcs so ha he nersecons can be n a sae o unnerruped low. Yamada [6] appled shores pah algorhm o assgn evacuees o her opmal shelers as well as he opmal roue. I ook no accoun o he capacy consrans o places o reuge. The obecve s o mnmze he oal ravelng dsance by all evacuees. In our paper, we propose wo roue gudance sraeges. The rs s o maxmze he oal number o vehcles evacuaed rom a hazard zone durng a xed perod o me. The second s myopc a all nersecons by deermnng urnng proporons based on local rac supples o downsream lnks. Boh sraeges are based on a knemac wave model o nework vehcular rac. Ths model s ecen, snce no pah normaon s consdered snce we are consderng he ask o evacuae vehcles ou o a hazard zone as as as possble. The remander o hs paper s organzed as ollows. Secon II wll dscuss he obecve o evacuaon and presen a knemac wave model o model rac dynamcs n road neworks. Secon III wll ormulae wo evacuaon sraeges. Secon IV wll presen numercal resuls wh a smple road nework. Secon V wll conclude our sudy wh some remarks and uure sudes. II. EVACUATION OBJECTIVE AND TRAFFIC FLOW MODEL Emergency evacuaon pus he road nework under exreme sresses. Whou ecen plannng, he evacuaon process can be pu no hal. Thus s mporan o undersand he eecveness o deren evacuaon plans. Many acors can nluence he eecveness o deren evacuaon plans, such as he opology o a road nework, weaher, and rac managemen and conrol mechansms. Dependng on emergency suaons, we can also have deren evacuaon obecves. Among many emergency suaons are hose when as many as vehcles should be evacuaed ou o a hazard zone durng a lmed amoun o me. For example, one wshes o maxmze he evacuaed number o vehcles

2 when a hurrcane s o srke a regon. Under such suaons, we can hen assume ha () vehcles do no have preerred desnaon zones, and () he road nework sel s nac and sae. A. Eecveness measure and obecve o evacuaon plans Gven a me perod o T, he evacuaed number o vehcles ou o a hazard zone s he cumulave low, N, a he orgn,.e., he hazard zone. Snce cumulave low s he oal number o cars evacuaed durng he me nerval o, hen he cumulave low a orgn s N([, ]) = ( s) ds (1) s= where () s he boundary lux ou o orgn a me. In he dscree orm, cumulave low can be compued by N([, ]) ( ) = s Δ, where s he sar o he s= Δ evacuaon perod, and Δ he end. The evacuaon obecve uncon may be ormulaed as Max N[, ] (2) subec o:, () = 1,, () (3) DL() where N [, ] s he cumulave low durng he me duraon o and ; DL () s he downsream lnks o nersecon ;, () s he urnng proporon o h downsream lnk o nersecon a me. Ths sudy ocus on he roung acors o, ().A good roung sraegy, wll evacuae as many evacuees as possble whn a lmed me s crcal durng emergences. B. Formulaon o rac low model To develop an eecve evacuaon plan, one has o choose an approach o modelng he rac nework and evoluon o rac dynamcs. In hs sudy, we employ a knemac wave model presened by Jn [7], bu whou consderng vehcles pahs, whch are rrelevan snce vehcles do no have preerred desnaons. Whou consderng commodes, he new knemac wave model s more ecen and more suable or emergency evacuaon. The evoluon o rac dynamcs on a undreconal road lnk can be modeled by he LWR model [8,9]. The LWR model s one o he mos acceped models o rac low. I akes no accoun o densy and low-rae and can capure some rac phenomena such as shock wave. The LWR model s ormulaed as ollowng: + ( q( )) x = (4) where s he densy, q( ) s he low-rae, and x and are he space and me varables. When he LWR model s used o smulae he densy wave a lnk boundary wh a sep uncon as nal condon, s called Remann problem. Numercally, we can use Godunov mehod [1] o solve he LWR model. In he Godunov mehod, each lnk s spl no M cells wh a cell lengh o Δ x, and he me nerval s dvded no K me seps wh a me sep o Δ. Δ/ Δ x should sasy he CFL condon whch means ha a car wh ree low speed canno surpass a cell durng a me sep. The Godunov mehod s llusraed as Fg 1.The Godunov-ype ne derence equaon or oal low n cell rom me sep o me sep + 1 s /2 1/2 + = (5) Δ Δx where s he average densy n cell a he me sep o, 1 and + s he average densy n cell durng he me sep o + 1 ; 1/ 2 ( + 1/2 ) denoes he lux hrough he upsream (downsream) boundary o cell. Gven rac condon o cell a me sep, we can oban rac condons o me sep Δ = + ( 1/2 + 1/2) (6) Δx We can see ha we wan o acqure rac condon o all me seps, s crcal o oban he boundary luxes. The boundary luxes are deermned by s upsream and downsream cells, can be compued by he ollowng supply-demand mehod [11,12], where demand s he maxmum sendng low o a cell, and supply s he maxmum recevng low o a cell. We dene rac demand as q( ) when s under crcal D = max q when s over crcal and rac supply as ( + 1) Δ Δ 1/ 2 1 ( 1) Δx /2 + 1 x Δ ( + 1) Δx Fg. 1. Godunov mehod or he LWR model x (7)

3 S = q( ) when s over crcal max q when s under crcal (8) should ake oher oubound lnks. For an nersecon wh U upsream cells and W downsream cells, he boundary luxes are compued as [7,13,14] D U 1/ 2 = d= 1 u 1 u d = d mn { D, S / } 1/ 2, d 1/ 2 d = d = 1, 2, L, W Du 1/ 2, u = 1/ 2 u 1, 2,, U U = L D (9) u= 1 u where d s he proporon o vehcles headng downsream 1/ 2 1/2, d cell d ; s he boundary lux, s he nlow o 1/2, u downsream cell d, s he oulow o upsream cell u. Compared wh he mul-commody knemac wave model n [7], he knemac wave model appled n hs paper s more ecen and suable or large-scale nework evacuaon applcaon, snce hs model does no rack he commodes whch are derenaed by pahs or OD pars. I a road nework has P pahs, hen here are P commodes o rac low on he road nework, and he model n [7] consders evoluon o rac lows o all commodes. Bu durng he evacuaon rac, rack o every commody s no needed, consderng ha vehcles do no have preerred desnaons. Thus n he revsed model, by gnorng he commodes, he compuaonal cos decreases sgncanly and road neworks are much easer o se up. III. FORMULATION OF EVACUATION PLANS Applyng he knemac wave model o nework vehcular rac low dscussed n he precedng secon o model he evoluon o rac dynamcs, we wll dscuss wo evacuaon sraeges, whch are deermned by he urnng proporons a all nersecons wh more han one oubound lnks. The rs one s an olne sraegy solved by a genec algorhm (GA). The second sraegy s a dynamc sraegy n whch unng proporons are deermned a each smulaon me nervals by supply o each downsream lnk. A. Olne opmzed evacuaon sraegy An olne evacuaon roung sraegy evacuaon plan can provde necessary normaon o communy agences and he publc n advance o an emergency suaon. Followng he olne approach, we should deermne a se o urnng proporons a each nersecon whch has several oubound lnks. The urnng proporons wll nsruc how many vehcles should ake one oubound lnk and how many Fg. 2. Flow-char or solvng olne evacuaon sraegy wh genec algorhms In hs sudy, we use a genec algorhm (GA) [15,16] o solve he opmzed evacuaon roung problem and nd a se o opmal urnng proporon. GA maps he decson varables o urnng proporons by a srng (chromosome) o bnary alphabes (genes). Frs, a generaon o srngs represens he decson varables are randomly creaed. Then he evaluaon procedure decodes he decson varables o calculae he obecve uncon value whch s names as he ness o a srng. Fness n hs sudy s he cumulave low. Once all srngs are evaluaed, hree basc genec operaors-selecon, crossover and muaon are used o creae an osprng generaon. Selecon s a process o selec members o orm he parens generaon ha o as he bass or creang he ollowng generaon. The selecon operaors have a bas oward hgher ness srngs. Ths bas ensures ha hgh ness ones should survve and breed. Then, he crossover operaors randomly selec wo paren members, and some bs (genes) o he wo members are swapped. Fg 2 represens he process n low-char orm. The muaon operaors play a secondary role wh respec o selecon and crossover operaors. The muaon operaors wll change gene o o 1 and 1 o. Once hree operaors process are execued, a new generaon s creaed. Ths populaon s hen evaluaed and he process begns agan. Aer a number o generaons, we can oban he opmal proporons by decode he bes sraegy genes.

4 B. Supply-proporonal evacuaon sraegy Evacuaon s que a sochasc process because every elemen, such as human behavor s unpredcable. Durng emergences, hgh demand level oen leads buldup o rac ams all over he ransporaon nework. Olne rac roung sraegy may al o cach on he congeson accuraely. In hs secon, we presen a dynamc evacuaon mehod ha urnng proporons are deermned a each smulaon me nerval. We can ormulae he dynamc roue gudance as ollowng: S, (), () = (1) S () k k, where, () s he proporon akng h downsream lnks o nersecon a me, S, () s he supply o h downsream lnk a me, and S k, () k s he oal supply o all downsream lnks o nersecon. We can see ha a downsream lnk has more supply, more vehcles wll ake hs lnk. IV. SIMULATION RESULTS In hs secon, we sudy he proposed evacuaon sraeges wh a road nework as shown n Fg. 3. In hs nework, lengh o lnk 4 (L4) s 2 meers, and ohers have he same lengh o 1 meers. Lnk 4, lnk 8 and lnk 11 have wo lanes, and ohers have only one lane. The undamenal dagrams or all lnks are rangular n he ollowng orm: o each lane, and v =65 mph=29.1 m/s he ree low speed. Tha s, all lanes have he same undamenal dagram wh capacy o qc = cv. Here we smulae rac dynamcs durng a me nerval o 2 seconds, wh a me sep o.45s, and cell lengh.1455m. We assume a sucen large ravel demand a he orgn durng he whole evacuaon perod. A. Olne opmzed evacuaon roung resul Wh he proposed GA, hs secon nends o show he resuls or maxmzng he cumulave low a orgns durng he gven evacuaon duraon. In hs nework, we ocus on he urnng proporons o uncon, uncon 1 and uncon 3. Juncon connecs wo oubound lnks o lnk 4 and lnk 5, uncon 1 connecs lnk 6 and lnk 7, and uncon 3 connecs lnk 9 and lnk 1. The genec algorhm s perormed wh he ollowng parameers: he number o bs or codng each varable s 12; populaon sze s 2; he maxmum number o generaons s 2; he probably o crossover s.75; and he probably o muaon s.65. The opmal resuls are shown n Fg 4 and Table 1. Fg 4 shows he maxmum cumulave low o every generaon. TABLE 1 shows he opmal proporons o he hree uncons. The opmal roue gudance proporons are ound a he 75 h eraon, where he oal number o evacuaon vehcles was maxmzed when perormng he GA. Resuled oal number o evacuaon vehcles s 5338 durng he 2 seconds. By applyng he opmal roue gudance proporons, he cumulave low a orgn o lnk 4 s 262, lnk 5 s 1134, lnk 6 s 688, and lnk 7 s 914 v, ac qa (, ) = c v ( a ), ac a c (11) where s he oal densy o all lanes, lanes, a he number o =18 veh/mle=.112 veh/m he am densy o each lane, c = 36 veh/mle=.22 veh/m he crcal densy Orgn L Orgn 1 L1 J1 L5 L6 L7 L4 L8 J J2 J4 J6 J3 L9 L1 L11 J5 Desnaon L2 Desnaon 1 L3 Max N o every generaon generaon number Fg. 4. Maxmum cumulave low or every generaon Fg. 3. Trac nework

5 TABLE Ι OPTIMAL TURNING PROPORTIONS AT INTERSECTIONS Orgn o Lnk 4.7 Juncon Orgn o Lnk 5.3 Juncon 1 Juncon 3 Orgn 1 o Lnk 6.43 Orgn 1 o Lnk 7.57 Lnk 4 o Lnk 9.5 Lnk 4 o Lnk 1.5 B. Dynamc evacuaon roung resul Ths secon explores he uncon o he proposed dynamc evacuaon roung sraegy. The supply o a lnk can be ormulaed as ollowng: v C ac S = c v ( a ) ac a c (12) For uncon, he proporon o ake lnk 4 s shown n Fg 5; or uncon 1, he proporon o ake lnk 6 s shown n Fg 6. For uncon 3, he proporon o ake lnk 9 and lnk 1 s always.5 versus.5, because he rac low a lnk 9 and lnk 1 are ree low all he me. We can dvde he evoluon o urnng proporons no several sages. The rs sage s ha begnnng o he evacuaon perod, rac low a all lnks are ree low. So or uncon, he urnng proporon o ake lnk 4 s 2/3, whch s v4c4 /( v4c4 + v5c5). And or uncon 1, he proporon o ake lnk6 s 1/2, whch s v 6 c 6 /( v 6 c 6 + v 7 c 7 ). A me 1 =1/ v =34.4s, vehcles on lnk 7 reach uncon 4, a me o 2 =2/ v =68.8s, vehcles on lnk6 and lnk 8 arrve a uncon 4. Aer ha, a he merge uncon 4, he demand o upsream lnks s 3q c, whch s more han he supply o downsream lnk o lnk 11. The ou-lux o lnk 7 s 3 qc D7 /( D7+ D8) = 4 q c /3, and ou-lux o lnk 8 s 3 qc D8 /( D7 + D8) = 2 q c /3. A hs me, par o vehcles can no pass uncon 4, hen a shock waves wll orm a uncon 4 and ravelng backward on lnk7 and lnk 8 a he speed o c =17mph.. The shock wave wll rs hs uncon 1 hrough lnk 7 a he me o 3 = 2 + 1/ c = 26 s, hen he rac supply o lnk 7 s reduced o 2 q c /3, and supply o lnk 6 s q c, so he urnng proporon o ake lnk 6 s ncreased o q /( q + 2 q / 3) =.6, as shown n Fg 6. A he c c c 3 same me o, he back-ravelng shock wave a lnk 8 h uncon 2. The ou-lux o lnk 5 and lnk 6 become o 2 q c /3, hen a shock wave orm a uncon 2 and back ravelng a lnk 5 and lnk 6. A 4= / c =344s, he shock wave ravelng hrough lnk 5 hs uncon, and he supply o lnk 5 decreases o 2 q c /3, he proporon ake lnk 4 becomes o 2 qc /(2qc + 2 qc / 3) =.75. A he same me o 4, shock wave ravelng a lnk 6 hs uncon 1, supply o lnk6 decrease o 2 q c /3, hen he proporon ake lnk 6 become o (2 qc / 3) / (2 qc / 3+ 2 qc / 3) =.5. Aer 4, he proporons wll seady a.75 and.5. The evacuaon cumulave low o our supply mehod s shown n Fg 7. Durng he perod o 2 seconds, he oal number o evacuees s 54. Cumulave low a he orgn o lnk 4 s 262, lnk 5 and lnk 6 s 943, and lnk 7 s 912. Turnng Proporon a Juncon Turnng Proporon a Juncon Tme(s) Fg. 5. Turnng proporon a uncon Tme(s) Fg. 6. Turnng proporon a uncon 1 Compared wh he rs sraegy, he supply-proporonal evacuaon sraegy can ecenly capure rac dynamcs n real me. Also perorms beer han he rs sraegy ha can evacuae more vehcles. Bu does no mean ha he rs sraegy s valueless, snce rac supply o each lnk should be obaned by he nellgen ransporaon sysem echnology. I here s no such sysem, hen he olne sraegy can be appled n hs suaon.

6 Cumulave Flow N o L4 N o L5 N o L6 N o L Tme(s) Fg. 7. Evacuaon cumulave low or our lnks TABLE II. NOTATION TABLE N([, ]) oal number o evacuaed car durng o () boundary lux ou o orgn a me DL () downsream lnks o nersecon, () urnng proporon o downsream lnk o nersecon a me rac low desy q x 1/ 2 rac low-rae space varable me varable average densy n cell a he me sep lux hrough he upsream boundary o cell a me sep D demand o a cell S `supply o a cell P a number o pahs number o lanes h am densy o each lane c crcal densy o each lane v ree low speed c shock wave speed V. CONCLUSION In hs paper, based on a knemac wave model, we presened wo emergency evacuaon roue gudance sraeges. The rs sraegy can be solved olne wh a genec algorhm and oban a global opmal soluon o urnng proporons a nersecons. The second sraegy can be mplemened onlne by deermnng urnng proporons rom local rac supples. Whou consderng vehcles pahs, we expec hese sraeges o be ecen or large road neworks. Wh he smple approaches developed n hs sudy, we could sudy more complcaed evacuaon scenaros. For example, when a srong earhquake srkes, some brdges n a roadway sysem may be desroyed. In hs case, road neworks have o be moded n real-me, and evacuaon sraeges also have o accommodae such changes. However, here are ceran lmaons o hese sudes. For he rs sraegy, because s an olne sraegy, als o cach he changes o rac nework durng evacuaon. In he uure, we wll develop a dynamc opmal sraegy. ACKNOWLEDGMENT Ths work was suppored n par by Naonal Naural Scence Foundaon o Chna (No ), H-Tech Research and Developmen Program o Chna (863 Proec) (No. 27AA11Z222), and Naonal Basc Research Program o Chna (973 Proec) (No. 26CB7556). REFENRENCE [1] Hobeka, A.G. and Jame, B., MASSVAC: A model or calculang evacuaon mes under naural dsasers, Emergency plannng, Smulaon Seres, 15, 23-28, [2] Saayhaewa, P. and Ran, B., Developng a dynamc rac managemen model or nuclear power plan evacuaon, Presened a he 79h Annual Meeng o he Transporaon Research Board, Washngon, D.C., 2. [3] She, Y. Mahmassan and Powell, W., A ransporaon nework evacuaon model, Transporaon Research A, vol. 16, n.3, pp , [4] Pdd, M., de Slva and Eglese, R., A smulaon model or emergency evacuaon, European Journal o Operaonal Research, vol. 9, n.3, pp , [5] Thomas, J. Cova and Jusn, P. Johnson., A nework low model or lane-based evacuaon roung, Transporaon Research Par A, 37(23), pp , 23. [6] Takeo Yamada, A nework low approach o a cy emergency evacuaon plannng, Inernaonal Journal o Sysems Scence, 1996, vol. 27, n. 1, pp , [7] Jn,W.-L., and Zhang H.M., A mul-commody knemac wave smulaon model o nework rac low. Transporaon Research Record: Journal o Transporaon Research Board 1883,59-67, 24 [8] Lghhll, M.J. and Whham, G.B., On knemac waves: II. A heory o rac low on long crowded roads, Proceedngs o he Royal socey o London A 229(1178), pp , [9] Rchards, P.I., Shock waves on he hghway, Operaons Research 4, pp.42-51,1956. [1] S.K.Godunov, A deren mehod or numercal calculaons o dsconnuous soluons o he equaons o hydrodynamcs, Maemachesk Sbornk 47, pp , [11] C.F. Daganzo, The cell ransmsson model par : Nework rac. Transporaon Research Par B 29 (2),79 93, [12] J.P. Lebacque, The godunov scheme and wha means or rs order rac low models. In: The Inernaonal Symposum on Transporaon and Trac Theory, Lyon, France, [13] W.-L. Jn. Knemac Wave Models o Nework Vehcular Trac. PhD hess, Unversy o Calorna, Davs, Sepember 23. [14] Ka-Fu Qu, Lang Chen, Zhe Wang and Wen-Long Jn, A knemac wave model or emergency evacuaon plannng, presened a 14h World Congress On Inellgen Transporaon Sysem, Beng, 27. [15] Coln R. Reeves and Jonahan E. prncples and perspecves: a gude o GA heory, Boson : Kluwer Academc Publshers, c23. [16] J.Sender, E. Hllebrand and J. Kngdon. Genec algorhms n opmsaon, smulaon and modelng, Amserdam: IOS Press, c1994.`