A KINEMATIC WAVE MODEL FOR EMERGENCY EVACUATION PLANNING

Transcription

1 A KINEMATIC WAVE MODEL FOR EMERGENCY EVACUATION PLANNING KAI-FU QIU, LIANG CHEN, ZHE WANG WEN-LONG JIN Deartment of Atomation Center for Intelligent Transortation Systems University of Science and Technology of China PO Box 4, Hefei, Anhi P.R. China Tel: +86(0) Fax: +86(0) Jne 28, 2007 FINAL PAPER SUBMITTED TO 14TH WORLD CONGRESS ON INTELLIGENT TRANSPORT SYSTEMS 1

2 ABSTRACT Emergency evacation lans are critical to redcing damages cased by natral or hman-made disasters. In this aer, we resent a framework for develoing and evalating evacation strategies based on a kinematic wave model of network vehiclar traffic. We then discss a measre of the effectiveness of evacation lans and aly the model to stdy a simle road network. This model is sitable for emergency evacations, since road networks can be easily set and calibrated, comtational cost is indeendent of the nmber of vehicles, and traffic congestions cased by caacities of road links and network strctre can be effectively simlated. KEY WORDS: Emergency evacation lanning, kinematic wave model of network vehiclar traffic, measre of effectiveness 2

3 INTRODUCTION To redce damages cased by natral or hman-made disasters, it is necessary to evacate eole ot of the dangeros zone. Emergency evacation lans are critical in these sitations, esecially for massive evacations involving as many as millions of vehicles on a regional road network. None or inefficient evacation lans cold make evacation itself a disaster. For examle, dring the massive evacation for Hrricane Rita [1], millions of eole got stck in traffic ams as long as 100 miles, and many vehicles ran ot of gasoline after being on road for hors. When evalating the effectiveness of different evacation strategies, it is imortant to simlate or redict time-deendent traffic atterns nder different evacation strategies, since heavy congestion sally occr with a sdden srge of traffic demand. In literatre, varios traffic flow models have been sed when develoing and evalating evacation lans. For examles, a macroscoic model was sed in [2], DYNASMART-P in [3], and VISSIM in [4]. In this stdy, we roose a new latform for emergency evacation lanning based on a kinematic wave model of network vehiclar traffic. This model is based on the Lighthill-Whitham-Richards [5,6] model and, different from microscoic models, only considers caacities of road links and network strctre bt ignores heterogeneity in drivers behaviors, vehicle tyes, lanes, etc. It is simler to bild a road network, since only a limited nmber of arameters sch as nmber of lanes and seed limit are needed in this model. In addition, the comtational cost of the model is indeendent of the nmber of vehicles. All these featres make this model very sitable for emergency evacation lanning. In the rest of the aer, we first introdce a kinematic wave model of network vehiclar traffic in Section 2. In Section 3, we discss how to model evacation with the model and resent a measre of effectiveness of different evacation lans with cmlative flow. In Section 4, we stdy evacation lans for a simle road network. We conclde or stdy with some ftre stdies in the conclsion art. A COMMODITY-BASED KINEMATIC WAVE MODEL OF NETWORK VEHICULAR TRAFFIC For each road link, we se the following trianglar fndamental diagram to model the relationshi between density ρ (veh/km) and flow-rate q (veh/hr): 3

4 ρv f 0 ρ ρc Q( ρ) = ρ ρ ρcv f( ) ρc < ρ ρ ρ ρc (1) where v f is free flow seed, three arameters for each road link: ρ c critical density, and ρ am density. Here we only need v f is sally determined by the seed limit of a road, ρ is roortional to the nmber of lanes of a link, and ρ c is the traffic density when a road link reaches its caacity flow and also roortional to the nmber of lanes. In the fndamental diagram (1), traffic caacity is ρ cv f. The evoltion of traffic dynamics on a nidirectional road link can be modeled by the LWR model [5, 6]: ρ + ( Q( ρ)) = 0 (2) t When LWR model is sed to simlate the density wave at link bondary with a ste fnction as initial condition, it is called Riemann Problem. Nmerically, we can se Godnov method [7] to solve the LWR model. In the Godnov method, each link is slit into N cells, with cell length of Δ x, and the time interval is divided into K time stes, with a time ste of Then the Godnov-tye finite difference eqation for total flow in cell i from time ste to time ste + 1 is x Δ t. ρ + 1 i ρi fi+ 1/2 fi 1/2 + = 0 Δt Δx (3) where i 1/2 i 1/ 2 f ( f + ) denotes the flx throgh the stream (downstream) bondary of cell i. Bondary flxes can be comted by the following sly-demand method [8] [9]. Here we consider for tyes of bondaries: link bondaries, merging nctions, diverging nctions, and general intersections. 1. Link bondaries. Given xi 1/2 as a link bondary, its stream cell is denoted by and downstream cell is denoted by d. ρ, ρ is the density of cells and d resectively. We define the stream demand as q( ρ) when ρ is nder critical D = max q whenρ isover critical and define the downstream sly as max qd when ρd is nder critical Sd = q( ρd) when ρd isover critical then the bondary flx can be simly comted as d 4

5 where. fi 1/2, d i 1/2, d = d i 1/2, = d f min{ D, S }, (4) f min{ D, S } is the inflow of downstream cell d, and fi 1/2, is the otflow of stream cell 2. Merging nctions. Given xi 1/2 as a merging nction with P stream cells and a downstream cell, its stream cell are denoted by ( = 1,2,, P), downstream cell by d, stream demand by D, and downstream sly by S d. The bondary flxes are comted as P i 1/2, d = = 1 d f min{ D, S } D fi 1/2, = f 1/2, 1, 2,, i d = P P D = 1 (5) where fi 1/2, d is the inflow of downstream cell d, and fi 1/2, is the otflow of stream cell 3. Diverging nctions. Given xi 1/2 as a diverging nction with P downstream cells and an stream cell, its stream cell is denoted by, downstream cell by d ( = 1,2,, P), stream demand by D, downstream sly by S, and roortion of commodity in total traffic byξ. The bondary flxes are comted as P i 1/2, = = 1 ξ f min { D, S / } i 1/2, d i 1/2, ξ f = f = 1, 2,, P (6) 4. General intersections. For a general intersection with U stream cells and D downstream cells, we combine merge and diverge models together. 5

6 U U D 1 ξ D =, d fi 1/2 = min d= 1{ D, / } 1 S = d U D = 1 U D, 1 ξ = d fi 1/2, d = fi 1/2 d 1, 2,, D U = D = 1 = 1 D fi 1/2, = fi 1/2 1, 2,, U U = D (7) where ξ d, is the roortion of vehicles heading downstream cell d in stream cell. EVACUATION PLAN AND A MEASURE OF EFFECTIVENESS Under non-emergency sitations, eole wold selfishly choose their rotes, and a road network aroaches the state of ser eqilibrim [10]. For non-emergency traffic management, varios means, sch as signal control and congestion ricing, are alied to minimize total travel time of all drivers so that a road network is in the state of system otimal. To better evacate eole from dangeros zones in emergency sitations (e.g. hrricane, hazard material leakage, or dirty bomb attack), however, all vehicles wold be strictly gided with their deartre times and rotes, and the obective of traffic management wold be to evacate eole ot of the dangeros zones as fast as ossible, bt not to minimize the total time for them to arrive their destinations. If a transortation network has sfficient large caacity or the nmber of eole to be evacated is small, the task wold be relatively easy. However, for massive evacation, an extraordinary level of traffic demands can case serios congestion or even totally clog the whole transortation network. Withot carefl lanning, the evacation rocess can be t into halt as in the aforementioned case of Hrricane Rita. Ths it is imortant to nderstand the effectiveness of different evacation lans. Many factors can inflence the effectiveness of different evacation lans, sch as the toology of a road network, weather, and traffic management and control mechanisms. Since travel demand sally srasses the caacity of a road network in emergency sitations, it is imortant to simlate the formation and roagation of traffic qees at varios network bottlenecks. In this stdy, we se the kinematic wave model to simlate traffic dynamics for different evacation lans. In addition, we roose a measre of the effectiveness of an evacation lan by evacated nmber of vehicles dring a time interval, which eqals the nmber of vehicles that can be loaded into a road network at its origin. After finding the maximm evacated nmber of vehicles which are restricted by caacity and strctre of a road network with the kinematic wave model of network vehiclar traffic, we can arrange the evacation times and rotes of individal vehicles accordingly. 6

7 The evacated nmber of vehicles is the cmlative flow, N, at the origin. Since cmlative flow is the total nmber of cars assing the origin dring the time interval t 0 to t, then the cmlative flow at origin is t N([ t, t]) = f( s) ds, (8) 0 s= t0 where f ( t) is the bondary flx ot of origin at time t. In the discrete form, cmlative flow can be comted by N([, ]) = f( s) Δt 0, where t 0Δ is the start of the evacation s= 0 eriod, and Δ t the end. Figre 1 shows a tyical cmlative flow crve, which is non-decreasing in time and strictly increasing with ositive bondary flx. N Figre 1. Cmlative flow t SIMULATION RESULTS In this section, we demonstrate the feasibility of the evacation model in the receding section with a simle road network. Here we assme a sfficient large travel demand at the origin dring the whole evacation eriod. A ROAD NETWORK 7

8 We stdy a simle road network shown in Figre 2. In this network, links 2, 3, and 5 have the same length of 1000 meters, and the length of link 4 is 2000 meters. Link 3 has only one lane, links 4, and 5 have two lanes, and link 2 has three lanes. The fndamental diagrams for all links are trianglar in the following form: vfρ,0 ρ aρc Qa (, ρ) = ρc vf ( aρ ρ ), aρc ρ aρ ρ ρc where ρ is the total density of all lanes, a the nmber of lanes, ρ =180 veh/mile=0.112 veh/m the am density of each lane, ρ c = 36 veh/mile=0.022 veh/m the critical density of each lane, and v f =65 mh=29.1 m/s the free flow seed. That is, all lanes have the same fndamental diagram with caacity of qc = ρcvf. At the origin, travel demand is always 3 q c, among which a roortion, ξ, of all vehicles take link 3, and 1-ξ take link 4. At the destination, traffic sly is always 2 q c. Here we simlate traffic dynamics of the road network dring a time interval from 0 to 2000s with time ste Δ t = 2.5 s and the cell length Δ x = 100m. Link 2 Link 4 Link 5 Origin Link 0 Destination Link 1 Link 3 Jnction 0 Jnction 1 Jnction 2 Jnction 3 Figre 2. The traffic network In this network, the evacation lan is determined by the roortions of vehicles sing different rotes. Since link 5 has lower caacity than link 2, traffic qees will form after some time and revent vehicles entering the network at the origin at the caacity of link 2. Here we are interested in how the roortionξ, affects the total nmber of evacated vehicles dring a time eriod. IMPACTS OF DIFFERENT EVACUATION PLANS 8

9 In Figre 3, we demonstrate the cmlative flow at the origin with two different roortions: ξ =0.2 or 0.6. From the figre, we can see that the cmlative flow is increasing with time, and, for different evacation lan, cmlative crves are different. That is, the two evacation lans have different effectiveness. Figre 3. Cmlative flows at the origin for two evacation lans In Figre 4, we demonstrate the total nmber of evacated vehicles dring the interested time interval of 2000 s with different vales of ξ. From the figre, we can see that there is an otimal evacation strategy at ξ = , even for a very small cell size of Δ x = m. That is, when 32.5% of vehicles take link 3, we can evacate the maximm nmber of vehicles ot of the origin dring a time eriod of 2000 s. Actally, we exected the reslt to be 1/3, which is the ratio of the nmber of lanes of link 3 to the total nmber of lanes of both links 3 and 4. We also solve the roblem for different time eriods and obtain different otimal roortions. As an extreme case, when the time interval is very small, congestion initiated at the diverge or the merge does not roagate back to the origin, and the evacation flow-rate can be as large as the caacity of link 2. In this case, the ercentage will not affect the reslt, and, for any ercentage between 0 and 1, we can have the same nmber of evacated vehicles. 9

10 Figre 4. Nmber of evacated vehicles for different ξ CONCLUSION In this aer, we roosed a model for evacation lanning based a commodity-based kinematic wave model of network vehiclar traffic. We also discssed the measre of the effectiveness of an evacation lan and its comtation with the traffic flow model. We alied this model to stdy evacation roblem of a simle road network and demonstrated the feasibility of the model. That is, we can aly the evacation model to find an otimal evacation strategy when a disaster strikes, and vehicles can be advised to evacate following the strategy. Attractive featres of this model inclde easy set- of road networks, efficient simlation of traffic dynamics, and effective catre of constraints of road caacities and network strctre. Therefore, we exect this model to be an ideal latform for emergency evacation lanning. In the ftre, we will be interested in stdying roerties of different evacation lans for more comlicated road networks and comaring this model with others regarding its comtational efficiency and other merits. Indeed, this stdy is only the first ste toward bilding a comrehensive latform for evacation lanning, and sch a latform will be of or research interest in the longer term. In reality, with constraints in familiarity, qality and timeliness of information dissemination mechanism, we might not be able to obtain the otimal evacation reslt. In the ftre, we will investigate how these factors affect the reslts. In addition, we will be interested in validating the roosed model by comaring reslts from the latform with those observed in real-world sitations. 10

11 ACKNOWLEDGEMENT Helfl comments from an anonymos referee are areciated. The views and reslts contained herein are the athors alone. REFERENCES (1) Stiles, M. and Mrhy, B. and Elliott, J., Hrricane Rita: The Aftermath; Hrricane Costs; Reayment Soght for Waived Tolls; Conty Wants $7 Million that Was Lost in Fees for Using Roads, Hoston Chronicle 27, (2) Sheffi, Y. and Mahmassani, H. and Powell, WB, Agent-Based Modeling and Analysis of Hrricane Evacation Procedres for the Florida Keys 16(3), 1982, (3) Yi-Chang Chi, Traffic Schedling Simlation and Assignment for Area-Wide Evacation, 2004 IEEE Intelligent Transortation System Conference,Washington, D.C., USA, October 3-6,2004. (4) Chen, X. and Meaker, J.W. and Zhan, F.B., Agent-Based Modeling and Analysis of Hrricane Evacation Procedres for the Florida Keys 38(3), Sringer, 2006, (5) Lighthill, M.J. and Whitham, G.B., On kinematic waves: II. A theory of traffic flow on long crowded roads, Proceedings of the Royal society of London A 229(1178), 1955, (6) Richards, P.I., Shock waves on the highway, Oerations Research 4,1956, (7) S.K.Godnov, A different method for nmerical calclations of discontinos soltions of the eqations of hydrodynamics, Matematicheskii Sbornik 47, 1959, (8) Jin,W.-L., and Zhang H.M.,A mlti-commodity kinematic wave simlation model of network traffic flow. Transortation Research Record: Jornal of Transortation Research Board 1883,59-67,resented at TRB 2004 Annal Meeting, Washington DC, 2004 (9) Jin, W.-L., Kinematic wave models of networks vehiclar traffic, Ph.D. thesis, University of California, Davis, Available at: htt://arxiv.org/abs/math.ds/ (10) Sheffi,Y., Urban Transortation Networks: Eqilibrim Analysis with Mathematical Programming Method. Prentice Hall, Englewood Cliffsm, NJ,