Real Time Traffic Management

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Real-Time Traffic Management under EmergencyEvacuation Based on Dynamic Traffic Assignment

Yue-ming Chen Department of Automation

Tsinghua University

Beijing, 100084, [email protected]

De-yun Xiao Department of Automation

Tsinghua University

Beijing, 100084, [email protected]

Abstract - Effective evacuation traffic management iscrucial to maximize the utilization of transportationsystems and to minimize the fatalities and losses. A newapproach for real-time traffic management underemergency evacuation is proposed in this paper. Distinctfrom the well-studied evacuation planning, real-timetraffic management for evacuation aims at dynamicallycontrolling traffic flow under evacuation in such a waythat certain system objective like minimization of evacuation time could be achieved. The proposed

approach is based on dynamic traffic assignment byconsidering the traffic network under evacuation as adynamic system. First, the dynamic system optimal trafficassignment model based on the shortest emergencyevacuation time is established. Second, the optimalsolution to the optimal problem is obtained by usingPontryagin minimum principle. Finally, a numericalexample is given to illustrate that the proposed approachcan be effectively carried out in emergency evacuation.

Index Terms - Traffic management. Emergencyevacuation. Dynamic traffic assignment. Pontryaginminimum principle.

I. I NTRODUCTIONAs a rapid increase of population and the number of

vehicles in China in recent years, the impact on the surfacetransportation system of traffic accidents becomes moreserious than ever before. Especially, man-made or naturaldisasters, either predictable or not, could result in severe lifelosses and property damages. Emergency evacuation, a massmovement of people and their properties fromdisaster-impacted areas to safe areas, has been studied and

practiced for decades as one major means of countermeasuresto mitigate these calamitous consequences. Undertaking thisdifficult task primarily relies on the efficient utilization of roadway capacities and Intelligent Transportation Systems

(ITS) technologies, effective coordination of trafficmanagement equipment and available emergency aidresources. Most of the existing emergency evacuationmanagement researches focus on the planning stage [1,2,3,4],Moreover, due to the distinct features of different types of disasters, specific planning models and approaches have beendeveloped for various evacuation scenarios, including nuclear

plant crisis [5], hurricane [6,7], flooding [8], and fire [9], etc.Although evacuation planning is important for

emergency evacuation management, it is practically

impossible to give good preparedness for each disaster scenario due to the highly dynamic and uncertain featuresinvolved in extreme events. Past experience has shown thatineffective traffic management during evacuation could resultin severe traffic jams and life losses. Therefore, effectivereal-time traffic management for emergency evacuation isimportant to maximize the utilization of the transportationsystem and minimize fatalities and losses. As a result, there isan urgent demand for emergency management agencies(EMAs) to search ways to manage the evacuation traffic

efficiently and effectively in real time. An EMA is oftenfaceed with control and routing strategies that typicallyinvolve four critical operational decisions [10], including:(1)decide where to evacuate people (destinations); (2)decideon the best routes to take (route); (3)determine how to regulateflow rates on these routes (traffic assignment); and(4)determine the rate at which evacuees are allowed to enter the network from different areas of the region (phaseddeparture schedule). Obviously, these decisions areinterdependent and it is methodologically and computationallychallenging to make such decisions simultaneously and in acoherent manner.

The major challenge presented in emergency evacuation

management is the optimal utilization of all the routes exitingthe disaster-impacted according to their limitation in number and capacity. In most cases, it is inconvenient to construct newroutes or increase roadway capacities, so identifying ways tomaximize the utilization of the existing transportation network

becomes more important. For this evacuation application, it is particularly important to establish an evacuation model whichis based on dynamic network modeling techniques byconsidering the traffic network under evacuation as a dynamicsystem. This model should also have a simple structure thatcan be solved efficiently and quickly so that the optimalsolution can be obtained soon after the disaster happens. TheDynamic Traffic Assignment (DTA) methodology can be usedto address this problem. Most of the existing analyticalformulations of DTA are extensions of their equivalent staticformulations and seem to have two main disadvantages [11]:(a)they cannot adequately capture all realities of streetnetworks due to simplifications; and (b) they tend to beadaptive for realistic networks. DTA modeling techniques can

be generally classified into simulation based and analyticalapproaches. Simulation based DTA refers to mathematical

programming based models in which vehicular trafficdynamics and the link/path travel time are estimated through

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(0)(0) , for initial values, ( , ) jk jk x x j k L= , (2)

[ ]0 ( ) 1, ( , ) , 0, jk t j k L t T , (3)[ ]

( )( ) 1, ( , ) , 0, jk

k O j

t j k L t T

= . (4)

The dynamic system optimal traffic assignment model isthen formulated as a continuous-time optimal problem, whichwill be referred to as the dynamic system optimal problem J :

0( , )

min ( )T jk j k L

J x t dt

= . (5)s. t. (1)-(4)

Problem J is an objective function which aims tominimize the total system travel time that would be spent bydelivering all the evacuees to safe zones under emergencyevacuation management. From the definitions of the notations,the constraints and the performance index J is just described, itshows that the network considered is to be operated in thefollowing manner: (a) all the traffic demands at modes in EZare allowed to enter the network, not to be removed or rejectedand the traffic flows outside EZ are blocked from entering EZ;(b) a vehicle arrived at each node would make a decision toselect the next link, such decision is given by the optimalsplitting rate; (c) the optimal splitting rate is obtained byminimizing the performance index J and which implied thatvehicle would obey the signal at each intersection to choosetravel path from current node to the destination; (d) such pathcost is calculated at the time when the decision is made.

The constraints (1)-(4) ensure that the flows at each link will exit from the link at a later time by its link travel time.However, the link travel time is not pre-determined and should

be updated in an iterative procedure such as the trafficmonitoring and sensing system called diagonalizationtechnique used by Ran [18] et al. Here, the link travel time iscalculated when determining the shortest instantaneous travel

cost path. Therefore, the flow conservation on a link that avehicle entering a link at a certain time must exit from the link at a later time is not required here but this has already beenimplied in the proposed model.

III. D ERIVATION OF OPTIMALITY CONDITIONS

In this section, the optimality conditions for an optimalsolution of the problem J will be derived. The usual procedureto apply the Pontryagin minimum principle, developed byBudelis and Bryson [19] for the case where the statedifferential equations contain a time delay in the controlvariables, will be followed. To this end, the Hamiltonianfunction for the problem J is first constructed as:

[ ] ( , ) ( , )

( )

( ), ( ), ( ) ( ) ( )

( ) ( )( ( ) ( ))

jk jk j k L j k L

jk jk jk jk j ij ij iji I j

H x t t t x t t

u g x t d t u g x

= +

+ +

. (6)

Where ( ) jk t is the adjoint variable associated with the

state differential equation (1) for link ( j, k ), [ ]0,t T .According to the Pontryagin minimum principal, the

optimal solution of problem J must satisfy (1)-(4) and the

following conditions (7)-(9):* * * * *Min [ ( ), ( ), ( )] [ ( ), ( ), ( )] H x t t t H x t t t

= , (7)

s. t. (3)-(4), [ ]* *( ), ( ) , 0, x t t t T .

[ ]

( )

( ) 1 ( ) ( )

( , ) , 0,

jk jk jk jk jk kl kl l O k jk

H t u g x t

x

j k L t T

= = +

, (8)

( ) 0, ( , ) jk T j k L = . (9)

Equation (8) is the costate equation for link ( j, k ). For thegeneral optimal problems, the solution procedure is as follows:

1). From (7), to obtain the optimal control variable

[ ]* ( ), ( )U x t t = ;2). Replacing the in (8) with * , solving a two-point

boundary-value problem (i.e.,the state and costate equations (1)and (8) with the conditions (2) and (9) in order to obtain

* ( ) x t and * ( )t ;3). The optimal control variable is defined as

* * *

( ), ( )U x t t =

when the convergence criterion is met.

However, the two-point boundary-value problem here isvery difficult to be solved because of the complexity of (8)and the necessity of integrating state and costate equations insuch a way that the condition (7) has to be satisfied.

From the structure of the Hamiltonian function (6) andthe minimization problem (7), it is found that the problem can

be reformulated on the node basis. Therefore, (7) can bere-written as

( )

( )

Min ( )

( ) ( ) ( ) ( )

jk jk

k O j

jk jk jk jk j ij ij iji I j

t

u g x t d t u g x

+ +

, (10)

for each node j N , and this is equivalent to

( ) ( )Min ( ) ( ) ( ) ( )

jk jk jk j ij ij i j

k O j i I j

t t d t u g x

+

. (11)

Since the value of the second bracket of (11) is notdepending on jk , (11) may become

( )Min ( ) ( )

jk jk jk

k O j

t t

. (12)

s. t. (3)-(4), [ ], 0, j N t T .Let { }

( )inf , jk jk O j j N

= ; from (3) and (4), then

[ ]( ) ( ) ( ) , , 0, jk jk jk O j t t j N t T = . (13)Thus, the solution of (11) is

0, if ;

( ) 1, if ;

undefined,otherwise.

jk j

jk jk jt

>

= =

(14)

In other words, when jk j = exists for many links

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(more than one) outgoing from node j, the values of their optimal control variables cannot be determined uniquely bysolving the linear programming problem (12).

The SO model formulation is advantageous in modelingemergency evacuations because its optimal solutionencompasses all multidimensional decisions needed for evacuation operation. This LP formulation, however, does not

prevent vehicle holding at nodes, which is cumbersome for

regular traffic operation, but has a particular meaning in thecontext of evacuation. Vehicle holding may be interpreted asthe utilization of control measures to regulate flow on

particular roadways and/or intersections by emergencymanagement officers in order to implement the evacuationsolution. By limiting access at strategic locations, it may befeasible to move the systems true performance towards thesolved objective function.

IV. N UMERICAL EXAMPLE

After computing the evacuation routes to dispatch theevacuees to safe zone, the next step should be finding anapproach to control the evacuation signals at the intersections,

for directing the traffic flow in the proper directions, in a safemanner. We adopt the model proposed by Yang [20] to controlthe signal in each intersection.

{ }min ( , ), ( , ) z d C x E C x= . (15)s. t.

2 2

2

( )( , )

(1 ) 2(1 )2r r

r

q x yC xd C x

y x x

= +

, (16)

2 2

( )( , )

2 ( )r r

r

y x y E C x

x y x

=

, (17)

( ) , 0.8 0.9C x Y Lx x = , (18)

minC C , (19)

where ( , )d C x and ( , ) E C x represent delay function and parking rate function; r q is flow rate and r y is flow ratio in phase r ; x denoting intersection saturation; C represent cycletime; minC is the minimum cycle time.

The design of this signal model consists of an arrow pointing in each of the possible outgoing directions. Thesearrows can take on any one of the three traditional trafficcolors, namely red, yellow and green, at different times. A redarrow is equivalent to a traditional do not enter sign; and theincoming traffic should halt in the corresponding direction.Similarly, a green arrow gives the incoming traffic the right of way to proceed in that direction. A yellow arrow specifies atransitional stage and the incoming traffic should slow downand prepare to stop.

In this section, a numerical example is presented to

Fig. 2 Example network illustrate a potential application of the modeling techniquesdiscussed in the preceding sections. The test network is aseven-node network as illustrated in Fig. 2. The first step of the modeling process is to define the evacuation zone (EZ),intermediate zone (IZ) and safe zone (SZ) perimeters. Thus,node A is defined as the evacuation source node at which thetotal number of evacuees is assumed known. Evacuees areconsidered safe when they reach node E, F, or G. The network topology characteristics are summarized inTab.1. All linkshave two lanes, with a total maximum flow from 3960 to 5400vehicle per hour (vph). It should be noted that the number of evacuees to be evacuated is assumed to be known and weassume that the total number is 128 here.

The optimal route-flow solution, as listed in Table II,indicates that after 245 iterations, all 128 evacuation flowunits reach the safe zone with an optimal total system traveltime of 223.797 seconds. The solution shows that sixevacuation routes have been constructed to deliver evacuees tosafe zone. More importantly, the model solution indicates howmuch flow should be assigned to each destination.

For evacuation purposes, proactively managing demandand implementing the corresponding traffic control strategiesis a more meaningful operational concept than passively

predicting and reacting to demand because the performancedegradation of a transportation network nonlinearly increaseswith the level of network traffic loading. Undoubtedly,controlling evacuation demand is a significant practicalchallenge; understanding how the extent of noncomplianceaffects the model effectiveness also requires further research.

Real-time traffic data information obtained from sensorsand other surveillance technologies are used to adjust andupdate the optimal solution. Updated information can always

be incorporated into the next calculation since thereoptimization of the model is cyclic. Updating the optimal

solution is essential and brings the following twoadvantages [21] ; first, it prevents congestion, by avoiding

TABLE I G EOMETRIC CHARACTERISTICS OF EXAMPLE NETWORK AB AC AD BE BF CE CG DF DG

Travel time function 15+ x 10+ x 10+ x 5+3 x 8+2 x 6+3 x 8+3 x 6+3 x 8+3 x No. of lanes 2 2 2 2 2 2 2 2

Speed limit (feet/second) 50 50 50 50 50 50 50 50 50Maximun flow(Vph) 5400 4680 4680 4320 3960 4200 4000 4200 4000

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TABLE II O PTIMAL R OUTE -FLOW SOLUTIONABE ABF ACE ACG ADF ADG

Evacuees in each route 19 27 21 20 21 20Total iterations 245

Total evacuation time 223.797

delivering the evacuees into those already crowded links; and

secondly, it decreases congestion by directing traffic oncurrently congested routes to disperse on alternative routes.

V CONCLUDING R EMARKS A ND FUTURE TOPICS

Due to the unexpected and stochastic characters of disasters, itis absolutely necessary to be able to respond with flexibilityand coordination whenever necessary. In this paper, we

present an approach for real-time traffic management under emergency evacuation based on dynamic traffic assignment.Distinct from previous planning models, the proposedapproach aims to operate in real time and dynamic so thatcurrent prevailing traffic states can be utilized to moreeffectively guide traffic under evacuation. Simulation studiesshowed that the proposed approach is capable of real timeevacuation. Moreover, this method is applicable to further considering such as reduced roadway capacities due toearthquake damage, road blocks, and out-of-service roads.Based on the updated data information obtained from sensorsand other surveillance technologies, evacuating vehicles can

be reassigned to new safe zone in order to continuouslyreoptimize the evacuation operation.

This paper assumes the dynamic OD demand matrix isgiven and fixed during the entire evacuation process. Due tohighly dynamic feature of evacuation, dynamic OD demandswill change as the evacuation evolves. With the trafficmonitoring and sensing system, prevailing traffic informationis expected to be available during the emergency evacuation

process. Therefore, there is a potential to dynamically updateevacuation OD demands based upon real-time collected trafficdata. Real-time operational issues such as reoptimizationimplementation and the update of network initial conditions ateach reoptimization are also currently being studied.

ACKNOWLDGMENT

This research was granted and supported by the NationalKey Technologies Research and Development Program(No.2005BA414B09 and No. 2007BAK12B15) of China.Meanwhile, we shall also acknowledge the help andcooperation of Beijing Traffic Management Bureau. Theauthors assume the responsibility for the facts presented,

viewpoints expressed, and accuracy of the data in this paper.R EFERENCES

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