a differential game modeling approach to dynamic traffic assignment and traffic signal control -...

8/8/2019 A Differential Game Modeling Approach to Dynamic Traffic Assignment and Traffic Signal Control - 2003

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A Differential Game Modeling Approach to Dynam icTraffic Assignment and Traffic Signal C ontrol*Li Zhen- long

The Intell igent Con trol and Systems Engineering Center,Institute of Automation, Chinese Academy of Sciences, Beijing, Chin [email protected]

Abstract - In this pap er, the combined &namic tra ficassignment and traflc signal control problems arediscussed. Afer a brief retrospection to dynamic trafficassignment and trafic signal control is addressed. theconflict and interaction between dynamic trafficassignment and trafic signal control are analyzed. Then,the dynamic characteristic o tra@ assignment andtraffic signal control is considered and the theory ofdifferential game is used to model. Dynamic traficassignment and tra fic signal control are formulated as aleader-follower direrential game, in which a leader andmulti-follower participate, under the open-loopinformation structure. Then, a leader-follower differentialgame model af dynamic traffic assignment and traficsignal control isproposed and the model is discreted anda simulated annealing algorithm is used to solve it. A t last,a simulation is carried on a simple trafic network andnumerical results demonstrate the effectiveness o thepro pos ed differential game mo del.Keywords: Dynamic traffic assignment, traffic signalcontrol, differential game.1 Introduction

Congestion is a daily occurrence on many portions oftraffic networks in u rban areas. Building new roadways isno longer a feasible option due to the bigh costs, as well asenvironmental and geographical limitation. Traditionaltraffic engineering techniques are proving unequal to thegrowing traffic demand. It bas led to the technologylabeled as Intelligent Transportation System (ITS) Thefundamental problems in ITS are the Dynamic TrafficAssignment and Traffic Signal Control. Dynamic trafficassignment systems are suites of software tools designedto s upp ott traffic management systems. Using current andhistorical data, dynamic traffic assignment systemsestimate traffic flow patterns and determine appropriatetraffic control and route guidance strategies. Traffic signalcontrol systems are designed to foster real-time sensing,communication, and control of urban networks. Theprimary objective for traffic signal control systems isreducing congestion effects.

While, conventional methods for setting trafficsignal assume given flow patterns, and traffic flows areassigned to networks assuming fixed signal setting. Thismethod is not fully satisfactory in the normal case inwhich traffic flows and signal settings are mutuallyinterdependent[l].Allsop was the first to address theinteraction between traffic control and traffic assignment,and he suggested that the effects of signa l settings on thetraffic flow patterns should be taken into accountexplicitly by com bining traffic control and route choice[2].There have been two app roaches to address this problem:the global optimization models and the iterativeoptimization and assignment procedure[3].The globaloptimization models seek the global optimality of thecontrol policies when travelers adjnst their route selectionsto the ch anges in signal setting. Fisk desc ribed the globaloptimal signal setting problem as a Stackelberg gamebetween network users and a traf ic agency[4]. S h e a andPowell formulated the optimal signal setting problem as amathematical program in which the traffic flow isconstrained to be user equilihrium[5]. Smith et a1combined traffic assignment with responsive signalcontro l in a static case[6]. Yang an d Yagar formnlated theproblem as a bi-level program in which the optimal trafficcontrol is one level and the user equilibrium assignment isanother[l]. The iterative optimization assignmentprocedure is to update alternatively the signal setting forfixed flows and solve the traffic equilibrium problem forfured signal setting until the solutions of th e two problemsare considered to be mutually consistent[3,7].Smithproposed a consistent control policy that ensures theexistence of traffic equilibrium[S]. AI-Malik investigatedthe Wardrop equilibrium under Webster control[9]. Allstudies in the literature only considered the static trafficcase but there are three exceptions. The fust is Gartnerwho proposed a framework to integrate dynamic trafficassignment with real-time control, but be did n ot give anyanalytical model formulation; and the sec ond is Chen whoformulated one particular model for th e traffic-responsivesignal timing scheme with user-optimal route cboice[lO].The third is Owen J. Chen who formulated the combineddynamic traffic assignment and dynamic traffic con trol asa one-level Cournot game between traffic authority andusers, and he also formulated the dynamic trafficassignment and dynamic traffic control as a Stackelberg

8490-7803-7952-7/03/$17.00 2003 IEEE.
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game between traffic authority and users, but he did notgive the solution algorithm of the m odel[l 11.Game theory provides a framework for m odeling adecision-making process in which more than one playersare involved. Differential game theory is a branch of game

theory [12,13]. It assumes that the behavior of the systemcan be modeled as a system of ordinary differentialequations. It is interesting for its dynamics. Differentialgame theory can be used to model situations where seve ralinteracting agents make strategic dynamic decisions.Differential game theory, which frequently deals withmultiple performance indices, is found to be directlyapplicable for the m ulti-objective control problems.In this paper, the combined traffic assignment andtraffic signal control problems are discussed from adifferential game-theoretic viewpoint. The relationshipand dynamic characteristic of dynamic traffic assignmentand traffic signal control are analyzed. Inspired from the

adaptability of the Complex Adaptive System(CAS), weregard signal controllers at the intersections as theadaptive agents and model the combined trafficassignment and traffic signal control problems using theidea that traffic flows are assigned on the proper roadnetwork by the traffic assignment and then the flows areadapted by signal controllers though changing the signalsetting. Trafiic manager, by which traffic flows areassigned, is acted as the leader and signal controllers at theintersections, by which the signals are controlled, areacted as the followers, then a leader-follower differentialgame model of dynamic traffic assignment and trafficsignal control is proposed. The model is discreted and asimulated annealing algorithm is used to solve it. At last, asimulation is carried on a simple traffic network andnumerical results demonstrate the effectiveness of theproposed differential game model.2 Model formulation

As above mentioned, traffic signal control andtraffic assignment are mutually interdependent. Becausetraffic signal settings are usually determined by the flowpatterns, flow patterns should be considered in the trafficsignal setting. And flow patterns, as described by the flowon each link, are influenced by signal setting,thus trafficassignment should take signal setting into account. Trafficassignment can cause a change in traffic flows and thechanged flows will subsequently make the signal settingnon-optimum and thus require a accordingly change insignal setting. Meanwhile, traffic signal control can causea change in travel time and the changed travel time willsubsequently make the M i c assignment non-optimumand also require a acco rdingly change in traffic assignment.Traffic assignment can change the flow patterns from themacroscopic, and signal control can change the flowpatterns fiom the microscopic. These two process arecurrently carried out and interdependent on each other,

therefore there exists a strong interdependence betweenthem, as shown in Figure 1.Signal Settings

1 tAssignment Strategy Traffic Signal Controlssignment Strategy Traffic Signal Control

Figure 1 Interaction between traffic assignment and trafficsignal controlBefore we present the combined dynamic trafficassignment and traffic signal control problems, let us firstaddress each of the dynamic traffic assignment and trafficsignal control individually.

2.1 Dynamic system optimum modelDynamic traffic assignment has been a topic ofsubstantial research recently and various trafflc assignmentmodels have been developed[l4-18]. Am ong those models,the dynam ic system optimum model is usually used for the

criterion and its significance lies in that it provides anupper bound in systematic search procedures for theoptimal network design problem. And the state of systemoptimum is state which the traffic manager expect. So,dynamic system optimum model is used in the paper.Traffic networks are made up of different l i odnodes, one OD pair can hav e different routes. Each routeincludes different links and nodes. Let a directed graphG( N, A) denote the road network, LetNbe the set of nodes in the network:A be the set of links in the network:x,,(t) be the number of vehicles on link U at time f :x : ( t ) be the number of vehicles arriving atU&) be the entry flow into link a at time t;u:(t) be the entry flow into link U arriving atv.(t) be the exit flow from link a at time f :v : ( t ) be the exit flow from link U arriving at

destinationn on link a at time t :

destination n at time f ;

destinationn at time t :

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S,,,(!) be the flow generated at the node ka t time I ;A&) be the set of links whose tail node is k;B(k) be the set of links whose head node is k:t , x , (t), , t)) be the travel time on link U ;do x . ( t) ,, (f), A, t)) be the signal delay and

queuing delay at the intersection.The travel time between the OD pair should beconsidered as a sum of travel time and travel delay insignalized networks. Travel delay should be divided into

two kinds, signal delay and queuing delay. The former isdue to interruption of traffic by th e traffic signal, whereasthe latter is due to limited capacity [l].So, The total traveltime spent on he network, to x, ( t ) , u , ( t) ,v,(t)), is thesum of flow-dependent running time, t, (x, (t),u,(t)) ,and delay due to the signal and queuing delay at theintersection,d, x , (t),v, t),2,(t)) ,namely,

t , x, ( t ) ,u , (0,, (9 t. (x, ( t )P ,(9 )+ d , (x , ( a v o X% 0)) ( )

Friesz et a1 model the dynamic traffic assignmentproblem as an optimal control formulation. A standardsystem optimal control formulation that seeks to minimizethe total system navel time can be represented as follows:

( 2 - a )x : ( f )= u . ( t ) - V : ( t ) (2-b)

X , ( t ) 2 0 U:@) 2 0 , (aE A , n E N , E [O,T])(2-d)

2.2 Traffic signal contro l modelWe assume that the phasing for each intersectioncontroller has been determined and the cycle length is alsogiven. The problem fo r a intersection controller is to fmd adynamic system optimal signal timing, namely, to allocatea green time for each phase during a time period. Considerthe intersection shown in Figure 2, with the time period

[O s TI.Let I, (t ) denote the number of queuing vehiclesat time f ; u ; ( t ) denote the entry rate into the waitingarea of link 1.

1 1 I Lm r 2Link 3ILink 1 .+i

iFigure 2 The intersection

2.2.1 Trafiic flow state equationas follows:At the congested situation, traffic flow state equation is

where, C(f) is the cycle length, S is the capacity ofthe intersection, Cl@) s the green time for the phase 1,~ ( t )s the green ratio for the phase 1.

2.2.2 ConstraintsG, < G ( t )2 G,0 5 1,( t )2 ,,,, , I , ( 0 )= 0 , =1,2,3,4.

(4)(5)(6), ( t )= C ( f ) L

mwhere, G-, G is the minimum and maximumgreen time respectively, 1 is the maximum queuinglength. C,(f) is the green time for the phase m. is losttime per cycle. Fqution (6) states that the summation ofgreen time over all phase and total lost time on eachintersection equals the cycle time.2.23 Objective function

The intersections objective is minitaize the queuingdelay and the problem is to allocate a green time for eachphase during a cycle period in order to minimize thequeuing d elay. So,the objective function is as follows:min Jc = min [(I(f))TQI(t)dt (7)KO = ( ~ ~ ( f ) , I ~ ( f ) , I ~ ( t ) , I ~ ( t ) ) ( 8 )

where ,Q s weighting matrix.For each intersection, any phase w ith a positive g reen

time expects to have a m inimal queuing delay. Namely, foreach intersection, green time can only be assigned to aphase with the minimal queuing delay. The traffic controlmodels can be obtained though extending above-mentioned traffic state equation, constraint and objectivefunction to k intersection, as formula (9) shown, and themeaning of model is to make total delay of eachintersection minimum.851


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OIG,, IG : ( t ) lG , ( k s N , e [ O , T ] )(9 -e)

C G : ( t ) = C ( t ) - L (9 - f )mwhere,

k s the index of the intersection:G: ( t ) is the green time for phase m at intersection k

at time I:1; t ) is the green ratio for phase m at intersection k

at time f;1: ( 1 ) is the number of queuing vehicle on link U atintersection ka t time I:

U,",, t ) s the entry flow into the waiting area of linkU at intersection k at time t :If the attraction of links to vehicle is not consideredand distance between intersections is not too far, the entryrate into the waiting area of link is equal to the de lay of theentry rate into Imk, as follows:

( t )= U: ( t- ) (10)where,Eqution(l0) is a concrete embodiment of interaction

between the dynamic traffic assignment and traffic signalcontrol.2.3 Model formulation

r is the travel time of freedom flow.

Traffic manager pursues the " a 1 total cost,namely, want to minimize the total travel time. Eachintersection aims to find a green time for each phase thatminimizes its queuing delay. Traffic manager andintersections have different objectives that each strives tosatisfy. The goal of the manager is to fmd a assignmentstrategy that drivers the traffic flows to the systemoptimum. Such strategy of the manager achieves theminimal total cost and ,therefore, leads to the mostefficient utilization of system resources. Each intersectionis mandated by the desire to minimize an individual costfunction, namely, his queuing delay. The traffic managerand signal controllers at the intersection are regarded asthe adaptive agents, and we assume that traffic flows areassigned on the proper road network by the trafficmanager and then the flows are adapted by signalcontrollers at the intersections though changing the signalsetting. Manager and intersections are regarded as theparticipant of game, the leader is traffc manager who has

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the first mover advantag e, and followers is intersec tions.Using the differential game framework, we consider acontinuous time model. So , dynamic traffic assignmentand traffic signal control can be formulated as a leader-follower differential game under the open-loopinformation structure. U:(/) is the control variable oftraffic manager, and ak,(t) is the control variable ofintersection k. J G s the payoff function of traffic manager,and J i is the payoff function of intersection k. Theleader-follower differential game m odel of dynam ic trafficassignment and traffic signal control is as follows:x: ( t ) = u : ( t ) - ."(t)

i," ( 1 ) = U: ( t - 7) a: (t)sfCa t ) s,w+ c v : oi f ( O ) = O x : ( t ) t O u : ( t ) > Oo e A ( k ) OSB ( L )

Z G : ( t )= C ( t )-L

The aims of player, manager and intersection s, are tomake their benefit maximnm though selecting thestrategies, namely, control variables.3 Solution algorithm

It is very difficult to solve the model directly, thus themodel is discreted and then solved in the paper.MJ , = m i n C C x , ( i ) t , ( x , ( i ) , u , ( i ) , v , ( i ) )

x : ( i + 1 ) = x: (i)+ U: (i)- :(i)asA i=li = 0,1,2;. .,M -

u : ( i ) > 0 a E A , n E N I x : ( i ) t 0MJ : =minCTi"(i)ei=l

i k ( ~ ) = ( i ~ ( i ) , i , " ~ ( i ) ; ~ ~ , i ~ ~ ( i ) ) , u lA ( k )k = 1,2; . ,Nit (i+ 1) = i," i) +U; (i) - A , i)Sm

i , " (O ) 2 0


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Where, Ti s the control period, e is unit column vector.The upper-level of this model is a decision-maker,and there are a lot of decision-makers in the lower-level.The decision-making problem that the model describe isthe leader announce its strateg y at fEst, which w ill influentthe follower's restraint-set and objective function, then the

follower choose the strategy which make the follower'sown objective function reach optimum. Both the leader'sand the follower's strategies will affect the overall systemperformance. The strategy ( u' (k) , l* (k) makes theoverall system performance optimal. u'(k) an d , l*(k)can be gotten by solving the model. Because the model isnonconvex, we use the Simulated h e a l i n g (SA) to solveit. As its name implies, the Simulated Annealing exploitsan analogy between the way in which a metal cools andfreezes into a minimum energy crystalline structure (theannealing process) and the search for a minimum in a moregeneral system. The algorithm of m odel is the following:Step1 : Determine an initial temperature TI and theinitial solution (U," m))'. olve the lower problem forgiven( U: (m)' ,and hence get (1:(m)' ,and set theiteration cou nterp = 1:the formula:Step2: New trial solutions be generated according to

(U:(~))~=(U,"m))P-'+Auwhere Au is a random number in neighbourhood of(U: (m)) -' . Compute the new objective function valueF((u:(m))9 nd the change A F in objective function,

+Au: if A F > O , . Compute the probability of acceptingnew trial solutionsL wp ( W )= exp(--) IP . T p

and generate a random number r, 0 < r < 1 , if theMetropolis criterion is satisfied,namely, p ( a r then(~,"(m))~=(U,"(m))~-'+Au;therwise,(U,"m))"=(u,"m) -' .

Stepl: If a convergence cr iterio n is satisfied, then(U: (m)p is the optimal solution of the upper problem

hence get(At(m)p, ( (U: (m)) p (U," m)p ) is theapproximate global optimal solution. Otherwise, go toStep5Steps: A new temperature is generated according tothe formula:

and solve the lower problem for given( u,"(m)p ,and

853

TIT =p+' ( p + l ) P

Let p=p+ 1 and go to Step2.4 Numerical example

In this section, a numerical example is presented toillustrate the application of the model and a lgorithm above.The test network is indicated in Figure 3 with seven nodesand eight links. Detailed link characteristics are shown inTable 1.

7

3 6Figure 3 Simulation network

Table 1 Link Information

It is assumed that there is one origin-destination pairfrom node 1 to node 7. The 0-D airs has four alternativeroutes available which are Routel(alla4/a5/aS) with1500m-lengtk1, Route 2(al/a4/a6/a7) with 1700m-length,The Route 3(a2/a3/aYaS) with 1300111 -length; Route4(a2/a3/a6/a7) with 1500m-length.During the simulation, Webster's delay formula isused for the intersection delay and the travel time modelof Bureau of Public Roads(BPR) is used for the traveltime. Cycle of assignment and control cycle is IOOS, hesample time is 50s. The signal phase is a s follows:

Figure 4 Four phase signa ls


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The simulation is carried on"traffic control only" and"the proposed model of this paper" respectively, the traveltime and the number of vehicles are be obtained on eachroute at the sample time, The results of the simulation areshown in Figure 5,6. I Route 1- - - Route2--c--6 Route 3-. ._. Route 4

5010 10 20 30 40 5 0 . Mf 70(5-a) The travel time on "traffic control only"

0 1 0 2 0 3 0 4 0 5 0 6 0 7 0(5-b)The travel time on "the proposed model of this paper"

Figure 5 The travel time

t

(6-a) The number of veh icles on "traffic control only"

60 1

(6-b) The number of vehicles on "the proposed model ofthis paper"Figure 6 The flows

In the case of "traffic contro l only", the ave rage traveltime of the fo ur routes is 105.71~,108.35s,l85.46~,140.38respectively, and total average travel time is 134.98s. Inthe case of " the proposed model of this paper ", theaverage travel time of the four routes is114.97s,116.62s,156.59s,106.21s,respectively, and totalaverage travel time is 123.60s, which is a reduction of 8.43%. From the Figure5,6, we can find out that in the case of"traffic control only", because m ost vehicles choose Rou te3,which is most short in static situation, the vehicles on theRoute 3 increase sharply and the network performancesde c li e subsequently. In the case of '' the proposed modelof this paper ", when the flows on the Route 3 areincreasing, the flow s are adjusted by the traffic assignmentand signal settings are changed accordingly. So, roadnetwork flows are equilibrium an d traffic jam arealleviated to a certain extent.5 Conclusion

In this paper, th e differential game theory is appliedto model the combined dynamic traffic assignment andtraffic signalcontrol problems. This approach is illustratedby a simple example network, in which the cases "trafficcontrol only" and I' the proposed model of this paper arecompared. The simulation results are satisfactory, andc o n f m the effectiveness of the proposed approach.ACKNOWLEDGEMENTS

This research was supported by the NationalOutstanding Youth Fund of China (No.60125310) .References[l ] Yang H,Yagar S, "Traffic assignment and signalcontrol in saturated road networks", TransportationResearch Val 29A,No 2,pp. 125-139,1995[2 ] Allsop R E, "Some possibilities for using trafficcontro l to influence trip distribution and route choice". Pro6th 1nt.Symp.on transportation and Traffic Theory,Elsevier, New York, pp.345-373,1974

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