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A DECENTRALIZED CONTROL APPROACH FOR TRANSPORTATION NETWORKS 1

Banu Ata§lar

and

Altug

iftar

Department of Electrical and Electronic8 Engineering Anadolu Univer8ity 26470 E8ki§ehir, Turkey Faz: 90-222-335 3616 [email protected]·mm.anadolu.edu.tr [email protected]·mm.anadolu.edu.tr Abstract: A decentralized control approach for transportation networks to prevent traffic congestion is presented. The approach was developed by considering flow control and routing control together. The control algorithm is based on a previously proposed congestion measure. In this paper, in order to illustrate the performance of the proposed controller, some simulation results are presented. Copyright © 1998 IFAC

Keywords: Transportation networks; Routing control; Flow control; Decentralized control.

1. INTRODUCTION

One of the important problems in transportation networks is routing control. The routing control problem is defined as to direct vehicles in a transportation network from a source node to a destination node through the links and other nodes. To find a solution to this problem provides only a partial solution. Because of the fact that congestion and excessive delay may occur, in the absence of control of rate of vehicles entering to the network. For this reason, it is also necessary to control the flow entering the network from each node. Many different algorithms have been proposed for routing control in the literature. A decentralized routing control algorithm, which is based on minimization of queue lengths and total travel time, was proposed by Sarachik and Ozgiiner (1982) for transportation networks which have only one destination node. This approach was extended to the multi-destination case by Sarachik (1982) byassuming that the traffic demands at each node are conresearch was supported by the Sciex:t~c a.nd Technica.l Research Council of Turkey (TUBITAK) research gra.nt EEEAG-173.

1 This

343

stant. Routing control and flow control algorithms are considered together by Sheu (1987) . However, the proposed decentralized routing controller may not produce desirable results under the heavy traffic conditions. A non-linear optimization approach, which requires excessive computation, was considered by Papageorgiou (1990). By simplifying network dynamics this non-linear optimization problem turns out to be a linear optimization problem (Messmer and Papageorgiou, 1994). A decentralized routing controller approach proposed by iftar (1996) is based on a dynamic conge8tion mea8ure derived from the solution of an optimal control problem. By removing two drawbacks (mentioned in iftar , 1997) o~ this approach, a new algorithm was suggested by Iftar (1997). This algorithm is defined in discrete-time and it routes a vehicle only if the vehicle which is in front of it has been routed. The controller is decentralized in the sense that all the on-line computations are done locally at the nodes with only information from the adjacent downstream nodes. On the other hand, many control approaches have been proposed for flow control. A control approach was proposed by using the augmentation

method (Isaksen and Payne, 1973). This method V(Pm,i(k)) = Vi,m exp [_~(pm'i(k))""'] obtains suboptimal controls based on the optimal a.,.. Per,m control laws for a set of overlapping subsystems derived from the original system's dynamic descrip- where: tion . Another control approach was proposed by p;",i(k): the density of vehicles per lane with Goldstein and Kumar (1982) by using the cascadthe destination node 1 on segment i ing technique. It was shown that the control pro(i = 1,2, .. . , Im) of link m at time kT. vided by the cascading technique is more effective than the control provided by the augmentation q!n,i(k) : the flow rate of vehicles with the destitechnique. Papageorgiou, et al. (1990) developed nation node 1 on segment i of link m at a coordinated control strategy, where a multivaritime kT. able regulator with integral parts, which appears to vm,i(k): the mean speed of the vehicles on seghave some advantages as compared to the classical ment i of link m at time kT. LQ-approach, was presented. Pcr,m: the critical traffic den"ity for link m . Ata.§lar and iftar (1997) considered routing convi ,Tn: the free flow "peed for link m. trol and flow control together and developed a deITn: the number of segments of link m . centralized control algorithm by using the congesLTn : the length of each segment of link m . tion measure approach proposed by iftar (1997) . Modeling approach and proposed control algorithm ATn: the number of lanes of link m. (Ata.§lar and iftar, 1997) will be presented in the J Tn : set of destination nodes which are reachfollowing sections of this paper. Simulation studable through link m. ies will also be presented to illustrate the perT: sampling time interval. formance of the proposed controller. For this aim, a simulation program is developed based on and a.,.., K, 11 and T are constant parameters which METANET (Messmer and Papageorgiou, 1990). are related to traffic and environment conditions METANET developed by Papageorgiou and Mess- (Papageorgiou, et al., 1989) . mer is a macroscopic simulation program for transportation networks. By changing many parts of the In order to evaluate above equations, the flow rates program a new simulation program is developed to and the mean speed at the beginning of the links simulate the proposed control algorithm. Some sim- (q!",o(k) and vTn,o(k)) and density at the end of the ulation results will be presented in the following sec- links (Pm,i ... +1 (k)) are required (these values are the boundary conditions of the dynamic equations tions of this paper. given above). These values are calculated as below (for each link m, which leave node n and which be2. MODELING NETWORK DYNAMICS long to one of the alternative paths from node n to The controller presented in Ata.§lar and iftar, 1997 the destination node 1): is based on the modeling approach proposed by Papageorgiou, et al. (1989) . In this model, the traffic behaviour in transportation networks is represented by using the traffic density P, the traffic flow rate q and the mean speed v. For modeling purposes, each link of the network is subdivided into segments with a particular length, and the traffic variables mentioned above are calculated for each time interval _ 2: IoIE o,. VIoI,l(k)) T: Vn (k) - max Vn , ,,(On)

(M

p;",i(k+1)

= p!",i(k) + LmTAm PmAk) =

L

[Pn(k)]2 + 2: IoIE O,,[PIoI,i(k)j2 Pn(k) + 2: IoIE O" PIoI,l(k)

[q!,.,i_l(k) - q!",i(k)]

p!",i(k)

IEJ...

q!",i(k) = p!",i(k) vm,i(k) Am vm,i(k

T

+ 1) = vm,i(k) + -; [V(Pm,i(k)) - vm,i(k)]

where: In : set of links entering node n .

T + Lm vm,i(k) [Vm,i-l(k) - vm,i(k)] _

11

T

4>~'Tn(k) : the splitting rate into link m of the ve-

hicles at node n with destination node 1 calculated (by the routing control algorithm) for the time period [kT, (k + 1)T).

T [Pm,i+l(k) - Pm,i(k)] Lm [Pm,i(k) + K] 344

r~ (k): the traffic flow rate of vehicles with destination node I entering the network from

of S~ respectively. The order of elements of S~ represents the preference of paths from node n to the node n calculated (by the flow control destination node I; i.e., the first node on the most algorithm) for the time period [kT , (k + preferred path from node n to node I is taken as the l)T) . first element of S~ , the first node on the next most preferred path from node n to node I is taken as Vn (k): the mean speed of vehicles entering the the second element of S~, and so on. The elements network from node n at time kT. of J~, on the other hand, are ordered arbitrarily. the maximum mean speed allowed for ve- Indeed, their ordering do not effect the outcome of hicles entering the network from node n . the proposed algorithm. Tn.mu; is the maximum allowed flow rate of vehicles entering the network On : set of links leaving node n . at node n. The following algorithm describes the s(·): the number of elements of the set (.). decentralized controller to be implemented at each Pn (k) : the traffic density on the off-ramp of node n at each time step k : node n at time kT. 1. Set t = 1, r~(k) 0, VI E J~ . v~ : the mean speed on the off-ramp of node 2. Set I = J~(t), h 1. n. 3. j = S~(h) . ).~: the number of lanes on the off-ramp of 4. If p~ (k) ~ p} (k), route vehicles with destinode n. nation node I from node n to node j : I , m E On nIj 3. CONTROL ALGORITHM

v;::

if>~.m(k) =

A decentralized control algorithm is proposed by using the mathematical model mentioned above. It is developed by considering the flow and routing control together and using a previously proposed congestion measure approach (iftar, 1997). For this control approach the congestion measure is defined for each node n and destination node I (I E J~ := U mEO...1m):

p~(k) =

I:

Q~ a~(k) + /3~ (~(k)

{

0,

!

PI'.!,. (

If h ~ s(S~), go to step 3. 6. Distribute the vehicles with destination node I among the elements of S~ with ratios inversely proportional to the respective congestion measure. 1

I

a,,(k) = ~

(~(k

pj(k) (

PI'.!,.

(k)

> Per.I' 0,

PI'.!,. (k) ~ Per.1'

(~(k)

+T

j E S~,

if>~.m(k) =

- Per.1' , PI'.!,. (k)

0

+ 1) =

)

n Ij

5. h = h+ 1.

I'EI ..

p~.!,.(k) k)

m >1. On

Go to step 7.

+

7. t = t

(T~(k) - r~(k))

1;

1ft

m E

On nIj

m >1.

OnnIs !,

< s(J~), go to step 2.

8. If queue length at node n is zero , STOP.

where Q~ > 0 (J.L E In) and /3~ > 0 are constant parameters to be chosen by the designer to reflect the relative contributions of high density (on the incomming link J.L) and the queue length (at node n) on the congestion at node n and where: (~( k): the number of vehicles at time

kT with destination node I waiting in queue to enter the network at node n at time kT.

T~(k): the traffic demand at time

kT from node n to destination node I at time kT.

Following control algorithm is based on the congestion measure defined above. It is assumed that J~ and S~ (S~: set of nodes which are the first nodes of the alternative paths from node n to the destination node I, (I E J~)) are ordered sets. J~(t) and S~ (h) are the tth element of J~ and the hth element

345

9. Obtain the destination node of the first vehicle in the queue at node n and denote that as node I. 10. Set h = 1. 11. j = S~(h). 12 . Ifp~(k) ~ p}(k), admit the first vehicle from node n to the network: 1

r~(k)

= r~(k) + T

Go to step 14.

13. h = h+ 1. If h ~ s(S~), go to step 11. Otherwise STOP. 14. If

I: r~(k) <

T n •max ,

IEJ~

Otherwise STOP.

go to step 8.

The above algorithm provides a solution to both routing control and flow control problems. The algorithm can be implemented locally at each node. The only information required from the other nodes is the congestion measure values from the adjacent downstream nodes. Thus, the proposed controller is a decentralized controller which requires very limited information exchange among the nodes. The controller does not require any synchronization among the nodes either (each node may simply use the last received congestion measure values). At each node, at each sampling instant, the algoritm first determines the splitting rates for the routing control based on the congestion measure values. Vehicles with node n are routed through the most preferred path provided that the congestion measure value at the first node of that path is not greater than the congestion measure value at the present node. If the congestion measure value at all downstream nodes leading to destination node 1 are greater than the congestion measure value at the present node, then the incomming vehicles to node n with destination node 1 are distributed among the downstream nodes leading to node 1 by using ratios inversely praportional to the congestion measure values of those nodes.

determined based on the congestion measure values. - The number of vehicles to be admitted to the network at each sampling instant is obtained based on the flow control algorithm given Section 3. In order to obtain the destinaton nodes of vehicles waiting in the queue, a random number generator was developed. The generator produces a destination node for a vehicle being at node n with probability equal to the ratio of vehicles with that destination in that queue; i.e., at node n probability (destination

=l) = (~(k)/ L

(~(k).

vEJ~

Besides the control module, some more changes were also required for the program: - The related parts of the original METANET program were changed to calculate the mean speed at the beginning of the links (vm.o(k)) and density at the end of the links (Pm.f ... +l(k)) as defined in Section 2. - The related parts of the original METANET program were changed to calculate the mean speed on the on-ramps and density on the offramps.

After determining the splitting rates, the algorithm determines the number of vehicles to be admitted to - The calculation of queue lengths was changed the network at that sampling instant; i.e., it applies based on the flow control presented in Section the flow control. The algorithm does not admit a 3. vehicle before it admits other vehicles in front of - Additional information (v~, v;;t, etc.) were that vehicle in any queue. The number of vehicles defined for the network description input file to be admitted is also determined based on the conwhich were not provided in METANET. gestion measure values. A vehicle is admitted to the network only if the congestion measure value of at least one of the downstream nodes leading to the By using the developed program, the proposed aldestination of that vehicle is less than or equal to gorithm is simulated for many different network topologies and traffic conditions. In this paper, the congestion measure value of the present node. one of these simulation studies will be presented (see, iftar and Ataljlar (1998) for further simulations). The proposed algorithm is applied to the 4. SIMULATION STUDIES example network shown in Figure 1 which is taken from Messmer and Papageorgiou, 1990. The netIn order to illustrate the performance of the prowork consists of 21 links (subdivided in 45 segposed controller presented in Section 3, a simulaments) and 19 nodes. The traffic enters the nettion program is developed. The program is written work from 5 source nodes (U1, .. . , U5) in order to in the C programming language under the UNIX reach the 5 destination nodes (Z1, ... , Z5). During operating system. It is based on METANET (Messthe simulation, the network is fed with the traffic mer and Papageorgiou, 1990). In order to simulate data for 6 hours. For the purpose of comparison, the proposed control algorithm, a control module is the control approach used in Messmer and Papaadded to METANET. By using this module: georgiou, 1990 is also applied to the same network under the same conditions as an alternate approach. - The congestion measure values are calculated The simulation results are presented in Figures 2-5. for each node n and destination node 1 (l E The graphics of simulation results for the proposed J~) at each sampling instant. The constant approach and the alternate approach are shown in parameters required for calculation of the con- same figure. The continuous lines indicate the simgestion measure (a~ and /3~) are defined in an ulation results of the proposed control and dashdot input file by the user. lines indicate those of the alternate approach. The - The splitting rates for the routing control are graphics of the total queue lengths are shown in

346

Figure 2. The other figures are graphics of the simulation results on the 4 th segment of the link L4 which have 3 lanes and 7 segments. The graphics of the flow rates (veh/h) , the mean speeds (km/h) and the densities (veh/km/lane) are shown in Figures 3-5 respectively. From Figure 2, it is seen that the maximum value of the total queue length produced by the proposed controller is about 64% of the maximum value of the total queue length produced by the alternate approach. The superiority of the proposed controller over the alternate controller can also be observed from the traffic conditions on the example link L4 (Figures 3-5). When the proposed controller is applied, congestion occurs on that link at about 08:20, which is about 2 hours later than when the alternate approach is applied.

Table 1: Performance indices

TTT TTW VD! VDO

The Proposed Approach

The Alternate Approach

9935 .61 38766.43 63462 60523

13989.50 55204.67 54914 51602

REFERENCES Ata§lar, B. and A. iftar (1997) . A decentralized routing control algorithm for transportation networks (in Turkish). Proceedings of the 7th National Electrical-Electronics-Computer Engineering Congress, vol. 2, pp. 380-383, Ankara, Turkey.

For the purpose of comprasion, some performance indices are also examined for both approaches during the simulation. The values of these indices are presented in Table 1. The meanings of the crite- Goldstein, N. B. and K. S. P. Kumar (1982) . A deria terms in the table are: TTT: total time spent centralized control strategy for freeway regfor traveling during the simulation period (veh x ulation. Transportation Research, Part B, h) , TTW: total time spent for waiting in queues vol. 16, pp. 279-290. during the simulation period (veh x h), VDI: the iftar, A. (1996). A decentralized routing connumber of vehicles driven into the network during trol strategy for semi-congested highways. the simulation period (veh), VDO: the number of Preprints of the 13th IFAC World Congress, vehicles driven out of the network during the simvol. P, pp. 319-324, San Francisco, CA. ulation period (veh). It is seen that the number of if tar , A. (1997). An intelligent control approach vehicles allowed to use the network by the proposed to decentralized routing and flow control in controller is about 16% more than that allowed by highways. Proceedings of the 12th IEEE Inthe alternate controller. In spite of the more vehiternational Symposium on Intelligent Control, cles, the total time comsumed during traveling in pp. 269-274, istanbul, Turkey. the case of the proposed controller is about 29% less than that in the case of the alternate controller iftar , A. and B. Ata§lar (1998). Routing controller and the total time spent during waiting in queues design to prevent traffic congestion in transin the case of the proposed controller is about 30% portation networks (in Turkish). Final Reless than that in the case of the alternate controller. port for Grant No: EEEAG-173, to be presented to the Electric, Electronics, and Infor5. CONCLUSION matics Research Grant Committee, Scientific and Technical Research Council of Turkey A decentralized control approach was proposed for (TUBiTAK), Turkey. (In preparation). transportation networks. Flow control and routing Isaksen, L. and H. J. Payne (1973). Subopticontrol were considered together and a control algomal control of linear systems by augmentarithm was developed by using the congestion meation with application to freeway traffic regulasure approach proposed by iftar (1997) . In order tion. IEEE Transactions on Automatic Conto illustrate the performance of the proposed control, vol. AC-18, pp. 210-219. trol approach, a simulation program has been deMessmer, A. and M. Papageorgiou (1990). veloped and the algorithm has been simulated for METANET: A macroscopic simulation promany different network topologies and traffic condigram for motorway networks. Traffic Engng . tions. Controller's performance was illustrated with and Control, vol. 31, pp. 466-470. a few simulation results in this paper. Further results can be found in iftar and Ata§lar (1998) . Messmer, A. and M. Papageorgiou (1994). Automatic control methods applied to freeway network traffic. Automatica, vol. 30, pp. 691702.

ACKNOWLEDGEMENT The authors would like to thank Prof. Markos Papageorgiou and Dr. Albert Messmer for providing the source code of METANET.

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Papageorgiou, M. (1990). Dynamic modeling, assignment, and route guidance in traffic networks. Transportation Research, Part B, vol. 24, pp. 471-495.

Papageorgiou, M., J. Blosseville, and H. HadjSalem (1989). Macroscopic modeling of traffic flow on the Boulevard Peripherique in Paris. Traruportation Research, Part B, vol. 23B, pp. 29-47. 15"00

Papageorgiou, M., J. Blosseville, and H. Hadj-Salem (1990). Modeling and real-time control of traffic flow on the southern part of Boulevard Peripherique in Paris: Part II: Coordinated on-ramp metering. Traruportation Research, Part A, vol. 24, pp. 361-370. Sarachik, P. E. (1982). An effective local dynamic strategy to clear congested multi-destination networks. IEEE Transactioru on Automatic Control, vol. AC-27, pp. 510-513. Sarachik, P. E. and U. Ozgiiner (1982). On decentralized dynamic routing for congested traffic networks. IEEE Traruactions on Automatic Control, vol. AC-27, pp. 1233-1238. Sheu, H. T . (1987). A coordinated decentralized flow and routing control algorithm for an automated highway system. Ph.D. Dissertation, Dept. of Electrical Eng., The Ohio State University, Columbus, Ohio.

'''''''

,<>0

.<>0

5,00

..

,

' .00

Figure 2: Total queue lengths. t"I"'lclbor""~)

~'----~----~----~~~----~----,

:t

. 1~\

:1 -~

=1

, ',-- ------ J

:~

,

' ' ' f-------~·.~..;===;...;=~~.=<>O----,~..da,~--~~OO~--~'=<>O--~IO~ tme cl IN

Figure 3: Traffic flow rates.

Z2

. :. :..):.

lL26

"o,r---~----~-.. =-~-::..:.:

@ 25

~ 17

(h:m)

~---------

Z5 '00

L24 10

60

«Il

20r I

" -------

;

,

""

,.,

6-,0 0

1'00 _ d . . clily[tl-'")

_-

,L'---~:....:....::..j

e-oo

..

,

Figure 4: Mean Speeds .

..

IOI I ----~--~----~--~--~====~

,J I I

"I !

sot

..,l )

i

30f

I

Figure 1:

The example network (Messmer and Papageorgiou, 1990).

348

l_.. __... - -_...

~~~==~,~~~.;~--,~OO~--~_~--~'~ ..--~I".. . . aI ... ..,(tI:tJI)

Figure 5: Traffic densities.