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Nonlinear Optimal Control Applied to Coordinated Ramp Metering in the Amsterdam Ring-Road
Copyright @ IFAC Large Scale Systems: Theory and Applications. Bucharest. Romania. 200 1
NONLINEAR OPTIMAL CONTROL APPLIED TO COORDINATED RAMP METERING IN THE AMSTERDAM RING-ROAD Apostolos Kotsialos· Markos Papageorgiou·
• {}ynarnic Systems and Simulation Laboratory, Technical lJni1!el'sity of Cl'Cte, Chania, Greece e-mail: Ilppie @ilssl.tuc.gr
Abstrart: Thp goal of t.his pappr is to dpscrihe a generic approach to the problem of optilllal coordinated ramp metering control in large-scalp motorway networks. In this approach, tilt' traffic flow procpss is macroscopically modelled by use of a second-order lIlacroscopic t.raffic flow modt'\. The overall prohlem of coordinatt'd ramp metering is formulat.t'd as a constrained discretc-t.im(, nonlint'ar optimal control prohlem, and a feasihh'-direction nonlincar opt.imisation algorithm is employed for its numerical solution. The control stratt'gy's t'fficienry is demonstrated through its application to the 32-km Amst.erdam ring-road for a numbt'r of differt'nt sct'narios. Copyright @20011FAC
Kt'ywords: Traffic control, Optilllal co IIt rol , Opt.irnisarion , Nonlinear systems.
1. INTRODUCTION
descrihed in this paper and the control software tool AMOC (Advanced Motorway Optimal Control). Results for coordinated ramp metering baspd on AMOC are also reported in (Kotsialos et al., 2001).
A numher of approaches have bcen dpveloped in the past for t.he design of co Iltro I stratcgies for motorway Hct.works that involvt' cont.rol nwasures such as !'Out.e recommendation wit.h t.he usp of Variahle Mcssage SigHS (VMS) or equipped vehicles , ramp metering, motorway-t.o-mot.orway (mtm) control, etc. In this paper the optimal control approach (discret<,-t.inw formulation) is applied to the design of larg<'-scale optimal coordinated ramp nl('tering control strat.egies.
The rest of this paper is struct.ured as follows. Section 2 briefly presents the macroscopic traffic flow modd which is used for the design of the control strategy. Section 3 formulates the problem of coordinated ramp mt'tering as a const.rained nonlinear discrt'te-tinw optimal control problem. Section 4 prespnts thp numerical solution algorithm for the formulated prohlem. Section 5 presents the application results for the Amsterdam ring-road. The maill conclusions and future work are reported in sect.ion 6.
Early applications of nonlinear collst.rain('d optimal control to coordinatpr! ramp IlIctPriHg were rpportt'd in (nIinkin, 19i6), and (Papageorgiou, 1983). In (Bhouri cl at.. 1990) alld (Bhomi . 1991) a similar a.pproach WilS applied to the Boulevard Ppripheriqut' ill Paris, while in (Zhallg cl al., 1996) a differelll ilpproach was applied to il IlwdiullIscale motorway stretch. In (Kotsialos et 0.1.,1999) J'(~sults rplat.l~d to th(' integrar.ed control of motorway IIPtworks. considering hoth ro1ltP J'('COTIIJl)en
2. TIlE MACROSCOPIC TRAFFIC FLOW MODEL A validat.pd spcolHl-order traffic flow model is used for t h(~ descript.ion of traffic flow on motorway networks to provide the modelling part of the optimal
451
-
control problem formulation (see (Messmer and Papageorgiou , 1990) for details). The network is represented by a directed graph whereby the links of the graph represent motorway stretches. Each mot.orway st.retch has uniform characteristics, i.e. no on-/off-ramps and no major changes in geometry. The nodes of t.he graph are placed at locations where a major change in road geometry occurs, as well as at junctions , onramps , and off-ramps.
Fig. 1. The origin-link queue model. a simple queue model is used (Fig1l1'e 1). The outflow q" of an origin link () depends on the traffic c:onditions of t.lw corresponding mainstream segment. (Il , 1) and t.he ex ist.ence of ramp met.ering c:ontrol measures. If ramp metering is applied, thcn t.he out.flow qo(k) that is allowed t.o leave origin () during period k , is a port.ion T,,(k) of t.he maximum outflow (I,,(k) that would leave ° and ('nter t.he mainst.ream in absence of ramp m('t('ring. Thus , To(!.:) E [Tlnin"" 1J is thc metering rate for tllf' origin link 0, i.e. a control variable, where Tlllin,,, is a millinlllm admissihle value; typic:ally, Tlllin,o > 0 is chos('n so as t.o avoid ramp closure. If 'ro(k) = 1, no ramp metering is applied , else 'ro(k) < 1. The queueillg model is described hy the following equatioll
The t.ime and space argument.s are discretised . The discrete-time step is denoted by T (t.ypically T = 10 ... 15s). A motorway link m is divided into N m segment.s of equal length Lm (typically Lm :::::; 500m). Each segment i of link m at time t = kT , k = 0, ... , J( , is macroscopically characterised via the following variables: The traffic density Pm ,i( k) (veh/lane-km) is the number of vehicles in segment i of link m at time t = kT divided by Lm and by the number of lanes Am; t.he mea.n speed vm,i(k) (km/h) is the mean speed of the vehicles included in segment i of link m at time kT; and the traffic volume or flow qm,i(k) (veh/h) is the number of vehicles leaving segment i of link m during t.he time period [kT, (k + 1)T], divided by T. For each segment i of link 111 at each time step k we have the following equations:
-(k+1) Pm,,_
m
(2)
qm ,i (k) = Pm,;(k)vm,i(k)A m 1)m ;
,
where 1110(k) is t.he Cjueu(' length (veh) in origin time kT, and d" (k) is t.he demand (veh /h) at () at. t.he same period. The outflow q,,(k) is determined as follows:
° at
= Pm ,l-( k)+ T[qm ,i-I( k) - qm.i(k) ] (1) L A 111
segment. #(/1,1) q,,(k)
(6)
'.)_ (k) T{V[Pm ,; (k)]-Vm ,i(k)} (,,+1 - Vm , i +--'----"--'-'-'-'--'---''-'---'-'-'-''-'-'T +T
[7)m ,i-1
wit.h
(k) - vm,i(k)] vm,i(k)
Lm
qo(k) = mill {q" ,1 ("~),q" , 2(k)}
_ liT Pm ,H ! (k) - Pm ,;(k)
Pm ,i(k) V[Pm,i(k)]
= VI,m exp [ -
1n
(3)
+K
1 (Pm; (k) ) -:;- - ' -
(7)
(j m
qo , 1
1 (4 )
,
= d,,(k)
q" ,2 =
Per,m
where vI,m denotes the free-flow speed of link 111 . Per,m denotes the critical density per lanf' of link m (the density where the maximum flow in the link occurs) , and am is a parameter of the fundamental diagram (eqn. (4)) of link m. Furt.hermore, T, a time constant, 11, an anticipation constant. and K., are constant parameters , which are equal for all net.work links. Additionally, the mean speed resulting from (3) is limited from below hy the minimum speed in the network Vmin.
+ 1I!0(k)/T
Q ' omm
{I
,
Pmax - Pll ,l (k)} Pmax - PC,.,ll
(8) (9)
Two additional terms are added to (3) in order t.o consider the speed decrease caused hy merging phenomena at a junction and hy lanes drops , respectively, see (Papageorgiou et a.l. , 1990) for details.
where Qo is t.he on-ramp's capacit.y (veh/h), i.e. t.he on-ramp 's maximum possible ou t.flow under free-flow t.raffic condit.ions in th(' mainst.ream; and Pmax (veh/lane-km) is t.he maximum density in t.he net.work segment.s. Thus t.he maximum outflow qo(k) is det.ermined hy the current origin demand if qo ,1 < Qo.2 (sec (7), (8)), or by the geometrical ramp capac:it.y Q" if t.he mainst.ream densit.y is underc:rit.ic:al , i.e. Pll ,! (k) < Per ,I' (see (9)) , or by t.he reduced capacity due t.o congest.ion of the mainstream , if Pll ,! (/.:) > Per,ll (s('e (9)). Thus , (9) models t.he reduct.ion of t.he origin link's capacit.y dIH' t.o mainst.ream c:ongest.ion. A model similar t.o (5)-(9) applies t.o mot.orway-tomotorway int.erchanges.
For origin links, i.e . links t.hat receive traffic d('mand and forward it into the motorway network ,
Motorway hifmcations and junctions (including on-ramps and off-ramps) are represent.ed by
452
nodes. Traffic ('nt.ers it node 11. t.hrough a number of input links aJl(1 is e!istrilmted to the output links
Q,,(I.:)
L
=:
,,( 'n
({",N,.
(1.:)
subject. to
x(k
UP,min::; uf(ktl::;
(11 )
=
If a nod e 11. has mOl'e t.han one leaving links , then t.he upstl'eam inHuence of densit.y (which is necessary fol' the modelling of congest.ion spillback) is t.aken int.o
=
The chosen cost criterion aims at minimising the Tot.al Time Spent. (TTS) of all vehicles in the network (inclue!ing t.he wait.ing time experiencee! inside t.he network queues) . More precisely, the cost crit.erion is as follows
=
J =
3 PIWBLEM FORMULATlOl\' + 0
The cool'dinated ramp metering cont.rol prohlem is formulated as a dynamic optimal cont.rol problem wit.h const.rained cont.rol variables which can 1)(' solved numerically over a given time horizon . The mot.orway traffic How is considered as a process under cont.rol via the various ralllp meters installed at. the' net.work ent.rances. The st.at.e of t.he' process is described hy t.he st.ate vector x E \RN. The input variables are dist.inguish ed into control variables 11 E ~RM ane! the uncont.rollable ext.ernal dist.urbances d E ~R/} In t.he following , we will int.roduce a general problelll formulat.ion whereby t.he value of each control mf'aS\ll'e may change I(~ ss frequent.ly than at. each model sample t.ime T. Assume that. th~' 111 differellt control measures have ]J distinct cont.rol sample t.imes which are assumed t.o he multiples of the model sample tillle T , i.e . Tp zpT , Zp E Ai, = 1, ... , ]1, 71 < AI dimll. Let J.:, integer [I.:j.::f ].
=
r,. .. .
=
and l'E {1, .. . ,p}(14)
Based on the previous section, it may be seen that. by substituting (2) , (10), (11) into (1); (4) into (3); (6)-(9) into (5), the traffic How model equat.ions take the form of eqn . (13). In this case t.he st.at.e vector x consists of the e!ensities Pm ,; and mean speeds V m ,; of every segment i of every link 1n , and t.he queues Wo of every origin o. The control vector 11 consists of the ramp metering rat.es roof every on-ramp 0 under control , with r" .mill ::; ro(k) ::; 1 according to (14). We assume here that all ramp meters have the same sample time TI ZI T. Finally, the disturbance vect.or consists of the demands do at every origin of t.he network, and all the turning rates j3~1 at the ne twork 's bifurcations.
where In is t.he set. of links entering node 11. , On is t.he set. of links leaving 11., Q,,(I.:) is t.he t.ot.al traffic volume ('nt.ering 11. at period k , Cfm.o(k) is t.he t.raffic volume t.hat leaves 11. vi
=
Uf ,max
(13)
where f{ is t.he considered time horizon, and 19, 'P are arbitrary, t.wice different.iable, nonlinear cost functions.
(10)
IIm.o(l.:) = /'J',;'(I.:)Qn(k) 'rim. E On
+ 1) = f[x(k), u(k) , d(k)], x(O) = Xo
f
T~ {~~ Pm ,;(k)LmAm + ~ 1/I,,(k)+ ~ [r,,(kIl-r,,(k
l
-1 )]2
+ aw ~ 1jJ [w o (k)]2 } (15)
with '1/1 [1/1 0
(/.:)]
= max {O, 11!o(k) -
(16)
1Uo ,max}
where (}f , a", are weighting factors . The first two t.erms of equat.ion (15) correspond exactly to the TTS. The term with weight o'f is included ill the cost criterion to suppress high-frequency oscillations of the control trajectories . The last penalty term is included in the cost criterion in order t.o enable the cont.rol st.rategy to limit t.he queue lengths at the origins if and to the level desiree!. The parameters 1Uo .max are predet.ermined constants t.hat express the maximum permissible number of vehicles at any time period in origin a 's queue.
r
Thell 1l(1.~ ) - = [Ill (1.: 1 UI'(kl, )T(. The gell('ral discrpt(,-t.illle formulat.ion of t.he' opt.imal control problelll reads:
4. OPTIMALlTY CONDITIONS Define t.he quant.ities [\- - 1
Minimise f \ -f =
"' - 1
.J = 1)[1\-] +
L
(12 )
k ~ ()
45.~
Zf
if f{
f{
mod Zr
{ int.eger ( zp ) otherwise .
=0 (17)
For a given admissible trajectory ut(k t ), kt = 0, ... , Kt, the state trajectory x(k + 1) can be found by solving (13) starting with the known initial state x(O), and hence the cost criterion can be regarded as depending on the control variables only, i.e. J = ](u). The gradient of] with respect to ut(k t ) on the equality constraints surface for the time period kt is given by
A8
035 A10-East
Amsterdam
z..boqer
...,.1
Al A4
029 030
A2 A20 32
034 033
A1
Fig. 2. The Amsterdam ring-road model. where the co-state vector A E A(k)
+
)RN
satisfies
flow model against real data taken from the motorways (Kotsialos et al., 1998).
= Of[X(k~:(~t(k)f A(k + 1) +
orp [x(k), u(k)d(k)) k = ox(k) ,
The ring-road was divided in 76 segments with average length 421m. This means that the state vector is 173-dimensional (including the 21 onramp queues). If ramp metering is applied to all on-ramps, the control vector is 21-dimensional, while the disturbance vector is 41-dimensional. With a time step T lOs we have, for a horizon of 4h , [( = 1440. This means that for a control sample time of 1 minute and all on-ramps metered, there are 254,160 variables included in the resulting nonlinear optimisation problem.
° ]' _ , ... , \
1(19)
and
'(l') A
\
= o{) [x([())
ox([()'
(20)
=
A projected gradient ~l(kl) is defined to have its components ~l,i(kl) equal to 9l,i(k l ) if none of the corresponding bounds (14) is active, and ~l,i(ktl = 0 else.
Four different scenarios are considered for this network . For all scenarios it is assumed that, even in case of long on-ramp queues , no re-routing takes place. Scenario 0 corresponds to the nocontrol case, i.e. when no ramp metering control measures are applied. In scenario 1 the maximum permissible storage wo,max in (16) is set to 40 vehicles for the urban on-ramps and 100 vehicles for the motorway-to-motorway ramps; in scenario 2 these storage capacities are equal to 80 and 120 vehicles, respectively, while in scenario 3 no maximum queue constraints are considered, i.e. a w = in eqn. (15) .
The necessary conditions of optimality are given by (13), (14), (19), (20), and ~l,i(kl) = 0 Vi , kl,f.. A well-known solution algorithm based on feasible directions is described by (Papageorgiou and Marinaki, 1995). 5. APPLICATION EXAMPLE 5.1 Site description
The previously described approach to networkwide optimal ramp metering has been applied to the Amsterdam ring-road with the use of AMOC. The Amsterdam Orbital Motorway (AlO) topological network model may be seen in Figure 2. There are four main connections with other motorways, the A8 at the North, the A4 at the SouthWest, the A2 at the South, and the Al at the South-East.
°
5.2 Optimal results
When no control measures are applied, the excessive demand coupled with the uncontrolled entrance of drivers in the mainstream causes congestion from the beginning of the time horizon (Figure 3a). This congestion originates at the junction of A2 with A10 and propagates upstream blocking the A4 and a large part of the A 10-West. By the time this congestion begins to dissolve, a new one appears at the junction of A 10 with A 1 which begins to propagate upstream until it reaches the first congestion whose trend of resolving is reversed and both are united into a single more severe congestion. This strong congestion keeps the A4 entrance to the AlO blocked, which results in the accumulation of many vehicles in the
For the purposes of our study, only the counterclockwise direction of the A10, which is about 32 km long, is considered. There are 21 on-ramps on this motorway, including the connections with the A8, A4, A2, and Al motorways, and 20 offramps, including the junctions with A4, A2, AI, and A8. It is assumed that ramp metering may be performed at each of the on-ramps. The model parameters for this network were determined from validation of the network traffic
454
motorway-to-motorway (mtm) on-ramp of A4 (i.e. a spilJback of the congestion from A 10 onto the A4 motorway) and in the surrounding on-ramps (Figure 3b). As a result the TTS becomes 13,226 veh ·h.
'''' "'"eo '"'"o 10
When optimal control is applied to the network under scenario 1, the TTS becomes 9,032 veh·h, a 31. 7% improvement. This improvement is evident in Figure 4 which depicts the density and queue evolution profiles for scenario 1. The large ramp queues that occur in the no-control case are not present any more, but queues are spread to many on-ramps so as to countermand the formation of congestion.
(a)
Scenario 2 assumes that there is more ramp storage capacity at the strategy's disposal (80 vehicles for urban on-ramps and 120 vehicles for mtm onramps). When optimal control is applied under these conditions, the TTS becomes 8,230 veh ·h, a 37.8% improvement over the no-control case. This larger improvement is evident in Figure Sa where the density profile is seen to be much flatter than that of scenario 1 (Figure 4a). Because larger storage space is available to the strategy, larger queues are formed , but less on-ramps are used for storage purposes, see Figure 5b.
Fig. 3. No control: (a) density, (b) on-ramp queues.
In the case of scenario 3, the absence of any queue constraints leads to a TTS equal to 7,589 veh ·h, an improvement of 42.6% over the no-control case. As can be seen from Figure 6a, the density evolution profile is completely flat in this case , indicating the fact that there is no congestion present in the ring-road. The control strategy achieves this impressive amelioration of traffic conditions by creating large queues at the onramps that are located in the critical bottleneck area where congestion originates, see Figure 6b. The amount of TTS reduction depends on the on-ramp storage capacity available to the control strategy. The behaviour exhibited by optimal control application in the three scenarios indicates that the most efficient way to deal with bottlenecks and potential congestion is to perform ramp metering at the on-ramps immediately upstream of the primary bottleneck location, as was the case in scenario 3. In each scenario the control strategy establishes an optimal trade-off between the delay reduction due to the decrease of the congestion extent , and the delay increase due to the metering of vehicles that exit before reaching the bottleneck.
Fig. 4. Scenario 1: (a) density, (b) on-ramp queues. strongly degrades the available capacity. An impressive amelioration of traffic conditions in motorway networks (including the ramps and motorway intersections) is possible with the use of optimal ramp metering by increasing the network throughput. The highest efficiency is achieved if only the on-ramps closest to the bottlenecks are strongly metered, which however creates long queues and disbenefits the corresponding users for the sake of the general efficiency.
6. CONCLUSIONS AND FURTHER DEVELOPMENTS
The formulation of the problem of coordinated ramp metering control as a discrete-time optimal control problem, allows the application of wellknown concepts from automatic control theory,
The reported results demonstrate that the uncontrolled utilisation of the motorway infrastructure
455
van/ f'iTiphenqu(' de f'aT·is . PIi.D . Dissert.ation, Ulliversit{' de Paris-Sud. C('ntn' d 'Orsay, rran("('. Uhouri, N., M. Papag('()rgiou and .T.M. B10sseville (1990) Opt.imal cOlltrol of traffic flow on perinrball ringways with applicatioll to the BOlllf'v;ml Periph{~riqu(' ill Paris. f'r'(~ Jirmt8 11th IFAC Wo1"id Conq1'l~ ss, Vol. 10, Talinll , Estollia, 23G 243 . l1Iinkin , M. (1976). Problelll of optimal control of traffic flow Oil highways. A nto"ll!ation and R(~ 11I.otc Control 37 , 662 6G7. K01 sialos, A. , M . Papagporgioll and A. Mpssm('r (1f)99). Opt.imal coordinatpd and int.egrau,d motorway network traffic control. Ill: f'mcf:f:d inqs of the 14 th Intenw.tiono.l Symposium 0"/1 T1'II.ns]!ortatum and Tmfir: Th.f:01"Y (ISTTT) (A. C('der, Ed.), ,kmsal('m, Israel , G21 644. I{otsialos, A., M. Papag(~orgiou ;md F. Middplham (2001). Optilllal coordinated ramp nJ('tering with AMOC. In: f'r'epnnts (CD-ROM) of thf: 80th. T1'II.nsp07'tatum Rr~ sf:o.rch Hoard A n7l.1w.l Mf:eii1l.Y , Pilper No 01-3125, Washington , DC , USA. I{otsialos , A., Y. Pavlis, r. Middelham, C. Diakaki , G. Vardaka alld M. Papag('orgioll (1998). Modelling of tll(' large scalP motorway llf'twork aroulld Alllst.erdarn. Ill: f'r~print. s of 8th IFA C SY17lposinm on I,(l7W Scale Systems ThUJ7"11 and Appiimtions, Patras , Greece, 354
Fig. 5. Scena rio 2: (a) density, (b ) on-ra mp queues.
120 l OO
eo eo
., 20
o
3GO. M( ~ss m('r,
7. ACKNOWLEDGEMENTS
A. alld M. Papaw'orgiou (1990). M ETA NET: A macroscopic silllulatioll program for mot.orway lle1.works. Tm.ffir: Enyilwenng (Jud ContmI31(8/9), 466 470; 549. Papag('orgiou, M. (1983). Application of Antol1WtU: Coutml Conc~pts to Tm.ffic 1"10111 Moddliny and Contml. Sprillgf~r Vcrlag. N. York, USA. Papageorgiou , M. alld M. Marillaki (1905). A Fe(J.8ihlt~ Di7"(~ction Alg07'ithm for the Nurner'ical Solll.l.ion of Optimal Clmtml f'ml!lf:lI!.~. DYllamic Syst.ems amI Simulation Laborat ory, Tecilllical Univ('J'sity of Cn~t<'. Chania, Gr('('c(' . Papageorgioll, M. , .J.M 13I0ssevill(' alld H. HadjSaknl (1990). Modpllillg awl rml-t.iTlle C0I1trol of traHic flow 011 1.lw south('rn part of Boulpvilrcl Prriph<"~riqll< ' ill Paris - Part I lIlodelling. Tm.nsportatio1/. H(~ Sf:I/.1'('h A 24 , 345
The authors would like to thank Mr. r. I\liddelham fram AVV-Rijkswiltersl.ilar. , Thf' NNhprlands , for providing the necessn.ry dilta for t hp Amsterdill1l ring-road.
Zhallg , H. , S. Ri1chi(' alld \\'. H('ckf'r (1996) . Some gf'll('ral results Oil 1.h(' optilllal ramp meterillg cOIITrol problplll. T{'(J. n~J!{)7"fI/.I.io n H(~ sm.1'I; h C 4 , .'il 69.
Fig. 6. Scenario 3: (a) density, (h) on-ramp q\J('ues. and allows the consideration of other control measures as well. Further control measures such as speed rantral or route guidance may 1)(' readily integrated cooperatively due to the flexible natnrf' of the prohlem formulation.
359.
8. REFERENCES Bhouri, N. (1991). Com.m.a.ndf: d'un Sy8th/l.(~ d~ Traffic A utoTOutif:r: Application (/.11. IJonle-
456
NONLINEAR OPTIMAL CONTROL APPLIED TO COORDINATED RAMP METERING IN THE AMSTERDAM RING-ROAD Apostolos Kotsialos· Markos Papageorgiou·
• {}ynarnic Systems and Simulation Laboratory, Technical lJni1!el'sity of Cl'Cte, Chania, Greece e-mail: Ilppie @ilssl.tuc.gr
Abstrart: Thp goal of t.his pappr is to dpscrihe a generic approach to the problem of optilllal coordinated ramp metering control in large-scalp motorway networks. In this approach, tilt' traffic flow procpss is macroscopically modelled by use of a second-order lIlacroscopic t.raffic flow modt'\. The overall prohlem of coordinatt'd ramp metering is formulat.t'd as a constrained discretc-t.im(, nonlint'ar optimal control prohlem, and a feasihh'-direction nonlincar opt.imisation algorithm is employed for its numerical solution. The control stratt'gy's t'fficienry is demonstrated through its application to the 32-km Amst.erdam ring-road for a numbt'r of differt'nt sct'narios. Copyright @20011FAC
Kt'ywords: Traffic control, Optilllal co IIt rol , Opt.irnisarion , Nonlinear systems.
1. INTRODUCTION
descrihed in this paper and the control software tool AMOC (Advanced Motorway Optimal Control). Results for coordinated ramp metering baspd on AMOC are also reported in (Kotsialos et al., 2001).
A numher of approaches have bcen dpveloped in the past for t.he design of co Iltro I stratcgies for motorway Hct.works that involvt' cont.rol nwasures such as !'Out.e recommendation wit.h t.he usp of Variahle Mcssage SigHS (VMS) or equipped vehicles , ramp metering, motorway-t.o-mot.orway (mtm) control, etc. In this paper the optimal control approach (discret<,-t.inw formulation) is applied to the design of larg<'-scale optimal coordinated ramp nl('tering control strat.egies.
The rest of this paper is struct.ured as follows. Section 2 briefly presents the macroscopic traffic flow modd which is used for the design of the control strategy. Section 3 formulates the problem of coordinated ramp mt'tering as a const.rained nonlinear discrt'te-tinw optimal control problem. Section 4 prespnts thp numerical solution algorithm for the formulated prohlem. Section 5 presents the application results for the Amsterdam ring-road. The maill conclusions and future work are reported in sect.ion 6.
Early applications of nonlinear collst.rain('d optimal control to coordinatpr! ramp IlIctPriHg were rpportt'd in (nIinkin, 19i6), and (Papageorgiou, 1983). In (Bhouri cl at.. 1990) alld (Bhomi . 1991) a similar a.pproach WilS applied to the Boulevard Ppripheriqut' ill Paris, while in (Zhallg cl al., 1996) a differelll ilpproach was applied to il IlwdiullIscale motorway stretch. In (Kotsialos et 0.1.,1999) J'(~sults rplat.l~d to th(' integrar.ed control of motorway IIPtworks. considering hoth ro1ltP J'('COTIIJl)en
2. TIlE MACROSCOPIC TRAFFIC FLOW MODEL A validat.pd spcolHl-order traffic flow model is used for t h(~ descript.ion of traffic flow on motorway networks to provide the modelling part of the optimal
451
-
control problem formulation (see (Messmer and Papageorgiou , 1990) for details). The network is represented by a directed graph whereby the links of the graph represent motorway stretches. Each mot.orway st.retch has uniform characteristics, i.e. no on-/off-ramps and no major changes in geometry. The nodes of t.he graph are placed at locations where a major change in road geometry occurs, as well as at junctions , onramps , and off-ramps.
Fig. 1. The origin-link queue model. a simple queue model is used (Fig1l1'e 1). The outflow q" of an origin link () depends on the traffic c:onditions of t.lw corresponding mainstream segment. (Il , 1) and t.he ex ist.ence of ramp met.ering c:ontrol measures. If ramp metering is applied, thcn t.he out.flow qo(k) that is allowed t.o leave origin () during period k , is a port.ion T,,(k) of t.he maximum outflow (I,,(k) that would leave ° and ('nter t.he mainst.ream in absence of ramp m('t('ring. Thus , To(!.:) E [Tlnin"" 1J is thc metering rate for tllf' origin link 0, i.e. a control variable, where Tlllin,,, is a millinlllm admissihle value; typic:ally, Tlllin,o > 0 is chos('n so as t.o avoid ramp closure. If 'ro(k) = 1, no ramp metering is applied , else 'ro(k) < 1. The queueillg model is described hy the following equatioll
The t.ime and space argument.s are discretised . The discrete-time step is denoted by T (t.ypically T = 10 ... 15s). A motorway link m is divided into N m segment.s of equal length Lm (typically Lm :::::; 500m). Each segment i of link m at time t = kT , k = 0, ... , J( , is macroscopically characterised via the following variables: The traffic density Pm ,i( k) (veh/lane-km) is the number of vehicles in segment i of link m at time t = kT divided by Lm and by the number of lanes Am; t.he mea.n speed vm,i(k) (km/h) is the mean speed of the vehicles included in segment i of link m at time kT; and the traffic volume or flow qm,i(k) (veh/h) is the number of vehicles leaving segment i of link m during t.he time period [kT, (k + 1)T], divided by T. For each segment i of link 111 at each time step k we have the following equations:
-(k+1) Pm,,_
m
(2)
qm ,i (k) = Pm,;(k)vm,i(k)A m 1)m ;
,
where 1110(k) is t.he Cjueu(' length (veh) in origin time kT, and d" (k) is t.he demand (veh /h) at () at. t.he same period. The outflow q,,(k) is determined as follows:
° at
= Pm ,l-( k)+ T[qm ,i-I( k) - qm.i(k) ] (1) L A 111
segment. #(/1,1) q,,(k)
(6)
'.)_ (k) T{V[Pm ,; (k)]-Vm ,i(k)} (,,+1 - Vm , i +--'----"--'-'-'-'--'---''-'---'-'-'-''-'-'T +T
[7)m ,i-1
wit.h
(k) - vm,i(k)] vm,i(k)
Lm
qo(k) = mill {q" ,1 ("~),q" , 2(k)}
_ liT Pm ,H ! (k) - Pm ,;(k)
Pm ,i(k) V[Pm,i(k)]
= VI,m exp [ -
1n
(3)
+K
1 (Pm; (k) ) -:;- - ' -
(7)
(j m
qo , 1
1 (4 )
,
= d,,(k)
q" ,2 =
Per,m
where vI,m denotes the free-flow speed of link 111 . Per,m denotes the critical density per lanf' of link m (the density where the maximum flow in the link occurs) , and am is a parameter of the fundamental diagram (eqn. (4)) of link m. Furt.hermore, T, a time constant, 11, an anticipation constant. and K., are constant parameters , which are equal for all net.work links. Additionally, the mean speed resulting from (3) is limited from below hy the minimum speed in the network Vmin.
+ 1I!0(k)/T
Q ' omm
{I
,
Pmax - Pll ,l (k)} Pmax - PC,.,ll
(8) (9)
Two additional terms are added to (3) in order t.o consider the speed decrease caused hy merging phenomena at a junction and hy lanes drops , respectively, see (Papageorgiou et a.l. , 1990) for details.
where Qo is t.he on-ramp's capacit.y (veh/h), i.e. t.he on-ramp 's maximum possible ou t.flow under free-flow t.raffic condit.ions in th(' mainst.ream; and Pmax (veh/lane-km) is t.he maximum density in t.he net.work segment.s. Thus t.he maximum outflow qo(k) is det.ermined hy the current origin demand if qo ,1 < Qo.2 (sec (7), (8)), or by the geometrical ramp capac:it.y Q" if t.he mainst.ream densit.y is underc:rit.ic:al , i.e. Pll ,! (k) < Per ,I' (see (9)) , or by t.he reduced capacity due t.o congest.ion of the mainstream , if Pll ,! (/.:) > Per,ll (s('e (9)). Thus , (9) models t.he reduct.ion of t.he origin link's capacit.y dIH' t.o mainst.ream c:ongest.ion. A model similar t.o (5)-(9) applies t.o mot.orway-tomotorway int.erchanges.
For origin links, i.e . links t.hat receive traffic d('mand and forward it into the motorway network ,
Motorway hifmcations and junctions (including on-ramps and off-ramps) are represent.ed by
452
nodes. Traffic ('nt.ers it node 11. t.hrough a number of input links aJl(1 is e!istrilmted to the output links
Q,,(I.:)
L
=:
,,( 'n
({",N,.
(1.:)
subject. to
x(k
UP,min::; uf(ktl::;
(11 )
=
If a nod e 11. has mOl'e t.han one leaving links , then t.he upstl'eam inHuence of densit.y (which is necessary fol' the modelling of congest.ion spillback) is t.aken int.o
=
The chosen cost criterion aims at minimising the Tot.al Time Spent. (TTS) of all vehicles in the network (inclue!ing t.he wait.ing time experiencee! inside t.he network queues) . More precisely, the cost crit.erion is as follows
=
J =
3 PIWBLEM FORMULATlOl\' + 0
The cool'dinated ramp metering cont.rol prohlem is formulated as a dynamic optimal cont.rol problem wit.h const.rained cont.rol variables which can 1)(' solved numerically over a given time horizon . The mot.orway traffic How is considered as a process under cont.rol via the various ralllp meters installed at. the' net.work ent.rances. The st.at.e of t.he' process is described hy t.he st.ate vector x E \RN. The input variables are dist.inguish ed into control variables 11 E ~RM ane! the uncont.rollable ext.ernal dist.urbances d E ~R/} In t.he following , we will int.roduce a general problelll formulat.ion whereby t.he value of each control mf'aS\ll'e may change I(~ ss frequent.ly than at. each model sample t.ime T. Assume that. th~' 111 differellt control measures have ]J distinct cont.rol sample t.imes which are assumed t.o he multiples of the model sample tillle T , i.e . Tp zpT , Zp E Ai, = 1, ... , ]1, 71 < AI dimll. Let J.:, integer [I.:j.::f ].
=
r,. .. .
=
and l'E {1, .. . ,p}(14)
Based on the previous section, it may be seen that. by substituting (2) , (10), (11) into (1); (4) into (3); (6)-(9) into (5), the traffic How model equat.ions take the form of eqn . (13). In this case t.he st.at.e vector x consists of the e!ensities Pm ,; and mean speeds V m ,; of every segment i of every link 1n , and t.he queues Wo of every origin o. The control vector 11 consists of the ramp metering rat.es roof every on-ramp 0 under control , with r" .mill ::; ro(k) ::; 1 according to (14). We assume here that all ramp meters have the same sample time TI ZI T. Finally, the disturbance vect.or consists of the demands do at every origin of t.he network, and all the turning rates j3~1 at the ne twork 's bifurcations.
where In is t.he set. of links entering node 11. , On is t.he set. of links leaving 11., Q,,(I.:) is t.he t.ot.al traffic volume ('nt.ering 11. at period k , Cfm.o(k) is t.he t.raffic volume t.hat leaves 11. vi
=
Uf ,max
(13)
where f{ is t.he considered time horizon, and 19, 'P are arbitrary, t.wice different.iable, nonlinear cost functions.
(10)
IIm.o(l.:) = /'J',;'(I.:)Qn(k) 'rim. E On
+ 1) = f[x(k), u(k) , d(k)], x(O) = Xo
f
T~ {~~ Pm ,;(k)LmAm + ~ 1/I,,(k)+ ~ [r,,(kIl-r,,(k
l
-1 )]2
+ aw ~ 1jJ [w o (k)]2 } (15)
with '1/1 [1/1 0
(/.:)]
= max {O, 11!o(k) -
(16)
1Uo ,max}
where (}f , a", are weighting factors . The first two t.erms of equat.ion (15) correspond exactly to the TTS. The term with weight o'f is included ill the cost criterion to suppress high-frequency oscillations of the control trajectories . The last penalty term is included in the cost criterion in order t.o enable the cont.rol st.rategy to limit t.he queue lengths at the origins if and to the level desiree!. The parameters 1Uo .max are predet.ermined constants t.hat express the maximum permissible number of vehicles at any time period in origin a 's queue.
r
Thell 1l(1.~ ) - = [Ill (1.: 1 UI'(kl, )T(. The gell('ral discrpt(,-t.illle formulat.ion of t.he' opt.imal control problelll reads:
4. OPTIMALlTY CONDITIONS Define t.he quant.ities [\- - 1
Minimise f \ -f =
"' - 1
.J = 1)[1\-] +
L
(12 )
k ~ ()
45.~
Zf
if f{
f{
mod Zr
{ int.eger ( zp ) otherwise .
=0 (17)
For a given admissible trajectory ut(k t ), kt = 0, ... , Kt, the state trajectory x(k + 1) can be found by solving (13) starting with the known initial state x(O), and hence the cost criterion can be regarded as depending on the control variables only, i.e. J = ](u). The gradient of] with respect to ut(k t ) on the equality constraints surface for the time period kt is given by
A8
035 A10-East
Amsterdam
z..boqer
...,.1
Al A4
029 030
A2 A20 32
034 033
A1
Fig. 2. The Amsterdam ring-road model. where the co-state vector A E A(k)
+
)RN
satisfies
flow model against real data taken from the motorways (Kotsialos et al., 1998).
= Of[X(k~:(~t(k)f A(k + 1) +
orp [x(k), u(k)d(k)) k = ox(k) ,
The ring-road was divided in 76 segments with average length 421m. This means that the state vector is 173-dimensional (including the 21 onramp queues). If ramp metering is applied to all on-ramps, the control vector is 21-dimensional, while the disturbance vector is 41-dimensional. With a time step T lOs we have, for a horizon of 4h , [( = 1440. This means that for a control sample time of 1 minute and all on-ramps metered, there are 254,160 variables included in the resulting nonlinear optimisation problem.
° ]' _ , ... , \
1(19)
and
'(l') A
\
= o{) [x([())
ox([()'
(20)
=
A projected gradient ~l(kl) is defined to have its components ~l,i(kl) equal to 9l,i(k l ) if none of the corresponding bounds (14) is active, and ~l,i(ktl = 0 else.
Four different scenarios are considered for this network . For all scenarios it is assumed that, even in case of long on-ramp queues , no re-routing takes place. Scenario 0 corresponds to the nocontrol case, i.e. when no ramp metering control measures are applied. In scenario 1 the maximum permissible storage wo,max in (16) is set to 40 vehicles for the urban on-ramps and 100 vehicles for the motorway-to-motorway ramps; in scenario 2 these storage capacities are equal to 80 and 120 vehicles, respectively, while in scenario 3 no maximum queue constraints are considered, i.e. a w = in eqn. (15) .
The necessary conditions of optimality are given by (13), (14), (19), (20), and ~l,i(kl) = 0 Vi , kl,f.. A well-known solution algorithm based on feasible directions is described by (Papageorgiou and Marinaki, 1995). 5. APPLICATION EXAMPLE 5.1 Site description
The previously described approach to networkwide optimal ramp metering has been applied to the Amsterdam ring-road with the use of AMOC. The Amsterdam Orbital Motorway (AlO) topological network model may be seen in Figure 2. There are four main connections with other motorways, the A8 at the North, the A4 at the SouthWest, the A2 at the South, and the Al at the South-East.
°
5.2 Optimal results
When no control measures are applied, the excessive demand coupled with the uncontrolled entrance of drivers in the mainstream causes congestion from the beginning of the time horizon (Figure 3a). This congestion originates at the junction of A2 with A10 and propagates upstream blocking the A4 and a large part of the A 10-West. By the time this congestion begins to dissolve, a new one appears at the junction of A 10 with A 1 which begins to propagate upstream until it reaches the first congestion whose trend of resolving is reversed and both are united into a single more severe congestion. This strong congestion keeps the A4 entrance to the AlO blocked, which results in the accumulation of many vehicles in the
For the purposes of our study, only the counterclockwise direction of the A10, which is about 32 km long, is considered. There are 21 on-ramps on this motorway, including the connections with the A8, A4, A2, and Al motorways, and 20 offramps, including the junctions with A4, A2, AI, and A8. It is assumed that ramp metering may be performed at each of the on-ramps. The model parameters for this network were determined from validation of the network traffic
454
motorway-to-motorway (mtm) on-ramp of A4 (i.e. a spilJback of the congestion from A 10 onto the A4 motorway) and in the surrounding on-ramps (Figure 3b). As a result the TTS becomes 13,226 veh ·h.
'''' "'"eo '"'"o 10
When optimal control is applied to the network under scenario 1, the TTS becomes 9,032 veh·h, a 31. 7% improvement. This improvement is evident in Figure 4 which depicts the density and queue evolution profiles for scenario 1. The large ramp queues that occur in the no-control case are not present any more, but queues are spread to many on-ramps so as to countermand the formation of congestion.
(a)
Scenario 2 assumes that there is more ramp storage capacity at the strategy's disposal (80 vehicles for urban on-ramps and 120 vehicles for mtm onramps). When optimal control is applied under these conditions, the TTS becomes 8,230 veh ·h, a 37.8% improvement over the no-control case. This larger improvement is evident in Figure Sa where the density profile is seen to be much flatter than that of scenario 1 (Figure 4a). Because larger storage space is available to the strategy, larger queues are formed , but less on-ramps are used for storage purposes, see Figure 5b.
Fig. 3. No control: (a) density, (b) on-ramp queues.
In the case of scenario 3, the absence of any queue constraints leads to a TTS equal to 7,589 veh ·h, an improvement of 42.6% over the no-control case. As can be seen from Figure 6a, the density evolution profile is completely flat in this case , indicating the fact that there is no congestion present in the ring-road. The control strategy achieves this impressive amelioration of traffic conditions by creating large queues at the onramps that are located in the critical bottleneck area where congestion originates, see Figure 6b. The amount of TTS reduction depends on the on-ramp storage capacity available to the control strategy. The behaviour exhibited by optimal control application in the three scenarios indicates that the most efficient way to deal with bottlenecks and potential congestion is to perform ramp metering at the on-ramps immediately upstream of the primary bottleneck location, as was the case in scenario 3. In each scenario the control strategy establishes an optimal trade-off between the delay reduction due to the decrease of the congestion extent , and the delay increase due to the metering of vehicles that exit before reaching the bottleneck.
Fig. 4. Scenario 1: (a) density, (b) on-ramp queues. strongly degrades the available capacity. An impressive amelioration of traffic conditions in motorway networks (including the ramps and motorway intersections) is possible with the use of optimal ramp metering by increasing the network throughput. The highest efficiency is achieved if only the on-ramps closest to the bottlenecks are strongly metered, which however creates long queues and disbenefits the corresponding users for the sake of the general efficiency.
6. CONCLUSIONS AND FURTHER DEVELOPMENTS
The formulation of the problem of coordinated ramp metering control as a discrete-time optimal control problem, allows the application of wellknown concepts from automatic control theory,
The reported results demonstrate that the uncontrolled utilisation of the motorway infrastructure
455
van/ f'iTiphenqu(' de f'aT·is . PIi.D . Dissert.ation, Ulliversit{' de Paris-Sud. C('ntn' d 'Orsay, rran("('. Uhouri, N., M. Papag('()rgiou and .T.M. B10sseville (1990) Opt.imal cOlltrol of traffic flow on perinrball ringways with applicatioll to the BOlllf'v;ml Periph{~riqu(' ill Paris. f'r'(~ Jirmt8 11th IFAC Wo1"id Conq1'l~ ss, Vol. 10, Talinll , Estollia, 23G 243 . l1Iinkin , M. (1976). Problelll of optimal control of traffic flow Oil highways. A nto"ll!ation and R(~ 11I.otc Control 37 , 662 6G7. K01 sialos, A. , M . Papagporgioll and A. Mpssm('r (1f)99). Opt.imal coordinatpd and int.egrau,d motorway network traffic control. Ill: f'mcf:f:d inqs of the 14 th Intenw.tiono.l Symposium 0"/1 T1'II.ns]!ortatum and Tmfir: Th.f:01"Y (ISTTT) (A. C('der, Ed.), ,kmsal('m, Israel , G21 644. I{otsialos, A., M. Papag(~orgiou ;md F. Middplham (2001). Optilllal coordinated ramp nJ('tering with AMOC. In: f'r'epnnts (CD-ROM) of thf: 80th. T1'II.nsp07'tatum Rr~ sf:o.rch Hoard A n7l.1w.l Mf:eii1l.Y , Pilper No 01-3125, Washington , DC , USA. I{otsialos , A., Y. Pavlis, r. Middelham, C. Diakaki , G. Vardaka alld M. Papag('orgioll (1998). Modelling of tll(' large scalP motorway llf'twork aroulld Alllst.erdarn. Ill: f'r~print. s of 8th IFA C SY17lposinm on I,(l7W Scale Systems ThUJ7"11 and Appiimtions, Patras , Greece, 354
Fig. 5. Scena rio 2: (a) density, (b ) on-ra mp queues.
120 l OO
eo eo
., 20
o
3GO. M( ~ss m('r,
7. ACKNOWLEDGEMENTS
A. alld M. Papaw'orgiou (1990). M ETA NET: A macroscopic silllulatioll program for mot.orway lle1.works. Tm.ffir: Enyilwenng (Jud ContmI31(8/9), 466 470; 549. Papag('orgiou, M. (1983). Application of Antol1WtU: Coutml Conc~pts to Tm.ffic 1"10111 Moddliny and Contml. Sprillgf~r Vcrlag. N. York, USA. Papageorgiou , M. alld M. Marillaki (1905). A Fe(J.8ihlt~ Di7"(~ction Alg07'ithm for the Nurner'ical Solll.l.ion of Optimal Clmtml f'ml!lf:lI!.~. DYllamic Syst.ems amI Simulation Laborat ory, Tecilllical Univ('J'sity of Cn~t<'. Chania, Gr('('c(' . Papageorgioll, M. , .J.M 13I0ssevill(' alld H. HadjSaknl (1990). Modpllillg awl rml-t.iTlle C0I1trol of traHic flow 011 1.lw south('rn part of Boulpvilrcl Prriph<"~riqll< ' ill Paris - Part I lIlodelling. Tm.nsportatio1/. H(~ Sf:I/.1'('h A 24 , 345
The authors would like to thank Mr. r. I\liddelham fram AVV-Rijkswiltersl.ilar. , Thf' NNhprlands , for providing the necessn.ry dilta for t hp Amsterdill1l ring-road.
Zhallg , H. , S. Ri1chi(' alld \\'. H('ckf'r (1996) . Some gf'll('ral results Oil 1.h(' optilllal ramp meterillg cOIITrol problplll. T{'(J. n~J!{)7"fI/.I.io n H(~ sm.1'I; h C 4 , .'il 69.
Fig. 6. Scenario 3: (a) density, (h) on-ramp q\J('ues. and allows the consideration of other control measures as well. Further control measures such as speed rantral or route guidance may 1)(' readily integrated cooperatively due to the flexible natnrf' of the prohlem formulation.
359.
8. REFERENCES Bhouri, N. (1991). Com.m.a.ndf: d'un Sy8th/l.(~ d~ Traffic A utoTOutif:r: Application (/.11. IJonle-
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