Optimal control of an isolated two-phase intersection

III IN T E R N A T IO N A L C O N F E R E N C E

T R A N S P O R T S Y S T E M S T E L E M A T I C S T S T '0 3

Z E S Z Y T Y N A U K O W E PO L IT E C H N IK I Ś L Ą S K IE J 2003

T R A N S P O R T z.5 1 , n r kol. 1608

traffic, o p tim a I contro!, B a n g -b a n g c o n tro !

Y o u ssef A H A D A R 1 R achid B O U Y E K H F A bdellah EL M O U D N I

O P T I M A L C O N T R O L O F AN IS O L A T E D T W O -P H A S E I N T E R S E C T IO N

In th is p a p e r w e c o n sid e r an iso lated tw o -p h a se in te rse ctio n streets w ith c o n tro lla b le tra ffic lights on each c o rn er. Iso lated im p lie s th a t w e w ill d e v e lo p sig n al co n tro l at th e in te rse ctio n w ith o u t co n sid e rin g a d ja c e n t in te rse ctio n s. W e c o n stru c t a m odel th a t d e sc rib e s th e ev o lu tio n o f th e q u e u e le n g th s in each lane as a fu n ctio n o f cy cle. W e d iscu ss h o w op tim al co n tro l can m in im ize q u eu es in the o v e rsatu ratio n c o n d itio n s. A n a lg o rith m w ill be p ro p o se d to illu strate the resu lts o b tain ed .

O P T Y M A L N E S T E R O W A N IE

D W U F A Z O W Y M I IZ O L O W A N Y M I S K R Z Y Ż O W A N IA M I

W a rty k u le ro z w a ż a s ię w y iz o lo w a n e d w u fa z o w e sk rz y żo w a n ia ulic w y p o sa ż o n e w sy g n aliza c ję św ietln ą. W y iz o lo w an ie im p lik u je , iż b ę d zie o p ra c o w an e stero w an ie sy g n ała m i n a s k rz y żo w a n iu bez u w zg lęd n ian ia s ą sia d u ją c y c h sk rz y żo w a ń . Z b u d o w a n o m o d e l, k tó ry o p isu je zm ia n ę d łu g o śc i k olejki na k a żd y m p a sie ja k o fu n k c ję cy k lu . R o zw a ż an e je s t o p ty m a ln e stero w an ie , k tó re m o że m in im a liz o w a ć k o le jk ę w w a ru n k a c h p rz e sy c e n ia . Z ap ro p o n o w a n o alg o ry tm do zilu stro w a n ia u z y sk a n y c h w y n ik ó w .

1. IN T R O D U C T IO N

A s the n u m b e r o f veh icles an d the n eed fo r transportation grow , cities around the w o rld face serious ro ad traffic co n g estio n p roblem s. C osts in clu d e lost w ork and leisure tim e, increased fuel co n su m p tio n , air p o llu tio n , health p ro b le m s... In general th ere e x ist different m ethods to tack le th e traffic congestion problem . T he m o st effectiv e m easures in the battle against traffic co n g estio n seem to be a selective construction o f n ew roads and a b etter control o f traffic th ro u g h traffic m an ag em en t. T raffic lig h t control can be used to au g m en t the flow o f traffic in urban en v iro n m en ts b y p ro v id in g a sm ooth circulation o f the traffic o r to regulate the access to hig h w ay s o r m ain roads.

T raffic light co n tro ls have been the subject o f several investigations. In particular, [1]

has show n h ow an optim al traffic light sw itch in g schem e fo r an in tersectio n o f tw o streets can

1 L ab o rato ire S e T -U n iv ersité d e T e c h n o lo g ie B e lfo rt-M o n tb é lia rd - U T B M 9 0 0 1 0 , B elfo rt c ed e x , F ran ce y o u s se f.a h a d a r@ e d u .u n iv -fc o m te .fr.

14 Y o u sse f A H A D A R , R achid B O U Y E K H F , A b d ellah EL M O U D N I be determ in ed . In g en eral th is leads to a m in im izatio n problem o v er th e so lu tio n set o f an ex ten d ed lin ear co m p le m e n ta rity p ro b le m (E L C P ). H ow ever, it is w ell k now n th a t th e E LC P is N P -h ard pro b lem . H en s, as c laim ed b y [1] this approach is n o t feasib le i f the n u m b e r o f sw itching cy cle s is large. [2] c o n stru cte d m o d els for o versaturation control. B u t th e ir m odels are all co n tin u o u s types an d do n o t address th e p roblem o f op tim izin g cycle length. T he only w o rk th a t m erits o u r atten tio n is [3], In th at paper the authors h ave d e v elo p e d a discrete dynam ic o p tim izatio n m o d els in term o f delay. T he optim al cycle len g th a n d the optim al assigned green tim e are d ete rm in e d fo r th e case o f tw o -p h ase control.

In this p ap er, firstly , b a se d on the w o rk o f [3]-[4]-[5] in w hich the au th o rs calcu late the d elay d u rin g a p e rio d o f o v e rsa tu ra te d co n d itio n s in a cro ssro ad s, w e ad o p t a m odel that describes the ev o lu tio n o f th e q ueue lengths in discrete tim e acco rd in g to th e cycles. This m odel p ro v id es an im p ro v e d level o f control b y using b etter m o d ellin g o f tra ffic variables.

Secondly, th e p ro b lem o f fin d in g th e set o f ad m issib le control to m in im ize the total length q ueue is co n sid ered in th e en tire o v ersatu rated period. In o rd er to ap p ly th e optim al control strateg y w ith re s p e c t to real c o m p o rtm en t o f th e system , an algorithm w ill be p roposed

2. D E F IN IT IO N O F T H E SY STE M

W e c o n sid e r an iso la te d tw o -p h ase in tersectio n as show n in F ig .l. T h e re are tw o lanes and on each c o m e r o f the in tersec tio n th ere is a traffic light. H ence, there are tw o traffic flow s o f veh icles to be serv ed in th e intersection.

Fig. 1. F o u r-le g in te rse ctio n w ith tw o -p h a se sig n al co n tro l

Optim al control o f an iso lated tw o-phase intersection 15 T o be able to p resen t the m odel, a certain n u m b er o f d efinitions are necessary :

Ef f e c t i v e g r e e n t i me

A s a v eh icle ap p ro ach es an intersection displaying a red signal the d riv er decelerates and stops eith er a t the sto p line o r at th e end o f a queue Fig.2. W hen the signal tu rn s green the driver accelerates until the v eh icle reach es its desired o r m ax im u m p o ssib le speed. It is usually [6], [4] assu m ed th a t after startup lo st tim e the saturation flow rate rem ain s constant until the b e g in n in g o f th e yellow change interval. T he effective green tim e is:

g c = g + y - w = g + y - ( W i + w 2) (1)

w here th e lo st tim e W is the sum o f startup lost tim e W { and clearan ce lo st tim e W 2.

W\ Effective green, & ÏÏ2

H i ■ i

Red Green

F ig.2. E ffectiv e g reen tim e

Yellow Red

R a t e o f s at urati on

R ate o f saturation s is d efin ed as being the m axim um n u m b er o f veh icles b ein g able to use the c o rrid o r w ith o u t in terru p tio n durin g the effective tim e o f the green light g c .

Ov e r s a t u r a t e d condi t i ons

If arrival rate ex ceed s cap acity , the intersection is oversaturated. A p h ase is saturated when a v eh icle at least is co n strain e d to aw ait m o re than one cycle to cro ss the crossroads.

The crossroads is satu rate d w h en at least one o f its phases is saturated. Form ally, the n u m b er o f arrivals A ( k ) du rin g cy cle k e N is d efined b y the equation:

A{ k) =A[ kc] ~A[ ( k- \ ) c\ (2)

A[kc] is defined as the n u m b e r o f th e arrivals at the end o f the cycle. T he dep artu re rate is s during the effectiv e tim e o f the g reen lig h t g e , so that:

D(t + c) - D(t) = s ■ g e (3)

w here D { t ) is the n u m b e r o f dep artu res at the instant t . Fig.3 displays a case w here dem and flow rate in stan tan eo u sly increases above the cap acity at th e b e g in n in g o f a cycle. T he capacity curve C (7 )is n o t the saw -to o th ed departure curve D ( t ) , so th at the area betw een A ( t ) and C ( t ) curves is th e overflow delay [7],

16 Y o u s s e f A H A D A R , R ach id B O U Y E K H F , A b d ellah EL M O UDN1

F ig .3 . M o d el fo r o v e rflo w d e la y

Let D j( c ) the n u m b e r o f th e dep artu res durin g the cycle k , then th e co n d itio n o f o versaturation is d efin ed by:

A(k)> D k(c) (4)

3. M O D E L IN G

C o n tin u o u s type m o d els are lim ited in th at the sw itch -o v er p o in t d oes n o t n ecessarily o ccu r at th e en d o f a cy cle, n e ith e r does th e term in atio n o f the o v ersatu rated p erio d o ccu r only at the end o f th e final cycle. O n the o th er hand, the sw itch -o v er points d eterm in ed b y a discrete m odel o c c u r e x a c tly at the term in atio n o f a cycle. D iscrete o p eratio n p ro v id es a sm ooth, reg u lar, an d o rd e re d tra n sfe r o f control. C alcu latin g queue is m o re reliab le. Fig.4 illu strates th e situ atio n o f q u eu e d u rin g o v ersatu ratio n and show s th e d u ality b etw een the tw o phases. T o k eep th is d u ality , it is n e cessary th a t the queue rep resen ted w hen th e effective green tim e term in ates at a certain cy cle state k .

X l(k + 1 )

X l( k )

k é m e cycle

F ig .4 . Q u e u e o f a fo u r-le g in te rse ctio n w ith tw o -p h a se c o n tro l

O ptim al control o f an iso la te d tw o -p h ase intersection 17

Let X2(k) be a q u eu e len g th o f approach 2 w hen the effective green tim e g^(k) term in ates at a certain cycle state k . A cco rd in g to Fig 4, the relation o f the q ueue len g th s betw een state

k and k +1 can b e re p re se n te d b y th e follo w in g equations:

X 2 (k+]) = X 2 (k) + A2 ( k ) - D 2 (k) (5)

w here A2(k) = q r c , rep resen ts the n u m b e r o f arrivals durin g cycle k and D 2(k) = S 2- g c2(k) the n u m b er o f d ep artu res d u rin g cycle k . T he equation (5) becom es:

X 2( k+l ) = X 2(k) +Q2- c - S 2-ge2(k)

(

)

Sim ilarly, let x , ( k ) be a q u eu e length o f approach 1 w hen the effectiv e green tim e

= c-g ,(/c) term in ates at a certain cycle state k . T hen

X ^ k + \ ) = X l ( k) +A]( k ) - D i(k) (7 ) w h e r e :

> A](k) = qy g e2( k - l ) + qr ( c - g e2(k)) : represents the n u m b er o f the arrivals at the end o f c - g e2W -

> D\(k) = S v ( c - g e2(k) ) : rep resen ts the n u m b er o f the d epartures at the en d o f c - g ,(& ) ■ The equation (7) becom es:

X\{k + \) = X\{k) + q y g e2{ k - \ ) + q y { c - g c2{ k ) ) ~ S y { c - g e2{k)) (8) L et x (k ) be th e v e c to r d efin ed as Xt®=Q(f® X,(k)J, and U{k) be the v ecto r o f control o f the system d efin es b y U ( k ) = ( g, , ( k ) g r2( k - 1 ) ) '. T hen w e can g ath er the tw o equations (6) an d (8) in the fo llo w in g m atrix form :

X ( k + \ ) = A - X ( k ) + B-U(k) + C (9) W here

' x m

,A = '1 0“

, B=s\ - Q Q

, U(k) = ga lik) - c =

( Q - s J - c

X 2(k) 0 1 -¡2 0 J S e l t t- 0 . Q c

N o tice th a t eq u atio n (9) is th e lin ear form w ith resp ect to the state variab le and control.

Hens, it is easily am en a b le to m athem atical analysis. T herefore, w e can e fficien tly com pute optim al traffic light co n tro l, w hich is the m ean concern in th e follow ing section.

4. O P T IM A L L IG H T S C O N T R O L

C o n sid er the p ro b lem o f fin d in g a control U(k) w hich m in im isin g on o f th e follow ing possible criterions (10) d efin ed as:

N

► J, = ^ x ( k ) - th e sum o f th e queues during oversaturated period.

k=o

18 Y o u sse f A H A D A R , R achid B O U Y E K H F , A b d ellah E L M O U D N 1

/V

> j 2 = Y ^ \ l 2 X 2(k) ' q u ad ratic form o f the queues length.

k=o N

> y 3 = max ( £ X ¡ ( k ) ) '■ the q u eu e length o v er the w o rst queue.

1 * = o

su b ject to th e d y n am ic eq u atio n :

X ( k + \) = A X ( k ) + B U ( k ) + C (11)

and the in eq u ality c o n stra in ts o f th e form :

n = {(U(k),X(k)) e (R2 x jR2 / Umn < U(k) < t/ _ , X(k) > o} (12) w here N is th e te rm in a tiv e cy cle o f the o v ersatu rated p erio d an d Q is a given set o f constraints.

M in im izin g (1 0 ) su b jected to (9), b ased on the optim al control th eo ry [8], involves eq u iv alen tly to m in im iz in g the H am iltonian fo rm u la (13). T he H am iltonian fo rm u la is d efined as:

H ( k ) = 0(k) + Ar (k + 1 ) [ X ( k ) + B U ( k ) + C ] (13) w here A ( k ) is the co state c o rresp o n d in g to x (k)- N ow , the task is to find an adm issible sequence U ( k ) e Q such th a t (1 3 ) is m inim al subject to (12). A cco rd in g to th e optim al control th eo ry , i f an e x trem u m o f H exists, it m u st satisfy th e follow ing con d itio n s:

= + + (14)

' ’ d X ( k ) d X ( k )

— — -- = X ( k + l ) = X ( k ) + B U ( k ) + C ( 15) 5/t(A + l)

_ d H _ = o (16)

8 U (k)

w here (j){k)\s the function w hich appears in the criterions form s. E quation (13) clearly reveals th at the relatio n b etw een H and the control v ariab le U { k ) is linear. T h is leads to:

- ^ — = At {k + 1 )5 * 0 (12)

d U{ k )

H ens, eq u atio n (17) o b v io u sly in d icates th at on ly the single control v a riab le U { k ) can m i n im iz e / / . In signal co n tro l, the control variable is related to the effectiv e g reen tim e w hich sh ould be taken as the v a lu e b etw een p red eterm in ed u pper and lo w er lim its. T h is p ro p o sitio n is clea rly a b a n g -b a n g control w ith a su ccessiv e tw o -stag e o peration U mirl and (7ni„ :

U ( k ) = U min i f k T (k + \ ) B > 0 V ( k ) = U m ax i f A T (k + ] ) B < 0

(18)

O ptim al control o f an iso lated tw o -p h ase intersection 19

A t first glance, (18) in d icates th at the im plem entation o f th e control strateg y is note problem atic. H ow ever, th e d iffic u lty arises in the determ ination o f the initial value o f A ( k ) w hich obeys th e re c u rre n t equation (14). T his difficulty can be av o id ed b y tak in g a set o f certain initials v alu es o f A ( k ) th a t respects the m axim al ad m issib le qu eu e len g th w hich is defined b y th e o v ersatu ratio n conditions. W ith this in m ind, the follow ing algorithm is proposed in ord er to tak e this consideration.

Step 1: Let * in itiate A (0 ).

Step 2: W hen Ar (£+1)5>0 D o U (k)=U mii:.

W hen / ( k + \ ) B < 0 D o l_J(k)=Umt>.

Step 3: G ive ( X M, X„, ) T, calcu late X ( k ) .

Step 4: If X i(k) = 0 o r

X i( k ) > X i

Step 5: C alcu late AT (A:) until k = N . End.

A cco rd in g to figure 4, step 4 indicates th at the cum ulative o u tp u t curves do n o t intersect the cu m u lativ e in p u t curves fo r an y o f the approaches. T his fact im plies th at no queue becom es n eg ativ e o r z ero b efo re the end o f the o v ersaturated p eriod. If a q ueue becom es negative w hile th e signal is green, the d esigned green tim e b ecom es in v alid due to the w aste o f control tim e. In Fig.5 w e have plotted the evolution o f queue lengths acco rd in g to the algorithm and the co n strain ts. C learly w e observe that X t(k) decreases ra p id ly w h ile XJ k ) increases slo w ly until 90 th cycle. O nce x , ( k ) becom es zero the control is ch an g ed according to step 4. A fter 90th cy cle x \ k ) increases until x l j m =24 w hich is th e m axim al value that we h ave p erm itted , o f c o u rs e x ,( k ) d ecreases in th is case. A ll th ese o b serv atio n s in d icate th a t the control strateg y forces the q ueue lengths to change according to co n strain ts an d m ak es a flexibility to get o u t the o v ersatu ratio n situation.

F ig.5. E v o lu tio n o f th e q u eu es

20 Y o u sse f A H A D A R , R achid B O U Y E K H F , A b d ellah EL M O U D N I

5. C O N C L U S IO N

W hen the traffic flow ex ceed s intersection capacity, th is situation cau se q u eu in g o f veh icles th a t can n o t b e e lim in a te d in one signal cycle. T o m in im ise th is c o n g estio n , firstly, w e ex p lain ed a m odel th a t d escrib es the evolution o f the qu eu e lengths at an in tersec tio n o f tw o tw o -w ay streets w ith co n tro llab le traffic lights on each co m er. S eco n d ly , w e h ave p ro ceed ed at the stag e o f th e o p tim isatio n b y d efin in g the appropriate control strategy. T h is resu lts in b an g -b an g like co n tro l, w h ic h is q u ite ap p ro p riate fo r o v ersatu ratio n co n tro l. F inally, an alg o rith m has b een p ro p o se d in o rd er to take into acco u n t th e constraints.

B IB L IO G R A P H Y

[1] S C H U T T E R , B .D ., M O O R , B .D : O p tim a l tra ffic light fo r a sin g le in te rse ctio n .

[2] M IC H A L O P O U L O S , P .G . a n d S T E P H A N O P O U L O S , G ., O v e rsa tu rate d sig n al sy ste m s w ith q u e u e len g th c o n strain ts - 1: S in g le in te rse ctio n . T ra n s p o rta tio n R ese arch 11, 4 1 3 -4 2 1 . (1 9 7 7 a).

[3 ] T a n g -H sie n C h a n g : O p tim a l sig n al tim in g fo r an o v e rsatu rated in te rse ctio n , D e p a rtm e n t o f T ra n s p o rta tio n S c ie n c e, T a m k a n g U n iv e rs ity (1 9 9 8 ).

[4] W E B S T E R , F .V ., 1958. T ra ffic sig n al settin g s, ro ad re search tech n ical p a p er, N o .3 9 , G re a t B rita in R oad R ese arch L a b o ra to ry , L o n d o n .

[5] M A Y , A .D ., J r., 1965. T ra ffic flo w th e o ry -th e tra ffic e n g in e er's c h allen g e . In: P ro c e e d in g s o f th e In stitu te T ra ffic E n g in ee rin g , pp. 2 9 0 ± 3 0 3 .

[6] C L A Y T O N , A .J .H ., 1941. R o ad tra ffic c alcu la tio n s. Jo u rn al o f th e In stitu tio n o f C iv il E n g in ee rs, vo l. 16, no. 7, p. 2 4 7 -2 8 4 . W ith d isc u ssio n and c o rre sp o n d e n c e in n o . 8, pp . 5 8 8 -5 9 4 .

[7 ] G A Z1S, D .C ., P O T T S , R .B ., 1965. T h e o v e rsatu rated in tersectio n . In: P ro c e e d in g s o f th e S eco n d In te rn a tio n a l S y m p o s iu m on th e T h e o ry o f R o ad T r a f c Flow . O rg a n iz atio n fo r E c o n o m ic C o o p e ra tio n and D e v elo p m en t, P a ris, pp. 221 ± 2 3 7 .

[8] S A G E , A n d re w P. an d W H IT E , C h e lse a C.,I1I. O p tim u m S y stem C o n tro l, S e c o n d e d itio n (1 9 7 7 ).

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