AN AGORITHM OR CACUATING THE CYCETIME AND GREENTIMES OR A SIGNAIZED INTERSECTION Henk Taale 1. Intoducton o a snalzed ntesecton wth a fedte contol state the cclete and eentes ae the vaables that nfluence the dela of vehcles. In the ast a lot of effot has been ut n fndn a ethod to calculate an otal cclete and otal eentes that nze dela. A faous eale s the foula of Webste (1958), whch s deved fo a foula descbn the aveae dela e vehcle. In ths atcle a ethod s descbed to calculate an otal cclete and eentes, based on a enealzed Webste foula and takn nto account the nu eente and au deee of satuaton fo each oveent. 2. Basc ncles and defntons o a oveent of a snalzed ntesecton cetan chaactestcs can be defned. If we call G the eente fo oveent, than the effectve eente can be defned as G t s, +t e,, whee t s s the stat la and t e the end an. Anothe useful aaete s the flow ato. If q s the flow and s s the satuaton flow fo oveent, than the flow ato s defned as q /s. The flow ato can also be consdeed as the facton of the te that s needed fo oveent to handle all taffc. The aaete whch elates the flow ato to the eente ato ( /C, whee C s the cclete) s called the deee of satuaton. Ths deee of satuaton s defned as ( C)/ (Akçelk, 1986). o an ntesecton cetan oveents have a conflct wth each othe, whch eans that the ae not allowed to eceve the een lht at the sae te. A set of oveents conflctn wth each othe, but whee no othe oveent, whch has a conflct wth all othe oveents n the set, can be added to, s called a au set of oveents. Wthn each au set of oveents an ntenal lost te can be detened fo a cetan sequence of oveents. The ntenal lost te s the su of the nteeen tes lus the su of the stat las and nus the su of the end ans of all oveents n the set. The nteeen te s deendent of the sequence of oveents and theefoe anothe sequence of oveents can ve anothe ntenal lost te. o the calculaton of the otal cclete the sequence of oveents, wthn a au set of oveents, wth nu ntenal lost te s used.. Condtons fo a contol state In desnn a (fedte) contol state thee condtons ae otant: the avalable te ust be lae o equal than the te needed [1], the eente of eve oveent ust be lae o equal than the nu eente [2] and the deee of satuaton of eve oveent ust be less o equal than the au deee of satuaton []. We consde the collecton K of au sets of oveents of all oveents of an ntesecton. o eve au set of oveents K the ntenal lost te and the load Y s known, whee Y s the su of the flow ato's of all oveents wthn K. Accodn to condton [1] the follown ust hold fo the cclete C : C 1 Y (A) Condton [2] leads to: C { + Y K } a (B) whee s the nu effectve eente (nu eente lus end an nus stat la). nall, condton [] ves:
, C { a a, Y K } a (C) whee a, s the au deee of satuaton fo oveent, whch s also known. Ths condton holds unde the assuton that a, > Y fo all oveents, othewse neatve ccletes would be equed. Because condton () s the sae as condton (1), but oe stct f a, <1 fo all K, the cobnaton of all thee condtons leads to the follown foula fo the nu cclete:, C = { { a a a, Y K }, a{ + Y K }} a (D) 4. Calculatons takn nto account the nu eente The faous foula of Webste fo the otal cclete s: C = 1.5 + 5 1 Y (E) Ths foula can be enealzed nto: 1 2 C = + 1 Y () whee 1, 2 and ae called the Webste coeffcents. The atchn effectve eentes fo all oveents K ae calculated wth: = Y ( C ) (G) Due to condton (4) t s ossble that the cclete becoes ve lae whch s caused b oveents wth a ve low flow ato. To coect ths oble, the flow ato of these oveents s nceased atfcall to a ont that the nu eente condton s satsfed. st the set N of oveents K s detened fo whch the follown condton holds: (1 Y ) Y ( 1 1+ Y 2 ) + (H) Than an abta oveent N s taken fo whch the "atfcal" flow ato s calculated wth:
a 1 = + N N b 1 = ( 1 + Y N c 1 = ( Y N Y 1 ) + 2( 1 2 )( N ) N (I) (J) Y N 1 ) (K) = b + b 4 a c 2 a 1 1 2 1 1 1 () The othe "atfcal" flow ato s fo N \{} and the new su of flow ato s Y ae calculated wth the follown foula s: = (M) + (N) Y = Y N N 5. Calculatons takn nto account the au satuaton flow Afte coectn fo the nu eente, also the condton fo the au satuaton flow has to be satsfed. The oveents K fo whch ths condton does not hold, can be detened wth (f K \N than = ): ( 1 1+ Y 2 ) +, ( 1 + 2 )Y a (O) Ths set of oveents s naed M and the cosssecton of the sets N and M s naed P. The net ste s to take an abta oveent q M and to calculate: a = 2 a,q a,q (P) M q a, b 2 = a,q (( 1 1+ 2 q P Y P M _ P )( M \ P c = (Y + ) 1 2 ) + )( a,q 2 1 + 2 ) M a, (Q) (R)
= b + q b 4 a c 2 a2 2 2 2 2 2 (S) Now, the othe flow ato s fo the oveents M \{q} and the new su of flow ato s ae calculated wth: = q q a,q a, (T) + P M _ P M Y = Y (U) Because fo the oveents M the flow ato s nceased, the su of flow ato s s also nceased, leadn to a eente lae than the nu eente fo the oveents N \M. Ths eente can becoe ve lae fo a oveent wth a ve low flow ato and a low au satuaton flow. To ovecoe ths oble, the flow ato fo these oveents has to be adjusted aan. Theefo t s necessa to detene the set of oveents N \M whch satf the follown condtons: > (1 Y ) Y ( 1 1+ Y 2 ) + (V) > a, ( + )Y 1 2 ( 1 1+ Y 2 ) + An abta oveent t N \M s taken and a new flow ato s calculated wth: a = + N \ M N \ M t (W) (X) b = ( 1 + Y Y N \ M N \ M 2 ) + ( 1 1 2 c = t ( Y N \ M )( ) N \ M Y N \ M 1 ) (Z) (Y) 2 t (AA) = b + b 4 a c 2 a The othe flow ato s fo the oveents N \(M {t}) and the new su of flow ato s ae calculated wth:
(BB) = t t + Y = Y N _ M N _ M (CC) nall, the cclete and effectve eentes ae calculated wth 1 2 C = + 1 Y = Y ( C ) (DD) (EE) fo K \(N M ) and = Y ( C ) () fo N \M and = Y ( C ) (GG) fo M. Afte ths, aan the set of oveents N s detened and all stes ae taken aan untl both the sets N and M ae et. Ths aloth s caed out fo all au sets of oveents K. If the cclete has a au fo the au set of oveents K h, so that: C h C ( = 1,...,n) (HH) than, fo the enealzed Webste foula, C h s the cclete that should be used and the accoann eentes G can be calculated wth +t s, t e,. 6. nal eak and lteatue The aloth descbed n ths atcle has been leented n the coute oa KRAAN, owned b DTV consultants the Beda, The Nethelands. 1. Webste,.V., Taffc Snal Settns, Road Reseach Techncal Pae No. 9, Road Reseach aboato, 1958 2. Akçelk, R., Taffc Snals: Caact and Tn Analss, Reseach Reot ARR 12, Austalan Road Reseach Boad, 1986 Autho:. Henk Taale Mnst of Tansot, Publc Woks and Wate Manaeent Tansot Reseach Cente (AVV) P.O. Bo 101 000 BA Rotteda The Nethelands