IMPEDANCE EFFECTS OF LEFT TURNERS FROM THE MAJOR STREET AT A TWSC INTERSECTION

Transcription

1 Citation: Brilon, W. (2009): Impedane Effets of Left Turners from the Major Street at A TWSC Intersetion. Transportation Researh Reord Nr. 2130, pp. 2-8 IMPEDANCE EFFECTS OF LEFT TURNERS FROM THE MAJOR STREET AT A TWSC INTERSECTION by Werner Brilon Ruhr-University Bohum D 780 Bohum Germany Tel: Fax: werner.brilon@rub.de

2 Brilon 1 ABSTRACT At a two-way stop ontrolled intersetion without turning lane on the major street all three movements (right, through, left turners) share the same lane. To over the effet of mutual impediments between these movements on the minor street apaity, the major street shared lane formula is provided by the HCM. This paper shows that this formula should be replaed by an improved equation to get more realisti results. This improved formula is derived. The theoretial onept an be extended to inlude also the onsequenes of a short turning lane on the major street. An equation is derived whih alulates the effet of the length of the left turn poket on the apaity of minor movements.

3 Brilon 2 1. INTRODUCTION The estimation of apaity and quality of flow at unsignalized intersetion is based on the gap aeptane theory in the guidelines of many ountries, e.g. in the HCM 2000 (1), hapter 17, or in the German HBS 2001 (2), hapter 7. For these methods, as they are doumented in the guidelines, a signifiant improvement should be made. This inorret part of the proedures onerns the treatment of left turners from the major (priority) street (LTPS). (eq of the HCM 2000). To explain the problem, first the theoretial bakground is highlighted. Here the abbreviations and symbols from the HCM are used. The explanations of the basis for the ritial gap theory are, however, only explained quite shortly. For further explanation the reader is referred to (3). The paper, initially, is onentrated on the probability of a queue-free state for LTPS movements and the onsequenes on the through traffi of the major (priority) street (TTPS). On this basis the apaity for left turners from the minor street (LTMS) is estimated. The solution then is extended towards an expression whih estimates the effet of the limited length of a left turn poket on LTMS movements. 2. SHARED LANES The purpose of alulations aording to hapter 17 of the HCM 2000 is the assessment of traffi performane at an unsignalized two-way stop ontrolled (TWSC) intersetion. As a preondition to this, the potential apaity p, k for eah of the twelve movements has to be estimated. This an be ahieved by Harder's () formula (eq in HCM 2000) or other useful apaity estimation formulas. Also orresponding graphs (exhibit 17-6 and 17-7 in HCM 2000) an be used. For priority movements 2, 3,, and 6 these apaities are defined to be 1800 veh/h, without assuming that this number is ompletely realisti (2). They ould also be estimated from field observations (1). These potential apaities p, k are equal to the movement apaities m, k if movement k is of rank 2 (eq. 17- in HCM 2000). Then the further ranking of priorities is taken into aount by using impedane fators to alulate the movement apaities m, k for eah movement k with a higher rank of priority than 2. An impedane fator p 0, j is the probability that a non-priority movement j is in a queuefree state (eq. 17- in HCM 2000): p v j 0, j = 1 g j = 1 (1) m, j J = p (2) m, k p, k j = 1 0, j p 0, j = probability of a queue-free state in minor movement j [-] v j = flow rate for minor movement j [veh/h] m, j = movement apaity for minor movement j [veh/h] g j = degree of saturation for minor movement j = v j / m, j [-] m, k = movement apaity for minor movement k [veh/h]

4 Brilon 3 p, k = potential apaity for minor movement k to be alulated by equation 17-3 or by any other useful apaity equation (f. (3) ) [veh/h] k = index for minor movements of rank 3 or [-] j = index for minor movements of rank 2 or 3 whih have priority over movement k [-] (Definition of indies: f: Figure 1) J = number of minor movements of rank 2 or 3 whih have priority over movement k [-] FIGURE 1 Definition of movements at an unsignalized intersetion Here, e.g. at a T-juntion for the LTMS-movement we have: k = 7, j = (J = 1) and m, 7 = p, 7 p 0,. For a rossroads intersetion these alulations beome more omplex due to movements of rank, whih is well desribed in (1) or (3). Up to this point the alulation assumes that eah movement has its own lane with unlimited length. The realisti intersetion design, however, usually provides only turning lanes of limited extension. Moreover, in many ases there are no turning lanes at all. Then several movements have to share the same lane. We all this a shared lane. Here, shared lanes on the major street and on the minor approah have a different funtionality whih leads to different algorithms. For minor street approahes the apaity of a shared lane is estimated by the shared formula after Harders (3), whih is also used in the HCM as eq SH vi + v j + vk = (3) v v i j vk + + m,i m, j m,k SH = apaity of the shared lane [veh/h] v i, v j, v k = traffi volume of movements i, j, and k [veh/h] m,i, m,j, m,k = movement apaities of movement i, j, and k [veh/h] i, j, and k = 7, 8, 9 or = 10, 11, 12

5 Brilon This equation is not ompletely realisti whih has been argued by (). The required slight orretions are, however, rather omplex and they are of minor importane for the result. Thus, eq. 3 is still quite an adequate solution for appliation in guidelines. 3. SHARED LANES ON THE MAJOR STREET This paper is mainly direted on the shared lane problem on the major street. Here we treat one approah of the major street ontaining movements i, j, and k (f. Figure 2a). As an example we use i =, j =, k = 6. a).. b) ) FIGURE 2 Shared lane (b) and short left turning lane (). Let us first onentrate on a shared lane without any turning bay spaes (f. Figure 2b). Then equation of the HCM 2000 (1) defines p g 0, = 1 () 1 ( g + g6 )

6 Brilon p 0, = probability of a queue-free state for the shared lane on the major street [-] g j = degree of saturation in movement j = v j / m, j [-] g 6 = 0, if movement 6 is operating on a separate lane (For the opposite diretion at a rossroads intersetion indies are replaed: 1, 2, 6 3 ) This equation also goes bak to Harders (3). In the German Manual HBS (2) even another equation is used: ( g + g ) p = + () 0, 1 g6 Both equations and are not a perfet solution for the TWSC-intersetion model whih will be demonstrated in the following. They should be replaed by the improved equation 12. For simpliity of the derivations let us first look at the movements as they are existing at a T-juntion (Figure 2b). The potential apaity of the TTPS-movement is denoted by p, (e.g. p, = 1800 pu/h). During a time period when a vehile from movement is queuing, the apaity of movement is equal to zero. Thus, the real movement apaity of movement is redued to m, p, 0, p, ( 1 g ) = p = (6) Due to this kind of queuing, over a time period of duration T (e.g. 1 hour) it has to be expeted that x vehiles from movement are impeded and are queued behind waiting left turners (movement ) x ( p ) v = g v = 0, 1 (7) The LTMS (movement 7) is inhibited to enter the intersetion in the subsequent ases: a) A vehile from movement is waiting (Figure 2b). This ase is treated by the usual impediment fators (eq. 1 and 2). This means the intersetion is losed for the LTMSmovement 7 during different time intervals within period T whih sum up to ( 1 p ) = g tlose, 0, = (8) b) A queue from the TTPS-movement (whih has been formed behind a waiting left turner; f. Figure 2b) is being disharged. Eah of these -vehiles needs t = 1/ m, seonds to ross the entrane to the intersetion. Sine there are x suh -vehiles there is a time during whih the intersetion is losed for 7-vehiles, of t lose, 1 1 = x = g v = g g m, m, ) A vehile from movement or from movement is approahing without any queue existing on the major street and the time to its arrival at the intersetion is less than the ritial headway t, 7 of the LTMS vehile. This ase is ompletely overed by the apaity alulation aording to eq. 17- of the HCM 2000 in onjuntion with the definition of onfliting flows by exhibit 17-. Summing up these times during whih the intersetion is bloked for movement 7, we get the sum of all bloked periods from eq. 8 and 9 as (9)

7 Brilon 6 tlose, + = tlose, + tlose, = g + g g (10) The proportion of time during whih no blokage ours, then is p t lose, + = 1 = 1 g 0, g g T At a rossroad intersetion movement 6 has also to be taken into aount whih is assumed to be operating on the same lane of the major street as movements and. With that eq. 11 an be transformed into the more general form of ( 1 g ) g ( g g ) (11) + i i j k p0, i = Max (12) 0 i = 1 or j = 2 or k = 3 or 6 p 0, i = probability of a queue free state on the shared lane of the major street [-] g i, g j, g k = degree of saturation for movements i, j, k [-] p0, 1,00 0,80 0,60 0,0 HBS 2001 / eq. HCM 2000 / eq. eq. 12 v 2 = 300 veh/h v = 300 veh/h 0,20 0, v [veh/h] p0, 1,00 0,80 0,60 0,0 HBS 2001 / eq. HCM 2000 / eq. eq. 12 v 2 = 300 veh/h v = 800 veh/h 0,20 0, v [veh/h] FIGURE 3 Probability that no left turner is impeding the through traffi on a shared lane at a T-juntion depending on LTPS volume v.

8 Brilon 7 This equation should be used instead of equation in the HCM To ompare this new equation with the former approahes (eq. and ) some numerial alulations have been performed. Figure 3 shows one example for the dependeny of p 0, on the volume of movement. It an be seen that eq. redues p 0, signifiantly ompared to the new eq. 11 or 12. Thus, eq. obtained from the German guideline (2), produes a signifiant underestimation of the apaity for the LTMS-movement 7, sine p 0, is introdued into the further apaity alulations in the sense of eq. 2. The HCM equation (f. eq. of this paper) is muh loser to the new equation. However, for larger major street volumes there is also quite a differene with the onsequene of an underestimation of the minor street apaity. There are many ases in reality this differene may beome rather relevant for the question if an unsignalized intersetion offers a suffiient quality of servie. This omparison, however, does not prove whih equation is the orret one. The desirable empirial evidene an not be reahed due to the enormous efforts to get suffiient sample sizes required for reliable measurements of this term. Thus, the only realisti option to onfirm the validity of one of the equations is mirosopi simulation. Here, a time-oriented Monte-Carlo tehnique has been applied to simulate apaities and delays at an unsignalized T- juntion. For demonstrating the results, in this paper two ases have been hosen: a) v = 300 veh/h b) v = 900 veh/h All other parameters remain onstant for both ases: v 3 = 0 veh/h v 2 = 300 veh/h Behavioral parameters for drivers have been obtained from the HCM 2000 (1), exhibit 17-: t, =,1 s t f, = 2,2 s t, 7 = 7,1 s t f, 7 = 3, s t, i = ritial headway for movement i [s] t f, i = follow-up time for movement i [s] Here the LTMS apaity m, 7 is looked for, depending on the LTPS volume v. The simulation uses onstant ritial headways and follow-up times and onsistent driver behavior. Figure shows the results. The urves illustrate that there is a good oinidene between equation 12 and the simulated results as equation and fail to meet the simulated results. In ases of a rather large apaity (e.g. FIGURE a) the differene of the new equation 12 to the urrent HCM approah (equation ) is very small. If however the apaity is on a low level (e.g. Figure b) the differene may beome quite signifiant. This result, whih was also obtained for other ombinations of parameters, underlines the desriptive quality of eq. 11 and 12 ompared to the existing solution. Remaining differenes between the analytial solution and simulation results are due to the approximate nature of the analytial apaity estimation proedures.

9 Brilon 8-7 [veh/h] 0,0 00,0 30,0 300,0 20,0 200,0 10,0 100,0 0,0 Eq. Eq. Eq. 12 Simulation a) v 2 = 300 veh/h v 3 = 0 veh/h v = 300 veh/h 7 [veh/h] 0,0 180,0 160,0 10,0 120,0 100,0 80,0 60, v- [veh/h] Eq. Eq. Eq. 12 Simulation b) v 2 = 300 veh/h v 3 = 0 veh/h v = 800 veh/h 0,0 20,0 0, v [veh/h] FIGURE LTMS apaity estimated by several types of impediment fator ompared to simulated results.. LEFT TURNING LANE OF LIMITED LENGTH Also a separate lane for left turners on the major street does not guarantee that the through traffi is ompletely unimpeded. There might be ases the queue of left turners is exeeding the length of the left turning poket (f. Figure 2). Here a method is needed to quantify this effet on the intersetion apaity. Again the derivation of the analytial solution is explained for a T-juntion. The following derivations are only valid if there is just one single lane to serve the TTPS movement. It is assumed that the turning lane offers spae to store n L left turning vehiles without impeding the through traffi. The probability that the queue of waiting left turners is shorter than n L is p F, ( nl + 1) ( N n ) = 1 g = p (13) L N = no. of waiting LTPS-vehiles [-] n L = no. of spaes on the left turn lane [-] For this equation it is assumed that the queuing system of LTPS-movements is operating like an M/M/1-queue. Also if reality is slightly different from this simplifying assumption, this

10 Brilon 9 approah has been suessfully used for many theoretial derivations in unsignalized intersetion researh. In analogy to eq. 1 and 2 we get from eq. 13 (f. eq. 6) for the apaity of the TTPS movement : m, p, ( nl + 1) ( 1 g ) = (1) The number of vehiles from movement whih are queued due to a spillbak of the LTPS-movement beyond the length of the left turn bay during a time period of duration T is (f. eq. 7) ( n + 1) L x = g v The LTMS-movement 7 is impeded by movements or from entering the intersetion in the following ases: a) like ase a) in setion 3 b) A queue from the TTPS movement is disharged (i.e. -vehiles whih have been queued behind an overflow of the left turning lane). With x suh vehiles during period T and with 1/ p, seonds per disharging movement--vehile the total duration of this disharge proess during period T will onsume a time of 1 1 L tlose, = x = g v p, p, ) like ase ) in setion 3. The sum of all these bloked times (ase a) - ) ) is ( n + 1) ( n + 1) = g ( n L + 1) ( n + 1) L g L ( g + g g ) tlose, + = g + g g = (17) Thus, the proportion of time during whih no blokage ours is (f. eq. 11) ( n + 1) p L 0, = 1 g g g Like in setion 3 we now generalize this equation (whih is only valid for a T-juntion) to a rossroads intersetion: n L p 0, i ( nl + 1 ) ( g g ) 1 gi gi j + = Max 0 = no. of spaes on the left turn lane Other variables are explained in onjuntion with eq. 12. k The orretness of this equation an also be heked using some boundary onditions whih must be fulfilled: 1. for g j + g k = 0 (i.e.: no through or right turning traffi): p = 1 ( ) 0, i gi 2. for n L = (i.e.: an extremely long left turning lane): p = 1 (sine g i < 1) ( ) 0, i gi (1) (16) (18) (19)

11 Brilon for v i = 0 (i.e.: no left turning traffi) 1 p, i 0 =. for n L = 0 (i.e. no left turning lane, i.e. a shared lane) p = 1 g g g + (= eq. 12) ( ) ( ) 0, i i i j g k 0,80 0,70 0,60 ase 1: v 2 = 00 veh/h v = 300 veh/h v = 900 veh/h p0, 0,0 0,0 0,30 0, ase1 ase2 n L [pu] ase 2: v 2 = 100 veh/h v = 800 veh/h v = 600 veh/h FIGURE Probability that the LTMS-movement is not impeded by waiting LTPSvehiles or by the disharging proess in the TTPS-movement depending on the length of the left turn poket at a T-juntion. A graph to illustrate the influene of n L (whih represents the length of the turn poket, ounted in number of spaes for passenger ars) on the LTMS apaity is given in Figure. The funtion starts from the result of eq. 12 for n L = 0 and it approahes (1 g i ) for large n L. If equation 19 would replae eq in the HCM (f. eq. ) then this equation would also over the limited length n L of the left turn poket automatially, whih is not possible in the HCM (1) algorithm for TWSC-intersetions up to now. For the German guideline HBS (2), the urrent solution is ompletely wrong, eq. 19 provides a very important orretion.. INCREASE OF CAPACITY CAUSED BY IMPEDED MOVEMENTS Usually we assume that the inrease of demand traffi volumes will derease apaity and, thus, traffi performane for all of the other movements. It an, however, ome to a point further inreasing traffi volumes of speifi movements will ause bloking effets (as desribed in setion 3 and of this paper). These an reate additional apaities for other movements. One example is illustrated in FIGURE 6a. In this ase a spillbak of LTPS (movement ) is impeding the TTPS-movement to pass through the intersetion. Due to this effet the opposing LTPS-movement 1 an freely disharge from the stop line. Also the RTMS (movement 12) would benefit from this bloking effet. One might be attempted to expand equation 19 to take into aount also this effet. This may be possible. However, before further derivations some onerns should be disussed:

12 Brilon 11 a) Resulting equations beome ompliated and are diffiult to apply. b) Of ourse, also in the profiting diretion the same bloking phenomenon ould our temporarily with the effet that then the inrease of apaity would move over to the opposite diretion. This means: the proess of bloking ould beome dynami. In these ases an iterative alulation might beome appropriate. This of ourse, would not longer be aessible for a straightforward analyti proedure to be inluded into Highway Capaity Manual proedures. ) The apaity inrease by bloking effets an only our if at least one movement is overloaded for some periods; i.e. this movement is operating on LOS E or even F. Thus, these bloking effets will never beome a means to raise the whole intersetion into an aeptable level of performane. d) The dynami effets may better be addressed by mirosopi simulation. Due to these reasons the analytial treatment of apaity inreasing effets by temporal bloking of mayor movements is postponed. Effets like these might, however, also be employed to aomplish better traffi performane at TWSC-intersetions in speifi ases. E.g. a zebrarossing (i.e. absolute priority for pedestrians) in one of the major street entries offers inreased apaities for the LTPS (movement ; see FIGURE 6 b ). The analytial treatment of these aspets ould beome a matter of further researh. a) b) FIGURE 6 Inrease of apaity for one left-turn movement by bloking effets in the opposing diretion at a rossroads juntion. E) CONCLUSION This paper provides a new theoretial derivation of the shared lane formula to be applied in the ase of shared lanes on the major street at TWSC intersetions. This formula ould be verified by a series of simulations. In addition, this formula an be extended to over also the effets of a short left turning lane on the major street. This extended formula fulfills all the restritions whih are typial for the problem.

13 Brilon 12 It is reommended to replae eq of the HCM 2000 by this new formula (eq. 19) for the new edition of the HCM. It is proposed to use this approah also for unsignalized intersetion guidelines in other ountries, e.g. for the German HBS (2). Finally, it should be noted that also these derivations are not a perfet mathematial solution to the problem of unsignalized intersetions. Like the whole gap aeptane theory, also the derivations submitted in this paper are more like an appliation of rather pragmati mathematis, sine a series of simplifying assumptions are needed to ome to a solution ready for use in pratie. REFERENCES 1. HCM (2000). Highway Capaity Manual. Transportation Researh Board, Speial Report 209, Washington D.C. 2. HBS (2001). Handbuh zur Bemessung von Strassenverkehrsanlagen (German Highway Capaity Manual), hapter 7: Unsignalized Intersetions, FGSV, Cologne 3. Troutbek, R. J., Brilon, W. (1999). Unsignalized Intersetion Theory. Chapter 8 in: Monograph on Traffi Flow Theory. Federal Highway Administration, Aessed Nov Harders, J.: Die Leistungsfaehigkeit niht signalgeregelter staedtisher Verkehrsknoten (Capaity of unsignalized urban intersetions). Series "Forshung Strassenbau und Strassenverkehrstehnik", no. 76, Bonn, Knote, T.: Die Kapazitaet von Mishstroemen in Nebenstrassenzufahrten von Kreuzungen und Einmuendungen mit Vorfahrtbeshilderung. (The apaity of shared lanes at unsignalized rossroads and T-juntions), Strassenverkehrstehnik, Nr. 2, S. 7-80, 2006