Bus control strategies in corridors with signalized intersections

Bus control strategies in corridors with signalized intersections Miquel Estrada*a, Josep Mensión*b, Josep M. Aymami*c and Laura Torres*d *a Associate Professor. Center for Innovation in Transport (CENIT) . Universitat Politècnica de Catalunya- BarcelonaTECH, C. Jordi Girona 1-3, B1-113 08034 Barcelona, Spain. [email protected]. Corresponding author. *b Director of Central Services and Deputy Chief Officer of Bus Network. Transports Metropolitans de Barcelona (TMB). C. 60, n.º 21-23, sector A, Pol. Ind. de la Zona Franca, 08040, Barcelona, Spain. [email protected] *c Regional manager. Business Development. Transport Simulation Systems (TSS). C. Ronda Universitat 22B, 08008 Barcelona, Spain. [email protected] *d Consultant engineer. Transport Simulation Systems (TSS). C. Ronda Universitat 22B, 08008 Barcelona, Spain. [email protected] This paper proposes a new dynamic bus control strategy aimed at reducing the negative effects of timeheadway variations on route performance, based on real-time bus tracking data at stops. In routes with high demand, any delay of a single vehicle ends up causing an unstable motion of buses and producing the bus bunching phenomena. This strategy controls the cruising speed of buses and considers the extension of the green phase of traffic lights at intersections, when a bus is significantly delayed. The performance of this strategy will be compared to the current static operation technique based on the provision of slack times at holding points. An operational model is presented in order to estimate the effects of each controlling strategy, taking into account the vehicle capacity constraint. Control strategies are assessed in terms of passenger total travel time, operating cost as well as on the coefficient of headway variation. The effects of controlling strategies are tested in an idealized bus route under different operational settings and in the bus route of highest demand in Barcelona by simulation. The results show that the proposed dynamic controlling strategy reduces total system cost (user and agency) by 15-40% as well as the coefficient of headway variation 53-78% regarding the uncontrolled case, providing a bus performance similar to the expected when time disturbance is not presented. Keywords: Bus bunching; dynamic controlling strategies; time-headway adherence, bus reliability 2 1. INTRODUCTION The reliability of transit modes is an important issue to ensure their competitiveness against the extended use of private cars in major cities. However, in overall surface transit services with partial right of way, route travel times are highly dependent to transit demand and traffic states. There are several reasons in these systems that cause service disruptions such as illegal freight loading/unloading operations, taxi stops, use of bus lanes by slow vehicles (bikes, street sweepers) or car merging operations due to right turns. These facts, combined with transit demand fluctuations at stops and traffic light settings, make it difficult to maintain time-headway adherence and control the transit system performance. In bus routes with high demand, when a single bus is delayed from its schedule, the number of waiting passengers will increase at the following stops, resulting in a higher vehicle delay. This local disruption propagates to the whole fleet producing vehicle bunching, irregular vehicle arrivals at stops, unstable time-headways and higher user waiting times. Some research has been done to describe the dynamic performance of the bus system operations. Newell and Potts (1964) and Osuna and Newell (1972) were the first contributions that described the unstable performance of a cyclic bus fleet operation. In order to tackle the bus bunching problem, several control strategies are available. Traditionally, the bus pairing has been mitigated allocating slack times in bus schedules at determined stops (holding points) along the route (Barnett, 1974; Turnquist, 1981; and Rossetti and Turitto, 1998). Slack times should compensate the delays of those buses experiencing random disruptions so that the schedule adherence would be still satisfied. Nevertheless, the obligation that all buses must remain a common slack time in a holding point represents a reduction of commercial speed. Indeed, it causes a significant inefficiency in the system’s productivity. Moreover, this control strategy for maintaining the schedule of a single bus, does not take into account the real performance of the others. Therefore, some studies propose dynamic control strategies to monitor the response of the whole fleet to random disruptions in a short time horizon (Eberlein et al. 2001; Dessouky et al. 2003; Adamski and Turnau, 1998). These contributions determine the location of a holding point and a specific amount of slack time for each bus, based on suboptimal procedures and the dynamic bus performance data. Real-time information is supposed to be available, as Automated Vehicle Location (AVL) and Automatic Passenger Counter (APC) systems will be equipped in the vehicles. In Yu and Yang (2007) an improved holding-point optimization procedure is presented based on genetic algorithms to minimize total passenger costs. Other contributions develop optimization models based on holding points and stop skipping strategy, where the performance of the bus system is predicted over a rolling horizon. This prediction is made considering that all variables are deterministic and known in advance (Delgado et al. 2012) or even stochastic (Sáez et al., 2012; Cortés et al., 2010). Fonzone et al. (2015) proposed bus overtaking at stops in order to accommodate better the waiting passengers in buses that didn’t reach its vehicle capacity constraint. Although the previous contributions are generally based on short term predictions of the system behavior, other approaches propose adaptive strategies to the real performance of buses. They actuate over the system variables in the interstation segments of a single bus route. Based on control theory principles, Daganzo (2009) defines an adaptive variable cruising speed patterns for public transportation vehicles. This control strategy may be conceived as dynamic holding times in a segment of the route: if a fast vehicle is catching up the vehicle ahead, the speed of the former vehicle is linearly reduced with regard to the difference between the target and the actual headway. The results provided by this method outperform the former static holding point strategies in terms of system productivity and regularity. Nevertheless, this procedure does not respond properly when the headway adherence is significantly poor. Daganzo and Pilachowski (2011) improved the determination of the cruising speed pattern when the time-headway variance is significant. In Xuan et al. (2011), a family of dynamic holding strategies are presented to improve both user and operating costs. This method improves the efficiency of existing control strategies since it minimizes the required slack times by 40% compared to conventional schedule-based methods. In Bartholdi and Eisenstein (2012), a method based on Markov-chains is presented where headways are dynamically self-equalized to a natural value. In addition to that, Argote-Cabanero et al. (2015) extends a dynamic control method for several interacting bus routes. The proposed method consists of a combination of dynamic holding and driver guidance that shows the proper cruising speed of buses along the route based on real-time data. As is stated in Muñoz et al. (2015), previous contributions based on control theory assume that buses have infinite capacity to accommodate all the passengers waiting at stops. However, the scalable reduction of bus speeds in high transit demand corridors may lead to a problem of vehicle capacity. Experience shows that some users cannot get on overcrowded buses arriving at the stop and need to wait to the following transit vehicles. Indeed, both holding point and dynamic speed strategies are aimed to guarantee the time-headway adherence at the expense of losing commercial speed (in the whole fleet or passenger travel time) and increasing 3 operating costs. Nevertheless, few contributions assessed the cost in which transit agencies will incur to deal with bus bunching. Indeed, transit agencies would take advantage of dynamic transit signal priority measures in order to minimize the reduction of the vehicle speed due to the time spent at holding points. In TRB (2013) there is an extended analysis of different techniques of transit vehicle actuated strategies that design off-line and online synchronization of traffic signals. The connection of buses to the transit control center (TCC) and the deployment of a coordinated Transit System Priority (TSP) system may significantly reduce the bus delay by 55-75% with regard to static transit priority systems (Hu et al. 2015). To our knowledge, there are no contributions analyzing how traffic signal priority may help the system to maintain a good regularity. Therefore, this paper proposes an adaptive dynamic bus control strategy, based on active signal priority for buses. Taking into account real-time headway information and traffic signal variables, we propose an adaptive transit speed pattern combined with a signal offset modification at specific intersections, to avoid the bus bunching effect. The adaptive transit speed pattern has been adapted from the contributions of Daganzo (2009) and Daganzo and Pilachowski (2011). All stops are conceived as check points where the timeheadway adherence control is estimated using AVL technologies. When the time-headway of one bus (with respect to the bus ahead) is larger than a targeted value, the green phase of downstream traffic lights may be extended (constrained to a maximum value) to allow the bus to pass through the signal without stopping. At the same time, the speed of buses showing smaller time-headways with regard to the target value with the vehicle ahead, will be reduced. However, in this paper, this speed reduction is lower than the presented in Daganzo and Pilachowski (2011). This strategy outperforms comparatively user costs and the coefficient of headway variation with regard to existing control procedures. Besides, it also improves the operating costs, since no additional vehicles are required in comparison to slack time strategies. Moreover, the modeling approach alleviates some of the drawbacks of the former contributions as stated in Muñoz et al. (2015): the occupancy of the vehicles is considered when activating the control criteria. However, it requires that APC systems should be deployed in vehicles to put in practice these control strategies. Although the largest bus transit agencies in developed countries have already deployed expensive AVL systems, in the recent years, affordable technology has arisen to trace each bus in the line. Last developments use simple smart phones equipped in each bus with a single ad-hoc application to implement coordinated dynamic speed control strategy in a bus corridor. On the other hand, the development of Radio Frequency Identification Devices (RFID) of large range, allows the communication between vehicles with the infrastructure. This cheap technology is currently able to recognize a specific bus at upstream sections of traffic light intersections and activate some modifications of signal settings (TSP). The integration of the former technologies would make it possible for any kind of transit agency all over the world to deal with the schedule adherence problem. Moreover, the time-headway adherence problem will be a crucial issue for those bus agencies that are willing to deploy full electric vehicles in routes to mitigate the local emissions and Green House Gases (GHG). In fact, European Union is fostering electromobility services in cities by means of several research projects (ZeEUS and Eliptic). Different bus technologies and charging infrastructure solutions will be analyzed in demonstration sites. Based on the experience gained by the authors in these projects, it can be stated that there is not any fullyelectric bus of 18 meters of length in the market (or even an articulated bus prototype) able to provide continuous service (15-16 hours per day) with an initial charge at the bus garage (September 2015). All of them need on-route charging operations at charging stations located along the route. The slack time reserved at specific holding stops should be sufficient to perform these charging operations under perfect time-headway adherence conditions. Nevertheless, if the bus arrivals at these points are irregular, vehicles cannot be charged at full capacity, unless some queues of vehicles at the charging stations appear (causing more disturbances and schedule variance) or redundant charging stations (more servers) are deployed. 2. MODELING FRAMEWORK A dynamic bus following model similar to that introduced in Daganzo (2009) is presented for describing the physics of bus behavior and their trajectories. This method may be adaptive to the actual performance of the bus system. It can reproduce strategies controlling the headway variation and oscillatory effects. We consider a straight bus route of length 2L as it is depicted in Figure 1. Buses run along the route in two directions (from A to B and from B to A). The route presents 2N bus stops, where the distance between stop s and s+1 is denoted by ls. Let J be the total number of buses operating the route in the two directions. Each bus is labeled by j=1,…, J and is supposed to travel the roundtrip, stopping at each stop s, s=1,..2N. It is considered that bus j=j*+1 is in the rear of bus j* (j*=1,.., J-1). Since buses may operate the route several cycles, the stops are labeled by s=1+(k2N), 2+(k2N),.., 2N+(k2N), where k (0≤k<∞) is an integer number that denotes the completed number of round trips made by the bus in the line. Stops s=1+(k2N) and s=N+1+(k2N) denote the starting s=N+1+ different W controlle time (Ci) we consi FIGUR E E q H as a f u route se g (Ts) and ( s  ). T h addition are alloc a mandato r mathem a T h the cruis suppose d on the m o T h s+1).Th e section b time tha t intersect i points for e k2 N ) represe n directions of W e also assu m d by the Traf f and the gree n der that the s i E 1 Schemati c q uation (1) d e u nction of the g ment betwee n the slack ti m h e fourth ter m to tha t , it is s u a ted to let dri v r y labor rule a tical operato r h e travel tim e ing speed of d to be the m a o dification o f h e first term e second term b etween stops t bus j wait s i on p when t h e ach route d i n t the same p service. m e that b use s f ic Control C e n offset with r i gnal cycle ti m c illustration e fines the nu m bus travel ti m n stop s and s m e introduced m allows us t h u pposed that v ers rest an a for each age n r   x denotes       J e of bus j b et w bus j in this a ximal, vb . W f the speed of b of Equation represents th e (s; s+1) enc o s at each inte r h e green pha s i rection trip. p hysical poi n s run along a e nte r . Each in t r egard to a g e m e consists of of the bus r o m ber of vehicl m e in a roundt r +1: running t i in schedule a h e assessment at terminals A m ount of tim e n cy and it is t he upper int e  2 1 ,   TT N s psr w een stops s a n section. If th e W e will see lat e b us j).   srj Ts T , )( (2) captures e delay cause d o mpasses a t o r section p∈Is s e is active, t h 4 Thus, stop n t (terminals o a corridor e q t ersection i=1 e neral referen c a green phas e o ute es needed (J) r ip. It is esti m i me, (Tr,s), ti m at stop s in o of “static ho A and B (s= N e b efore cont i independent e ger of the ter m  ,  H T sss p  n d s+1 (Tj,(s) ) e time head w e r that one o f   j s sp s v l T , ( the running d by traffic si g o tal amount o f in the route h e variable d j pairs (s=1+ o r headers) b q uipped with , 2, .., I is ch a c e clock  i ( 0 e time (gi) fol l in the route t m ated as the s u m e spent at in t o rder to com p lding point s t N +k2N and s= 2 i nuing the se r of the slack t m x.         BA  ) can be calc u w ay is perfect l f the strategie s    s I p pj d s 1 , ) time of bus j g nals on the b u f Is intersecti o section betw e j ,p will be eq u k 2 N ; s=2N+ k ut refers to b a set of I s a racterized b y ≤  i< Ci). Fo r l owed by the r o maintain th e u m of four ti m t ersections (T p p ensate poten t t rategies” to t a 2N +k2N) lay o r vice. This ex t t ime devoted  u lated by Equ a l y regula r , th e s to tackle bu s j in the seg m u s performan c o ns (I s ||). e en stops (s,s + u al to 0. In o t k2 N ) and (s b us stops bel s ignalized int y the traffic si g r the sake of s r ed phase tim e e targeted bu s m e componen t T p ,s), time spe n n tial service d t ackle bus bu n o ver times A  t ra time is de f at holding p o a tion (2), wh e e bus cruisin g s bunching w m ent betwee n c e. We suppo s The variable +1). If bus j ther situation = N +k2N; o nging to ersections g nal cycle s implicity, e (ri). s headway t s for each n t at stop s d isruptions n ching. In A and B  f ined by a o ints. The (1) e re vj (s) is g speed is w ill consist (2) n stops (s; s e that the dj,p is the arrives at s, the bus 5 trajectory needs to be modified. Equation (3) allows the evaluation of the arrival time at intersection p, a pj t,, based on the departure time at the previous intersection or stop ( d pj t1, ) and the location of intersections (p-1,p), as it is depicted in Figure 2. It is supposed that the length xp between the location of intersection p with regard to the first stop is known. Equation (4) establishes the number of signal cycles of Cp time ( * ,pj n) that have been completed before the arrival of bus j at intersection p, where [x]- denotes the mathematical operator estimating the lower integer of x. From this value, it is possible to determine the departure time at intersection p as well as the total signal delay time by Equation (5) and (6) respectively. The first case of Equation (5) determines that bus j arrives at intersection p when the green phase is activated; consequently, there is no vehicle delay. Otherwise, the second case represents that the traffic signal is red when this bus arrives at this intersection. Therefore, its departure must be postponed to the green phase of the next signal cycle Cp. Finally, the delay at intersection p is assessed in Equation (6) as the difference between the departure and arrival time of bus j at this intersection. The time spent in accelerating/ deaccelerating the vehicles up to/from the cruising speed due to a stop or a traffic light is neglected. s j pp d pj a pj Ip sv xx tt ,..,2 )( 1 1,,             (3)           p p a pj pj C Δt n, * , (4)           casesother in )1( * if * , ,,, , ppjp pppjp a pj a pj d pj ΔnC gΔnCtt t (5) a pj d pjpj ttd ,,,  (6) The estimation of the arrival time of bus j to the first intersection of the section (s;s+1), i.e. p=1, is made by Equation (3). In this Equation, the term d pj t1,  should be replaced by the departure time of the last stop s ()(st d j) and xp-1 by the coordinate of stop s (xs) . Furthermore, the time that each bus j spent at each stop s is evaluated as a function of the number of boarding and alighting passengers. We assume that  is the O-D matrix which defines the passenger flow at time interval t that boards at stop o and alights at stop d (o=1,.., N-1; d=o+1,..N in direction A-B; o=N,..2N-1; d=o+1,..,2N for direction B-A). The total passenger flow in one direction of service can be calculated as ∑∑       . Therefore, the percentage of passengers travelling between stops (o,d) in time interval t is evaluated by  󰇛1/󰇜  . In Section 3, the performance of the bus route will be assessed, keeping the percentage of the passenger flow distribution constant between stops ( 󰇜, and scaling the total passenger demand q in the route. The time interval t may have different time lengths, from minutes to several hours. It depends on how the information has been obtained from the real world (on-board O-D survey, boarding alighting counters). Although real implementations usually have estimations for the passenger O-D matrix, aggregated in hours or even for the whole day, the variation of  over short domains of time intervals implies a significant disturbance of the dwell time at stops and consequently of the headway adherence. Hence, the number of passengers alighting (aj(s)) and boarding (bj(s)) at stop s (s=1,..,N-1) for each bus j when the headway adherence is perfect, may be estimated using Equation (7a) and (7b). The term (Hq) captures the total number passengers that have got on bus j in the whole direction of service (A-B). We sum the passenger flow percentages from all potential origins (k=1,..s-1) to the stop s, when the alighting passengers of bus j at stop s is addressed in Equation (7a). The boarding passengers are addressed in a similar way, adding the flow percentage from stop s to all potential downstream destinations (k=s+1,..,N). The term gjk is equal to 1 if the arrival of vehicle j at stop k is made in the time interval t* (1≤t*≤F) and 0 otherwise. The parameter F is the number of subsets of stationary time periods in which the passenger flow distribution among stops is evaluated. To be consistent, the boarding and alighting demand values at starting and ending stops of a route must defined (a1=aN+1=bN=b2N=0). The calculation in direction B-A can be done easily adapting Equations (7) to the corresponding passenger flows and demand patterns. If we do not find empirical data to estimate boarding and alighting passengers, Equations (7a) and (7b) may be substituted by stochastic functions according to the assumption of probabilistic distributions of these va r FIGUR E T h Equatio n vehicle a states th a  O stops an d variable s respecti v study j a Aj(s-1) d When th e 1)=aj (s- 1 H the stop u with reg a Equatio n should t a the endi n satisfyin g time. Th e cycle, w h of Equa t wouldn’ t r iables. E 2 Schemati c b h us, the time n (8)- The pa r a nd p aramete r a t the boardin g nce the b us t r d holding po i s tj d(s-1), t j a( s v ely. The vari a a t the previou etermine the e system is t o 1 ) respectivel y owever, the c u nde r analysi s a rd to the tr a n 10). N evert h a ke into acco u n g stop is m a g a perfect c o e refore, we a d h ere the time t ion (10). Fi n t be enough s l c representa t )(  j H sa )(  k j Hqs b s p ent by ea c r ameter t oc is a r s  ,  are re s g and alightin g r avel time ha i nts) in a reg u s ) refer to th e a ble tj d(s-1) is s stop (s-1) a total number o tally regular y . c alculation of s is not one o a vel time fro m h eless, when u nt the layove r a de without a o rrespondence d d kJ times t h disturbance i s n ally, if that b l ack time in t t ion of the tr a 1 11 t    s k F t j ks gy H q 1 t 1   N s k j sk F t gy c h bus at sto a constant ti m s pectively th e g operations a ocs tT   s been estim a u lar state, th e e departure t estimated thr a nd the corres p of boarding p (perfect time - arrival times a f the first sto p m the previo u we analyze t r and slack ti m relevant del a to the target h e headway H s still not pre s b us has an i m h e terminal t o 6 aj ector y of b u direcin jk di r in j k o p s when sy s m e devoted to e unit boardi n a re performe d  s a b   ;max  a ted for each e b us motion t ime of bus j r ough Equatio s ponding dwe l p assengers a n - headway ad h at stops need s p s of the two u s stop and t h t he variable t j m es. If the ar r a y, this bus j headway. It m H to the arriv a s ent in the sys m portant arri v o compensate u s j B-A t ion B -A r ection s tem is totall y the door op e n g and alighti n independent l  s a  phase of the model is des c j from stop ( n (9) as a fun c l l time at thi s n d alighting p h erence), we a s to identify t h r oute directio h e potential d t j a(s) at termi n r ival of bus j a may start ru n m eans that b u a l time at this tem. This sit u v al delay at t this irregula r B y regular (Ts ) e ning and clo s n g time per p y by differen t route roundtr c ribed by Eq u ( s-1) and its c tion of the a r s stop s-1. T h p assengers at s a ssume that B h ree different n s (terminals ) d elays at inte r n als (s=1+2k N a nd its passen g n ning in the o u ses should d e stop of each u ation is repre t he last stop o r ity. Therefor e ) can be cal c s ing operatio n p assenge r Eq u t iated doors. r ip (links, int e u ations (9) - arrival time r rival time of h e variables B stop (s-1) re s B j(s-1)=bj(s-1) cases of stud i ) , arrivals are e rsections (fir s N or s=N+1 + ger alighting p opposite dire c e part at each H bus j in the f e sented in the of one direc t e , bus j woul d (7a) (7b) c ulated by n s of each u ation (8)  e rsections, (11). The at stop s the bus of B j (s-1) and s pectively. and Aj(s- i es. When estimated s t case of + 2kN), we p rocess at c tion trip, H units of f irst round third case t ion, there d not start 7 from the first stop at the target time headway H (second case of Equation 10) so that the disturbances would still be propagating in the opposite direction trip. In that case, the vehicle starts the service in the opposite direction just after being held the mandatory lay-over time ( min  ) at this terminal. The model only needs the insertion time of each vehicle j (j=1,..,J) in the system to properly characterize its trajectory along the route. This information can be defined by Equation (11).  )1();1(max)1()1(  sAsBtstst jjoc a j d j  (9)                         21;21 and )()1( if )2( 21 ;21 and )()1( if )1( 21 and 21 if )1( )1( )( 1min 1minmin kNNkNsHststkJHkNst kNNkNsHststst kNNskNst sv L st st a j d j a j a j d j d j p r p j d j a j   (10) ,...,JjHjt a j1 )1()1(  (11) 2.1. Modelling the unstable motion of buses The analysis of bus system performance under service disruptions is conducted by the insertion of an extra time Uj(s) in the arrival time of specific bus j to stop s (Equation 10). The exogenous variable Uj(s) represents the potential delay that bus j may experience during the trip between stops (s-1; s). It would cause the headway variation among the whole fleet. Moreover, the model takes into account the traffic signal settings along the corridor. When the time-headway of the bus route is not multiple of the signal cycle time (H/Cp is not an integer number), buses will find a different sequence of green-red phases at intersections. This is an additional source of instability in the bus performance. The proposed dynamic model will analyze the performance of the system and passengers behavior because of this alteration Uj(s), considering the current signal settings in the route. In this state of service irregularity, the assumption regarding the estimation of terms Bj(s-1)= bj(s-1) and Aj(s-1)= aj(s-1) is not valid. The number of boarding passengers of vehicle j at stop (s-1) will directly depend on the real headway with the bus operating ahead. As we track the arrival and departure time of all vehicles at the overall bus stops, the evaluation of terms Bj(s-1) and Aj(s-1) can be easily done by Equations (12)-(13). The term   )1()1( 1 stst a j a j represents the current time-headway between buses (j,j-1). On one hand, the boarding passengers on bus j at stop s will depend on the waiting passengers at this stop, term )1( 1 sDj in Equation (12). The model here also improves the existing contributions in bus bunching because it takes into account the vehicle capacity constraint. This constraint is addressed in the following Equations (14)-(15). Therefore, the total number of passengers that cannot get on the previous bus (j-1), )1( 1  sDj, also contributes to the number of boarding passengers on bus j at stop s-1, calculated in Equation (12). The summation of Equation (12) represents the number of trips carried out, in the same direction of service, between stop (s-1) and all potential downstream destinations (m, m>s-1). Therefore, parameter k* denotes the number of roundtrips completed from the initial time of study. Parameter  refers to the route direction in the roundtrip where bus j is running (  =0 for direction A-B and  =1 for direction B-A). On the other hand, the number of alighting passengers at stop s-1, Aj(s-1), does not depend on the headway between buses but, on the current onboard passengers of bus j alighting at this stop. In equation (13), we assume that the number of alighting passengers at stop s is proportional to the ratio of the hourly passenger flow to stop destination s divided by the total demand of the static hourly O-D matrix.  1 )1( )1()1()1( 1 )*22( 111        ssDqyststsB j Nk sm t ms a j a jj  otherwise 0 221 if 1 *;*22*21 * where      k*NNsk*N kNkNsNkk  (12) 8 )N(k*Np s y y mBsA Np mr rm sm s pm jj          12 where 1 )()1( 1 , 1, 2 1 (13) The occupancy Mj (s) of the bus j during the segment between stops s-1 and s can be evaluated by Equation (14) taking into account the vehicle capacity, C. For formulation consistency, we state that Mj(0)=0. Equation (15) evaluates the total amount Dj(s) of waiting passengers at stop s that cannot get on the bus j (if they exist) and may board on the following bus j+1. It is supposed that the definition of the targeted headway H (input of this model) is properly defined in order to accommodate the passenger demand in the static system with buses of capacity C.   )()()()1(;min)( 1sDsAsBsMCsM jjjjj   (14)   CsAsBsMsD jjjj  )()()1(;0max)( (15) 2.2. Bus headway control strategies All strategies aimed at controlling fixed bus intervals are based on the real-time headway monitoring of bus departures from stops. Equation (16) evaluates the real time headway between two consecutive buses ()(st d j ) considering the departure time from stop s. Therefore, it is necessary that each time any bus j is going to depart from one stop s, the variable )(st d j is updated. It should be noted in Equation (16) that the real-time evaluation of current headway of bus j is made with regard to the bus j-1 ahead, at stop s (forward comparison). This information should be used to calculate the adherence with regard to the targeted headway H (see Equation 17a). The headway analysis of bus j with regard to bus j+1 (backwards) is infeasible because bus j+1 has not arrived yet at stop s. Hence, the backward comparison of headway with the following bus (Equation 17b) will be made taking into account the difference of departure time of bus j and j+1at the last stop s* visited by bus j+1 (*)( 1st d j ) up to this moment. sststst d j d j d j )()()( 1  (16) forward comparison to bus j-1 Hsts d jjj  )()( ,1  backward comparison to bus j+1 Hsts d jjj   *)(*)( 11,  (17a) (17b) 2.3. Control strategies The model developed in section 2.1 only considers the usual practice of bus agencies, defined here as Strategy S0. It consists of the allocation of slack time  s at the holding points (terminals) in order to tackle the lack of regularity. However, as is reported in Daganzo (2009), this solution presents several problems. The control strategy is not adaptive, since slack times are not dependent to the deviation of targeted time headways. In fact, these slacks represent an unproductive allocation of time in the cycle time of buses when the performance of the system is regular. Therefore, this paper analyzes two complementary fleet management strategies that may overcome the limitations of the previous operation. On one hand, Strategy S1 will be based on “dynamic holding points”, so that the cruising speed of buses will be varied depending on the time-headway between buses. The control scheme of this strategy is practically similar to that presented in Daganzo (2009). On the other hand, Strategy S2 presented here will encompass the previous variable bus speed pattern combined with an additional measure based on signal priority for buses. 9 Strategy S1 This strategy obliges drivers to adapt the cruising speed of their bus when the headway adherence is irregular. The motion law that modifies the speed of bus is defined by Equation (18) when the vehicle capacity constraint of vehicles ahead and at rear are not achieved. Therefore, this formulation is only valid when Mj+1(s*)<  C and Mj-1(s’)<  C, where    and s' is the last visited stop of vehicle (j-1) in the route.                                      otherwise 0)( )1( and 0 ; if )( )1( ;min 0 ; if )( )1( )( ,11,,11,,1 ,11, 1,,11, ,11, b jjjjb j s jjjjjj jjjjb j s s b jjjjjj jjjjf j s s j v f sv l f sv l l v f sv l l sv     (18) C)(sC or M(s')Mvsv jj-bj    * if )( 11 (19) Tthe first case of Equation (18) reduces the actual cruising speed when: i) bus j is getting further away from bus j+1 than the desired headway H (i.e.  j,j+1(s*)>0); and ii) this time spacing is greater than the corresponding value with the vehicle ahead at stop s. It is desirable that this bus j will operate the stretch up to the next station at a cruising speed below the maximal value in order to reestablish the desired time headway. The parameter ff is a speed adjusting factor (ff>0). Therefore, the speed reduction of vehicle j is proportional to the difference between the total headway deviation and the vehicle at rear and ahead )( ,11, jjjj     . On the other hand, the second case of Equation (18) increases the current cruising speed when: i) bus j presents a higher headway with bus j-1 than the target value H in stop s (i.e.  j-1,j(s)>0); and ii) this headway is higher than the corresponding with the vehicle at rear at stop s*. The reason is that bus j will find more passengers at stops than the expected (increase in dwell time). These additional passengers would be supposed to get on bus j+1 if bus regularity would be perfect. If no control measures are implemented, one may suppose that this tendency will be amplified until bus j+1 reaches bus j (bus pairing phenomena). In order to tackle this problem, it is recommended that bus j will run at a higher speed than the previous segment )(svj > )1(  svj. As defined in Equations (18), the speed modifications are proportional to the difference of the time headway adherence between buses (j,j+1) and buses (j-1, j). Note that this difference )( ,11, jjjj     has a negative value in the second case of Equation (18), where fb (fb >0) is the speed adjusting factor in this situation. If we set a value of the speed adjusting parameter fb that produces 0)())1(/( ,11,     jjjjbj fsvL   ; the corresponding cruising speed will present a negative value too. That possibility is constrained in the second case of Equation (18) since the system will require a higher cruising speed and de facto we will use bj vsv  )( in these situations. As it is pointed out in Daganzo (2009), the dimensionless speed adjusting parameters ff and fb represent the marginal increase in expected bus delay caused by a unit increase in headway. It may be considered as the expected number of passenger arrivals at one stop during the average marginal delay induced by one boarding move. Therefore, ff is also conceived as a speed factor to reestablish the desired headway between two consecutive buses. If ff=1, the current headway will be close to the targeted headway in the next stop, but it will produce a significant reduction of bus speeds. Otherwise, if ff→0, it will maintain the modified speed close to vb and it will take a great number of stops to overcome the deviation from desired headway. Similar statements can be provided for the adjusting parameter fb. However, when the occupancy of the bus ahead or at rear of bus j is equal or slightly lesser than the vehicle capacity, it is preferable that bus j runs at the maximal speed as it is defined in Equation (19). The reason of this statement is justified as the vehicle ahead or at rear will also present a passenger load similar to the vehicle capacity. It will experience shorter dwell times at stops since boarding operations are not made. Therefore, it will run at a maximal speed vb (it will follow the 3rd case of Equation 18). FIGUR E in Probl e FIGUR E in Probl e T h compens b ase line S0  s= 3 m E 4. Total pa s e m 1.1 E 5. Total pa s e m1.2 h e provision o ate the time d case. The ex t m in and redu c s sen g er trav e s sen g er trav e o f a slack tim e d isturbance U 3 t ension of sla c c es the head w e l time and c o e l time and c o e s of  s= 3 mi n U 3 (42)= 2 min. c k time to  s = w ay variation u 16 o efficient of h o efficient of h n (strategy S 0 The total co s = 6 min impr o u p to cv= 0.1 6 h eadwa y vari a h eadwa y vari a 0 ) to control t h s t is now 1.0 8 o ves the user t 6 . However, t h a tion for eac h a tion for eac h h e headways p 8 times the co t ravel time in h e operating c h controllin g h controllin g p rovides goo d o rresponding c comparison t c ost is increa s strate gy strate gy d results to c ost in the t o strategy s ed due to the prov i than larg T h 1.1 since Uncontr o adaptive constrai n correspo n graded a s in the ti m strategy S travel ti m are slig h strategy S those of p roduce s greater t h remarka b only 1.0 0 manage m recurren t cruising s FIGUR E in Probl e P r (q=1400 of D=30 consider e p asseng e vehicle c i sion of 2 ad d er slacks (  s= h e total pass e e the capacity o lled case). T controlling s p n t is not consi d n ding figures s LoS E (cv= 0 m e headway a S 2 is able to c m e savings at h tly highe r t h S 1. However, Strategy S0 s the minima l h an the corre s b le that the h y 0 7 times grea t m ent. The ide a t disturbance s peed pattern E 6. Total pa s e m1.3. r oblem 1.3 is pax/h in bot h vehicle depa r e d is summar i e r load in the c apacity (C= 7 d itional vehicl 6min), in ter m e nger time sa v constraint als o T his fact is c o p eed propose d d ered. The t o of strategy S 0 .72). If we a d a dherence an a c ompensate th the traffic lig h h an the basel i as this gree n with slack t i l total cost ( Z s ponding to s t y brid controll i t er than the b a a is to allocat e value in the and green tra f s sen g er trav e aimed at str e h directions) a n r tures have b e i zed by TPT= route Omax= 7 7 5 pax/veh). es. Now, the m s of total co s v ings with im p o helps to sta b nsistent with d by Daganzo o tal cost of th i S 0. Addition a d d a slack ti m a lysis whose e passenger t i h ts. If the gr e i ne case. Ne v extension is mes. If this t Z T =25,275 € r ategy S0  s= i ng strategy ( S a se line and c v e a minimal s route) and m f fic lights ext e e l time and c o e ssing the pe r n d time distu r e en complete d 1547 h, ZO= 1 7 0.56 pax/ve h When the di s 17 strategy S0 w st. p lementation b ilize the per f the statemen t o does not wo r i s strategy is 1 a lly, t he time m e of  s= 3 m i metric is re d i me lost due t o e en extension v ertheless, th i determined t o t raffic light c € ). However, = 6min, due to S 2 with slac k v =0.17) witho lack time at h m itigate larg e e nsion). o efficient of h r formance of r bance (U=4 m d . The perfor m 1 4,701 €, ZT = h is detected b s turbance of U w ith short sla c of strategy S 1 f ormance (15 % t in Muñoz e t r k in routes w 1 .27 times th e headway ad h i n, strategy S 1 d uced to cv=0 . o disturbance time is mini m i s strategy o u o be G> 10 s e c ontrolling p a the coeffici e the discrete t k time  s= 3m i u t imposing h h eaders (esti m e r disturbanc e h eadwa y vari a the previous m in). Figure 6 m ance in the b = 37,912 €, cv = b etween stops U = 4 min oc c c k times (  s= 3 1 are not as i m % of improve m t al. (2015) w w ith high dem a e b ase line ca s h erence is no t 1 outperforms . 22. Finally, and the natur a m al (G≤10 se c u tperforms th e e conds, the re s a rameter is s e e nt of headw a t ime saving a t i n) gives goo d h uge dynamic m ated as a fun c e s p erformin g a tion for eac h bus route wi t plots the res u b aseline case w 0 and J=10 v e #11 and #12 , c urs, when n o 3min) is mor e m portant as i n m ent with re g w here it is sa i a nd and wher e s e, even high e t acceptable s s strategy S0, the impleme n a l motion of b c onds), the c o h e metrics pr o sults are com p e t to G=20 s e a y variation i t traffic lights d results (var i changes in t r ction of an e x g strategy S2 h controllin g t h higher de m u lts when the w here no dist u eh. The high e , and almost e o controlling s e efficient n Problem g ard to the i d that the e capacity e r than the s ince it is e specially n tation of b uses with o st metrics o vided by p arable to e conds, it i s slightly . It is also i able ZT is r affic light x pected or (variable strate gy m and flow r ound trip u rbance is e st vehicle e quals the s trategy is 18 activated, the TPT, ZO and ZT are increased respectively by 60%, 7% and 40% with regard to the baseline case. The time headway adherence reaches cv=0.96. Strategy S0 is able to stabilize the system. When slack times are  s=6 min at terminals, the results in terms of total cost are quite similar to the baseline case (ZT is increased by 9.3%) and the level of service of time headway adherence (cv=0.18) is stated as LoS A. In this problem, strategy S1 without slacks is not effective since the metrics are comparable to the corresponding values of the uncontrolled case. It is noticeable that only by reducing the speed of vehicles in systems with high demand, low vehicle capacity and short headways; we cannot maintain a good performance of the route. On the contrary, strategy S2 outperforms the results of the previous controlling strategies. The total cost of the system (ZT metric) is increased by 3.9% (G=5 sec), 1.8% (G=10 sec) and -0.3% (G= 20 sec) compared to the baseline case. In these situations, the performance in terms of time headway adherence can be considered as LoS B. In this problem, the implementation of strategy S2 with G= 10 seconds and slack times  s= 3 min even improves the total travel time with regard to the baseline case and cv=0.15 (LoS A). However, the total cost is slightly greater than baseline case and strategy S2 with G=10s due to the inclusion of an additional vehicle. 3.3.2. Unstable motion created by traffic lights and exogenous disturbances (Problem Set 2) A sensitivity analysis of the performance of each control strategy has been done in Problem Set 2 with regard to the traffic light settings. We considered three different intersection spacings along the route: l ={100; 210; 300} meters. Moreover, we have also generated problems with different green time at signaled intersections, ranging among g ={22.5; 45; 67.5} sec. In this case, the traffic light cycle length has been considered to be Cp =90 seconds. Since the time headway (H= 5min) is not multiple of the cycle time of traffic lights, buses arrive at intersections at different times of the red-green signal sequence. This fact may worsen the bus headway adherence, even when no exogenous disturbance is generated (base line scenario). The results are summarized in Figure 7 considering the departure of 27 consecutive buses from bus stop s=1. All instances with equal green time (g) are presented together. The relative increment of all performance indicators (TPT, ZO, ZT, cv) are roughly equal in those instances with the same green time allocation. The spacing between intersections seldom affects the behavior of the control strategies for a given traffic light setting. The variations of all indicators are lower than 5%, except for Strategy S1, where these differences are up to 10%. Therefore, the differential behavior of control strategies is only identified for instances with different green time settings. When the green time is equal to g=67.5 s (g/Cp=0.75), the bus delays at intersections generate by themselves low headway variations. In this situation (base line scenario), the level of service can be stated as LoS B (cv=0.28). The generation of a time disturbance of U=4 min to vehicle j=3 (Uncontrolled scenario) makes the system more unstable, increasing the variation of headways up to cv=0.72 (LoS D). Strategy S0 significantly reduces travel time of users with regard to no control scenario, at expenses of increasing operating costs, deploying more vehicles. Strategy S1 with no slacks is not effective, since the total cost of the system (agency and users) is 1.15 times greater than the base line scenario. Vehicles often arrive at intersection when the green phase is active (g/Cp=0.75), so that the efficiency of strategy S2 is still limited. This strategy cannot improve the values of all performance indicators in the base line scenario. The best control criteria is strategy S2 with slack of  s=3 min, characterized by cv=0.24 and a total cost of 1.019 times greater than baseline scenario. It is remarkable that Strategy S0 with  s =3 min presents similar results as the former one. The analysis of the bus performance when g= 45 sec (g/Cp=0.5) is essentially different. In the baseline case, the stoppings of vehicles at intersections significantly increase the bus bunching phenomena, presenting cv=0.75. It can be stated as LoS F. Therefore, the creation of an exogenous service disruption U= 4 min in the Uncontrolled scenario just worsens the total cost by 6.5% and seldom increases the bus bunching effect (cv=0.82). FIGUR E I n b ase line at the ex E 7. Sensitivi t n strategy S0, ) and the vari x penses of in c ty anal y sis o f a slack time ation of head w c reasing the o p f the perform o f  s = 3 mi n w ays at cv=0. 4 p erating cost, 19 m ance of buse s n is sufficient 4 2. Strategy S , so that this s s in Problem to reduce th e S 0 with  s = 6 s trategy is m o 1.3 when g= 2 total cost by min reduces o re expensive 2 2.5; 45 and 6 y 17.6% (wit h the travel ti m e considering 6 7.5 sec. regard to m e of users t otal cost. 20 Strategy S1 with no slacks is able to enhance the system performance with regard to the uncontrolled scenario, but the cost savings achieved are far away from those of strategy S0. Finally, strategy S2 really outperforms the previous strategies. When the green extension length is G= 5sec, this strategy is able to reduce both operating and user costs with regard to baseline scenario. As a result, the total cost is diminished by 20% and the service regularity can be graded as LoS=B (cv=0.35). If we continuously increase the green extension length, the performance of this strategy is outstanding, with total cost savings ranging from 20-34% and cv<0.39 (LoS B or C). Nevertheless, the hybrid strategy S2 with slacks (  s = 3min) presents the lowest coefficient of headway variation at the expenses of introducing one extra vehicle, increasing operating costs with regard to strategies without slack times. Eventually, the results obtained when the green time at intersections is g=22.5 sec are fairly similar to those presented when g= 45sec. Strategy S2 outperforms the indicators of other available strategies. However, the total cost savings are not as high as the former ones, ranging among 13-16% while the variation of headways are maintained between cv= 0.30-0.36. This strategy is able to reduce user cost, whereas maintaining the number of vehicles needed concerning the baseline scenario. Vehicles can be speeded up avoiding potential delays at intersections. This fact causes, in some instances, even lesser operating cost than baseline scenario. 3.3.3. Unstable motion created by traffic light settings, traffic flows and demand rates at stops (Problem 3) If we consider the test instance representing the local bus route of highest demand in Barcelona (H6 route), the results generally follow the same pattern explained above. The simulation is carried out during the peak morning time (6.00-9.00 AM). In that case, there is no disturbance artificially-generated since real traffic light control, heterogeneous user arrival rates at stops and car traffic flow, tend to make the headway adherence unstable. In this case, we implement hybrid strategies based on the provision of slack time at headers together with the dynamic implementation of strategies S1 or S2. Figure 8 summarizes the simulated metrics in that route for different slack times at last stops (  s=1, 3 and 6 minutes). The implementation of higher slack times generally improves both travel times and the headway adherence of the corridor, when only the static control strategy is considered (strategy S0). However, the case with  s=6 min does not provide any benefits compared to the instance when  s= 3min. It reflects that this strategy is neither scalable nor adaptable. Both travel times and headway adherence remain roughly constant for slack times greater than a minimum threshold. It is worth mentioning that the coefficients of headway variation when strategy S0 is implemented are greater than cv=0.75, which corresponds to the Level of Service F (taking into account the classification of TRB, 2013). The introduction of dynamic controlling strategies significantly outperforms the performance of the bus network. Strategy S1 reduces the total cost by 14-28% and cv by 27-58% with regard to the static controlling strategy with low slacks (strategy S0  s=1 min). The level of service in that situation with regard to the headway adherence criterion is E (  s=1min) and D (  s=3 or 6 min). However, when traffic light priority is activated for buses (strategy S2), the results are outstanding and outperform those provided by the speed modification controlling strategy (strategy S1). Strategy S2 improves TPT, ZT and coefficient of headway variation by 40-41%, 38-39%, 72-80% respectively, with regard to the strategy S0. The time adherence variable can be controlled with strategy S2 below the threshold cv<0.3, which corresponds to level of service A or B. The regularity effects of dynamic controlling strategies on the bus service can be observed in Figure 9 a-c. Bus trajectories in the route direction A-B (Zona Universitària-Fabra i Puig) are depicted when the slack time at the ending stop is  s=1 min. Although the inclusion of higher slack times (  s ≥6 min) would improve the results of strategy S1, the actual performance of the service with strategy S2 does not get better with those slack times. Therefore, full dynamic bus controlling strategy (strategy S2) does not need any unproductive slack time at holding points in order to guarantee good headway adherence and user travel times. FIGUR E times at FIGUR E s=1 mi n E 8. Total us e endin g stops E 9. Simulati o n . a) Strate gy e r travel tim e o n of bus tra j y S0, b) Strat e e and headw a j ectories in r o egy S1, c) St r 21 ay adherence o ute H6 (dir e r ate gy S2. in bus route e ction Zona U H6 consider i U niversitària - i n g different - Fabra i Pui g slack g ) when 22 4. CONCLUSIONS In this paper we analyzed the efficiency of control strategies of bus bunching based on the user performance and operating costs incurred by the transit agency to run the service. An operational model was presented to reproduce how time disruptions propagate along the bus route when no controlling strategy is implemented. The model advances the contributions of Daganzo (2009) since it takes into account the vehicle capacity constraint in the formulations. Moreover, the modeling formulations could also estimate the deployment of existing controlling strategies to compensate disruptions: S0 introduced slack times at holding points (bus headers) and S1 modified the cruising speed of buses at each stop to maintain the targeted headway in a similar way presented in Daganzo (2009). Since strategy S1 actively reduced the speed of buses (i.e. it is impossible that buses run above the maximal cruising speed v), several buses would experience a higher round trip cycle time than the theoretical one. To overcome this problem, we proposed a new strategy (S2) that allowed only delayed buses (i.e. those with a higher time-headway with the vehicle ahead than the target value H) to be benefited by traffic light priority so that they could speed up. The results showed that the propagation of even a small disturbance in a specific bus produces irregular vehicle arrivals at stops, causing extra operating costs, user costs and coefficient of headway variation increment. Indeed, when the vehicle departures from terminals are not synchronized with traffic lights, the system does not need any exogenous to present an unstable performance. In these situations, the time headway adherence worsens as the percentage of green phase at intersections is reduced. The coefficients of headway variation may rise to cv≈0.75 when the percentage of green time at intersections is g/Cp ≤0.45. Therefore, transit managers should define a target time-headway compatible with the light cycle times in the bus corridor to reduce bus bunching. Vehicle capacity is an endogenous attribute of the system that contributes to mitigate the propagation of delays and the unstable motion of buses without exogenous controlling measures. When the delayed vehicle has enough capacity to accommodate the overall waiting passengers, the system tends to be more unstable. Any disturbance produces a dramatically increase of passenger travel times. However, the effects in operating costs are softened. On the other hand, if we consider vehicle capacity constraint, the total cost increases by almost 50% with regard to the idealized performance with perfect regularity. Strategy S2 resulted to be the best control method in terms of total passenger travel time, operating costs and total costs. This fact justifies the need of speeding up delayed buses when adaptive cruising speed modification is performed. The effectiveness of this strategy does not depend on the number of intersections but on the traffic light settings: the green time (g) and the green extension (G). The total cost savings of this control strategy are essentially more relevant when vehicle departures are not synchronized with signal settings and for corridors with low percentage of green time at intersections (i.e. the most unstable motion of buses considered). The coefficient of headway variation can be slightly higher to the minimal one obtained with strategy S0 with larger slacks, unless headway variations range among level of service A or B. Nevertheless, Strategy S2 only provided competitive results when the green extension time was significant (G≥ 10 sec). This fact would produce negative effects on the traffic and passenger flow in the streets near to the intersections. Therefore, when the bus route under analysis runs along a corridor with important traffic volumes in the crossing streets, we recommend hybrid control strategies. They consist of providing minimal slack times (less than 2-3 minutes) in the bus schedule as well as implementing dynamic strategy S2 with a green extension time of G=10 seconds, to recover larger and unpredictable disruptions. Hybrid strategies were the second best control alternative, only outperformed by Strategy S2 with G=20 sec. They generally increase the total cost of the system by at most 4% in problems with vehicle capacity constraint, with regard to Strategy S2 with no slacks. The control strategy S0 based on holding points is effective to maintain the bus performance at the same level of service as in the baseline case in ideal problems. However, it requires much slack time to control the system performance (6 minutes at terminals) for routes with high disruptions as well as high passenger flow. For medium-demand problems, lower slacks (3 minutes) seem to be sufficient to guarantee a similar total passenger travel time when no disturbance takes place. Therefore, it is a myopic, not-adaptive strategy since bus managers can not define in advance a minimal slack time to tackle the potential deviations that can appear. This strategy can keep total travel times stable at the expense of increasing operating cost. The effectiveness of the control strategy S1 is significantly dependent on the stabilization parameters ff and fr that reduce the cruising speed proportional to the headway deviation. Although this strategy keeps the number of resources constant in comparison to strategy S0, it was unable to alleviate the effects of bunching on the total travel time of users. Consequently, in capacitated problems, this strategy presents higher total costs than strategies with slack time. Strategy S1 only outperformed the results provided by Strategy S0 with low slack 23 times when the vehicle capacity was supposed to be unlimited. This situation corresponds to the idealistic hypothesis considered in Daganzo (2009), where this cruising speed modification strategy was presented. Throughout the paper we assumed that: i) a delayed bus could not be overtaken by other buses and ii) the boarding time per passenger was considered to be constant and independent to the vehicle occupancy and the number of passengers waiting at stops. If assumption ii) is considered true and if we allow the model to consider overtaking, the results will not differ from the presented in the paper. However, if assumption ii) is substituted by variable unit boarding times, considering the crowdedness of stops and vehicles, the overtaking of buses may produce better results in passenger travel and waiting times. Moreover, another important assumption that deserves mentioning is that the passenger arrival rate at each stop was deterministic and constant during a predetermined stationary period of time. Therefore, the source of instability analyzed in this paper was any exogenous disruption that incremented the running time between two consecutive stops and the delays at signaled intersections. Here, the model can be further improved if stochastic passenger arrivals are considered, as presented in Bowman and Turnquist (1981) or Fonzone et al. (2015). ACKNOWLEDGEMENTS This paper is partially supported by the Spanish Economic and Competitiveness Ministry project, ENTROPIA TRA 2012-39466-C02-01. REFERENCES Adamski, A., Turnau, 1998. A. Simulation support tool for real-time dispatching control in public transport. Transportation Research Part A, vol. 32 (2), pp.73-87. Argote-Cabanero, J., Daganzo, C.F., Lynn J.W. 2015. Dynamic control of complex transit systems. 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Guinardó - Cartagena 5 15 91 16% Rda. Guinardó - Padill - Túnels 5 16 108 15% Rda Guinardó - Tunels Rovira - Lepant 5 25 91 27% Rda. Guinardó - Pi i Maragall - Pl. Alfons el Savi 5 44 91 48% Trav. Dalt - Escorial 5 66 120 55% Trav. Dalt - Massens - St. josep muntanya 5 75 120 63% Trav. Dalt - Verdi 5 84 120 70% Trav. Dalt - Torrent de l'Olla 5 84 120 70% Mitre - Vallirana - Padua 5 63 108 58% Mitre - Balmes 5 41 108 38% Mitre -Muntaner 5 35 108 32% Mitre - Mandri 5 39 108 36% Mitre - Ganduxer - Freixa 5 55 108 51% Mitre - Augusta 5 50 108 46% Mitre - Fleming 5 50 108 46% Mitre -Dr Roux 5 60 108 56% Pl. Prat de la Riba 5 14 108 13% Pg. Manuel Girona - Capita Arenas 7 37 85 44% Diagonal / Maria Cristina 1 26 120 22% Puis XII est 1 34 120 28% Pius XII oest 1 34 120 28% Diagonal - Gregorio Marañon 1 78 150 52%