- Email: [email protected]
Dynamical Modeling and Predictive Control of Bus Transport Systems: A Hybrid Systems Approach
Proceedings of the 20th World The International Federation of Congress Automatic Control Proceedings of the 20th9-14, World The International Federation of Congress Automatic Control Toulouse, France, July 2017 Available online at www.sciencedirect.com The International of Automatic Control Toulouse, France,Federation July 9-14, 2017 Toulouse, France, July 9-14, 2017
ScienceDirect
IFAC PapersOnLine 50-1 (2017) 7499–7504 Dynamical Modeling and Predictive Dynamical Modeling and Predictive Control of Modeling Bus Transport Systems: Dynamical and Predictive Control of Bus Transport Systems: A Hybrid Systems Approach Control of Bus Transport Systems: A Hybrid Systems Approach A Hybrid Systems Approach ∗ ∗∗
I.I. Sirmatel N. Geroliminis ∗∗ I.I. Sirmatel ∗∗∗ N. Geroliminis ∗∗ I.I. Sirmatel N. Geroliminis ∗∗ ∗ Urban Transport Systems Laboratory, EPFL ∗ ∗ Urban Transport Systems Laboratory, EPFL (e-mail: [email protected]) ∗ Systems ∗∗Urban (e-mail: [email protected]) Urban Transport Transport Systems Laboratory, Laboratory, EPFL EPFL ∗∗ [email protected]) ∗∗ Urban(e-mail: Transport Systems Laboratory, EPFL (e-mail: [email protected]) ∗∗ Urban Transport Systems Laboratory, EPFL (e-mail: [email protected]) (e-mail: [email protected]) Abstract: Bus operations, due to their unstable nature, are inefficient when left uncontrolled Abstract: due to their unstable nature, inefficient when with respectBus to operations, retaining headways. Irregularities such asarebus bunching leadleft to uncontrolled loss of time Abstract: Bus operations, due to their unstable nature, are inefficient when left with respect to retaining headways. Irregularities such as bus bunching lead to uncontrolled loss of time and decrease bus service quality. Development of bus transport system management schemes to with respect to retaining headways. Irregularities such as importance, bussystem bunching loss of time and decrease bus service Development ofare bus transport management schemes to avoid bus bunching and quality. improve performance of high andlead hastothus been the and bus service quality. Development ofare bus system management schemes to avoiddecrease busmany bunching and improve performance of transport high importance, and by hasthe thus been the focus of works in the public transport systems literature. Motivated importance avoid bus bunching and improve performance are of high importance, and has thus been the focus of many works in the public transport systems literature. Motivated by the importance of developing bus control strategies for improving performance, and specifically by the lack of of many works in the public transport systems literature. Motivated the of detailed developing control strategies for improving performance, and specifically by importance the system lack of afocus butbus computationally efficient mathematical model describing busbytransport of developing bus control strategies for improving performance, and specifically by lack of a detailed butthe computationally efficient mathematical model describing bus transport system dynamics in literature (which can facilitate model-based control design), we the propose a a detailed but computationally efficient mathematical model describing bus transport system dynamics in the literaturemodel (which facilitate propose mixed logical dynamical of can a single loop model-based bus transportcontrol system,design), which we involves botha dynamics in(e.g., the literature (which facilitate design), propose a mixed logical dynamical model of can a binary single loop bus which we involves both continuous bus positions) and (e.g., model-based the transport state of control a system, bus regarding whether it is mixed logical dynamical model of a single loop bus transport system, which involves both continuous (e.g., bus positions) and binary (e.g., the state of a bus regarding whether it is holding at a certain stop or not) states. Furthermore, we develop a hybrid model predictive continuous busactuation positions) binary (e.g., the of a bus regarding itbus is holding scheme at a(e.g., certain stop or not) states. Furthermore, we develop a hybrid model predictive control with viaand bus speeds, which canstate regularize headways andwhether improve holding at a certain stop or not) states. Furthermore, we develop a hybrid model predictive control quality. scheme Performance with actuation speeds,controller which canis regularize and improve bus service of via the bus predictive evaluated headways via simulation experiments control scheme withmodel, actuation bus speeds,controller which canis regularize headways andare improve bus servicethe quality. Performance of via the predictive evaluated viabus simulation experiments using proposed where the passenger demands and maximum speeds extracted service quality. Performance of the predictive controller is evaluated via simulation experiments using the proposed model, where the passenger demands and maximum bus speeds are extracted from data collected from the bus network of the city of Fribourg. Results indicate the potential using the proposed model, where the passenger and maximum busindicate speeds are extracted from collected from the network thedemands city and of Fribourg. Results thetimes. potential of thedata proposed controller in bus avoiding bus of bunching decreasing passenger travel from data collected from the bus network of the city of Fribourg. Results indicate the potential of the proposed controller in avoiding bus bunching and decreasing passenger travel times. of the proposed controller Federation in avoiding bus bunching andHosting decreasing passenger travel times. © 2017, IFAC (International of Automatic Control) by Elsevier Ltd. All rights reserved. Keywords: Automatic control, optimization, real-time operations in transportation; Modeling Keywords: Automatic control, optimization, real-timetransportation operations in transportation; and simulation of transportation systems; Intelligent systems; Model Modeling predictive Keywords: Automatic control, optimization, real-timetransportation operations in transportation; and simulation of systems. transportation systems; Intelligent systems; Model Modeling predictive control of hybrid and simulation of systems. transportation systems; Intelligent transportation systems; Model predictive control of hybrid control of hybrid systems. 1. INTRODUCTION Station control methods include those that involve taking 1. INTRODUCTION Station control methods include those involve control decisions at a subset of stops of that the bus loop.taking Some 1. INTRODUCTION Station control methods include those involve taking control decisions at a subset of stops of that the bus loop. Some methods on this direction focus on regularizing headways It is well known in the public transport systems literature control decisions at a subset of stops of the bus loop. Some methods on this direction focus on regularizing headways via holding, with the assumption that this would lead to It is well known in the transport literature that bus operations arepublic inefficient when systems left without man- methods on this direction focus on regularizing headways It is well known in the public transport systems literature via holding, with the assumption that this would lead to efficient operation and decreased travel times (Abkowitz that bus operations are inefficient when left without agement (Daganzo and Pilachowski (2011)). Buses manthat via holding, with the assumption that this would lead to efficient operation and decreased travel times (Abkowitz thatbehind bus operations aremore inefficient when left without manLepofsky (1990); Daganzo (2009); Xuan et al. (2011)). agement (Daganzo and Pilachowski (2011)). Buses that and lag encounter passengers waiting for them, efficient operation and decreased travel times (Abkowitz and Lepofsky (1990); Daganzo (2009); Xuan et al. (2011)). agement (Daganzo and Pilachowski (2011)). Buses that situations where there is high variability in the delag behind encounter more passengers forslightly them, In leading to them lagging more, and buseswaiting that are andsituations Lepofsky (1990); Daganzo (2009); Xuan et al. (2011)). lag behind encounter more passengers waiting for them, In where there is high variability in demands, passenger waiting times need also to be considered leading to themless lagging more, This and buses that are slightly fast encounter passengers. positive feedback loop In situations where there is high variability in the the deleading to them lagging more, and buses that are slightly mands, passenger waiting times need also to be considered in the problem formulation alongside headways (Ibarrafast less passengers. This positive leadsencounter to the undesirable phenomenon thatfeedback is knownloop as mands, passenger waiting times need also to be considered in the et problem formulation alongside fast encounter less passengers. positive feedback loop al. (2015)). Studies on this headways direction (Ibarrainclude leads to the undesirable phenomenon that is known as Rojas bus bunching. Instabilities in This the bus transport system in the problem formulation alongside headways (IbarraRojas et al. (2015)). Studies on this include leads to the undesirable phenomenon that is known as et al. (2001) and Delgado et al.direction (2012). Holding bus bunching. Instabilities in thespatiotemporal bus transportvariabilsystem Eberlein (BTS) operation, resulting from Rojas etbe al.used (2015)). Studies on this direction include bus bunching. Instabilities in the bus transport system Eberlein et al. (2001) and Delgado et al. (2012). Holding can also to improve timing of passenger transfers (BTS) operation, from variabil- Eberlein et al. (2001) and Delgado et al. (2012). Holding ity of both traffic resulting congestion andspatiotemporal stop-to-stop passenger (BTS) operation, resulting from variabilbe(2001); used toDelgado improveettiming of passenger (Hallalso et al. al. (2013)). Anothertransfers subclass ity of both traffic congestion andspatiotemporal stop-to-stop passenger demands, and manifesting themselves as irregularity in can canstation also becontrol used to improveet timing of passenger transfers (Hall et al. (2001); Delgado al. (2013)). Another subclass ity of both traffic congestion and stop-to-stop passenger of methods is the stop-skipping strategies, demands, and manifesting themselves as irregularity in headways ultimately as bus bunching, lead to inef- (Hall et al. (2001); Delgado et al. (2013)). Another subclass demands, and manifesting themselves as irregularity in of station control methods is the stop-skipping strategies, the decisions are realized by forcing buses to headways and ultimately as bus leadofto inef- where ficient operations, loss of time, andbunching, degradation service of station control methods the stop-skipping strategies, headways andtoultimately as bus bunching, leadofto inefthe control decisions are realized forcing buses(Fu to skip some stops, to increaseis speed and by thus efficiency ficient operations, loss reasons, of time, and degradation service quality. Due these research on modeling and where where the control decisions are realized by forcing buses to ficient operations, loss of time, and degradation of service skip some stops, to increase speed and thus efficiency (Fu et al. (2003); Cort´ e s et al. (2010); Delgado et al. (2012)). quality. Due to these reasons, research on modeling and control of BTSs is of high importance. skip some stops, to increase speed and thus efficiency (Fu et al. (2003); Cort´ e s et al. (2010); Delgado et al. (2012)). quality. Due to these reasons, research on modeling and Although station control strategies can be effective in control of BTSs is of high importance. et al. (2003); Cort´econtrol s etinal. (2010); Delgado et effective al. (2012)). Considerable research hasimportance. been directed, especially in Although control of BTSs is of high station strategies can be in regularizing headways moderate demand situations and Although station control strategies can be effective in Considerable research has been directed, especially in the last 4 decades, to developing real-time bus control regularizing headways in moderatethey demand situations and leading to improved performance, have adverse affects Considerable research has been directed, especially in regularizing headways in moderate demand situations and the last 4 decades, to developing real-time bus control methods, with the goal of avoiding bus bunching and leading improved performance, they via haveholding adversethe affects on BTStoperformance as they actuate bus the last 4efficient decades, to reliable developing real-time bus control leading toperformance improved performance, theyavia have adverse affects methods, with theand goal of avoiding busofbunching and at on aBTS as they holding the bus ensuring operation BTSs. Realstop or making the busactuate skip stop. Under some methods, with theand goal of avoiding buscan and on BTS performance as they actuate via holding the bus ensuring efficient reliable operation ofbunching BTSs. Realtime control methods for bus operations be classified at a stop or making themake bus buses skip asignificantly stop. Underslower, some circumstances they can ensuring efficient and1) reliable operation of BTSs. Real- at a stop or making the bus skip a stop. Under some time two control methods for bus operations be classified into categories: Station control, can 2) inter-station circumstances they can make buses significantly slower, which will influence the quality of in-vehicle service, but time control methods for bus operations can be classified circumstances they can make significantly slower, into two categories: 1) Station control, 2) inter-station control (see Ibarra-Rojas et al. (2015) for an extensive which will influence the quality of in-vehicle service, but also increase the operating costbuses and required fleet size. into twoa(see categories: Station control, 2) section). inter-station will influence the quality of in-vehicle service, control Ibarra-Rojas et isal. (2015) for an extensive which also increase the operating cost and required fleet size.but review, summary of1) which given in this control (see Ibarra-Rojas et al. (2015) for an extensive also increase the operating cost and required fleet size. review, a summary of which is given in this section). review, a summary of which is given in this section).
Copyright © 2017 IFAC 7770 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2017, 2017 IFAC 7770Hosting by Elsevier Ltd. All rights reserved. Peer review under responsibility of International Federation of Automatic Copyright © 2017 IFAC 7770Control. 10.1016/j.ifacol.2017.08.1067
Proceedings of the 20th IFAC World Congress 7500 I.I. Sirmatel et al. / IFAC PapersOnLine 50-1 (2017) 7499–7504 Toulouse, France, July 9-14, 2017
Another category of real-time bus control methods is the inter-station control, where decisions are taken while the bus is moving between stops. Traffic signal priority methods belong to this category, where the aim is to manage traffic flow efficiently via prioritizing certain circulations of an intersection with actuation over traffic lights (Liu et al. (2003); Van Oort et al. (2012)). Methods based on the control of bus speed constitute another subclass of the inter-station control strategies. The idea in bus speed control is to continuously manipulate the speed of each bus during its movement via feedback control mechanisms to avoid bunching and increase the efficiency of the BTS. On this direction, a control strategy combining bus speed control and signal priority is developed by Chandrasekar et al. (2002), where control actions are taken to ensure that the buses operate with headways equal to a desired headway, which is shown to be able to regularize headways. A bus speed control method is proposed by Daganzo and Pilachowski (2011), where the controller manipulates the speed of each bus according to its front and rear headways, while at the same time respecting bus speed bounds, and is able to prevent bus bunching. A more recent study by Ampountolas and Kring (2015) develops a combined state estimation and LQR control scheme to achieve coordination between the buses, leading to higher headway regularity and improved service. Considering future information about number of passengers waiting to board at each stop and the number of passengers within each bus can significantly improve the performance under congested conditions and decrease the occurrence of bus bunching, but also decrease the number of times passengers are unable to board a bus due to full buses. Motivated by the goal of developing a detailed mathematical model describing the dynamics of a single-loop BTS, in this paper we propose a novel mixed logical dynamical (MLD) model that considers the interaction between bus motion and passenger accumulation. The MLD modeling framework is proposed by Bemporad and Morari (1999) as a systematic approach to the mathematical modeling of dynamical systems which involve the interaction of physical laws, logic rules, and constraints. The proposed MLD BTS model is detailed enough to allow in-depth simulation-based analysis of BTSs, but at the same time computationally lightweight to allow for fast execution. Furthermore, the proposed model enables development of hybrid model predictive control (MPC) schemes, since prediction models based on the proposed MLD model enable the formulation of MPC problems in the form of mixedinteger programs. In the control design aspect, the paper contributes by proposing a hybrid MPC (HMPC) scheme (based on a simplified MLD model), with the goal of regularizing headways and improving BTS performance. A simple PD-like controller and a linear MPC (LMPC) scheme are also developed for comparison purposes. A case study is provided, which uses line 2 of the bus network of the Swiss city of Fribourg. The study is based on simulations with the proposed MLD model, with the information on origin-destination passenger flow demands and maximum bus speeds extracted from bus data collected from the real bus network. Results indicate that control is required for headway regularization. Furthermore, the proposed hybrid MPC scheme shows substantial potential in regularizing headways and improving BTS efficiency.
2. HYBRID MODELING AND SIMULATION OF BUS TRANSPORT SYSTEMS 2.1 Bus Loop as a Hybrid System We consider a single loop BTS with Kb buses and Ks stops, and assume that (i) buses operate on the loop with always positive speed (i.e., they never change direction), (ii) a bus always holds at a stop if there are passengers onboard that want to alight at that stop, (iii) passengers do not differentiate between buses when boarding, since any bus they board will hold at their destination stop, (iv) the position of the first stop of the loop is assumed to be 0 and the bus positions are reset to 0 when they complete the loop and reach the first stop. Dynamics of Continuous States The continuous states xi (t), ni,j (t), and mh,j (t) keep track of how bus positions, passenger accumulations inside buses, and those at stops, evolve over time. (a) Dynamics of bus position can be expressed as follows xi (t + 1) =
Ks
γi,j (t) (xi (t) j=1 Ks
+
j=2
+ T vi (t)) (1)
δi,j (t)xi (t) + δi,1 (t) · 0,
for i = 1, . . . , Kb , where t (–) is the time step counter, T (s) is the sampling time, xi (t) (m) and vi (t) (m/s) are the position and speed of bus i, respectively, γi,j (t) is a binary state that is equal to 1 if bus i is cruising towards stop j and 0 otherwise, and δi,j (t) is a binary state that is equal Ks to 1 if bus i is holding at stop j. The term j=1 γi,j (t) is equal to 1 if bus i is cruising, thus its position will increase proportional to its speed. If it is holding at a stop Ks other than stop 1, i.e., if j=2 δi,j (t) is equal to 1, its position will stay constant. If it is holding at stop 1 (i.e., if δi,1 (t) = 1) this means that it has completed the loop and its position will be reset to 0. (b) Dynamics of bus accumulation can be written as follows Ks out in (t) − qi,j (t), (2) qi,h,j ni,j (t + 1) = ni,j (t) + h=1,h =j
for i = 1, . . . , Kb and j = 1, . . . , Ks , where ni,j (t) (person) is the accumulation in bus i with destination stop j, in qi,h,j (t) (person) is the number of people with destination stop j that board bus i at stop h inside a sampling period out (t) (person) is the number (i.e., in T seconds), whereas qi,j of people that alight from bus i at stop j inside a sampling period. (c) Accumulation at a stop evolves according to the following equation mh,j (t + 1) = mh,j (t) + T βh,j (t) −
Kb
in qi,h,j (t),
(3)
i=1
for h, j = 1, . . . , Ks , h = j, where mh,j (t) (person) is the accumulation at stop h with destination stop j, βh,j (t) (person/s) is the rate of passengers arriving at stop h
7771
Proceedings of the 20th IFAC World Congress I.I. Sirmatel et al. / IFAC PapersOnLine 50-1 (2017) 7499–7504 Toulouse, France, July 9-14, 2017
with destination stop j (i.e., the time varying origindestination passenger flow demand from stop h to stop j). With each time step the accumulation at stop h will increase with T βh,j (t), whereas if there are bus(es) holding Kb in at stop h (i.e., if i=1 qi,h,j (t) is nonzero) it will decrease as passengers board the bus(es). Dynamics of Binary States The binary states γi,j (t) (cruising state) and δi,j (t) (holding state) keep track of information about buses that cannot be expressed as continuous variables. The reason is that, the information they contain can be interpreted as an answer to a yes or no question, and its answer thus can only be stored as a binary variable. (a) Dynamics of γi,j (t) are γi,j (t + 1) = δi,j−1 (t) ∧ em j−1 (t)∨ eci (t) ∧ eni,j−1 (t) ∨ γi,j (t) ∧ ¬exi,j (t) , (4) for i = 1, . . . , Kb , j = 1, . . . , Ks , where the binary event em j (t) is 1 if there are no passengers at stop j and 0 otherwise, eci (t) is 1 if bus i is full and 0 otherwise, eni,j (t) is 1 if there are no passengers on bus i wanting to alight at stop j and 0 otherwise, exi,j (t) is 1 if bus i has reached stop j and 0 otherwise. Equation (4) states that bus i will start cruising to stop j if it is holding at stop j − 1 (δi,j−1 (t) = 1), and there are no passengers wanting to alight at stop j − 1 (eni,j−1 (t) = 1), and either there are no passengers at stop j − 1 wanting to board (em j (t) = 1) or the bus has no more vacant places (eci (t) = 1). It continues to cruise if it is cruising (γi,j (t) = 1) and it has not yet reached stop j (exi,j (t) = 0). With this formulation, it is possible to define skipping a stop as a control input, and let the controller force a bus to skip a stop if there are no passengers wanting to alight. (b) Dynamics of δi,j (t) are x δi,j (t + 1) = γ i,j (t) ∧ ei,j (t) ∨ δi,j (t)∧ c n ¬ em j (t) ∨ ei (t) ∧ ei,j (t) ,
(5)
for i = 1, . . . , Kb , j = 1, . . . , Ks . Equation (5) states that bus i will start holding at stop j if it is cruising to stop j (γi,j (t) = 1) and it reaches stop j (exi,j (t) = 1). It continues to hold if it is holding (δi,j (t) = 1), and there are passengers on-board wanting to alight (eni,j (t) = 0), or there are passengers at the stop wanting to board c (em j (t) = 0) and the bus has vacant places (ei (t) = 0). Constraints Defining Binary Events The binary events are defined such that their value depends on the value of a continuous variable and a fixed threshold: (a) The event exi,j (t) expresses whether bus i has passed stop j or not, and is defined as follows: 0 if xi (t) ≤ Dj x (6) ei,j (t) = 1 otherwise for i = 1, . . . , Kb , j = 1, . . . , Ks , where Dj is the circular distance of stop j from stop 1 (with D1 specially defined as the length of the whole loop). (b) The event em h (t), storing information on whether there is at least 1 passenger waiting at stop h or not, is defined via the following constraints:
em h (t) =
0
1
if 0 <
Ks
7501
mh,j (t)
j=1
for h = 1, . . . , Ks . (7)
otherwise
(c) The event eci (t) stores the information on whether bus i is full of passengers or not, and is as follows: Ks 0 if n (t) < n i,j max c for i = 1, . . . , Kb , ei (t) = j=1 1 otherwise (8) where nmax is the bus passenger capacity. (d) The event eni,j (t), storing information on whether there are passengers on bus i that want to alight at stop j or not, is defined via the following constraints: 0 if 0 < ni,j (t) eni,j (t) = (9) 1 otherwise, for i = 1, . . . , Kb , j = 1, . . . , Ks . Constraints Defining Passenger Flows The auxiliary in out variables qi,h,j (t) and qi,j (t) expressing the flow of passengers between the buses and stops need to be properly constrained. (a) Total number of passengers boarding bus i at stop h cannot exceed the physical passenger transfer limit αT and is only allowed to take positive values when bus i is holding at stop h, which can be expressed via the following constraint: Ks in qi,h,j (t) ≤ δi,h (t)αT, (10) j=1
for i = 1, . . . , Kb , h = 1, . . . , Ks , where α (person/s) is the passenger flow parameter between stops and buses, expressing the maximum number of passengers that can alight from or board a bus in one second.
(b) Total number of passengers transferring from stop h to bus i cannot exceed the number of vacant places on bus i, which can be formulated via the following constraint: Ks Ks in qi,h,j (t) ≤ nmax − ni,j (t), (11) j=1
j=1
for i = 1, . . . , Kb , h = 1, . . . , Ks .
(c) Total number of passengers transferring from stop h to the bus(es) holding at stop h with destination stop j cannot exceed the number of passengers wanting to board at stop h with destination stop j, which we express with the following constraint: Kb in (12) qi,h,j (t) ≤ mh,j (t), i=1
for h, j = 1, . . . , Ks , h = j.
(d) Total number of passengers alighting from bus i at stop j cannot exceed the physical passenger transfer limit αT and is only allowed to take positive values when bus i is holding at stop j, which can be expressed via the following constraint: out qi,j (t) ≤ δi,j (t)αT (13) for i = 1, . . . , Kb , j = 1, . . . , Ks .
7772
Proceedings of the 20th IFAC World Congress 7502 I.I. Sirmatel et al. / IFAC PapersOnLine 50-1 (2017) 7499–7504 Toulouse, France, July 9-14, 2017
(e) Number of passengers alighting from bus i at stop j cannot exceed the number of passengers on bus i wanting to alight at stop j, which can be formulated via the following constraint: out qi,j (t) ≤ ni,j (t) (14) for i = 1, . . . , Kb , j = 1, . . . , Ks . 2.2 Simulation of Bus Transport System Using the MLD model defined via equations (1)-(5) and the events (6)-(9), it is possible to simulate the behavior of a single loop BTS. The bus speeds vi (t), passenger flow demands βi,j (t), and the initial values of the continuous and binary states are inputs to the simulator. The binary events depend on states and model parameters; thus they can simply be evaluated by considering the corresponding states and the threshold values (e.g., at time t, if ni,j (t) has a nonzero value, then eni,j (t) is set to 0, otherwise it is set to 1). The passenger flows, on the other hand, depend on the states and model parameters in a more complicated way, which can be formalized via the following linear program (LP): K Kb Ks s in out maximize qi,h,j (t) + qi,j (t) in out qi,h,j , qi,j
i=1 j=1
h=1
subject to constraints (10)–(14) in (t), 0 ≤ qi,h,j for i = 1, . . . , Kb , h, j = 1, . . . , Ks , h = j out 0 ≤ qi,j (t), for i = 1, . . . , Kb , j = 1, . . . , Ks , (15) where, at time t, the states δi,h (t), ni,j (t), and mh,j (t) are given constants. The physical interpretation of the LP in (15) is that the passenger flows will take their maximum possible values, which are constrained by maximum flow rates, bus capacity constraints, number of people present in buses and at stops, and, by the presence of buses at stops (as modeled via the binary state δi,h (t)). The procedure for simulating the BTS is given in algorithm (1). Algorithm 1 Simulation algorithm for mixed logical bus transport system dynamics 1) At time step t, given the previous states xi (t), ni,j (t), c mh,j (t), and δi,j (t), evaluate exi,j (t), em j (t), ei (t), and n in out ei,j (t), and qi,h,j (t) and qi,j (t) by solving the optimization problem (15) (or, via heuristics); 2) Given the values obtained at step 1, the previous states xi (t), ni,j (t), mh,j (t), δi,j (t), and γi,j (t), and the exogenous inputs vi (t) and βh,j (t), evolve system dynamics by evaluating the difference equations (1)(5) to obtain the updated states xi (t + 1), ni,j (t + 1), mh,j (t + 1), δi,j (t + 1), and γi,j (t + 1). 3) Repeat steps 1 and 2 for t = 1, . . . , tfinal .
3. CONTROL OF BUS TRANSPORT SYSTEMS 3.1 PD-like Bus Speed Controller We propose here a simple PD-like control law for regularizing headways. It can be assumed that the ideal position
for a bus is the middle point between the bus ahead of it and the bus behind it (i.e., the situation where the front and rear headways are equal). Then, the position error can defined as the difference between the ideal and current positions: exi (t) = 0.5(xai (t) + xbi (t)) − xi (t), (16) where exi (t) is the position error of bus i at time t, whereas xai (t) and xbi (t) are the positions of buses ahead of and behind bus i at time t, respectively. Furthermore, speed error can be defined as the difference between the maximum bus speed (which is dependent on traffic conditions and needs to be measured/estimated in realtime) and current speed as follows: (17) evi (t) = v˜i (t) − vi (t), where evi (t) is the speed error, whereas v˜i (t) and vi (t) are the maximum and actual speeds of bus i at time t, respectively. The PD-like bus speed controller (PD-BSC) then updates the speed of bus i based on its position and speed errors: (18) vi (t) = vi (t − 1) + K1 exi (t) + K2 evi (t − 1), where K1 and K2 are controller gains. For K2 = 0, the PD-BSC is similar to the controller developed by Daganzo and Pilachowski (2011). 3.2 Linear Model Predictive Control We formulate the problem of finding the bus speed values that regularize the headways and lead to fast BTS operation as the following discrete time finite horizon constrained optimal control problem: Kb N −1 2 2 f r xi,k − xi,k + σ (v i,k − vi,k ) minimize vi,k
subject to
i=1 k=0
for i = 1, . . . , Kb : xfi,1 = xfi (t) xri,1 = xri (t) for k = 1, . . . , N : xfi,k+1 = xfi,k + T (vi+1,k − vi,k ) xri,k+1 = xri,k + T (vi,k − vi−1,k ) 0 ≤ vi,k ≤ v i,k ,
(19) where the subscript i, k denotes a variable for bus i at prediction step k, xfi,k and xri,k are the predicted front and rear headways, xfi (t) and xri (t) are their measured values at time t, v i,k and vi,k are the maximum and actual speeds, vi+1,k and vi−1,k are the speeds of buses ahead of and behind bus i (the ordering of buses are assumed to remain the same for the prediction horizon), σ is the weighting factor for the fast operation term, N is the prediction horizon, whereas T is the sampling period. The LMPC scheme expressed via (19) obtains information related to the traffic conditions, dwell times (i.e., the time spent by a bus at a stop for allowing passenger flows), and the times for arriving at the upcoming stops in the form of bound on speed of bus i, namely v i,k , defined as: v˜i (t), if k ≤ cat (t) (20) v i,k = 0, if cat (t) < k ≤ cat (t) + cdt (t) v˜i (t), otherwise, for k = 0, . . . , N − 1 and i = 1, . . . , Kb , where v˜i (t) is the measurement on maximum speed for bus i
7773
Proceedings of the 20th IFAC World Congress I.I. Sirmatel et al. / IFAC PapersOnLine 50-1 (2017) 7499–7504 Toulouse, France, July 9-14, 2017
(which depends on the traffic congestion around the bus) at time t, cat (t) and cdt (t) are the estimated arrival time to, and the estimated dwell time at, the upcoming stop, respectively, for bus i at time t, which can be obtained using the bus speeds vector from the previous times step and measurements on number of passengers at the stops and inside the bus.
defined above (in contrast to the case with the LMPC, where the arrival time cat (t) is a measurement): Kb N −1 2 2 f r minimize xi,k − xi,k + σ (v i,k − vi,k ) vi,k
subject to
The problem (19) is a convex quadratic program (QP), thus well-suited for real-time control purposes, since powerful software packages are available for solving convex optimization problems in a fast and reliable manner. 3.3 Hybrid Model Predictive Control The LMPC scheme given in (19) considers the arrival time to the upcoming stop as a measurement, which is an approximation since in reality it is a function of the control input (i.e., the bus speeds). To capture the dynamics of the bus traveling to the upcoming stop, hold for the dwell time, and leave, we develop a simple hybrid prediction model. The event related to a bus reaching the upcoming stop is defined as follows: xui,k ≤ Mx 1 − exi,k (21) xui,k ≥ ε + (mx − ε)) exi,k ,
where xui,k is the predicted distance of bus i to the upcoming stop, Mx and mx are the upper and lower bounds of xui,k , respectively, whereas exi,k is the binary variable expressing whether bus i has reached the upcoming stop at prediction step k or not.
Furthermore, we define the event related to the dwell time of a bus passing the estimated dwell time as follows: cdt (t) − ci,k ≤ Mc 1 − edi,k (22) cdt (t) − ci,k ≥ ε + (mc − ε)) edi,k , where Mc and mc are upper and lower bounds on the cdt (t) − ci,k term, respectively, edi,k is the binary variable expressing whether the dwell time of bus i has passed the estimated dwell time cdt (t) at prediction step k or not, whereas via ci,k a discrete-time clock is defined for bus i, the dynamics of which we express as follows: (23) ci,k+1 = ci,k + exi,k ,
where ci,k expresses the time (in prediction steps) that has passed since the bus has arrived at the upcoming stop. Considering the events expressed with the binary variables exi,k and edi,k together, bus i will be required to hold at the upcoming stop if it has reached the stop (i.e., if exi,k = 1) and if its dwell time has not yet reached the estimated dwell time (i.e., if edi,k = 0). We can express this with a third event ehi,k related to whether the bus should be holding at the upcoming stop or not, defined as follows: ehi,k <= exi,k ehi,k <= 1 − edi,k exi,k − edi,k <= ehi,k .
7503
i=1 k=0
for i = 1, . . . , Kb : xfi,1 = xfi (t) xri,1 = xri (t) xui,1 = xui (t) ci,k = 0 for k = 1, . . . , N : xfi,k+1 = xfi,k + T (vi+1,k − vi,k ) xri,k+1 = xri,k + T (vi,k − vi−1,k ) xui,k+1 = xui,k − T vi,k ci,k+1 = ci,k + exi,k 0 ≤ vi,k ≤ v i,k (1 − ehi,k ) constraints (21), (22), and (24),
(25) where xui (t) is the measured distance of bus i to the upcoming stop at time t. The problem (25) is a mixed-integer quadratic program (MIQP), which, although nonconvex, can be solved via software packages specially developed for mixed-integer programs in an efficient manner. 4. SIMULATION RESULTS The HMPC controller is compared with the no control (NC) case, where bus speeds are fixed to their maximum values v˜i (t), together with the LMPC and PD-BSC, using the algorithm (1) for simulating BTS. The setting is line 2 of the bus network of the city of Fribourg, which has 9 buses and 44 stops and a length of 15 km. Passenger demands (i.e., βh,j (t)) and maximum bus speeds (i.e., v˜i (t)) are extracted from real bus data. Simulation covers a time period of a single day of BTS operation (from 06:00 to 20:00) and the sampling period is 10 s, resulting in a simulation length of tfinal = 5040. The prediction horizon of the MPC schemes is chosen as 10. Through trial-anderror, a weighting factor of σ = 104 and controller gains of K1 = 0.01 and K2 = 0.25 are found to yield a good balance between headway regularization and fast operation. Bus positions on the loop as a function of simulation time and headway distributions are shown in Fig. 1, whereas results are summarized in table 1, showing mean time spent per passenger (TSPP; time spent by a passenger for waiting at a stop and traveling on the bus), mean speed of buses, standard deviation of headways (std. of hws.; a metric of headway irregularity), and mean/maximum CPU times (which are obtained by calling CPLEX from the YALMIP toolbox by L¨ofberg (2004), in MATLAB 8.5.0 Table 1. Control Performance Evaluation
(24)
We formulate the HMPC problem, which builds on the basis of the LMPC problem (19), but also includes the process of buses reaching the upcoming stop as a function of control inputs in the prediction model via the events 7774
Control scheme
mean TSPP (min)
mean speed (km/h)
std. of hws. (min)
mean/max. CPU time (s)
NC PD-BSC LMPC HMPC
25.7 31.2 24.6 18.6
25.7 17.6 19.3 18.7
13.3 0.95 1.02 1.03
– – 0.16/0.21 0.47/0.68
bus positions (NC) frequency
15 10 5 0 7:00
7:30
8:00
8:30
9:00
7:30
8:00
8:30
frequency mean
2000 0
0
9:00
1000 0
0
8:00
8:30
7:30
8:00
8:30
9:00
9:00
simulation time (hh:mm)
20
2000 1000 0
0
simulation time (hh:mm) bus positions (HMPC)
15 10 5 0 7:00
10
time headway (minutes) headway dist. (LMPC) frequency
7:30
20
2000
simulation time (hh:mm) bus positions (LMPC)
15 10 5 0 7:00
10
time headway (minutes) headway dist. (PD-BSC) frequency
15 10 5 0 7:00
headway dist. (NC)
4000
simulation time (hh:mm) bus positions (PD-BSC)
10
20
time headway (minutes) headway dist. (HMPC) frequency
distance (km)
distance (km)
distance (km)
distance (km)
Proceedings of the 20th IFAC World Congress 7504 I.I. Sirmatel et al. / IFAC PapersOnLine 50-1 (2017) 7499–7504 Toulouse, France, July 9-14, 2017
2000 1000 0
0
10
20
time headway (minutes)
Fig. 1. Time-space diagrams (left column; each color is for one bus) and headway distributions (right column) for the NC case and the three controllers. (R2015a) on a 64-bit Windows PC with 3.6-GHz Intel Core i7 processor and 16-GB RAM). The results indicate that: (a) PD-BSC achieves good headway regularity, but lacking coordination between bus movements, it causes sustained variations in speed (see fig. 1) and a general slowing down of buses, leading to a larger TSPP, (b) LMPC is able to coordinate bus movements, and thus performs better than PD-BSC, (c) owing to a prediction model capturing simple hybrid dynamics, the proposed HMPC is superior to both PD-BSC and LMPC. Moreover, the CPU times of both MPC schemes are roughly negligible in comparison to the sampling period of 10 s, showing their computational tractability. 5. CONCLUSION We developed a dynamical model for single loop BTSs based on the mixed logical dynamical systems approach. The model is detailed enough to allow in-depth analysis and simulation of BTS operation, whereas also computationally lightweight, providing fast execution. Furthermore, we proposed a hybrid MPC scheme based on a simple hybrid BTS model for regularizing headways and improving BTS performance, and evaluated its performance via simulations with the proposed MLD model. Future work could include: (a) Comparison of the proposed schemes with other bus speed control methods from the literature, (b) consideration of passenger accumulation in MPC prediction model, (c) extension to multi-loop BTSs. REFERENCES Abkowitz, M.D. and Lepofsky, M. (1990). Implementing headway-based reliability control on transit routes.
Journal of Transportation Engineering, 116(1), 49–63. Ampountolas, K. and Kring, M. (2015). Mitigating bunching with bus-following models and bus-to-bus cooperation. In Intelligent Transportation Systems (ITSC), 2015 IEEE 18th International Conference on, 60–65. IEEE. Bemporad, A. and Morari, M. (1999). Control of systems integrating logic, dynamics, and constraints. Automatica, 35(3), 407–427. Chandrasekar, P., Long Cheu, R., and Chin, H.C. (2002). Simulation evaluation of route-based control of bus operations. Journal of transportation engineering, 128(6), 519–527. Cort´es, C.E., S´aez, D., Milla, F., N´ un ˜ez, A., and Riquelme, M. (2010). Hybrid predictive control for real-time optimization of public transport systems’ operations based on evolutionary multi-objective optimization. Transportation Research Part C: Emerging Technologies, 18(5), 757–769. Daganzo, C.F. (2009). A headway-based approach to eliminate bus bunching: Systematic analysis and comparisons. Transportation Research Part B: Methodological, 43(10), 913–921. Daganzo, C.F. and Pilachowski, J. (2011). Reducing bunching with bus-to-bus cooperation. Transportation Research Part B: Methodological, 45(1), 267–277. Delgado, F., Contreras, N., and Munoz, J.C. (2013). Holding for transfers. In Transportation Research Board 92nd Annual Meeting, 13-2913. Delgado, F., Munoz, J.C., and Giesen, R. (2012). How much can holding and/or limiting boarding improve transit performance? Transportation Research Part B: Methodological, 46(9), 1202–1217. Eberlein, X.J., Wilson, N.H., and Bernstein, D. (2001). The holding problem with real–time information available. Transportation science, 35(1), 1–18. Fu, L., Liu, Q., and Calamai, P. (2003). Real-time optimization model for dynamic scheduling of transit operations. Transportation Research Record: Journal of the Transportation Research Board, (1857), 48–55. Hall, R., Dessouky, M., and Lu, Q. (2001). Optimal holding times at transfer stations. Computers & industrial engineering, 40(4), 379–397. Ibarra-Rojas, O., Delgado, F., Giesen, R., and Mu˜ noz, J. (2015). Planning, operation, and control of bus transport systems: A literature review. Transportation Research Part B: Methodological, 77, 38–75. Liu, H., Skabardonis, A., and Zhang, W.b. (2003). A dynamic model for adaptive bus signal priority. In Preprint CD-ROM, 82nd Transportation Research Board Annual Meeting, Washington, DC. L¨ofberg, J. (2004). YALMIP: A toolbox for modeling and optimization in MATLAB. In IEEE International Symposium on Computer Aided Control Systems Design, 284–289. IEEE. Van Oort, N., Boterman, J., and Van Nes, R. (2012). The impact of scheduling on service reliability: triptime determination and holding points in long-headway services. Public Transport, 4(1), 39–56. Xuan, Y., Argote, J., and Daganzo, C.F. (2011). Dynamic bus holding strategies for schedule reliability: Optimal linear control and performance analysis. Transportation Research Part B: Methodological, 45(10), 1831–1845.
7775
ScienceDirect
IFAC PapersOnLine 50-1 (2017) 7499–7504 Dynamical Modeling and Predictive Dynamical Modeling and Predictive Control of Modeling Bus Transport Systems: Dynamical and Predictive Control of Bus Transport Systems: A Hybrid Systems Approach Control of Bus Transport Systems: A Hybrid Systems Approach A Hybrid Systems Approach ∗ ∗∗
I.I. Sirmatel N. Geroliminis ∗∗ I.I. Sirmatel ∗∗∗ N. Geroliminis ∗∗ I.I. Sirmatel N. Geroliminis ∗∗ ∗ Urban Transport Systems Laboratory, EPFL ∗ ∗ Urban Transport Systems Laboratory, EPFL (e-mail: [email protected]) ∗ Systems ∗∗Urban (e-mail: [email protected]) Urban Transport Transport Systems Laboratory, Laboratory, EPFL EPFL ∗∗ [email protected]) ∗∗ Urban(e-mail: Transport Systems Laboratory, EPFL (e-mail: [email protected]) ∗∗ Urban Transport Systems Laboratory, EPFL (e-mail: [email protected]) (e-mail: [email protected]) Abstract: Bus operations, due to their unstable nature, are inefficient when left uncontrolled Abstract: due to their unstable nature, inefficient when with respectBus to operations, retaining headways. Irregularities such asarebus bunching leadleft to uncontrolled loss of time Abstract: Bus operations, due to their unstable nature, are inefficient when left with respect to retaining headways. Irregularities such as bus bunching lead to uncontrolled loss of time and decrease bus service quality. Development of bus transport system management schemes to with respect to retaining headways. Irregularities such as importance, bussystem bunching loss of time and decrease bus service Development ofare bus transport management schemes to avoid bus bunching and quality. improve performance of high andlead hastothus been the and bus service quality. Development ofare bus system management schemes to avoiddecrease busmany bunching and improve performance of transport high importance, and by hasthe thus been the focus of works in the public transport systems literature. Motivated importance avoid bus bunching and improve performance are of high importance, and has thus been the focus of many works in the public transport systems literature. Motivated by the importance of developing bus control strategies for improving performance, and specifically by the lack of of many works in the public transport systems literature. Motivated the of detailed developing control strategies for improving performance, and specifically by importance the system lack of afocus butbus computationally efficient mathematical model describing busbytransport of developing bus control strategies for improving performance, and specifically by lack of a detailed butthe computationally efficient mathematical model describing bus transport system dynamics in literature (which can facilitate model-based control design), we the propose a a detailed but computationally efficient mathematical model describing bus transport system dynamics in the literaturemodel (which facilitate propose mixed logical dynamical of can a single loop model-based bus transportcontrol system,design), which we involves botha dynamics in(e.g., the literature (which facilitate design), propose a mixed logical dynamical model of can a binary single loop bus which we involves both continuous bus positions) and (e.g., model-based the transport state of control a system, bus regarding whether it is mixed logical dynamical model of a single loop bus transport system, which involves both continuous (e.g., bus positions) and binary (e.g., the state of a bus regarding whether it is holding at a certain stop or not) states. Furthermore, we develop a hybrid model predictive continuous busactuation positions) binary (e.g., the of a bus regarding itbus is holding scheme at a(e.g., certain stop or not) states. Furthermore, we develop a hybrid model predictive control with viaand bus speeds, which canstate regularize headways andwhether improve holding at a certain stop or not) states. Furthermore, we develop a hybrid model predictive control quality. scheme Performance with actuation speeds,controller which canis regularize and improve bus service of via the bus predictive evaluated headways via simulation experiments control scheme withmodel, actuation bus speeds,controller which canis regularize headways andare improve bus servicethe quality. Performance of via the predictive evaluated viabus simulation experiments using proposed where the passenger demands and maximum speeds extracted service quality. Performance of the predictive controller is evaluated via simulation experiments using the proposed model, where the passenger demands and maximum bus speeds are extracted from data collected from the bus network of the city of Fribourg. Results indicate the potential using the proposed model, where the passenger and maximum busindicate speeds are extracted from collected from the network thedemands city and of Fribourg. Results thetimes. potential of thedata proposed controller in bus avoiding bus of bunching decreasing passenger travel from data collected from the bus network of the city of Fribourg. Results indicate the potential of the proposed controller in avoiding bus bunching and decreasing passenger travel times. of the proposed controller Federation in avoiding bus bunching andHosting decreasing passenger travel times. © 2017, IFAC (International of Automatic Control) by Elsevier Ltd. All rights reserved. Keywords: Automatic control, optimization, real-time operations in transportation; Modeling Keywords: Automatic control, optimization, real-timetransportation operations in transportation; and simulation of transportation systems; Intelligent systems; Model Modeling predictive Keywords: Automatic control, optimization, real-timetransportation operations in transportation; and simulation of systems. transportation systems; Intelligent systems; Model Modeling predictive control of hybrid and simulation of systems. transportation systems; Intelligent transportation systems; Model predictive control of hybrid control of hybrid systems. 1. INTRODUCTION Station control methods include those that involve taking 1. INTRODUCTION Station control methods include those involve control decisions at a subset of stops of that the bus loop.taking Some 1. INTRODUCTION Station control methods include those involve taking control decisions at a subset of stops of that the bus loop. Some methods on this direction focus on regularizing headways It is well known in the public transport systems literature control decisions at a subset of stops of the bus loop. Some methods on this direction focus on regularizing headways via holding, with the assumption that this would lead to It is well known in the transport literature that bus operations arepublic inefficient when systems left without man- methods on this direction focus on regularizing headways It is well known in the public transport systems literature via holding, with the assumption that this would lead to efficient operation and decreased travel times (Abkowitz that bus operations are inefficient when left without agement (Daganzo and Pilachowski (2011)). Buses manthat via holding, with the assumption that this would lead to efficient operation and decreased travel times (Abkowitz thatbehind bus operations aremore inefficient when left without manLepofsky (1990); Daganzo (2009); Xuan et al. (2011)). agement (Daganzo and Pilachowski (2011)). Buses that and lag encounter passengers waiting for them, efficient operation and decreased travel times (Abkowitz and Lepofsky (1990); Daganzo (2009); Xuan et al. (2011)). agement (Daganzo and Pilachowski (2011)). Buses that situations where there is high variability in the delag behind encounter more passengers forslightly them, In leading to them lagging more, and buseswaiting that are andsituations Lepofsky (1990); Daganzo (2009); Xuan et al. (2011)). lag behind encounter more passengers waiting for them, In where there is high variability in demands, passenger waiting times need also to be considered leading to themless lagging more, This and buses that are slightly fast encounter passengers. positive feedback loop In situations where there is high variability in the the deleading to them lagging more, and buses that are slightly mands, passenger waiting times need also to be considered in the problem formulation alongside headways (Ibarrafast less passengers. This positive leadsencounter to the undesirable phenomenon thatfeedback is knownloop as mands, passenger waiting times need also to be considered in the et problem formulation alongside fast encounter less passengers. positive feedback loop al. (2015)). Studies on this headways direction (Ibarrainclude leads to the undesirable phenomenon that is known as Rojas bus bunching. Instabilities in This the bus transport system in the problem formulation alongside headways (IbarraRojas et al. (2015)). Studies on this include leads to the undesirable phenomenon that is known as et al. (2001) and Delgado et al.direction (2012). Holding bus bunching. Instabilities in thespatiotemporal bus transportvariabilsystem Eberlein (BTS) operation, resulting from Rojas etbe al.used (2015)). Studies on this direction include bus bunching. Instabilities in the bus transport system Eberlein et al. (2001) and Delgado et al. (2012). Holding can also to improve timing of passenger transfers (BTS) operation, from variabil- Eberlein et al. (2001) and Delgado et al. (2012). Holding ity of both traffic resulting congestion andspatiotemporal stop-to-stop passenger (BTS) operation, resulting from variabilbe(2001); used toDelgado improveettiming of passenger (Hallalso et al. al. (2013)). Anothertransfers subclass ity of both traffic congestion andspatiotemporal stop-to-stop passenger demands, and manifesting themselves as irregularity in can canstation also becontrol used to improveet timing of passenger transfers (Hall et al. (2001); Delgado al. (2013)). Another subclass ity of both traffic congestion and stop-to-stop passenger of methods is the stop-skipping strategies, demands, and manifesting themselves as irregularity in headways ultimately as bus bunching, lead to inef- (Hall et al. (2001); Delgado et al. (2013)). Another subclass demands, and manifesting themselves as irregularity in of station control methods is the stop-skipping strategies, the decisions are realized by forcing buses to headways and ultimately as bus leadofto inef- where ficient operations, loss of time, andbunching, degradation service of station control methods the stop-skipping strategies, headways andtoultimately as bus bunching, leadofto inefthe control decisions are realized forcing buses(Fu to skip some stops, to increaseis speed and by thus efficiency ficient operations, loss reasons, of time, and degradation service quality. Due these research on modeling and where where the control decisions are realized by forcing buses to ficient operations, loss of time, and degradation of service skip some stops, to increase speed and thus efficiency (Fu et al. (2003); Cort´ e s et al. (2010); Delgado et al. (2012)). quality. Due to these reasons, research on modeling and control of BTSs is of high importance. skip some stops, to increase speed and thus efficiency (Fu et al. (2003); Cort´ e s et al. (2010); Delgado et al. (2012)). quality. Due to these reasons, research on modeling and Although station control strategies can be effective in control of BTSs is of high importance. et al. (2003); Cort´econtrol s etinal. (2010); Delgado et effective al. (2012)). Considerable research hasimportance. been directed, especially in Although control of BTSs is of high station strategies can be in regularizing headways moderate demand situations and Although station control strategies can be effective in Considerable research has been directed, especially in the last 4 decades, to developing real-time bus control regularizing headways in moderatethey demand situations and leading to improved performance, have adverse affects Considerable research has been directed, especially in regularizing headways in moderate demand situations and the last 4 decades, to developing real-time bus control methods, with the goal of avoiding bus bunching and leading improved performance, they via haveholding adversethe affects on BTStoperformance as they actuate bus the last 4efficient decades, to reliable developing real-time bus control leading toperformance improved performance, theyavia have adverse affects methods, with theand goal of avoiding busofbunching and at on aBTS as they holding the bus ensuring operation BTSs. Realstop or making the busactuate skip stop. Under some methods, with theand goal of avoiding buscan and on BTS performance as they actuate via holding the bus ensuring efficient reliable operation ofbunching BTSs. Realtime control methods for bus operations be classified at a stop or making themake bus buses skip asignificantly stop. Underslower, some circumstances they can ensuring efficient and1) reliable operation of BTSs. Real- at a stop or making the bus skip a stop. Under some time two control methods for bus operations be classified into categories: Station control, can 2) inter-station circumstances they can make buses significantly slower, which will influence the quality of in-vehicle service, but time control methods for bus operations can be classified circumstances they can make significantly slower, into two categories: 1) Station control, 2) inter-station control (see Ibarra-Rojas et al. (2015) for an extensive which will influence the quality of in-vehicle service, but also increase the operating costbuses and required fleet size. into twoa(see categories: Station control, 2) section). inter-station will influence the quality of in-vehicle service, control Ibarra-Rojas et isal. (2015) for an extensive which also increase the operating cost and required fleet size.but review, summary of1) which given in this control (see Ibarra-Rojas et al. (2015) for an extensive also increase the operating cost and required fleet size. review, a summary of which is given in this section). review, a summary of which is given in this section).
Copyright © 2017 IFAC 7770 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2017, 2017 IFAC 7770Hosting by Elsevier Ltd. All rights reserved. Peer review under responsibility of International Federation of Automatic Copyright © 2017 IFAC 7770Control. 10.1016/j.ifacol.2017.08.1067
Proceedings of the 20th IFAC World Congress 7500 I.I. Sirmatel et al. / IFAC PapersOnLine 50-1 (2017) 7499–7504 Toulouse, France, July 9-14, 2017
Another category of real-time bus control methods is the inter-station control, where decisions are taken while the bus is moving between stops. Traffic signal priority methods belong to this category, where the aim is to manage traffic flow efficiently via prioritizing certain circulations of an intersection with actuation over traffic lights (Liu et al. (2003); Van Oort et al. (2012)). Methods based on the control of bus speed constitute another subclass of the inter-station control strategies. The idea in bus speed control is to continuously manipulate the speed of each bus during its movement via feedback control mechanisms to avoid bunching and increase the efficiency of the BTS. On this direction, a control strategy combining bus speed control and signal priority is developed by Chandrasekar et al. (2002), where control actions are taken to ensure that the buses operate with headways equal to a desired headway, which is shown to be able to regularize headways. A bus speed control method is proposed by Daganzo and Pilachowski (2011), where the controller manipulates the speed of each bus according to its front and rear headways, while at the same time respecting bus speed bounds, and is able to prevent bus bunching. A more recent study by Ampountolas and Kring (2015) develops a combined state estimation and LQR control scheme to achieve coordination between the buses, leading to higher headway regularity and improved service. Considering future information about number of passengers waiting to board at each stop and the number of passengers within each bus can significantly improve the performance under congested conditions and decrease the occurrence of bus bunching, but also decrease the number of times passengers are unable to board a bus due to full buses. Motivated by the goal of developing a detailed mathematical model describing the dynamics of a single-loop BTS, in this paper we propose a novel mixed logical dynamical (MLD) model that considers the interaction between bus motion and passenger accumulation. The MLD modeling framework is proposed by Bemporad and Morari (1999) as a systematic approach to the mathematical modeling of dynamical systems which involve the interaction of physical laws, logic rules, and constraints. The proposed MLD BTS model is detailed enough to allow in-depth simulation-based analysis of BTSs, but at the same time computationally lightweight to allow for fast execution. Furthermore, the proposed model enables development of hybrid model predictive control (MPC) schemes, since prediction models based on the proposed MLD model enable the formulation of MPC problems in the form of mixedinteger programs. In the control design aspect, the paper contributes by proposing a hybrid MPC (HMPC) scheme (based on a simplified MLD model), with the goal of regularizing headways and improving BTS performance. A simple PD-like controller and a linear MPC (LMPC) scheme are also developed for comparison purposes. A case study is provided, which uses line 2 of the bus network of the Swiss city of Fribourg. The study is based on simulations with the proposed MLD model, with the information on origin-destination passenger flow demands and maximum bus speeds extracted from bus data collected from the real bus network. Results indicate that control is required for headway regularization. Furthermore, the proposed hybrid MPC scheme shows substantial potential in regularizing headways and improving BTS efficiency.
2. HYBRID MODELING AND SIMULATION OF BUS TRANSPORT SYSTEMS 2.1 Bus Loop as a Hybrid System We consider a single loop BTS with Kb buses and Ks stops, and assume that (i) buses operate on the loop with always positive speed (i.e., they never change direction), (ii) a bus always holds at a stop if there are passengers onboard that want to alight at that stop, (iii) passengers do not differentiate between buses when boarding, since any bus they board will hold at their destination stop, (iv) the position of the first stop of the loop is assumed to be 0 and the bus positions are reset to 0 when they complete the loop and reach the first stop. Dynamics of Continuous States The continuous states xi (t), ni,j (t), and mh,j (t) keep track of how bus positions, passenger accumulations inside buses, and those at stops, evolve over time. (a) Dynamics of bus position can be expressed as follows xi (t + 1) =
Ks
γi,j (t) (xi (t) j=1 Ks
+
j=2
+ T vi (t)) (1)
δi,j (t)xi (t) + δi,1 (t) · 0,
for i = 1, . . . , Kb , where t (–) is the time step counter, T (s) is the sampling time, xi (t) (m) and vi (t) (m/s) are the position and speed of bus i, respectively, γi,j (t) is a binary state that is equal to 1 if bus i is cruising towards stop j and 0 otherwise, and δi,j (t) is a binary state that is equal Ks to 1 if bus i is holding at stop j. The term j=1 γi,j (t) is equal to 1 if bus i is cruising, thus its position will increase proportional to its speed. If it is holding at a stop Ks other than stop 1, i.e., if j=2 δi,j (t) is equal to 1, its position will stay constant. If it is holding at stop 1 (i.e., if δi,1 (t) = 1) this means that it has completed the loop and its position will be reset to 0. (b) Dynamics of bus accumulation can be written as follows Ks out in (t) − qi,j (t), (2) qi,h,j ni,j (t + 1) = ni,j (t) + h=1,h =j
for i = 1, . . . , Kb and j = 1, . . . , Ks , where ni,j (t) (person) is the accumulation in bus i with destination stop j, in qi,h,j (t) (person) is the number of people with destination stop j that board bus i at stop h inside a sampling period out (t) (person) is the number (i.e., in T seconds), whereas qi,j of people that alight from bus i at stop j inside a sampling period. (c) Accumulation at a stop evolves according to the following equation mh,j (t + 1) = mh,j (t) + T βh,j (t) −
Kb
in qi,h,j (t),
(3)
i=1
for h, j = 1, . . . , Ks , h = j, where mh,j (t) (person) is the accumulation at stop h with destination stop j, βh,j (t) (person/s) is the rate of passengers arriving at stop h
7771
Proceedings of the 20th IFAC World Congress I.I. Sirmatel et al. / IFAC PapersOnLine 50-1 (2017) 7499–7504 Toulouse, France, July 9-14, 2017
with destination stop j (i.e., the time varying origindestination passenger flow demand from stop h to stop j). With each time step the accumulation at stop h will increase with T βh,j (t), whereas if there are bus(es) holding Kb in at stop h (i.e., if i=1 qi,h,j (t) is nonzero) it will decrease as passengers board the bus(es). Dynamics of Binary States The binary states γi,j (t) (cruising state) and δi,j (t) (holding state) keep track of information about buses that cannot be expressed as continuous variables. The reason is that, the information they contain can be interpreted as an answer to a yes or no question, and its answer thus can only be stored as a binary variable. (a) Dynamics of γi,j (t) are γi,j (t + 1) = δi,j−1 (t) ∧ em j−1 (t)∨ eci (t) ∧ eni,j−1 (t) ∨ γi,j (t) ∧ ¬exi,j (t) , (4) for i = 1, . . . , Kb , j = 1, . . . , Ks , where the binary event em j (t) is 1 if there are no passengers at stop j and 0 otherwise, eci (t) is 1 if bus i is full and 0 otherwise, eni,j (t) is 1 if there are no passengers on bus i wanting to alight at stop j and 0 otherwise, exi,j (t) is 1 if bus i has reached stop j and 0 otherwise. Equation (4) states that bus i will start cruising to stop j if it is holding at stop j − 1 (δi,j−1 (t) = 1), and there are no passengers wanting to alight at stop j − 1 (eni,j−1 (t) = 1), and either there are no passengers at stop j − 1 wanting to board (em j (t) = 1) or the bus has no more vacant places (eci (t) = 1). It continues to cruise if it is cruising (γi,j (t) = 1) and it has not yet reached stop j (exi,j (t) = 0). With this formulation, it is possible to define skipping a stop as a control input, and let the controller force a bus to skip a stop if there are no passengers wanting to alight. (b) Dynamics of δi,j (t) are x δi,j (t + 1) = γ i,j (t) ∧ ei,j (t) ∨ δi,j (t)∧ c n ¬ em j (t) ∨ ei (t) ∧ ei,j (t) ,
(5)
for i = 1, . . . , Kb , j = 1, . . . , Ks . Equation (5) states that bus i will start holding at stop j if it is cruising to stop j (γi,j (t) = 1) and it reaches stop j (exi,j (t) = 1). It continues to hold if it is holding (δi,j (t) = 1), and there are passengers on-board wanting to alight (eni,j (t) = 0), or there are passengers at the stop wanting to board c (em j (t) = 0) and the bus has vacant places (ei (t) = 0). Constraints Defining Binary Events The binary events are defined such that their value depends on the value of a continuous variable and a fixed threshold: (a) The event exi,j (t) expresses whether bus i has passed stop j or not, and is defined as follows: 0 if xi (t) ≤ Dj x (6) ei,j (t) = 1 otherwise for i = 1, . . . , Kb , j = 1, . . . , Ks , where Dj is the circular distance of stop j from stop 1 (with D1 specially defined as the length of the whole loop). (b) The event em h (t), storing information on whether there is at least 1 passenger waiting at stop h or not, is defined via the following constraints:
em h (t) =
0
1
if 0 <
Ks
7501
mh,j (t)
j=1
for h = 1, . . . , Ks . (7)
otherwise
(c) The event eci (t) stores the information on whether bus i is full of passengers or not, and is as follows: Ks 0 if n (t) < n i,j max c for i = 1, . . . , Kb , ei (t) = j=1 1 otherwise (8) where nmax is the bus passenger capacity. (d) The event eni,j (t), storing information on whether there are passengers on bus i that want to alight at stop j or not, is defined via the following constraints: 0 if 0 < ni,j (t) eni,j (t) = (9) 1 otherwise, for i = 1, . . . , Kb , j = 1, . . . , Ks . Constraints Defining Passenger Flows The auxiliary in out variables qi,h,j (t) and qi,j (t) expressing the flow of passengers between the buses and stops need to be properly constrained. (a) Total number of passengers boarding bus i at stop h cannot exceed the physical passenger transfer limit αT and is only allowed to take positive values when bus i is holding at stop h, which can be expressed via the following constraint: Ks in qi,h,j (t) ≤ δi,h (t)αT, (10) j=1
for i = 1, . . . , Kb , h = 1, . . . , Ks , where α (person/s) is the passenger flow parameter between stops and buses, expressing the maximum number of passengers that can alight from or board a bus in one second.
(b) Total number of passengers transferring from stop h to bus i cannot exceed the number of vacant places on bus i, which can be formulated via the following constraint: Ks Ks in qi,h,j (t) ≤ nmax − ni,j (t), (11) j=1
j=1
for i = 1, . . . , Kb , h = 1, . . . , Ks .
(c) Total number of passengers transferring from stop h to the bus(es) holding at stop h with destination stop j cannot exceed the number of passengers wanting to board at stop h with destination stop j, which we express with the following constraint: Kb in (12) qi,h,j (t) ≤ mh,j (t), i=1
for h, j = 1, . . . , Ks , h = j.
(d) Total number of passengers alighting from bus i at stop j cannot exceed the physical passenger transfer limit αT and is only allowed to take positive values when bus i is holding at stop j, which can be expressed via the following constraint: out qi,j (t) ≤ δi,j (t)αT (13) for i = 1, . . . , Kb , j = 1, . . . , Ks .
7772
Proceedings of the 20th IFAC World Congress 7502 I.I. Sirmatel et al. / IFAC PapersOnLine 50-1 (2017) 7499–7504 Toulouse, France, July 9-14, 2017
(e) Number of passengers alighting from bus i at stop j cannot exceed the number of passengers on bus i wanting to alight at stop j, which can be formulated via the following constraint: out qi,j (t) ≤ ni,j (t) (14) for i = 1, . . . , Kb , j = 1, . . . , Ks . 2.2 Simulation of Bus Transport System Using the MLD model defined via equations (1)-(5) and the events (6)-(9), it is possible to simulate the behavior of a single loop BTS. The bus speeds vi (t), passenger flow demands βi,j (t), and the initial values of the continuous and binary states are inputs to the simulator. The binary events depend on states and model parameters; thus they can simply be evaluated by considering the corresponding states and the threshold values (e.g., at time t, if ni,j (t) has a nonzero value, then eni,j (t) is set to 0, otherwise it is set to 1). The passenger flows, on the other hand, depend on the states and model parameters in a more complicated way, which can be formalized via the following linear program (LP): K Kb Ks s in out maximize qi,h,j (t) + qi,j (t) in out qi,h,j , qi,j
i=1 j=1
h=1
subject to constraints (10)–(14) in (t), 0 ≤ qi,h,j for i = 1, . . . , Kb , h, j = 1, . . . , Ks , h = j out 0 ≤ qi,j (t), for i = 1, . . . , Kb , j = 1, . . . , Ks , (15) where, at time t, the states δi,h (t), ni,j (t), and mh,j (t) are given constants. The physical interpretation of the LP in (15) is that the passenger flows will take their maximum possible values, which are constrained by maximum flow rates, bus capacity constraints, number of people present in buses and at stops, and, by the presence of buses at stops (as modeled via the binary state δi,h (t)). The procedure for simulating the BTS is given in algorithm (1). Algorithm 1 Simulation algorithm for mixed logical bus transport system dynamics 1) At time step t, given the previous states xi (t), ni,j (t), c mh,j (t), and δi,j (t), evaluate exi,j (t), em j (t), ei (t), and n in out ei,j (t), and qi,h,j (t) and qi,j (t) by solving the optimization problem (15) (or, via heuristics); 2) Given the values obtained at step 1, the previous states xi (t), ni,j (t), mh,j (t), δi,j (t), and γi,j (t), and the exogenous inputs vi (t) and βh,j (t), evolve system dynamics by evaluating the difference equations (1)(5) to obtain the updated states xi (t + 1), ni,j (t + 1), mh,j (t + 1), δi,j (t + 1), and γi,j (t + 1). 3) Repeat steps 1 and 2 for t = 1, . . . , tfinal .
3. CONTROL OF BUS TRANSPORT SYSTEMS 3.1 PD-like Bus Speed Controller We propose here a simple PD-like control law for regularizing headways. It can be assumed that the ideal position
for a bus is the middle point between the bus ahead of it and the bus behind it (i.e., the situation where the front and rear headways are equal). Then, the position error can defined as the difference between the ideal and current positions: exi (t) = 0.5(xai (t) + xbi (t)) − xi (t), (16) where exi (t) is the position error of bus i at time t, whereas xai (t) and xbi (t) are the positions of buses ahead of and behind bus i at time t, respectively. Furthermore, speed error can be defined as the difference between the maximum bus speed (which is dependent on traffic conditions and needs to be measured/estimated in realtime) and current speed as follows: (17) evi (t) = v˜i (t) − vi (t), where evi (t) is the speed error, whereas v˜i (t) and vi (t) are the maximum and actual speeds of bus i at time t, respectively. The PD-like bus speed controller (PD-BSC) then updates the speed of bus i based on its position and speed errors: (18) vi (t) = vi (t − 1) + K1 exi (t) + K2 evi (t − 1), where K1 and K2 are controller gains. For K2 = 0, the PD-BSC is similar to the controller developed by Daganzo and Pilachowski (2011). 3.2 Linear Model Predictive Control We formulate the problem of finding the bus speed values that regularize the headways and lead to fast BTS operation as the following discrete time finite horizon constrained optimal control problem: Kb N −1 2 2 f r xi,k − xi,k + σ (v i,k − vi,k ) minimize vi,k
subject to
i=1 k=0
for i = 1, . . . , Kb : xfi,1 = xfi (t) xri,1 = xri (t) for k = 1, . . . , N : xfi,k+1 = xfi,k + T (vi+1,k − vi,k ) xri,k+1 = xri,k + T (vi,k − vi−1,k ) 0 ≤ vi,k ≤ v i,k ,
(19) where the subscript i, k denotes a variable for bus i at prediction step k, xfi,k and xri,k are the predicted front and rear headways, xfi (t) and xri (t) are their measured values at time t, v i,k and vi,k are the maximum and actual speeds, vi+1,k and vi−1,k are the speeds of buses ahead of and behind bus i (the ordering of buses are assumed to remain the same for the prediction horizon), σ is the weighting factor for the fast operation term, N is the prediction horizon, whereas T is the sampling period. The LMPC scheme expressed via (19) obtains information related to the traffic conditions, dwell times (i.e., the time spent by a bus at a stop for allowing passenger flows), and the times for arriving at the upcoming stops in the form of bound on speed of bus i, namely v i,k , defined as: v˜i (t), if k ≤ cat (t) (20) v i,k = 0, if cat (t) < k ≤ cat (t) + cdt (t) v˜i (t), otherwise, for k = 0, . . . , N − 1 and i = 1, . . . , Kb , where v˜i (t) is the measurement on maximum speed for bus i
7773
Proceedings of the 20th IFAC World Congress I.I. Sirmatel et al. / IFAC PapersOnLine 50-1 (2017) 7499–7504 Toulouse, France, July 9-14, 2017
(which depends on the traffic congestion around the bus) at time t, cat (t) and cdt (t) are the estimated arrival time to, and the estimated dwell time at, the upcoming stop, respectively, for bus i at time t, which can be obtained using the bus speeds vector from the previous times step and measurements on number of passengers at the stops and inside the bus.
defined above (in contrast to the case with the LMPC, where the arrival time cat (t) is a measurement): Kb N −1 2 2 f r minimize xi,k − xi,k + σ (v i,k − vi,k ) vi,k
subject to
The problem (19) is a convex quadratic program (QP), thus well-suited for real-time control purposes, since powerful software packages are available for solving convex optimization problems in a fast and reliable manner. 3.3 Hybrid Model Predictive Control The LMPC scheme given in (19) considers the arrival time to the upcoming stop as a measurement, which is an approximation since in reality it is a function of the control input (i.e., the bus speeds). To capture the dynamics of the bus traveling to the upcoming stop, hold for the dwell time, and leave, we develop a simple hybrid prediction model. The event related to a bus reaching the upcoming stop is defined as follows: xui,k ≤ Mx 1 − exi,k (21) xui,k ≥ ε + (mx − ε)) exi,k ,
where xui,k is the predicted distance of bus i to the upcoming stop, Mx and mx are the upper and lower bounds of xui,k , respectively, whereas exi,k is the binary variable expressing whether bus i has reached the upcoming stop at prediction step k or not.
Furthermore, we define the event related to the dwell time of a bus passing the estimated dwell time as follows: cdt (t) − ci,k ≤ Mc 1 − edi,k (22) cdt (t) − ci,k ≥ ε + (mc − ε)) edi,k , where Mc and mc are upper and lower bounds on the cdt (t) − ci,k term, respectively, edi,k is the binary variable expressing whether the dwell time of bus i has passed the estimated dwell time cdt (t) at prediction step k or not, whereas via ci,k a discrete-time clock is defined for bus i, the dynamics of which we express as follows: (23) ci,k+1 = ci,k + exi,k ,
where ci,k expresses the time (in prediction steps) that has passed since the bus has arrived at the upcoming stop. Considering the events expressed with the binary variables exi,k and edi,k together, bus i will be required to hold at the upcoming stop if it has reached the stop (i.e., if exi,k = 1) and if its dwell time has not yet reached the estimated dwell time (i.e., if edi,k = 0). We can express this with a third event ehi,k related to whether the bus should be holding at the upcoming stop or not, defined as follows: ehi,k <= exi,k ehi,k <= 1 − edi,k exi,k − edi,k <= ehi,k .
7503
i=1 k=0
for i = 1, . . . , Kb : xfi,1 = xfi (t) xri,1 = xri (t) xui,1 = xui (t) ci,k = 0 for k = 1, . . . , N : xfi,k+1 = xfi,k + T (vi+1,k − vi,k ) xri,k+1 = xri,k + T (vi,k − vi−1,k ) xui,k+1 = xui,k − T vi,k ci,k+1 = ci,k + exi,k 0 ≤ vi,k ≤ v i,k (1 − ehi,k ) constraints (21), (22), and (24),
(25) where xui (t) is the measured distance of bus i to the upcoming stop at time t. The problem (25) is a mixed-integer quadratic program (MIQP), which, although nonconvex, can be solved via software packages specially developed for mixed-integer programs in an efficient manner. 4. SIMULATION RESULTS The HMPC controller is compared with the no control (NC) case, where bus speeds are fixed to their maximum values v˜i (t), together with the LMPC and PD-BSC, using the algorithm (1) for simulating BTS. The setting is line 2 of the bus network of the city of Fribourg, which has 9 buses and 44 stops and a length of 15 km. Passenger demands (i.e., βh,j (t)) and maximum bus speeds (i.e., v˜i (t)) are extracted from real bus data. Simulation covers a time period of a single day of BTS operation (from 06:00 to 20:00) and the sampling period is 10 s, resulting in a simulation length of tfinal = 5040. The prediction horizon of the MPC schemes is chosen as 10. Through trial-anderror, a weighting factor of σ = 104 and controller gains of K1 = 0.01 and K2 = 0.25 are found to yield a good balance between headway regularization and fast operation. Bus positions on the loop as a function of simulation time and headway distributions are shown in Fig. 1, whereas results are summarized in table 1, showing mean time spent per passenger (TSPP; time spent by a passenger for waiting at a stop and traveling on the bus), mean speed of buses, standard deviation of headways (std. of hws.; a metric of headway irregularity), and mean/maximum CPU times (which are obtained by calling CPLEX from the YALMIP toolbox by L¨ofberg (2004), in MATLAB 8.5.0 Table 1. Control Performance Evaluation
(24)
We formulate the HMPC problem, which builds on the basis of the LMPC problem (19), but also includes the process of buses reaching the upcoming stop as a function of control inputs in the prediction model via the events 7774
Control scheme
mean TSPP (min)
mean speed (km/h)
std. of hws. (min)
mean/max. CPU time (s)
NC PD-BSC LMPC HMPC
25.7 31.2 24.6 18.6
25.7 17.6 19.3 18.7
13.3 0.95 1.02 1.03
– – 0.16/0.21 0.47/0.68
bus positions (NC) frequency
15 10 5 0 7:00
7:30
8:00
8:30
9:00
7:30
8:00
8:30
frequency mean
2000 0
0
9:00
1000 0
0
8:00
8:30
7:30
8:00
8:30
9:00
9:00
simulation time (hh:mm)
20
2000 1000 0
0
simulation time (hh:mm) bus positions (HMPC)
15 10 5 0 7:00
10
time headway (minutes) headway dist. (LMPC) frequency
7:30
20
2000
simulation time (hh:mm) bus positions (LMPC)
15 10 5 0 7:00
10
time headway (minutes) headway dist. (PD-BSC) frequency
15 10 5 0 7:00
headway dist. (NC)
4000
simulation time (hh:mm) bus positions (PD-BSC)
10
20
time headway (minutes) headway dist. (HMPC) frequency
distance (km)
distance (km)
distance (km)
distance (km)
Proceedings of the 20th IFAC World Congress 7504 I.I. Sirmatel et al. / IFAC PapersOnLine 50-1 (2017) 7499–7504 Toulouse, France, July 9-14, 2017
2000 1000 0
0
10
20
time headway (minutes)
Fig. 1. Time-space diagrams (left column; each color is for one bus) and headway distributions (right column) for the NC case and the three controllers. (R2015a) on a 64-bit Windows PC with 3.6-GHz Intel Core i7 processor and 16-GB RAM). The results indicate that: (a) PD-BSC achieves good headway regularity, but lacking coordination between bus movements, it causes sustained variations in speed (see fig. 1) and a general slowing down of buses, leading to a larger TSPP, (b) LMPC is able to coordinate bus movements, and thus performs better than PD-BSC, (c) owing to a prediction model capturing simple hybrid dynamics, the proposed HMPC is superior to both PD-BSC and LMPC. Moreover, the CPU times of both MPC schemes are roughly negligible in comparison to the sampling period of 10 s, showing their computational tractability. 5. CONCLUSION We developed a dynamical model for single loop BTSs based on the mixed logical dynamical systems approach. The model is detailed enough to allow in-depth analysis and simulation of BTS operation, whereas also computationally lightweight, providing fast execution. Furthermore, we proposed a hybrid MPC scheme based on a simple hybrid BTS model for regularizing headways and improving BTS performance, and evaluated its performance via simulations with the proposed MLD model. Future work could include: (a) Comparison of the proposed schemes with other bus speed control methods from the literature, (b) consideration of passenger accumulation in MPC prediction model, (c) extension to multi-loop BTSs. REFERENCES Abkowitz, M.D. and Lepofsky, M. (1990). Implementing headway-based reliability control on transit routes.
Journal of Transportation Engineering, 116(1), 49–63. Ampountolas, K. and Kring, M. (2015). Mitigating bunching with bus-following models and bus-to-bus cooperation. In Intelligent Transportation Systems (ITSC), 2015 IEEE 18th International Conference on, 60–65. IEEE. Bemporad, A. and Morari, M. (1999). Control of systems integrating logic, dynamics, and constraints. Automatica, 35(3), 407–427. Chandrasekar, P., Long Cheu, R., and Chin, H.C. (2002). Simulation evaluation of route-based control of bus operations. Journal of transportation engineering, 128(6), 519–527. Cort´es, C.E., S´aez, D., Milla, F., N´ un ˜ez, A., and Riquelme, M. (2010). Hybrid predictive control for real-time optimization of public transport systems’ operations based on evolutionary multi-objective optimization. Transportation Research Part C: Emerging Technologies, 18(5), 757–769. Daganzo, C.F. (2009). A headway-based approach to eliminate bus bunching: Systematic analysis and comparisons. Transportation Research Part B: Methodological, 43(10), 913–921. Daganzo, C.F. and Pilachowski, J. (2011). Reducing bunching with bus-to-bus cooperation. Transportation Research Part B: Methodological, 45(1), 267–277. Delgado, F., Contreras, N., and Munoz, J.C. (2013). Holding for transfers. In Transportation Research Board 92nd Annual Meeting, 13-2913. Delgado, F., Munoz, J.C., and Giesen, R. (2012). How much can holding and/or limiting boarding improve transit performance? Transportation Research Part B: Methodological, 46(9), 1202–1217. Eberlein, X.J., Wilson, N.H., and Bernstein, D. (2001). The holding problem with real–time information available. Transportation science, 35(1), 1–18. Fu, L., Liu, Q., and Calamai, P. (2003). Real-time optimization model for dynamic scheduling of transit operations. Transportation Research Record: Journal of the Transportation Research Board, (1857), 48–55. Hall, R., Dessouky, M., and Lu, Q. (2001). Optimal holding times at transfer stations. Computers & industrial engineering, 40(4), 379–397. Ibarra-Rojas, O., Delgado, F., Giesen, R., and Mu˜ noz, J. (2015). Planning, operation, and control of bus transport systems: A literature review. Transportation Research Part B: Methodological, 77, 38–75. Liu, H., Skabardonis, A., and Zhang, W.b. (2003). A dynamic model for adaptive bus signal priority. In Preprint CD-ROM, 82nd Transportation Research Board Annual Meeting, Washington, DC. L¨ofberg, J. (2004). YALMIP: A toolbox for modeling and optimization in MATLAB. In IEEE International Symposium on Computer Aided Control Systems Design, 284–289. IEEE. Van Oort, N., Boterman, J., and Van Nes, R. (2012). The impact of scheduling on service reliability: triptime determination and holding points in long-headway services. Public Transport, 4(1), 39–56. Xuan, Y., Argote, J., and Daganzo, C.F. (2011). Dynamic bus holding strategies for schedule reliability: Optimal linear control and performance analysis. Transportation Research Part B: Methodological, 45(10), 1831–1845.
7775