Multi-objective optimal control formulations for bus service reliability with traffic signals

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Multi-objective optimal control formulations for bus service reliability with traffic signals Andy H.F. Chow a,∗, Shuai Li a, Renxin Zhong b a b

Centre for Transport Studies, University College London, London WC1E 6BT, United Kingdom Research Center of Intelligent Transportation System, Sun Yat-sen University, Guangzhou 510006, China

a r t i c l e

i n f o

Article history: Received 18 July 2016 Revised 21 February 2017 Accepted 22 February 2017 Available online xxx Keywords: Bus service reliability Kinematic wave model Hamilton–Jacobi formulation Multi-objective optimal control Transit signal priority

a b s t r a c t This paper presents a set of optimal control formulations for maximising bus service reliability through deriving optimal adjustments on signal timings. The traffic dynamics is captured by an underlying kinematic wave model in Hamilton–Jacobi formulation. With traffic data collected through loop detectors and bus positioning devices, the control actions are carried out through adjusting signal timing plans according to short-term estimations of traffic flows and bus arrivals. We derive the optimality conditions of multi-objective control formulations and present an open loop solution algorithm. The proposed control system is applied to a test arterial developed based upon a real-world scenario in Central London, UK. It is found that the model is capable of regulating bus service reliability through utilising traffic signals while managing delays induced to surrounding traffic. The study generates new insights on managing bus service reliability in busy urban networks. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction Having an effective and reliable green transport system is the prerequisite for sustainable and resilient urban development. A recent empirical study reveals that 25%–30% of congestion observed in Central London (UK) could be reduced by effective traffic management (Chow et al., 2014). One of an effective means to promote green transport is to reduce car use through encouraging use of public transport. Among all performance indicators, reliability is recognised as the most important one determining whether buses are the preferred mode of transport to private vehicles (Bowman and Turnquist, 1981). Reliability of bus services can be measured by either punctuality or regularity (Daganzo, 2009). Punctuality refers to the percentage of buses arriving on time with respect to a predefined timetable or schedule, while regularity refers to the derivation with respect to the scheduled headways. It is known that bus operations are inherently unstable and will invariably deviate from the planned schedule without proper control. Newell and Potts (1964) show that this is due to the time a bus spends serving passengers at a stop generally increases with the time between consecutive bus arrivals to that stop. Consequently, a bus arriving early (or late) at a stop spends less (or more) time serving passengers and arrive even earlier (or later) at its next stop. The eventual result of this mechanism is the infamous bus bunching and the resultant uneven distribution of passengers among buses. One customary strategy, holding, is to insert an amount of slack time into the bus schedule, and mitigate the bus bunching issue by holding early arriving buses at specific control points (e.g. bus stops) for a certain amount of time ∗

Corresponding author. E-mail address: [email protected] (A.H.F. Chow).

http://dx.doi.org/10.1016/j.trb.2017.02.006 0191-2615/© 2017 Elsevier Ltd. All rights reserved.

Please cite this article as: A.H.F. Chow et al., Multi-objective optimal control formulations for bus service reliability with traffic signals, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.02.006

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Fig. 1. Signal-based control actions for bus service reliability.

(Eberlein et al., 2001; Hadasa and Shnaidermanb, 2012). Daganzo (2009), and later Daganzo and Pilachowski (2011), Xuan et al. (2011), and Bartholdi III and Eisenstein (2012), further present a set of headway-based holding strategies for stabilising the variance in bus headways subject to stochastic disturbances (Hans et al., 2015). Delgado et al. (2012) investigate the performance of various holding strategies. Berrebi et al. (2015) also presents a real-time bus dispatching policy that minimises passenger waiting time with appropriate implementation of bus holds. Most of these previous approaches operate as decentralised rule-based control systems. A weakness of these decentralised and rule-based systems is that they treat buses as a separated system and those studies do not consider the influence of surrounding traffic and signal control settings. It is known that prevailing traffic conditions can have a significant effect on the bus operations. Gu et al. (2013) investigate the queueing process of buses and their interaction with surrounding traffic at bus stops. Guler and Menendez (2014) further analyse the effect of pre-signals on the performance of bus priority, and Gu et al. (2014) look at the impacts of bus stops near signalised intersections: models of car and bus delays. Hans et al. (2015) further investigate the irregularity of bus journey times an their relationship with the underlying traffic conditions and signal timing settings. It is known that bus schedule control actions can be realised through adjusting traffic signal timings which will be the focus of the present paper. Following the most conventional transit signal priority (TSP) strategies (see example Hounsell et al., 2007), the signal-based bus schedule control is implemented through adjusting the start time and/or end time of the stage or phase giving the buses right away in each signal cycle. In practice, there are two common strategies used for adjusting signal timings to the benefit of buses: green extension and green recall (see Fig. 1). A green extension is used on a bus approaching a junction along a traffic phase that is due to lose its right of way. The signal controller may extend its current green phase during such that the bus may pass through the junction without being stopped by the red signal. If the bus is detected at a time when the traffic phase in which it travels does not have right of way, then the signal plan can be readjusted such that right of way can be given to the bus more quickly by using a green recall. This can be done by reducing the durations of any remaining stages in the signal cycle to their minimum values and then changing to the first stage in which buses have right of way. The green extension and green recall are conventional strategies used to provide priority to buses in many control systems (e.g. SCOOT). It is found that the two strategies help to improve journey times and associated reliability of buses while it may not help to improve the headway regularity through simply granting priority. This paper presents a set of multi-objective optimal control formulations which derive centralised and predictive strategies for maximising bus service reliability through deriving optimal adjustments in traffic signal settings. Following Chow et al. (2015), the traffic dynamics is captured by an underlying Hamilton–Jacobi formulation of kinematic wave model. The Please cite this article as: A.H.F. Chow et al., Multi-objective optimal control formulations for bus service reliability with traffic signals, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.02.006

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control strategies are implemented through adjustments made to signal timings with short-term prediction of bus arrivals surrounding traffic conditions. Optimality conditions are derived for each combination of objective functions, and are associated with an open-loop solution algorithm. In contrast to these two conventional signal-based strategies (green extension and recall), the optimal controller presented herein generates new strategies including the red extension and red recall (see Fig. 1, also in Ma et al., 2010) for regulating bus schedules or headways with consideration of the corresponding impact to traffic delays. The red extension and recall can be regarded as creating an extra red duration for slowing down early arriving buses. The red extension and red recall can be interpreted as one of the bus holding approaches (Daganzo, 2009), except that the red extension and red recall are carried out at junctions through traffic signals instead of at bus stops. The contribution of the paper also includes a set of sensitivity analyses of bus running times and traffic delays with respect to the control actions (adjustments in signal timing plan) for use in solving the optimality conditions. The paper is organised as follows: Section 2 reviews the representation of bus operations and analysis of headway deviations. Section 3 presents the Hamilton–Jacobi formulation of kinematic traffic model which we use in the optimal control framework. These two sections (Sections 2 and 3) present the existing modelling framework which we need for developing our proposed control system. The core contribution of the present paper is presented in Section 4, which include the multiobjective location-based optimal control formulation with different combinations of objective functions (Section 4.1), the corresponding optimality conditions (Section 4.1), a set of sensitivity analyses of the state variables (bus running times and traffic delays) with respect to control actions Section 4.2), and an open-loop solution algorithm (Section 4.3). Section 5 applies the control system to an arterial in Central London (UK) as an illustration of real world application. Finally, Section 6 presents some concluding remarks including discussion of how it can be applied to a closed-loop rolling horizon. Considering the importance of bus service efficiency and reliability when determining its public acceptance, this paper contributes to the implementation and promotion of green transport mode. 2. Model of bus operations Each bus run can be regarded as a series of ‘events’ of arrivals (τ = [τn,s ]) and departures (θ = [θn,s ]) which specify the arrival and departure times of each bus n at each stop s along the service route. The evolution of θ n, s is subject to the boundary condition at the first stop s = 1 (e.g. the terminal):

θn,1 = θn−1,1 + hˆ n ,

(1)

where hˆ n denotes the scheduled bus headway between bus pair (n − 1, n ), which is typically 3–10 min in urban areas. It is noted that the scheduled headway hˆ does not need to be constant and it can be changed over different time of the day with respect to the demand level. The quantity θ 1, 1 will be a predefined value (e.g. zero) for the departure time of the first bus from the first stop. The departure times θ n, s are related to the arrival times τ n, s as

τn,s+1 = θn,s + ηn,s ,

(2)

where ηn, s is the running time for bus n proceeding from stop s to its next stop s + 1. As discussed in previous literature (see Daganzo, 2009; Daganzo and Pilachowski, 2011; Xuan et al., 2011), the running times of buses could be subject to variations due to congestion on surface networks. Given the arrival time τn,s+1 , the corresponding departure time θ n, s of the same bus n can be determined as

θn,s = τn,s + κn,s ,

(3)

where κ n, s is the dwell time of bus n at stop s. Newell and Potts (1964) and Daganzo (2009) consider the dwell time to be a stochastic variable that is related to headway deviations. It is considered that the dwell time of a bus will be shorter (or longer) than its nominal value if the headway between the bus and its predecessor is shorter (or longer) than the planned value, as that would imply the bus will have less (more) passengers to pick up than expected on its arrival. Given the arrival times τ = τn,s , we can derive the actual headway between bus pair (n − 1, n ) as

hn,s = τn,s − τn−1,s .

(4)

Apart from defining the scheduled headway hˆ , the bus service can also be specified by an underlying timetable or schedule through defining the timetable τˆ = τˆn,s which represents the scheduled clock time at which the bus n has to arrive at stop s. This also gives rise to the scheduled headway:

hˆ n = τˆn,s − τˆn−1,s .

(5)

With the scheduled arrival time τˆn,s , we can first define

n,s = τn,s − τˆn,s ,

(6)

as the measure of the deviation of the actual arrival time of each bus n from its schedule over all stops s. Moreover, it can also be shown (see Daganzo, 2009) that the headway deviation from the scheduled value for each bus pair (n − 1, n ), i.e. hn,s − hˆ n , can be determined as

hn,s − hˆ n = n,s − n−1,s .

(7)

Please cite this article as: A.H.F. Chow et al., Multi-objective optimal control formulations for bus service reliability with traffic signals, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.02.006

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Given an initial perturbation n0 ,s0 caused by a specific bus n0 at stop s0 , the propagation of  n, s over n and s follows the recursive function (Daganzo, 2009):

n,s+1 = (ηn,s − ηn−1,s ) + n,s + σκs ,

(8)

where σκs = κn,s − κˆ s is dwell time deviation of bus n at stop s from its nominal value κˆ s . 3. Hamilton–Jacobi formulation of kinematic wave Following the bus operational model presented in Section 2, this section presents the Hamilton–Jacobi formulation of kinematic wave model that will be used as the underlying traffic model in the optimal control formulation to be presented in Section 4. The modelling framework consists of two main components: the basic framework (Section 3.1) calculates the spatio-temporal distribution of network flow and density (and hence the corresponding (space-) mean speed) of general traffic, the bus model (Section 3.2) then derives the bus trajectory and the corresponding influence on its surrounding traffic. 3.1. Basic framework The modelling framework starts with a node model where given a node m with Im incoming links and Jm outgoing links, we define a split matrix βk = [βi j (k )], which has a dimension of (Im × Jm ), for the node. The split ratio can be updated regularly (say, every 15-min) over a sampling period k based on field observations. The element β ij (k) in the split matrix βk specifies the proportion of traffic on each incoming link i, where i = 1, 2, . . . , Im , that is heading to each outgoing link j, where j = 1, 2, . . . , Jm , through node m during time interval k. The principle of conservation requires that Jm 

βi j (k ) = 1,

(9)

j=1

for all i = 1, 2, . . . , Im . Without consideration of excessive blocking (i.e. traffic can be freely flowing through node without restraint due to downstream queue), the traffic volume qij (k) during time k flowing from incoming link i to outgoing link j is determined as:

qi j (k ) = βi j (k ) i (k ),

(10)

where i (k) is total traffic volume on link i wanting to get through node m during time k. If the node is controlled by a traffic signal, the signalling effect can be captured by associating a binary variable γ ij (t) with the term on the right-hand-side in Eq. (10) (Chow et al., 2015), i.e.

qi j (k ) = βi j (k ) i (k )γi j (t ), in which:

(11)



γi j (t ) =

1, if movement from i to j is given a green signal 0, if movement from i to j is given a red signal

(12)

This node model can further be enhanced to capture downstream blockage, where detailed description can be found in Chow et al. (2010), Zhang et al. (2013), and Chow et al. (2015). Given the flows estimated flowing into each link by the node model, the link model estimates the flow propagation along the link in which we adopt a variational formulation of kinematic wave model (Daganzo, 2005a). Similar to the link transmission model proposed by Yperman (2007), the solutions to this variational formulation are expressed in terms of cumulative count of traffic N(x, t) at x and t, where by definition:

 ∂N  = q(x, t ), ∂ t (x,t )  ∂N  − = ρ (x, t ), ∂ x (x,t )

(13) (14)

in which q(x, t) and ρ (x, t) are respectively the flow and density at location x and time t. Following (13) and (14), the corresponding mean speed v¯ (x, t ) of traffic at (x, t) can also be derived as:

v¯ (x, t ) =

q(x, t ) . ρ (x, t )

(15)

that will be used for incorporating the bus movement in latter section (Section 3.2). Daganzo (2005a) presents the following formulation:

∂N ∂N | = [− | ], ∂ t (x,t ) ∂ x (x,t )

(16)

Please cite this article as: A.H.F. Chow et al., Multi-objective optimal control formulations for bus service reliability with traffic signals, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.02.006

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Fig. 2. A triangular fundamental diagram.

in which the function represents the flow-density relationship, or fundamental diagram, of traffic. Provided that the fundamental diagram function is concave in ρ Daganzo (2005b), the formulation (16) can be solved through introducing the  ∂q  following cost function in wave speed w(x, t ) = over (x, t): 

∂ρ

(x,t )

R(w, x, t ) = sup( (ρ , x, t ) − w(x, t )ρ ).

(17)

ρ

This cost function R is the convex conjugate (Boyd and Vandenberghe, 2004) of the fundamental diagram function . Physically this function R can be interpreted as the maximum rate at which traffic can pass an observer moving with speed w at (x, t). To illustrate this variational method, we now consider a link which starts at location x0 and ends at x1 . We also define B be the set of initial condition N(x, 0) at time t = 0 for all locations x; boundary condition N(x0 , t) at upstream end x0 ; boundary condition N(x1 , t) at downstream end x1 . Given this set of boundary values B, Eq. (16) can be solved iteratively over discrete time as:

N (x, t ) = min{N (x − wt, t − t ) + tR(w )},

(18)

w∈W

where W is the set of all admissible wave speeds in the fundamental diagram . Consider a simple case where is triangular as the one depicted in Fig. 2 in which the wave speed u can only take two possible values: vf (forward) and vb (backward), the cost function R becomes:



R(w, x, t ) =

0,

if w = v f vb ρ¯ , if w = vb

(19)

for all (x, t). Eq. (18) can then be reduced to:

N (x, t ) = min{N (x − v f t, t − t ), N (x − vb t, t − t ) + ρv ¯ b t }.

(20)

x∗

Effect of a traffic signal controller at a specific location can be captured in Eq. (18) through introducing ‘short-cuts’ (Daganzo, 2005b). A traffic light at x∗ can be modelled by introducing the following revised cost function R∗ (0, x∗ , t) associated with wave speed w = 0 at x∗ :



R ( 0, x , t ) = ∗

∗

qmax , t ∈ G 0, t∈R

(21)

where qmax is the maximum discharge rate of traffic during green phases; G and R refer to the green and red phases of the traffic light respectively. When the traffic light at x∗ is in green, traffic is allowed to pass through at the maximum rate qmax . When the traffic light turns red, the passing rate R∗ is set to be zero and hence all traffic will have to queue behind x∗ . Finally, given the N(x, t) function values over space x along a link and time t, the total traffic delay (dimension: [vehtime]) along the link between the two points x0 and x1 over a given time horizon T can be determined as the total area enclosed between N (x0 , t˜) and N(x1 , t) within the period of interest T as shown as an example in Fig. 3, where t˜ = t − ( x1v−x0 ). Adopting the translated time t˜ is to take into account the time shift due to the free-flow travel time along the link f

when calculating delay (Daganzo, 1997). Fig. 3 shows the N functions under an example of signal control operations, where t = r0 and t = r1 denote respectively the start and end times of the red phase. The vertical separation between the two functions N(x1 , t) and N (x0 , t˜) at any time t gives the queue length at that time (see Daganzo, 1997; Chow and Lo, 2007, and many others). During the red phase where r0 ≤ t ≤ r1 , the discharge flow from the signal controller is zero and hence Please cite this article as: A.H.F. Chow et al., Multi-objective optimal control formulations for bus service reliability with traffic signals, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.02.006

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Fig. 3. Calculation of traffic delay with N−functions.

the cumulative flow function N (x0 , t˜) remains constant at N(x0 , r0 ). For r1 ≤ t ≤ t∗ , traffic is being discharged at a flow qmax and hence N(x1 , t) increases at the rate of qmax until the time t∗ when the queue dissipates. After time t∗ , the two functions N (x0 , t˜) and N(x1 , t) overlap which reflects there is no residual queue in the system thereafter. The total vehicle delay D is given by the shaded area in the figure, i.e.

 D=

T 0

(N (x1 , t ) − N (x0 , t˜))dt.

(22)

Moreover, we can also derive the average delay per vehicle d from dividing D by the number or vehicles within the time period T, i.e.

d=

D

(N (x1 , T ) − N (x0 , 0 ))

.

(23)

3.2. Incorporation of buses The node-link kinematic wave model presented in Section 3.1 forms the foundation on which the buses run. This section presents how the movement and impact of the buses can be integrated into the framework. We first have the bus trajectory function xn (t) of each bus n over time t which is derived from the fundamental kinematic law:

xn (t ) =



t

t0

vn (x, t )dt ,

(24)

where xn (t) represents the location of the bus n along the link. Or in discrete form (24) can be written as:

xn (t ) =

t 

vn (x, t )t ,

(25)

t=t0

where t0 is the time when the bus enters the link of interest, vn (x, t) is the speed of the bus at location x and time t, and t the sampling time step adopted in updating the bus location. Given (25), apparently the key now is to determine the bus speed vn (x, t) from which its trajectory (and hence running time ηn, s which will be used in the optimal control formulation presented in latter section) can be derived accordingly. Inspired by Laval (2004), considering the influence of surrounding traffic, the speed function vn (x, t) of bus n is determined as:

vn (x, t ) = min(v∗n , v¯ (x, t ) ),

(26)

where is the nominal speed of the bus; the notation v¯ (x, t ) denotes the mean speed of surrounding traffic at the bus’s ) location x at current time t, and this v¯ (x, t ) is determined as ρq((x,t through (15) with the underlying variational method x,t ) of kinematic wave model as discussed in Section 3.1. Eq. (26) states that each bus n could proceed with its nominal speed v∗n as long as it is not constrained by the prevailing speed of surround traffic v¯ (n, t ) (e.g. duration congestion or when the traffic is held by a red signal in urban networks). In case if v∗n ≥ v¯ (x, t ), the bus speed vn (x, t) will simply take the value of v¯ (x, t ) according to (26). Consequently, the bus will act like a particle flowing along the ‘velocity field’ v¯ (x, t ) in the same speed as other vehicles and behaving as other vehicles in the system as depicted in Gibbens and Saatci (2008) and Chow et al. (2010). ) In case if v∗n < ρq((x,t , the bus will then be incorporated as a ‘moving bottleneck’ (Newell, 1998) or an ‘internal bottleneck’ x,t ) (Mazare et al., 2011). Vehicles following the bus will be incurred a delay as they are forced to proceed with the bus’s speed v∗ that is lower than the prevailing traffic speed v¯ (x, t ). The effect of the moving bottleneck effect induced by the slower bus can be captured in the variational method of kinematic wave as follows: Based on the prevailing velocity field of traffic v¯ (x, t ) derived from the underlying kinematic wave model, we first determine the speed profile vn (x, t) of the bus n through

v∗n

Please cite this article as: A.H.F. Chow et al., Multi-objective optimal control formulations for bus service reliability with traffic signals, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.02.006

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Fig. 4. Movement of bus under influence of signal setting and its impact on surrounding traffic.

(26), and hence the corresponding trajectory xn (t) through (25) with vn (x, t). Following Daganzo (2005b); Daganzo and Laval (2005), and Daganzo and Menendez (2005) (also Chow et al., 2015), we then introduce the following ‘short-cut’ cost function along the bus trajectory xn (t) in the variational framework:

 R∗ (w, x, t ) =

Rxn (t ) ,

if x = xn (t ), w =

R(w, x, t ), otherwise,

dxn , dt

(27)

n where 0 ≤ Rxn (t ) ≤ R( dx ), and Rxn (t ) physically represents the maximum passing rate of traffic that can pass through the bus dt trajectory xn (t). This ‘short-cut’ cost function R∗ in (27) essentially governs the traffic flow passing through the bus as long as it is present, and hence the state of traffic following the bus. With the short-cut cost function R∗ defined, the variational solver presented in Section 3.1 will be re-run to determine the state of traffic with the presence of the bus. As an illustration, we now consider a 1-lane road section with saturation flow 1800 veh/h, jam density 250 veh/km, free-flow speed of general traffic (vf ) 36 km/h, and backward congestion propagation speed (vb ) 9 km/h. Traffic is being loaded into the road section at the upstream end at a rate of 800 veh/h. A traffic signal is located at x = 180 m, which starts a red phase at time r0 =12s and ends at time r1 =30s. Suppose a bus enters the road from a bus stop s located at x = 25m and time t = 15s, and exits the road at the next bus stop s + 1 located at x = 230m. While the bus is present and if it is obstructing the surround traffic in case it has a lower speed than general traffic, the maximum passing rate (relative to the bus) of traffic around the bus (i.e. Rxn (t ) in (27)) is considered to be 500veh/h. Fig. 4 shows the corresponding solutions derived from the hybrid kinematic wave traffic-bus model in which we consider two different nominal bus speeds. In Fig. 4a, the nominal bus speed (v∗n ) is taken as 36 km/h which is the same as the nominal free-flow speed (vf ) of general traffic. In Fig. 4b, the nominal bus speed (v∗n ) is taken as 10 km/h which is lower than the nominal speed of general traffic (vf ) and hence the bus will act like a moving bottleneck under the free-flow condition. The colour scale in both plots represent the level of calculated traffic density (i.e. ρ (x, t)) over time and space. The area in jam density (250veh/km) characterises the space-time domain of the traffic queue due to the red signal as determined by q(x, t ) the kinematic wave model. Based upon the predefined speed v∗n of the bus and the underlying v¯ (x, t ) = of general ρ (x, t ) traffic determined by the kinematic wave model, the actual speed and hence trajectory of the bus can be derived through (25) and (26). The thick arrowed line in the plots in Fig. 4 highlights the bus trajectory in both cases. The horizontal portion shows the location and time when bus is in the queue induced by the red signal in which the bus speed is zero. Hence, the length of this horizontal line segment also reveals the time delay incurred by the bus due to the red signal. Given the bus trajectory, its impact on surrounding traffic is captured by regarding the bus as a moving bottleneck through incorporation of the short-cut cost function (27). In Fig. 4a, the bus is proceeding at the same speed as other vehicles and hence the bus will impose no impact on surrounding traffic. In Fig. 4b, with the bus nominal speed v∗n lower than that of surrounding traffic under free-flow, the slower moving bus creates a moving platoon of a density ρ (x, t)=125.7veh/km following the bus during its presence under free-flow (in which we assume the passing rate of traffic through the slow moving bus is 500veh/h as aforementioned). It can be verified that the characteristics of the solution obtained herein is consistent with previous analysis presented in the literature (see example Newell, 1998; Laval, 2004; Daganzo and Menendez, 2005). Finally, the running time ηs of the bus between the stops s and s + 1 (see Fig. 4) is determined as the difference between the time when enters of the road section, which gives estimates of ηs being 17s and 22s respectively in the two settings in Fig. 4a and b.

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4. Optimal control This section presents the optimisation formulations and solution algorithm for maximising the bus service reliability through deriving optimal adjustments in signal timing. 4.1. Formulation The first objective considered herein aims to minimise the bus schedule discrepancies s = [n,s ] with and adjustments u made to signal timings through the following objective function:

minZ = (1 − α )Z0 + α Z1 u

(28)

where

Z1 =

S nall 1  2 n,s 2

(29)

s=1 n=1

represents the total squared schedule deviations over the predefined set of stops s = 1, 2, . . . , S for all buses n = 1, 2, . . . , nall , and

Z0 =

nall sM S   

dn,m (u )

(30)

s=1 n=1 m=s1

where dn, m (u) is the average traffic delay per vehicle (including both general traffic and buses) induced by the signal adjustment u for the buses as they pass through the intersection m, where m = s1 , s2 , . . . , sM denotes all intersections lying between stops s − 1 and s. The notation u represents the tuple of the control variables u = [um (r0 , c ), um (r1 , c )] which refers to the control action applied at intersection m at signal cycle c, in which r0 and r1 refer respectively to the start and end times of red for the bus route in cycle c. Specifically, u refers to the time shift introduced to r0 and/or r1 of the bus phase at intersection m. In such case a positive um (r0 , c) (or um (r1 , c)) implies shifting the corresponding r0 (or r1 ) to the right, while a negative um (r0 , c) (or um (r1 , c)) implies shifting the corresponding r0 (or r1 ) to the left. As a general practice, the duration of cycle c (or cycle time) is considered to be externally determined based upon prevailing network-wide degree of saturation (see example Chow and Sha, 2016) before making adjustments to other cyclebased timing variables such as splits. Changing the cycle time will definitely have an impact on the system performance (e.g. reduce the system-wide traffic delay). Nevertheless, the focus of this paper is about optimising bus service regularity, while the cycle time variable is mainly used for regulating network degree of saturation instead of carrying out transit related operations. For simplicity, this study regards the cycle time as predefined (and hence fixed) within the optimal control framework proposed herein. We should however emphasise that the proposed optimal controller can work no matter what the background cycle time is. The cycle time only needs to be predetermined as an input (or parameter) to be fed into the proposed bus schedule / traffic delay controller in each signal cycle. Finally, the parameter α ∈ [0, 1] connecting Z0 and Z1 in (28) determines the trade-off between the two objectives of minimising bus schedule deviations and traffic delays. Objective function (28) focuses on the punctuality of the bus service which aims to restore the bus arrival times to the scheduled values. It has been shown that (e.g. Xuan et al., 2011) it will be more practical to implement control actions aiming to minimise bus headway discrepancies instead of schedule discrepancies (29) for high frequency service (say, service headway lies between 35 min). Considering this, we also present an alternative objective function:

minZ = (1 − α )Z0 + α Z2 u

(31)

where

Z2 =

S nall 1  (n,s − n−1,s )2 2

(32)

s=1 n=1

represents the total squared headway deviations over the predefined set of stops s = 1, 2, . . . , S for all buses n = 1, 2, . . . , nall . The objective function Z2 can be used to replace Z1 in (29) should the system manager aims to minimise headway discrepancies instead of schedule discrepancies. The objective function, either (28) or (31), presented above is subject to the state equation on  n, s :

n,s+1 = [ηn,s (u ) − ηn−1,s (u )] + n,s .

(33)

which is essentially (8) except that the running times ηn, s are now expressed as function of the associated control variable u. The running time function η is determined through the underlying kinematic wave model given the prevailing traffic condition and control setting u. Moreover, we do not include the dwell time discrepancy, σκs , in (33) as we are considering the expected value in the formulation and we assume that the expected value of σκs is zero for all n. Please cite this article as: A.H.F. Chow et al., Multi-objective optimal control formulations for bus service reliability with traffic signals, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.02.006

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9

To derive the optimality conditions, the objective function Z is augmented with the state Eq. (33) to form a Lagrangian function (Chow, 2009b). We start with objective function (28), the Lagrangian function is written as:

L1 = ( 1 − α )

nall sM S   

dn,m (u ) + α

s=1 n=1 m=s1

+

nall S  

S nall 1  2 n,s 2 s=1 n=1

(34)

λn,s {[ηn,s (u ) − ηn−1,s (u )] + n,s − n,s+1 }

s=1 n=1

where λ = λn,s is the costate variable associated with each of state equation in n and s. We also define the Hamiltonian function from (34) as:

H1 = ( 1 − α )

nall sM  

dn,m (u ) + α

n=1 m=s1

+

nall 

nall 1 2 n,s 2 n=1

(35)

λn,s+1 {[ηn,s (u ) − ηn−1,s (u )] + n,s }

n=1

Likewise, if (32) is used as the objective function instead of (28), then Z2 will be used in place of Z1 and the Lagrangian function in (34) becomes

L2 = ( 1 − α )

nall sM S   

dn,m (u ) + α

s=1 n=1 m=s1

+

nall S  

S nall 1  (n,s − n−1,s )2 2 s=1 n=1

(36)

λn,s {[ηn,s (u ) − ηn−1,s (u )] + n,s − n,s+1 }

s=1 n=1

and the corresponding Hamiltonian function will be

H2 = ( 1 − α )

nall sM  

dn,m (u ) + α

n=1 m=s1

+

nall 

nall 1 (n,s − n−1,s )2 2 n=1

(37)

λn,s+1 {[ηn,s (u ) − ηn−1,s (u )] + n,s }

n=1

Following Pontryagin et al. (1962), we can restore the state equation from the Hamiltonian function as:

∂ H1 = n,s+1 = [ηn,s (u ) − ηn−1,s (u )] + n,s ∂λn,s+1 ∂ H2 = . ∂λn,s+1

(38)

which is the same for either objectives (28) or (31). If the schedule discrepancy is used as the indicator of the bus service reliability (i.e. (28) is used), then the corresponding costate equation on λ is derived as:

∂ H1 = λn,s = n,s + λn,s+1 ∂n,s

(39)

where the costate variable λ = [λn,s ] represents the change in the value of the objective function with respect to change in the state variable  = [n,s ]. Physically, the costate variable λn, s in (39) can be interpreted as the cumulative schedule deviations of bus n from current stop s onward the terminal S. On the other hand, if the headway discrepancy is used instead as the objective of the bus service reliability optimisation (i.e. (32)), then the corresponding costate equation on λ is determined as:

∂ H2 = λn,s = (n,s − n−1,s ) − (n+1,s − n,s ) + λn,s+1 ∂n,s = 2n,s − n−1,s − n+1,s + λn,s+1 .

(40)

The costate variable λn, s in (40) can then be interpreted as the cumulative headway deviations of bus n from current stop s onward until it reaches the terminal S. Please cite this article as: A.H.F. Chow et al., Multi-objective optimal control formulations for bus service reliability with traffic signals, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.02.006

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Fig. 5. Effect of shifting r1 .

Finally, we can also write down the optimality condition on the control variable u with objective function (28):

⎧ sM  ⎪ ∂ H1 ∂ηn,s (u ) ∂ dn,m (u ) ⎪ ⎪ = λ + ( 1 − α ) = 0, n,s +1 ⎨ ∂ um ( r0 , c ) ∂ um ( r0 , c ) ∂ um ( r0 , c ) m=s1 sM  ⎪ ∂ H1 ∂ηn,s (u ) ∂ dn,m (u ) ⎪ = λ + ( 1 − α ) = 0, ⎪ n,s +1 ⎩ ∂ um ( r1 , c ) ∂ um ( r1 , c ) ∂ um ( r1 , c ) m=s1

(41)

∂η

(u )

∂η

(u )

for all intersections m = s1 , s2 , . . . , sM over all bus stop pairs (s − 1, s ) and bus runs n. The derivatives ∂ u n,s(r ,c ) and ∂ u n,s(r ,c ) ; m 0 m 1 ∂ dn,m (u ) ∂ dn,m (u ) and in (41) are sensitivities of bus running times and traffic delays with respect to the control variable u. ∂ u (r ,c ) ∂ u (r ,c ) m

0

m

1

These derivatives will be discussed and derived in latter section. Likewise, we can also have the optimality condition when (31) is used instead as the objective function.

⎧ sM  ⎪ ∂ H2 ∂ηn,s (u ) ∂ dn,m (u ) ⎪ ⎪ = 0, ⎨ ∂ um (r0 , c ) = λn,s+1 ∂ um (r0 , c ) + (1 − α ) ∂ um ( r0 , c ) m=s1 sM  ⎪ ∂ H2 ∂ηn,s (u ) ∂ dn,m (u ) ⎪ = λ + ( 1 − α ) = 0, ⎪ n,s+1 ⎩ ∂ um ( r1 , c ) ∂ um ( r1 , c ) ∂ um ( r1 , c ) m=s1

(42)

The optimality condition (42) is essentially the same as (41) but associated with different costate equations (see (39) v.s. (40)) with respect to the corresponding objective functions (i.e. (28) v.s. (32)). ∂η (u ) ∂η (u ) The optimality condition (41) (and (42)) can be interpreted and implemented as follows: given ∂ u n,s(r ,c ) = 0 or ∂ u n,s(r ,c ) = m

0

m

1

0, one should determine a control policy u along the bus route for each bus n such that its perspective cumulative schedule (or headway, depending which objective function is adopted) deviations λn,s+1 is minimised while taking into account of the induced delay dn, m through the trade-off factor (1 − α ). The control rule (41) (and (42)) can be regarded as similar to the feedback and decentralised control strategies presented in the literature such as Daganzo (2009); Daganzo and Pilachowski (2011), and others. A key difference is that the decentralised strategies only consider local deviations at the current stop, while the control system presented herein considers propagation of deviations through the costate variables. Furthermore, the control system also considers associated impact to the surrounding traffic delays through the multi-objective formulation. 4.2. Sensitivity analysis of signal-based control actions This section derives the sensitivity of traffic delays (d = [dn,m ] ) and bus running times (η = [ηn,s ] ) with respect to the control action u for use in the optimality conditions (41) and (42) under the kinematic wave model. We start with analysing the bus running times (η = [ηn,s ] ). Fig. 5 shows the effect of shifting r1 to r1 on bus running time ηs from stop s to stop s + 1, where here we drop the bus index n in the figure for brevity, under two circumstances: bus experiencing a delay due to the red signal and bus experiencing no delay (i.e. the bus proceeds through without being stopped). In the figure, the notation φ s denotes the undelayed running time of the bus from s to s + 1, while ϕ s is the running time delay induced by the red signal. In the figure, if ϕ s > 0, we can determine that

ηn,s = r1 .

(43)

Please cite this article as: A.H.F. Chow et al., Multi-objective optimal control formulations for bus service reliability with traffic signals, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.02.006

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Fig. 6. Effect of shifting r0 .

where r1 is the time shift in r1 , and ηn, s is the corresponding change in bus running time. Eq. (43) should approximately hold if we do not consider explicitly the acceleration and deceleration dynamics of the bus, which are not considered in the kinematic wave model. On the other hand, if ϕs = 0 (i.e. the bus is undelayed), then we will determine that

ηn,s = max{0, r1 − (τn∗ − r1 )},

(44)

τn∗

where is the undelayed arrival time of the bus at the stop line. Eq. (44) states that the running time of the bus will be insensitive to r1 as long as its magnitude does not exceed the difference between the undelayed arrival time of the bus at the stop line and original r1 . In addition, Fig. 6 shows the changes in ηs due to a shift of the start of red from r0 to r0 under different circumstances. Fig. 6a shows the situation when the bus n is incurred with a running time delay ϕ n, s due to the red signal as it travels from s to s + 1. Similar to (43), we can determine the corresponding change in ηn, s due to r0 as:

ηn,s = −r0 ,

(45)

τn∗

by neglecting the acceleration and deceleration effect of the bus, and provided that ϕ n, s > 0 and r0 < − r0 . The condition r0 < τn∗ − r0 is to ensure the new start of red r0 = r0 + r0 does not occur after the undelayed arrival time τn∗ of bus n. If r0 > τn∗ , i.e. r0 > τn∗ − r0 , the bus which was originally delayed will now be able to proceed through the stopline without delayed (see Fig. 6b). For such case, we will have

ηn,s = −ϕn,s ,

(46)

in which we consider the saving in running time for the bus will be its original delayed time. The conditions for (46) to hold is ϕ n, s > 0 and r0 > τn∗ − r0 . If the bus is initially undelayed (Fig. 6c, where ϕn,s = 0), then

ηn,s = 0,

(47)

provided that ϕn,s = 0 and r0 > τn∗ − r0 . The second condition here, r0 > τn∗ − r0 , is to ensure the new start of red r0 = r0 + r0 does not occur before the undelayed arrival time of bus n. Please cite this article as: A.H.F. Chow et al., Multi-objective optimal control formulations for bus service reliability with traffic signals, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.02.006

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Fig. 7. Effect of timing plan adjustment on delays.

If r0 < τn∗ , i.e. r0 < τn∗ − r0 , the bus which was originally undelayed will now be stopped by the red signal (see Fig. 6d), and we will have

ηn,s = (r1 − r0 ) + r0 ,

(48)

in which we consider the increase in running time for the bus will be the red duration (r1 − r0 ) now it has to encounter. Eq. (48) can be regarded as a reverse of Eq. (46). The conditions for Eq. (48) to hold are ϕn,s = 0 and r0 < τn∗ − r0 . In addition to the bus running times, introducing timing adjustments u will also have an effect on general traffic delays and its impact will need to be estimated for solving (41) and (42). Fig. 7 shows the changes in queueing pattern, in the form of cumulative arrival and departure curves (see examples in Wong, 1995 and Guler and Menendez, 2014), due to changes made in timing plan setting. Following Chow and Lo (2007), the sensitivity of the total system delay D with respect to changes in r0 and r1 can be estimated respectively as

∂D = ∂ r0



t∗ r0

q∗ (r0 )dt

(49)

= q∗ (r0 )(t ∗ − r0 ) and

∂D = ∂ r1



t∗ r1

q∗ (r1 )dt

(50)

= q∗ (r1 )(t ∗ − r1 ). in which q∗ (r0 ) and q∗ (r1 ) respectively represent the discharge flow from the stopline at time r0 and r1 . Both q∗ (r0 ) and q∗ (r1 ) take the value of qmax (i.e. the saturation flow) if the discharged flow is unrestricted from downstream congestion. If there is downstream blockage (see Chow and Lo, 2007), q∗ (r0 ) and q∗ (r1 ) will be determined by the available space at the downstream through the underlying kinematic wave model. Incorporating q∗ (r0 ) and q∗ (r1 ) in the derivatives (49) and (50) automatically captures both free-flow and congested conditions. This is similar to the analysis adopted in Chow and Lo (2007) which use the cell transmission model instead of the variational method as presented herein. Following (49), (50), and the definition of average delay (23), the corresponding change in average delay d of each vehicle can be derived by dividing the total system delay D by the total number of vehicles, G(c), arriving within the associated signal cycle c, i.e.

∂d 1 ∗ = q (r0 )(t ∗ − r0 ) ∂ r0 G ( c )

(51)

∂d 1 ∗ = q (r1 )(t ∗ − r1 ). ∂ r1 G ( c )

(52)

and

where G(c ) = [N (x1 , c1 ) − N (x0 , c0 )], times c0 and c1 are respectively the start and end times of the current cycle c. Please cite this article as: A.H.F. Chow et al., Multi-objective optimal control formulations for bus service reliability with traffic signals, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.02.006

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4.3. Solution algorithm With the optimality conditions and the derivatives, we now present an open-loop solution algorithm for solving the optimal control formulations. The kinematic wave model is used to estimate the bus running times and traffic delays as well as their likely changes with respect to timing adjustments u as derived in the previous section. The control algorithm is implemented on a fixed space-time horizon for all buses n = 1, 2 . . . , nall , and is solved by a gradient search (see some examples in Yang and Huang, 1997; Chow, 2009b). ∂η (u ) As can be seen in the previous section, the derivatives of bus journey times with respect to signal settings (i.e. ∂ u n,s(r ,c ) ∂η

m

(u )

0

and ∂ u n,s(r ,c ) ) are discontinuous. Hence we may only be able to come up with a solution with which the corresponding m 1 ∂ H1 ∂ H1 ∂ H2 ∂ H2 ∂ u (r ,c ) and ∂ u (r ,c ) in (41) (or ∂ u (r ,c ) and ∂ u (r ,c ) in (42), depending on which objective function is used) can be made m

m

0

1

m

0

m

1

as close as zero, instead of being zero (see Step 5 below). The solution procedure is listed as follows: (0 ) (0 ) 1. Set um (r0 , c ) = 0 and um (r1 , c ) = 0 for all intersections m and cycles c as the initial solution, and set the iteration index k = 0; 2. Run the underlying traffic model, with prevailing traffic demand, signal settings, and initial locations of all buses n. This is to obtain the estimates of running times η = ηn,s (u(k ) ) and delays d = dn,m (u(k ) ) under current u(k ) = (k ) (k ) [um ( r0 , c ), um (r1 , c )] in the current iteration k; 3. Use the state Eq. (33), calculate iteratively forward over s the trajectory of the state variable  = n,s with η = ηn,s (u(k ) ); 4. Use the costate Eq. (39) (or (40) depending on the objective function) to calculate iteratively backward in s the trajectory of the costate variables λ = λn,s for all n based on the estimated state variables  = n,s ; 5. Define

A1 (u ) = λn,s+1

sM  ∂ηn,s (u ) ∂ dn,m (u ) + (1 − α ) , ∂ um ( r0 , c ) ∂ um ( r0 , c ) m=s1

(53)

A2 (u ) = λn,s+1

sM  ∂ηn,s (u ) ∂ dn,m (u ) + (1 − α ) , ∂ um ( r1 , c ) ∂ um ( r1 , c ) m=s1

(54)

and

respectively as the searching direction for um (r0 , c) and um (r1 , c) in u(k) . Do the following iteratively for bus from n = 1 to nall : For each stop pair (s, s + 1 ) and signal cycle c, if

∂ηn,s (u(k ) ) (k+1 ) = 0, perform a line search for um (r0 , c ) in [umin , 0] in case (k ) ∂ um (r0 ,c ) (k+1 )

of A1 (u ) < 0; or in [0, umax ] in case of A1 (u ) > 0, such that the new solution ui zero. Here, the admissible values of u is specified by the range [umin , umax ]. Likewise, for each stop pair (s, s + 1 ) and signal cycle c, if

(r0 , c ) will give A1 (u ) closest to

∂ηn,s (u(k ) ) (k+1 ) = 0, perform a line search for um (r1 , c ) in [umin , (k ) ∂ um (r1 ,c ) (k+1 )

0] in case of A2 (u ) > 0; or in [0, umax ] in case of A2 (u ) < 0 such that the new solution um closest to zero;

(r1 , c ) will give A2 (u )

6. Check if either ||u(k+1 ) − u(k ) || ≤ χ where χ is a predefined convergence criterion, or if the predefined maximum number of iterations K has been met. If so, stop the procedure and return u(k+1 ) as the final solution. If not, return to Step 2 and repeat the procedure with the updated solution u(k+1 ) . The algorithm presented above focuses on the use of traffic signals and their timings can be adjusted to improve bus service reliability with consideration of traffic delay. However, it should be noted that the algorithm presented herein is generic. By substituting the control variables u and the corresponding derivatives presented in Section 4.3, the algorithm can also be applied to derive optimal bus holding, stop skipping, and other bus schedule control strategies. Given the fact the non-signal based strategies (e.g. bus holding and stop skipping) will not alter the signal timing and hence will not affect the distribution of traffic delays in the network. Consequently, the delay objective (i.e. objective function Z0 ) may be dropped when dealing with these strategies. Hence, the multi-objective optimisation problem proposed in this paper can be reduced to a standard single objective problem aiming to minimise bus headway or schedule deviations. The resulting problem can then be solved by a number of existing models and algorithms (e.g. Hickman, 2001; Sun and Hickman, 2015). Moreover, considering the independence between the non-signal based strategies and signal timings, the integrated optimal control strategies can actually be determined through a two-stage procedure. One may first determine the optimal signal timing adjustments through solving the multi-objective optimal control formulation as discussed above. Based upon the signal timing adjustments and the corresponding bus headway or schedule deviations determined, we can then run the single objective bus holding / stop skipping optimisation problem using the proposed solution procedure or existing algorithms (e.g. Hickman, 2001; Sun and Hickman, 2015). Please cite this article as: A.H.F. Chow et al., Multi-objective optimal control formulations for bus service reliability with traffic signals, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.02.006

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Fig. 8. Tottenham Court Road (TCR), London, UK.

5. Numerical experiments 5.1. Setting The control system developed in Section 4 is applied to a test arterial which mimics a real world scenario of a 0.9mile long section of Tottenham Court Road (TCR) in Central London, UK (Fig. 8). The road section consists of three lanes and five signal-controlled intersections: Bayley Street (N02/056), Goodge Street (N02/058), Torrington Place (N02/062), University Street (N02/060), and Grafton Way (N02/059). The intersections are equipped with loop detectors (where the indices in the brackets are the IDs of these detectors adopted by Transport for London) from which we can obtain data of traffic volumes and signal timings. Following the field measurements, we take the saturation flow along the main arterial as 1750veh/h/lane, while the mean hourly inflow to the main arterial through Bayley Street (detector: N02/056) is 755veh/h/lane over the simulation period. The incoming flow from the cross streets, which all consist of one lane, into the main arterial through each of the five intersection is taken as 450veh/h. The total outflow rate from the main arterial into the cross streets at each intersection is taken as 325veh/h. The intersections are running at a common cycle time at 90s. The figure also shows the locations of the six bus stops in between the intersections along the corridor. In reality, there are seven bus routes (No. 10, 14, 24, 29, 73, 134, and 390) passing through the corridor and stopping at these stops. Again, in this study we focus on bus route #73 which is the most frequent service (@3min) among all. Our methodology, however, can be applied to accommodate multiple bus service lines without loss of generality. The traffic dynamics along the arterial is represented by a variational formulation of kinematic wave model with triangular fundamental diagram, where we take the free-flow speed vf as 35 km/h, congestion shockwave speed vb as 10 km/h, and Q as 1750veh/h/lane following field observations. The nominal running times φ s (i.e. running times with no congestion) between each stop pair (s, s + 1 ) are determined from the field as 33s, 49s, 51s, 48s, and 23s respectively for s = 1, 2 . . . , 5. The simulation period is to be 70min with a set of 20 buses entering the arterial. The nominal headway between the buses is supposed to be 3 min. To mimic the variations in headways in real world, Fig. 9 shows the randomly generated bus headways between the each bus pair at which they are loaded into the TCR arterial. The realised bus headways in Fig. 9 has an average value of 176s and a standard deviation 21s. To simplify the calculations, we assume that the buses will have the same nominal speed as surrounding traffic under free-flow (i.e. v∗n = v f in (26)), while this assumption can be easily relaxed through the variational method of kinematic wave model presented in the previous section. This paper tests two control objectives: one aims to restore the bus service back to the planned schedule taking traffic delays into account (i.e. based upon objective function (28)), the other one aims to restore the nominal headway rather than referring to the underlying service schedule (i.e. based upon objective function (31)). Based on the bus operational state Eqs. (1)–(3), we construct the scheduled arrival time τˆn,s recursively over bus n and stop s as

θˆn,1 = θˆn−1,1 + hˆ n ,

(55)

τˆn,s+1 = θˆn,s + φs ,

(56)

and

Please cite this article as: A.H.F. Chow et al., Multi-objective optimal control formulations for bus service reliability with traffic signals, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.02.006

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Fig. 9. Realised headways between buses entering TCR arterial. Table 1 Timetable of 20 buses over the TCR arterial (τˆn,s , in sec).

Bus Bus Bus Bus Bus Bus Bus Bus Bus Bus Bus Bus Bus Bus Bus Bus Bus Bus Bus Bus

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Stop 1

Stop 2

Stop 3

Stop 4

Stop 5

Stop 6

180 360 540 720 900 1080 1260 1440 1620 1800 1980 2160 2340 2520 2700 2880 3060 3240 3420 3600

233 413 593 773 953 1133 1313 1493 1673 1853 2033 2213 2393 2573 2753 2933 3113 3293 3473 3653

302 482 662 842 1022 1202 1382 1562 1742 1922 2102 2282 2462 2642 2822 3002 3182 3362 3542 3722

373 553 733 913 1093 1273 1453 1633 1813 1993 2173 2353 2533 2713 2893 3073 3253 3433 3613 3793

441 621 801 981 1161 1341 1521 1701 1881 2061 2241 2421 2601 2781 2961 3141 3321 3501 3681 3861

484 664 844 1024 1204 1384 1564 1744 1924 2104 2284 2464 2644 2824 3004 3184 3364 3544 3724 3904

where θˆn,s is the nominal departure time of bus n from stop s, which is determined as

θˆn,s = τˆn,s + κˆ s .

(57)

This scheduled arrival time τˆn,s is constructed and regarded as a timetable to measure the schedule deviations of each bus in (28). When solving (56) and (57), we take the initial value θˆ1,1 = 200s (i.e. the departure time of the first bus from the first stop), and κˆ s = 20s as the nominal dwell time for buses at each stop s. We set θˆ1,1 = 200s (which implies the corresponding

arrival times τˆ1,1 = θˆ1,1 − κˆ 1 = 180) after the start time of the simulation (t = 0) as a warm-up period to ensure the traffic flow in the network has reached the equilibrium distribution (Gomes et al., 2008) before loading in the first bus. Table 1 shows the corresponding timetable constructed, in which the numbers are the scheduled arrival times τˆn,s (in sec) of each bus at each stop. It can be seen from the table that the arrival time of the last bus n = 20 at the last stop s = 6 is 3904s, which is well before the end time of the simulation 4200s. We can hence ensure all buses can be cleared by the end of simulation. Finally, the convergence criterion, χ of the optimisation process presented in Section 4.3 is set to be 1s and the maximum number of iterations K in the optimisation process is set to be 10. Due to the lack of road space for storing buses on hold at stops, as well as the high passenger demand along the bus route for allowing buses to skip stop(s), the non-signal based bus schedule control strategies (e.g. bus holding, stop skipping, etc) will not be feasible in this scenario. The dwell time of each bus at each stop is subject to a coefficient of variation of 15% with respect to this nominal value in order to represent Please cite this article as: A.H.F. Chow et al., Multi-objective optimal control formulations for bus service reliability with traffic signals, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.02.006

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A.H.F. Chow et al. / Transportation Research Part B 000 (2017) 1–21 Table 2 Root mean square of bus headway deviations (unit: sec) under different α . a) Headway control

α=0 α = 0.25 α = 0.5 α = 0.75 α=1

Stop 2

Stop 3

Stop 4

Stop 5

Stop 6

Overall

25.88 24.96 24.27 23.69 21.72

36.03 32.72 30.01 28.33 26.34

27.27 25.62 24.59 24.23 23.11

35.90 33.37 31.62 30.78 28.89

26.97 26.37 27.05 26.01 23.92

30.41 28.61 27.51 26.61 24.80

b) Schedule control

α=0 α = 0.25 α = 0.5 α = 0.75 α=1

Stop 2

Stop 3

Stop 4

Stop 5

Stop 6

Overall

25.88 25.88 25.88 25.88 26.94

36.03 33.44 33.71 34.93 33.82

27.27 28.13 24.70 24.54 25.01

35.90 33.61 33.50 33.61 33.86

26.97 31.48 25.26 24.70 24.97

30.41 30.51 28.61 28.73 28.92

Table 3 Root mean square of bus schedule delays (unit: sec) under different α . a) Headway control

α=0 α = 0.25 α = 0.5 α = 0.75 α=1

Stop 2

Stop 3

Stop 4

Stop 5

Stop 6

Overall

108.61 107.81 106.95 106.89 106.11

99.41 103.18 103.11 102.78 102.53

82.62 90.53 89.96 89.38 89.13

78.16 84.26 83.48 82.84 82.13

73.51 75.34 74.96 73.95 73.13

88.46 92.22 91.69 91.17 90.61

Stop 4

Stop 5

Stop 6

Overall

82.62 79.03 78.95 75.87 71.23

78.16 73.57 71.10 69.08 68.95

73.51 71.02 68.83 65.78 61.76

88.46 85.89 84.76 82.37 79.28

b) Schedule control

α=0 α = 0.25 α = 0.5 α = 0.75 α=1

Stop 2

Stop 3

108.61 107.41 106.53 104.34 99.83

99.41 98.44 98.37 96.76 94.63

the stochastic variations observed in the real world. Following the experiment design in Chow and Li (2014), each derived control policy will be evaluated over 100 Monte-Carlo simulation runs to incorporate the dwell time variability. 5.2. Results and discussion The nature of the multi-objective optimal solutions can be examined through a Pareto analysis which derives optimal solutions with different settings of the parameter α Reilly et al. (2016). Table 1 shows the root mean square (RMS)of bus headway deviations (unit: sec) measured over the 100 Monte Carlo runs at Stops s = 2, 3, . . . , 6 along the arterial under the two control strategies over different α , where α = 0, 0.25, 0.5, 0.75 and 1. If α = 0, the timing plan will be adjusted in order to minimise the average delay d while taking no consideration of bus schedule or headway deviations. On the other hand, if α = 1, the control system will simply aim at minimising bus schedule or headway deviations with no consideration of its impact to surrounding traffic delay. Different value of α represents the trade-off taken into account between bus service reliability and surrounding delay under the control system, in which bus service reliability and surrounding delay are regarded as equally important if α = 0.5. The effect of setting different α can be revealed from Table 2 in which the RMS of bus headway deviations decrease as α increases under the headway based control system. It also reflects the effectiveness of the proposed optimal control formulation which aims to improve bus service reliability with fine adjustment of timing plans. Nevertheless, the schedule based control strategy does not appear to bring in significant effect on headway deviations. However, it should not be surprising due to the difference in the objective function setting. As revealed from other previous studies (e.g. Xuan et al., 2011), minimising schedule and headway deviations are not necessarily equivalent. To illustrate this, Table 3 shows the RMS of bus schedule delays (unit: sec) measured over the Monte Carlo runs at Stops s = 2, 3, . . . , 6 under the two control strategies over different α . Unlike the results shown in Tables 2 and 3 shows the effectiveness of the proposed schedulebased controller in terms of minimising the schedule deviations over different α , while the headway based control is not as effective in the latter case. One can also note that the magnitudes of the schedule deviations (generally in the range of 80s to 90s) is significantly larger than those of the headway deviations (20s to 30s). It is because schedule deviation is Please cite this article as: A.H.F. Chow et al., Multi-objective optimal control formulations for bus service reliability with traffic signals, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.02.006

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17

Table 4 Induced average traffic delays (unit: veh-sec/veh) under different control scenarios (A: main arterial; C: cross streets). a) Headway control

α=0 α = 0.25 α = 0.5 α = 0.75 α=1

A C A C A C A C A C

Bayley

Goodge

Torrington

University

Grafton

Total

Overall

12.92 12.34 13.44 12.09 14.22 11.98 16.24 10.72 19.13 7.99

12.15 14.67 12.89 13.97 14.35 12.45 16.49 10.74 18.35 8.99

13.26 12.57 14.08 11.83 15.18 10.71 17.36 9.37 19.07 8.12

12.88 14.92 12.59 15.33 12.12 16.16 10.97 17.94 10.36 18.48

13.40 18.11 13.52 17.97 14.61 16.89 17.16 14.93 18.92 13.08

12.92 14.52 13.30 14.24 14.10 13.64 15.64 12.74 17.17 11.33

13.71 13.75 13.92 14.21 14.29

b) Schedule control

α=0 α = 0.25 α = 0.5 α = 0.75 α=1

A C A C A C A C A C

Bayley

Goodge

Torrington

University

Grafton

Total

Overall

12.92 12.34 13.61 11.85 14.41 11.28 16.68 10.43 19.78 7.68

12.15 14.67 11.54 15.31 10.82 16.17 10.11 17.13 9.86 17.93

13.26 12.57 14.32 11.39 15.86 10.18 18.12 8.92 20.01 7.02

12.88 14.92 12.39 15.48 11.77 16.44 11.16 17.18 10.23 19.06

13.40 18.11 13.12 18.52 12.79 19.32 11.87 22.08 10.59 24.09

12.92 14.52 13.00 14.51 13.13 14.68 13.59 15.15 14.09 15.16

13.71 13.76 13.97 14.37 14.71

an absolute measure looking at the discrepancies between actual arrival times and fixed predefined arrival times in which discrepancies can be accumulating over buses and stops. On the other hand, headway deviations are relative measures which only consider discrepancies in headways between two successive buses. In fact, as shown in Daganzo (2009), the schedule discrepancy can theoretically grow arbitrarily large over bus routes, while headway discrepancies do not. In addition to the bus service reliability (in terms of both schedule and headway), the control system also takes into account of its impact to the surrounding traffic delay. Table 4 shows the impact of the two proposed control systems on the arterial (i.e. bus route) (A), the cross streets (C), and the overall average delays (unit: veh-sec/veh) on the TCR test corridor over different value of α . As expected, the system delivers the lowest delays when α = 0, i.e. no bus control is implemented. The delays increase with the value of α as the control systems are more willing to trade delay for bus service reliability. Moreover, it can be seen that the schedule-based controller causes more delay to surrounding traffic in particular when α is large (say, at 0.75 and 1). It is because of the larger magnitudes of the bus schedule deviations than the headway deviations, which induces more adjustments in the timing plans and hence induces more additional delays to surrounding traffic. Table 4 also reveals the characteristics and differences between the schedule based and headway based strategies. As can be seen from Fig. 9, the buses are entering the corridor at shorter headways than planned, which also implies most buses would arrive earlier than the time when they were scheduled. Consequently, both strategies operate by allocating less green to the bus route (A) at the first intersection (Bayley) as the buses enter the corridor in order to slow down the buses and hence spread out their headways or restore their schedule. This can be reflected by the increased delays on the bus route (A) at Bayley for all α > 0 compared with the case when α = 0 (i.e. no bus control applied). An interesting observation is that the schedule based controller would operate in an opposite sense at the following intersection (Goodge), at which it would increase the green allocated to the main street (A) (reflected by the decreased delays with α > 0) in order to speed up the buses and hence minimise their schedule discrepancies at stop s = 3, 4, ... Such policy however can induce instability in headway distributions due to the inconsistency between the control actions at two successive intersections. On the other hand, the headway based strategy focuses on balancing the bus headways instead of restoring the nominal schedule, and hence it keeps reducing green allocated to the bus route (A). As a graphical illustration, Fig. 10 presents the Pareto frontiers by plotting the optimal results from Tables 2–4, in which each data point of Z0∗ , Z1∗ , and Z2∗ on the frontiers represents respectively the optimal values of average delay, bus schedule deviations, and bus headway deviations obtained from different setting of α as shown in Tables 2–4. Combinations of (Z0∗ , Z1∗ ) in Fig. 10a and (Z0∗ , Z2∗ ) in Fig. 10b below the frontier corresponding the infeasible region in which those values of (Z0∗ , Z1∗ ) and (Z0∗ , Z2∗ ) are not attainable by a feasible u given the current network, demand, and timing plan settings. It is also noticed that the data points in Fig. 10 do not fall on a smooth curve. It is due to various reasons including the stochasticity (e.g. the dwell time variability) involved in the simulations, discrete nature of the solution space (the signal timing adjustments have to be in integers), as well as the discontinuity of the derivatives which prevent the solution process from reaching a continuous optimal solution. Please cite this article as: A.H.F. Chow et al., Multi-objective optimal control formulations for bus service reliability with traffic signals, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.02.006

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Fig. 10. Pareto frontiers derived from the optimal solutions.

Fig. 11. Green allocated to bus route under headway based control (α = 0.5).

To gain further insight, Figs. 11 and 12 shows respectively the profiles of green allocated to the bus route (A) over cycle under headway-based control and schedule-based control at the three intermediate intersections: Goodge, Torrington, and University when α is set to be one. The nominal green times at Goodge, Torrington, and University are 44s, 44s, and 50s respectively. At each intersection, a green duration higher than the nominal green implies that the traditional green extension and recall are used to facilitate the progress of a bus. In contrast, a green duration lower than the nominal green implies the red extension and recall are carried out to slow down an early arriving bus. Fig. 12 reveals that the adjustments made under the schedule-based control tend to be monotonic, which means timing adjustments are either all in green extension and recall mode (at Goodge and University streets) or all in red extension and recall mode (Torrington Place). The reason of this is that, in urban areas where buses are running at short headways and if there is no significant shortPlease cite this article as: A.H.F. Chow et al., Multi-objective optimal control formulations for bus service reliability with traffic signals, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.02.006

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Fig. 12. Green allocated to bus route under schedule based control (α = 0.5).

term variations in prevailing traffic flows and signal timings, an early (late) running bus will tend to be keep running early (late) should its service reliability is measured against a fixed clock time schedule (Daganzo, 2009). Consequently, the signal controller hence will keep making the same (and frequent) adjustments to slow down the early running buses (or speed up the late running buses) such that they can catch up with the predefined fixed clock time schedule. In contrast, Fig. 11 shows that the headway-based control adopts more evenly both green and red extensions (and recalls) in order to regulate the headway deviations instead of simply catching up the predefined schedule. Because of this different nature of the two control settings, the results also reveal that the schedule-based control will induce more disruptions to the surrounding traffic and result in more extra delays as shown in Table 4. This also supports a number of previous studies (e.g. Daganzo, 2009; Xuan et al., 2011) that headway based control strategy is more practical than the schedule based strategy when being applied to high frequency bus service in urban areas. The bus service reliability presented herein may further be enhanced by incorporating other non-signal based control policies, such as bus holding and stop skipping, in addition to the signal timing adjustments. It is noted these non-signal based strategies will not alter the signal timing setting, and hence they will not affect the distribution of general traffic delays. Consequently, the optimisation process for these strategies can be reduced from the proposed multi-objective formulation to the conventional one with single objective that aims to minimise the headway or schedule deviations with the appropriate definition of the control variables and use of the derivatives as discussed in Section 4.3. The ultimate optimisation setting will be similar to those proposed by Eberlein et al. (2001); Hickman (2001), and others. 6. Concluding remarks This paper presents a multi-objective optimal control framework for maximising bus service reliability in congested urban networks through adjusting signal timings. The objectives considered herein include both deviations with respect to predefined bus schedule and headway, as well as corresponding additional delays induced to general traffic. The interaction between buses and its surrounding traffic is captured by the variational formulation of kinematic wave model. The control system is applied to a test arterial constructed based upon a real case in Central London, UK. This paper generates new insights on managing bus reliability with consideration of its impact to surrounding traffic. In particular, the results highlight the performances of both schedule based and headway based strategies, and also justify the use of headway based approach for regulating reliability of high frequency bus service in urban areas with consideration of surrounding traffic delay. The kinematic wave model is used here due to the consideration of their capability of capturing formation and dissipation of traffic queues, as well as the physical dimension of traffic (Chow and Lo, 2007). Nevertheless, it should be noted that the control framework presented herein is generic. This implies the system control designers can resort to other kinds of traffic model such as the point queue models (Zhang et al., 2013), travel time functions (Chow, 2009a), or even empirical time series models (Cheng et al., 2014) for estimating bus running times and headway deviations. The choice of underlying models should not affect the applicability of the control framework presented in the paper. Further developments also include extending the current open-loop offline solution algorithm to real time control through the rolling horizon framework (Aboudolas et al., 2010). For real time operations, the control algorithm will require online feeding of traffic and bus data through loop detector, GPS (Global Positioning System) devices, and/or other sensing systems (see examples in Chow et al., 2014; Chow, 2016; Ottaviano et al., 2017). The algorithm can be called when a bus n0 approaches an controllable intersection upstream of a stop s0 . Given the initial value of n0 ,s0 −1 measured for the bus at previous stop s − 1, the control problem aims to determine the green time adjustments to be made at that intersection Please cite this article as: A.H.F. Chow et al., Multi-objective optimal control formulations for bus service reliability with traffic signals, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.02.006

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such that the estimated total schedule deviation over stops s0 , s0 + 1, . . . ,s0 + S is minimised:

Z1− =

s0 +S 1 π s−(s0 +1) n20 ,s 2 s=s

(58)

s0 +S 1 π s−(s0 +1) (n0 ,s − n0 −1,s )2 2 s=s

(59)

0

or:

Z2− =

0

if the objective is to minimise headway deviation. The objectives (58) and (59) are both subject to traffic delay calculated by Z0 and bus dynamics described by the underlying traffic model. Compared with the offline objective function (29) and (32), a discount factor π ∈ (0, 1) is added to the objective functions (58) and (59) in order to ensure stability and convergence of the solution algorithm (Bertsekas, 2007). It is noted the a larger(smaller) α implies more(less) consideration will be given to  at further s away from s0 Bertsekas (2007). Finally, there are a number of other strategies proposed to mitigate the schedule deviations or irregular headways in addition to signal timings. Existing examples include deadheading (Ceder and Stern, 1981; Eberlein et al., 1998), stop skipping (Liu et al., 2013; Sun and Hickman, 2015), and driver guidance (ArgoteCabanero et al., 2016). Further discussion regarding the implementation and testing of the rolling horizon concept as well as integration with other control strategies are beyond the present study and will be reported in the future. Acknowledgements We acknowledge Transport for London (TfL) for providing the ANPR journey times, SCOOT loop detectors, and bus operational data. The contents however do not reflect the official views or policies of TfL or any other organisations. The project is funded by a UK Royal Society (Ref: IE131317) and National Natural Science Foundation of China (NSFC, Ref: 513111163) research exchange scheme. References Aboudolas, K., Papageorgiou, M., Kouvelas, A., Kosmatopoulos, E., 2010. A rolling-horizon quadratic-programming approach to the signal control problem in large-scale congested urban road networks. Transp. Res. Part C 18, 680–694. Argote-Cabanero, J., Daganzo, C.F., Lynn, J., 2016. Dynamic control of complex transit systems. Transp. Res. Part B 81 (1), 146–160. Bartholdi III, J., Eisenstein, D., 2012. A self-coordinating bus route to resist bus bunching. Transp. Res. Part B 46, 481–491. Berrebi, S., Watkins, K., Laval, J., 2015. A real-time bus dispatching policy to minimize passenger wait on a high frequency route. Transp. Res. Part B 81, 377–389. Bertsekas, D., 2007. Dynamic Programming and Optimal Control Vol. I. Athena Scientific, Nashus, NH. Bowman, L., Turnquist, M., 1981. Service frequency, schedule reliability and passenger wait times at transit stops. Transp. Res. Part A 15 (6), 465–471. Boyd, S., Vandenberghe, L., 2004. Convex Optimization. Cambridge University Press, Cambridge, UK. Ceder, A., Stern, H., 1981. Deficit function bus scheduling with deadheading trip insertion for fleet size reduction. Transp. Sci. 15 (4), 338–363. Cheng, T., Wang, J., Haworth, J., Heydecker, B.G., Chow, A.H.F., 2014. A dynamic spatial weight matrix and localized space-time autoregressive integrated moving average for network modeling. Geogr. Anal. 46, 75–97. Chow, A.H.F., 2009a. Dynamic system optimal traffic assignment - a state-dependent control theoretic approach. Transportmetrica 5 (2), 85–106. Chow, A.H.F., 2009b. Properties of dynamic system optimal assignment and its solution method. Transp. Res. Part B 43 (3), 325–344. Chow, A.H.F., 2016. Heterogeneous urban traffic data and their integration through kernel-based interpolation. J. Facil. Manage. 14 (2), 165–178. Chow, A.H.F., Gomes, G., Kurzhanskiy, A.A., Varaiya, P., 2010. Aurora RNM - a macroscopic tool for arterial traffic analysis. In: Proceedings of 89th Annual Meeting of Transportation Research Board. Washington, DC, USA. Chow, A.H.F., Li, S., Szeto, W.Y., Wang, D., 2015. Modelling urban traffic dynamics based upon the variational formulation of kinematic waves. Transportmetrica B 3 (3), 169–191. Chow, A.H.F., Li, Y., 2014. Robust optimization of dynamic motorway traffic via ramp metering. IEEE Trans. Intell. Transp. Syst. 15 (3), 1374–1380. Chow, A.H.F., Lo, H., 2007. Sensitivity analysis of signal control with physical queuing: delay derivatives and an application. Transp. Res. Part B 41 (3), 462–477. Chow, A.H.F., Santacreu, A., Tsapakis, I., Tanasaranond, G., Cheng, T., 2014. Empirical assessment of urban traffic congestion. J. Adv. Transp. 48 (8), 10 0 0–1016. Chow, A.H.F., Sha, R., 2016. Performance analysis of centralised and distributed systems for urban traffic control. Transp. Res. Rec. 2557, 66–76. Daganzo, C.F., 1997. Fundamentals of Transportation and Traffic Operations. Emerald, Bingley, UK. Daganzo, C.F., 2005a. A variational formulation of kinematic waves: basic theory and complex boundary conditions. Transp. Res. Part B 39 (2), 187–196. Daganzo, C.F., 2005b. A variational formulation of kinematic waves: solution methods. Transp. Res. Part B 39 (10), 934–950. Daganzo, C.F., 2009. A headway-based approach to eliminate bus bunching: systematic analysis and comparisons. Transp. Res. Part B 43, 913–921. Daganzo, C.F., Laval, J., 2005. Moving bottlenecks: a numerical method that converges in flows. Transp. Res. Part B 39, 855–863. Daganzo, C.F., Menendez, M., 2005. A variational formulation of kinematic waves: bottleneck properties and examples. In: Transportation and Traffic Flow Theory, pp. 345–364. Daganzo, C.F., Pilachowski, J., 2011. Reducing bunching with bus-to-bus cooperation. Transp. Res. Part B 45, 267–277. Delgado, F., Munoz, J.C., Giesen, R., 2012. How much can holding and/or limiting boarding improve transit performance. Transp. Res. Part B 46, 1202–1217. Eberlein, X., Wilson, N., Barnhart, C., Bernstein, D., 1998. The real-time deadheading problem in transit operations control. Transp. Res. Part B 32 (2), 77–100. Eberlein, X., Wilson, N., Bernstein, D., 2001. The holding problem with real-time information available. Transp. Sci. 35 (1), 1–18. Gibbens, R., Saatci, Y., 2008. Data, modelling and inference in road traffic networks. Philos. Trans. R. Soc. A 366, 1907–1919. Gomes, G., Horowitz, R., Kurzhanskiy, A., Varaiya, P., Kwon, K., 2008. Behavior of the cell transmission model and effectiveness of ramp metering. Transp. Res. Part C 16 (4), 485–513. Gu, W., Cassidy, M., Gayah, V., Ouyang, Y., 2013. Mitigating negative impacts of near-side bus stops on cars. Transp. Res. Part B 47, 42–56. Gu, W., Gayah, V., Cassidy, M., Saade, N., 2014. On the impacts of bus stops near signalized intersections: models of car and bus delays. Transp. Res. Part B 68, 123–140. Guler, I., Menendez, M., 2014. Analytical formulation and empirical evaluation of pre-signals for bus priority. Transp. Res. Part B 64, 41–53. Hadasa, Y., Shnaidermanb, M., 2012. Public-transit frequency setting using minimum-cost approach with stochastic demand and travel time. Transp. Res. Part B 46 (8), 1068–1084.

Please cite this article as: A.H.F. Chow et al., Multi-objective optimal control formulations for bus service reliability with traffic signals, Transportation Research Part B (2017), http://dx.doi.org/10.1016/j.trb.2017.02.006

JID: TRB

ARTICLE IN PRESS A.H.F. Chow et al. / Transportation Research Part B 000 (2017) 1–21

[m3Gsc;March 14, 2017;17:7] 21

Hans, E., Chiabaut, N., Leclercq, L., 2015. Applying variational theory to travel time estimation on urban arterials. Transp. Res. Part B 78, 281–313. Hickman, M., 2001. An analytic stochastic model for the transit vehicle holding problem. Transp. Sci. 35 (3), 215–237. Hounsell, N., Shrestha, B., Head, J., Palmer, S., Bowen, T., 2007. The way ahead for Londons bus priority at traffic signals. IET Intel. Transport Syst. 2 (3), 193–200. Laval, J., 2004. Hybrid Models of Traffic Flow: Impacts of Bounded Vehicle Accelerations. University of California Berkeley PhD thesis. Liu, Z., Yan, Y., Qu, X., Zhang, Y., 2013. Bus stop-skipping scheme with random travel time. Transp. Res. Part C 35, 46–56. Ma, W., Yang, X., Liu, Y., 2010. Development and evaluation of a coordinated and conditional bus priority approach. Transp. Res. Rec. 2145, 49–58. Mazare, P., Dehwah, A., Claudel, C.G., Bayen, A.M., 2011. Analytical and grid-free solutions to the lighthill-whitham-richards traffic flow model. Transp. Res. Part B 45, 1727–1748. Newell, G., Potts, R., 1964. Maintaining a bus schedule. In: Proceedings of the 2nd Australian Road Research Board, 2, pp. 388–393. Newell, G.F., 1998. A moving bottleneck. Transp. Res. Part B 32, 531–537. Ottaviano, F., Cui, F., Chow, A.H.F., 2017. Modelling and fusion of dynamic highway traffic data. Transp. Res. Rec. In press. Pontryagin, L.S., Boltyanski, R., Gamkrelidze, R., Mischenko, E., 1962. The mathematical theory of optimal processes. Interscience. Reilly, J., Martin, S., Payer, M., Bayen, A., 2016. Creating complex congestion patterns via multi-objective optimal freeway traffic control with application to cyber-security. Transp. Res. Part B 91, 366–382. Sun, A., Hickman, M., 2015. The real-time stop skipping problem. Transp. Syst. 9 (2), 91–109. Wong, S.C., 1995. Derivatives of performance index for the traffic model from TRANSYT. Transp. Res. Part B 29 (5), 303–327. Xuan, Y., Argote, J., Daganzo, C.F., 2011. Dynamic bus holding strategies for schedule reliability: optimal linear control and performance analysis. Transp. Res. Part B 45, 1831–1845. Yang, H., Huang, H.J., 1997. Analysis of the time-varying pricing of a bottleneck with elastic demand using optimal control theory. Transp. Res. Part B 31 (6), 425–440. Yperman, I., 2007. The Link Transmission Model for Dynamic Network Loading. Katholieke Universiteit Leuven, Belgium PhD thesis. Zhang, H., Nie, Y., Qian, Z., 2013. Modelling network flow with and without link interactions: the cases of point queue, spatial queue and cell transmission model. Transportmetrica B 1 (1), 33–51.

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