Model formulation and solution algorithm of traffic signal control in an urban network

Computers, Environment and Urban Systems 24 (2000) 355±377 www.elsevier.com/locate/compenvurbsys

Model formulation and solution algorithm of trac signal control in an urban network Wann-Ming Wey * Graduate School of Architecture and Urban Design, Chaoyang University of Technology, 168 Gifeng E. Road, Wufeng, Taichung County, Taiwan, ROC

Abstract The existing network trac signal optimization formulations usually do not include trac ¯ow models, except for control schemes such as SCOOT (Split, Cycle, and O€set Optimization Technique) system that uses simulation for heuristic optimization. Other conventional models normally use isolated intersection optimization with trac arrival prediction using detector information, or optimization schemes based on green bandwidth approach such as MAXBAND (Maximal Bandwidth). In this paper we present a complete formulation of the problem that includes explicit constraints to model the movement of trac along the streets between the intersections in a time-expanded network, as well as constraints to capture the permitted movements from modern signal controllers. The platoon dispersion model used is the well-known Robertson's model, which forms linear constraints. Thus it is a rare example of a trac simulation being analytically embedded in an optimization formulation. The formulation is an integer-linear program, and does not assume ®xed cycle lengths or phase sequences. It assumes full information on external inputs, but can be incorporated in a sensorbased environment, as well as in a feedback control framework. The formulation is an integerlinear program that may not be eciently solved with standard simplex and branch and bound techniques. We discuss network programming formulations to handle the linear platoon dispersion equations and the integer constraints at the intersections. A special-purpose network simplex algorithm for fast solution is also mentioned. # 2000 Elsevier Science Ltd. All rights reserved. Keywords: Network programming formulation; Network simplex algorithm; Rolling horizon; Platoon dispersion

* Tel.: +886-4-3323000 ext. 7153; fax: +886-4-3742339. E-mail address: [email protected] (Wann-Ming Wey). 0198-9715/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S0198-9715(00)00002-8

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Nomenclature N n In Oi i Li T K k
T H E qiin …k† qis …k† qigo …k† Sgni

Syni

number of intersections index used to refer to one of the N intersections set of signal phases at the control intersection n; n  N set of signal phases at the start node of the from_link of a given phase i that feeds trac for phase i signal phase at a given node n, i  In from_link of phase i the time horizon under consideration speci®ed in seconds number of discrete
T time intervals in the optimization period time interval index the sample time interval of duration (s) number of links in the network, including entrances number of entrances and exits upstream in¯ow for phase i over a period [k
T, (k+1)
T] (veh/s) the ¯ow which arrives at the end of the waiting queue or at the stopline (veh/s) the capacity ¯ow for green trac light (veh/s) saturation ¯ow for green time of phase i at intersection n (veh/s)

saturation ¯ow for yellow time of phase i at intersection n (veh/s) 0 if signal state is green for phase i at intersection n and time step k, and 1 if signal state is red for phase i at intersection n and time step k ni …k† 1 if existing signal state of phase i at intersection n is switched at end of time step k, and 0 otherwise minimum green for phase i (s) Gimin maximum green for phase i (s) Gimax Uni(k) green time used by phase i at intersection n, at the end of time step k (s) qiout …k† out¯ow of phase i at the downstream end over period [k
T, (k+1)
T] (veh/s) number of vehicles of phase i queued up at the end of time interval k at lni(k) intersection n side entry ¯ow during phase i on the corresponding approach link (veh/s) di(k) side exit ¯ow during phase i on the corresponding approach link (veh/s) si(k) exit rates within phase i i0 fraction of queued vehicles of movement i that uses bu€er p lip p the queue bu€er number, p  Bn Bn set of separate queue bu€ers of node n set of phases that share the bu€er p Qp the storage capacity of the queue bu€er p Cp  a travel time coecient between the upstream and downstream of an intersection

uni(k)

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F TD Zi M

357

a dispersion parameter total delay (veh-intervals) 1 when phase i is oversaturated, and 0 when phase i is undersaturated very large positive value, called Big-M

1. Introduction Increasing urban road trac congestion in major US cities signi®cantly undermines the mobility of urban America (Lindley, 1987). With little space for construction of new roads, many e€orts towards congestion relief are focused on better utilization of existing transportation facilities through Advanced Transportation Management Systems (ATMS), a key component of the Intelligent Transportation Systems of the future. Traditionally, the congestion problem on surface streets was dealt with on the supply side by providing increased capacity by adding more lanes to existing roads or adding new links to the existing transportation network. Such a solution is no longer considered viable because of the prohibitive construction cost and the negative environmental impact. Instead, greater emphasis is placed on trac management. The management of trac on surface street networks is achieved primarily via signalized control of intersections. A promising enhancement to the current control systems of pre-timed and actuated signals is the application of ATMS technologies and techniques to isolated intersections, and to arterial and network signal systems. Conventional signal control strategies based on methods developed during the 1970s and 1980s are not considered uniformly e€ective under all possible conditions (Tarno€ & Gartner, 1993). Many strategies fail to provide improvements over well-timed ®rst-generation systems for congested conditions, and some of these systems exhibit degraded performance during speci®c sets of undersaturated conditions (Boillot et al., 1992). For example, current systems are relatively slow to respond to sudden changes in trac ¯ow caused by incidents or large ¯uctuations in demand. Such systems have been designed to implement small changes over time to overcome the problem of frequent transitioning (see Michalopoulos, 1992, for an excellent discussion of the de®ciencies in the current practice). Real-time stochastic control based on detected trac is an option which has not been applied in an integrated fashion at the network level due to the lack of complete analytical network-wide optimization formulations. This paper develops such a formulation, which also includes analytically embedded models for trac ¯ow between intersections so that such a network-wide optimization can be attempted. The intent is to use this optimization in the future as part of real-time control schemes with possibilities for stochastic feedback update of the state variables. The focus in the paper is on the optimization formulation, and its adaptation to a network

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programming form for ecient solution, because such speedy solutions are essential for it to be applicable in control schemes. We also discuss the specialpurpose network simplex algorithm we developed since the network form is non-standard. There is a tremendous variety of problems associated with optimal intersection control, some of which have undergone theoretical treatment (see Gazis, 1964, for a classical optimization formulation for the oversaturated conditions). Current practices, reviewed in the next section, however, are often based on quasi-optimal techniques, thanks to the ineciencies of the formulations. Application of linear programming for intersection signal control, however, shows a potentially practical approach (Eddelbuttel & Cremer, 1992, demonstrate an example for this). Using the linear models available for platoon dispersion along the links, our research develops such a linear optimization formulation. Mathematical optimization is applied to address two interrelated problems of trac systems: trac ¯ow modeling and optimal trac signal control. The objectives of the paper are three-fold. The primary objective is to develop a network-wide signal optimization formulation, including link-level trac movement models as constraints. The second objective is to develop it for application in a realtime control scheme, brie¯y discussed, but not yet implemented. The third objective is to provide a special-purpose solution algorithm for the formulation which has been shown to perform extremely eciently, so that practical future application within real-time control frameworks is possible. Another motivation in this paper is to show that the network-wide signal optimization problem when viewed on a time-expanded network form has an inherent structure that is suitable for network programming, and thus can be solved faster. Network programming is used extensively for inventory, cash-¯ow and other networks and shows orders of magnitude better solution eciency, as well as applicability to orders of magnitude of larger problems, compared to standard linear programming. Our research is perhaps the ®rst attempt to apply this to the signal control problem. Perhaps the reason why such a technique has not been used for this problem is that the resulting network problem is in a speci®c non-standard form which is considered to be in a dicult class of network optimization problems. We, however, develop some techniques to handle the non-standard nature without a€ecting the solution eciency. The paper provides a detailed review of the trac control types and existing trac control schemes in Section 2. The new formulation for network-wide optimization is provided in Section 3. Adaptation of the formulation to the network programming form, the non-standard structure of the network form, as well as the special-purpose solution algorithm are discussed in Section 4, which also provides a brief discussion of a rolling-horizon control scheme within which the optimization can be used. The paper concludes with a comparison of the computational performance of the special-purpose algorithm to that of a standard linear programming algorithm for a simulated case. The paper does not include any results based on ®eld data due to the extensive e€ort involved in implementing it in the ®eld, an activity planned for the future.

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2. An overview of trac signal control 2.1. Classi®cation of trac control methods 2.1.1. Fixed-time control Trac signals in use today typically operate based on a pre-set timing schedule. The most common trac control system used in the USA is the Urban Trac Control System (UTCS), developed by the Federal Highway Administration in the 1970s. UTCS generates timing schedules o€-line using manual or computerized techniques. These predetermined timing schedules are implemented by the system according to the time of the day. The timing schedules are typically obtained by either maximizing the bandwidth (which means the width of the through-band in seconds indicating the period of time available for trac to ¯ow within the band) on arterial streets or minimizing a disutility index that is generally a measure of delay and stops. Computer programs such as MAXBAND (Little, Kelson & Gartner, 1981) and TRANSYT (i.e., Trac Network Study Tool) (Robertson, 1969) are well established means for performing such optimization. The o€-line approach used by UTCS cannot respond adequately to unpredictable changes in trac demand. 2.1.2. Trac-responsive control without optimization These are the adaptive control schemes where the signals are changed based on the actuation of stop-line detectors and minimum/maximum green times. This type of control responds to trac but attempts no optimization, network-wide or local. 2.1.3. Trac-responsive control with optimization These techniques calculate control parameters according to prevailing trac conditions. They typically respond to changing trac demand by performing incremental optimization. The most notable of these are SCATS (i.e., Sydney Coordinated Adaptive Trac System) (Lowrie, 1982, 1990; Luk, 1984; Sims, 1979) developed in Australia, and SCOOT (Split, Cycle, and O€set Optimization Technique; Hunt, Robertson, Bretherton & Royle, 1982; Robertson & Bretherton, 1991) developed in England. SCATS is installed in several major cities in Australia, New Zealand, and parts of Asia; recently the ®rst installation of SCATS in the USA was completed near Detroit, MI. SCOOT is installed in even more cities around the world, including some in the USA (e.g. cities of Oxnard and Anaheim). SCOOT uses steady-state ¯ow patterns found by the TRANSYT models and attempts a heuristic optimization of cycle lengths and splits. Other notable methods under development over the past decade include UTOPIA, PRODYN and OPAC (i.e., Optimized Policies for Adaptive Control). These are all trac-responsive optimization schemes, with various levels of trac modeling capabilities and network-wide optimization capabilities. UTOPIA system developed in Italy was tested at Torino in 1985±86. The French system PRODYN (Henry & Farges, 1990; Henry et al., 1983) developed in 1982, experimentally operated in Toulouse (France) and was recently commercialized. In the meantime, the American system OPAC (Gartner et al., 1991) had the ®rst experimental tests in 1990. OPAC

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perhaps has one of the most comprehensive optimization formulations among these. It solves the formulation using dynamic programming for an isolated intersection. The extension for network-wide operation is under development, but is e€ectively based on optimization of isolated intersections with real-time prediction of arrival ¯ows, rather than based on an integrated network-wide formulation. One of the recent e€orts at developing comprehensive trac control systems was the RHODES (i.e., Real-Time Trac Adaptive Signal Control Logic: Prototype Control Description) prototype developed by Head and Mirchandani (1997), which does include a network control formulation that is promising. 2.2. Review of intersection control algorithms The most common approach to signalization design is to determine settings for a ®xed-cycle light that minimizes the average delay per car assuming constant arrival rates (Miller, 1963; Webster, 1958). For pre-timed signals the most well-known research was performed by Gazis and Potts (1963) and by Gazis (1964) for a system of two oversaturated intersections in succession. Later researchers (Burhardt, 1971; Gartner, 1983) based their work on Gazis' theory and further extended it for more intersections. Dunne and Potts (1964) developed time-varying control algorithms for an undersaturated intersection with constant arrivals which guarantee that, for any initial state, the system eventually reaches a limit cycle for which the equilibrium average delay per car is a minimum. In all these models, the control policy is not responsive to the dynamics of the trac ¯ow process since there is no trac ¯ow model or real-time trac ¯ow information involved. For real-time control, several algorithms have been proposed (Cremer & Schoof, 1990; Gartner et al., 1992; Gordon, 1969; Green, 1968; Lee, Crowley & Pigantaro, 1975; Michalopoulos & Stephanopolos, 1977; Miller, 1965; Papageorgiou, 1983; Ross, Sandys & Schlae¯l, 1970). For example, Miller (1965) considered an intersection with heavy trac and assumed that at time t the signal is green on primary approach. At this time the controller can make a binary decision, i.e. to change the signals immediately, or after an extension of one unit of time. However, Miller did not consider the intersection of adjacent intersections, and thus did not include the downstream delays in determining an optimal extension strategy. Ross et al. (1970), basing their work on a philosophy similar to that of Miller, developed a computer control scheme for trac-responsive control of a critical intersection that not only minimizes the total delay of all users of the intersection, but also minimizes the total delay accumulated at downstream intersections. Moreover, Longley (1968) proposed a control scheme for a two-phase congested intersection employing a `queue balancing' strategy. This strategy seeks to hold a particular linear function of the intersection queues to a value of zero by adjustment of the green time split. Lee et al. (1975) also considered queues rather than delays as the objective of the control and developed another semi-empirical strategy called ``Queue Actuated Signal Control''. This is a control policy where an approach receives green automatically when the queue on that approach becomes equal to or greater than some predetermined length, regardless of the conditions on the con¯icting approaches.

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The policy assumes that no two con¯icting approaches reach the upper bound speci®ed for them simultaneously. Another approach to critical intersection control has been suggested by Gordon (1969). Gordon did not attempt to minimize delay at the intersection but rather to maintain a constant ratio of the queue lengths on opposing approaches. The cycle length is assumed constant and the splits are changed according to the demand so that the ratio of the actual queues to the maximum link storage space on both phases are equal. Finally, Michalopoulos and Stephanopolos (1977) proposed an optimal control policy for both pre-timed and real-time control. His control policy was to minimize total system delay subject to queue length constraints. However, it should be noted that most of the control algorithms mentioned above su€er from complex computational requirements described as above, and from the lack of intersection-to-intersection trac ¯ow models. 3. Optimization model formulation We describe our formulation of the network-wide signal optimization problem, and provide the solution algorithm in Section 4. The formulation has several concepts borrowed from the above algorithms, and perhaps does not consider some constraints included in some of the above algorithms. We mention here that the network programming solution algorithm provides a very intuitive way to incorporate the constraints of the formulation and to add or delete additional constraints that could be considered. Brie¯y stated, the reason is that the solution uses network paths, and even the most complicated constraints of the ¯ow and signal problem at an intersection become intuitive and simpler to handle using the paths on the timeexpanded network. This will become clearer later, but it is useful to remember while reading the formulation given next. 3.1. Introduction In this research, the trac ¯ow and dynamic reduction of queues are described by an appropriate linear model with linear capacity constraints for both road links and ¯ows in intersections. It is also assumed that origin±destination relationships are known a-priori. As far as the network trac ¯ow is concerned, the network control is a superposition of control at individual junctions. Because the control of neighboring intersections determines the arrival process, the entire system is strongly interactive. The optimization model proposed here is a dynamic model with multiple-time steps, as it is meant for real-time operations. Conventional static models like PASSER II (i.e., Progression Analysis and Signal System Evaluation Routine) (Chang, Lei & Messer, 1988) and TRANSYT-7F (Wallace et al., 1988) which use one set of demand data during the entire control period are not appropriate for dynamic signal control. A dynamic model should not only obtain minimum delay subject

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to queue length constraints, but also should include formulas to de®ne relationships of queue transition between time slices. In the dynamic model, the green/red lights can change with every time slice. The signal intervals are adjusted toward minimizing delay and permitting queues to build up to a predetermined upper bound. The optimization model uses mixed integer linear programming (MILP) to mathematically model the above requirements. Some MILP formulations of signal control already exist (Chang et al., 1988; Kim, 1990; Messer, Hogg, Chaudhary & Chang, 1986; Tsay & Lin, 1988), though not for complete networks. An MILP problem could be solved using software packages available for mathematical programming such as GAMS (i.e., General Algebraic Modeling Systems) (Brooke, Kendrick & Meeraus, 1988) and LINDO (i.e., Linear, Interactive aNd Discrete Optimizer) (Schrage, 1984), which normally use the simplex algorithm and branch and bound solution techniques. We, however, develop much faster algorithms as given in Section 4, for use within a standard branch and bound solution scheme. 3.2. Road trac models and problem assumptions A well-known platoon dispersion model (Hunt et al., 1982) is used to provide dynamic interaction constraints among individual intersections which have their own set of signalization constraints. This model has been empirically validated in several urban areas around the world (since it is part of the SCOOT control system implemented around the world). Though it is only an empirical model, it is generally considered to represent (interrupted) trac ¯ow in signalized networks better than other models. The model is presented here to point to its linear structure, which makes it particularly useful in solvable network optimal control formulations. The dynamic evolution of trac ¯ow on the jth road section (approach) can be modeled by the following set of equations:  j q …k†
T j …1† qout …k†
T ˆ min go qsj …k†
T ‡ lj …k ÿ 1† j …k††
T lj …k† ˆ lj …k ÿ 1† ‡ …qsj …k† ÿ qout

qinj …k† ˆ

X i

i bij qout …k†

…2† …3†

A discrete time approach with sample time
T (the
T is selected adaptively, i.e. the length of
T can be set optionally depending on how adaptive the control system needs; here the
T is chosen to be 5 s) is adopted. The following variables apply for each road section j ( j=1, . . ., Js): qinj …k† is the volume (veh/s) entering the jth road section during k
T4t4…k ‡ 1†
T; which depends on the turning fractions bij from other links, qsj …k† is the volume which arrives at the end of the waiting queue or j …k† is the leaving volume at the downstream end of the jth road at the stop-line, qout

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363

j section, qgo …k† is the capacity ¯ow during green trac light and lj …k ÿ 1† is the number of vehicles waiting in the queue. qsj and qinj are connected through the platoon dispersion equation (Eq. (4)) stated as follows:

qsj …k ‡ † ˆ Fqinj …k† ‡ …1 ÿ F†qsj …k ‡  ÿ 1†

…4†

where F ˆ 1=…1 ‡ †; it is a smoothing factor determined by a speci®c platoon dispersion factor  and the travel time coecient between the upstream and downstream of an intersection (F=1 means no dispersion). And  is a constant parameter (normally around 0.5) which may have a value ranging from 0.2 to 0.5, depending upon whether there is very little or extensive platoon dispersion along the road link. The other parameter  which is equal to 0.8(average travel time in
T units). However, one thing that needs to be noted is that many studies have focused on the analysis and the calibration of the F or  factor. Most of these investigations have suggested that platoon dispersion should not be generalized with the standard default parameter settings that are suggested for this model, but that instead these parameters should each time be customized to match the unique road condition on each link. In order to control trac during a limited period of interest eciently, detailed information about the demand structure and the factors determining intersection capacities, which often are the bottlenecks causing the congestion, can be useful. The trac control model assumes that the trac assignment is known a-priori, which means that the demand can be speci®ed for individual routes. Rather than using the traditional concept of turning fractions at intersections, we use actual volumes for each movement at the intersection. Note that it is easy to ®nd these by multiplying approach arrivals by turning fractions as well. An additional reason is our pathbased approach. Recently proposed path-based assignment algorithms (Jayakrishnan, Tsai, Prashker & Rajadhyaksha, 1994; Sun, Jayakrishnan & Tsai, 1996) have been found to perform much faster than the link-¯ow-based assignment algorithms, which points to the attractiveness of our approach if path-based static or dynamic assignment is used in real time for prediction of path ¯ows in an ATMS. Roads entering the network have to be included in the model, because their queue lengths are determined by the signals at the intersections. On the other hand, it is assumed that vehicles can leave the networks as soon as they pass the last intersection on their routes. The entering links (or external approaches) are numbered 1, . . ., E while interior links (or internal movement) receive indices E+1, . . ., H. Each exit gets the same index as the associated entrance. We also assume that: 1. the lost time (yellow and all red) of intersection N is given; 2. the saturation ¯ows Si, I  In, are known; and 3. the turning movement fractions are known and may be time variant. With regard to the turning movement fractions (last assumption above), it should be noted that they may be estimated in real time by known algorithms (e.g. Cremer, 1991).

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3.3. Model formulation of network trac signal control Notations used in the model formulation are given in the nomenclature. Fig. 1 shows the main variables on a from_link of phase i connecting two intersections m, n such that i  Om and i  In. We de®ne qiin …k† and qiout …k† to be the in¯ow and out¯ow, respectively, of phase i over a period [k
T, (k+1)
T] where
T is the sample time interval and k=1, 2,. . . is a discrete time index. Similarly we de®ne di(k) and si(k) to be the demand ¯ow and the exit ¯ow, respectively, occurring within phase i. The control objective of the dynamic model is to minimize total delay in the network. Total delay is the sum of delays on all phases. The full formulation of the mathematical program for minimization of delay is below, and the description of the constraint set follows: Minimize Delay TD ˆ
T

N XX K X lni …k†

…5†

nˆ1 i2In kˆ1

subject to lni …k†50

; 8iIn; nN; kK

lni …k†5lni …k ÿ 1† ‡ …qis …k† ÿ qiout …k††
T lni …k†4MZni …k†

…6† ; 8iIn; nN; kK; Li4E

; 8iIn; nN; kK; where M is some large number

lni …k† ÿ flni …k ÿ 1† ‡ …qis …k† ÿ qiout …k††
Tg4M…1 ÿ Zni …k††; 8iIn; nN; kK; Li4E

Fig. 1. An urban road link.

…7† …8† …9†

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lni …k†5lni …k ÿ 1† ‡
Tf‰F  …1 ÿ i0 † 

X j2Oi

…k ÿ 1†Š ‡ ‰di …k† ÿ

qiout …k†Šg

365

j qout …k ÿ † ‡ …1 ÿ F†  qis

…10†

;

8iIn; nN; kK; E ‡ 14Li4H lni …k† ÿ flni …k ÿ 1† ‡
Tf‰F  …1 ÿ i0 † 

X j2Oi

qiout …k†Šgg4M…1

…k ÿ 1†Š ‡ ‰di …k† ÿ 8iIn; nN; kK; E ‡ 14Li4H

j qout …k ÿ † ‡ …1 ÿ F†  qis

ÿ Zni …k††

…11†

;

qiout …k† ˆ …1 ÿ uni …k††‰Sgni …1 ÿ ni …k†† ‡ Syni  ni …k†Š‡

Sgni  ni …k†  uni …k†

…12†

; 8iIn; nN; kK

uni …k† ˆ ni …k ÿ 1† ‡ uni …k ÿ 1† ÿ 2ni …k ÿ 1†uni …k ÿ 1† ; 8iIn; nN; kK

…13†

Uni …k† ˆ …Uni …k ÿ 1† ‡
T†…1 ÿ ni …k ÿ 1†† ; 8iIn; nN; kK

…14†

Gimin 4Uni …k†4Gimax

; 8iIn; nN; kK

…15†

; 8iIn; nN; kK; PBn

…16†

X

liP lni …k†4CP

i2QP

un1 …k† ‡ un2 …k†51; un1 …k† ‡ un3 …k†51; . . . un7 …k† ‡ un8 …k†51 8nN; kK

;

…17†

The number of the vehicles discharged during the green time depends on whether the corresponding phase is oversaturated or not. If the phase is oversaturated, then it is equal to the capacity ¯ow during green. If the phase is undersaturated, then it depends on the sum of the vehicle arrivals at the end of the waiting queue and the existing queue lengths at the end of time interval kÿ1 (li(kÿ1)). It should be noted that whether a phase is oversaturated or undersaturated cannot be predetermined. The model automatically determines the state of saturation during the optimization procedure. The above equation also implies that the queue lengths occurring at the end of each time interval must be non-negative. These considerations result in Eqs. (6)±(11). Note that integer variables, Zi are introduced for modeling the nonnegativity of the queue lengths. Unfortunately, the model now becomes a complex MILP problem because of the integer variables Zi, and ui described next.

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For an internal link, the queue length lni(k) is obtained by adding the balance between the new arrivals and the departures to the queue length lni(kÿ1) at the end of the previous interval. This gives us Eqs. (10) and (11), which includes the platoon dispersion variables described next. For the external approaches in the network (i.e. link number Li4E), the queue lengths at the end of time interval k, lni(k), can be represented as the sum of any queues transferred from the previous time interval and the di€erence between input and output at the current time interval, which results in Eqs. (7) and (9). The most important factor to be considered in a network of signals is the movement of vehicles from upstream intersections to downstream intersections, which requires the use of a reliable trac model that can accurately re¯ect the movement of vehicles in the network. Research has already been conducted on the applicability of platoon dispersion as a reliable trac movement model in urban street networks. Most of the research has shown that Robertson's model of platoon dispersion is reliable, accurate, and robust (Axhausen & Korling, 1987; Castle & Bonniville, 1985). While the arrival patterns for the external approaches of the intersections are derived exogenously, the arrival patterns on the internal movements are obtained from the departures of the upstream intersections using Robertson's platoon dispersion equations. The F variable in Eqs. (10) and (11) incorporates the platoon dispersion model into the constraints. In the presence of long queues, the assumption of a constant travel time may lead to some inaccuracy with regard to the queue evolution in each time step, but the delay calculation should be suciently accurate, as the time spent by the vehicles will still be captured over a number of time steps. Note that for consideration of further control measures such as route guidance and variable message signs control, the turning movements are externally speci®ed, through variables appearing in Eqs. (10) and (11). The signal state of any phase i, uni(k), at time step k, is given by Eq. (13). The ®rst term represents the control decision at the end of time step kÿ1. The second term signi®es the signal state of phase i at time step kÿ1. uni(k) is a binary variable. If the signal state was red for time step kÿ1, i.e. uni(kÿ1)=1, and the control decision at the end of the time step was to switchover, i.e. ni(kÿ1)=1, then the signal state for time step k must be 0, which corresponds to a green state. The green time already used up by phase i at intersection n, at the end of time step k, is computed by Eq. (14). The ®rst term denotes the green time by the end of step kÿ1. Based on the control decision, ni(kÿ1), green time is either increased by a duration of
T seconds or is increased by none. The standard minimum and maximum green time requirements are re¯ected in Eq. (15). The maximum queue length (capacity) constraint is ®xed in Eq. (16). Constraints are also needed to ensure that con¯icting phases are not given the green indication during any split. Since us are constrained to be binary variables in the formulation, it is easy to ensure that no more than one of any combination of two non-permissible phases has its corresponding u set to 0 (i.e. green time). In this formulation, the standard NEMA (National Electrical Manufactures Association) numbering convention for the di€erent movements are used

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to identify the di€erent phases. Thus, the inadmissible combinations of NEMA phases are phases numbered 1 and 2 (EW left and opposing through), 1 and 3 (EW left and SN left), 1 and 4 (EW left and NS through), and so forth, through 7 and 8 (NS left and opposing through). These constraints are shown in Eq. (17). 4. Development of solution process In this section we present an ecient solution approach, based on a modi®cation of the network simplex algorithm, to solve the model formulation proposed in the previous section. Large-scale MILP problems such as above are normally solved with branch and bound techniques with repeated simplex solutions of the linear programming problem within the MILP. We next explain the ecient network simplex algorithm that can replace the standard simplex for linear programming problems of network form. 4.1. Network simplex algorithm In the network simplex algorithm, we need not explicitly maintain the matrix representation (known as the simplex tableau) of the linear program and can perform all the computations directly on the network. If the dynamic optimal trac signal model is solved by a simplex tableau, there are some disadvantages. In the simplex tableau approach, we need to check each possible pivot to ®nd out the minimum objective value. As a result, the enormously large number of pivots can make the computation prohibitively expensive when the number of time periods and the number of intersections on the network increase signi®cantly. A specialized network simplex algorithm is used to eciently operate at any given time interval. It is shown that the algorithm is more ecient in those problems for which its structure is a large-scale network form (for more detailed description about the eciency of this method and its illustrated examples, see Ahuja, Magnanti & Orlin, 1993, and Kennington & Helgason, 1980). The network simplex algorithm maintains a feasible spanning tree structure and moves from one spanning tree structure to another until it ®nds an optimal structure. At each iteration, the algorithm adds one arc to the spanning tree in place of one of its current arcs. The entering arc is a nontree arc violating its optimality condition. The algorithm: (1) adds this arc to the spanning tree, creating a negative cycle loop; (2) sends the maximum possible ¯ow in this cycle until the ¯ow on at least one arc in the cycle reaches its lower or upper bound; and (3) drops an arc whose ¯ow has reached its lower or upper bound, giving us a new spanning tree structure. Because of its relationship to the primal simplex algorithm for the linear programming problem, this operation of moving from one spanning tree structure to another is known as a pivot operation. The network simplex algorithm maintains a feasible basis structure at each iteration and successively modi®es the basis structure via pivots until it becomes an

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optimum basis structure. The special structure of the basis enables the simplex computations to be performed eciently. 4.2. Modi®ed network simplex algorithm and the system control logic 4.2.1. Graphical presentation of network programming formulation The model presented in Section 3 provides the coupling constraints in the network optimization formulation. Each intersection has its own constraints in terms of the phase sequences and con¯icting movements. The arrival ¯ows in each time period at each intersection are dependent on the departure ¯ows from other intersections through these platoon dispersion coupling constraints. The research has identi®ed a key aspect of the formulation, namely that the linear optimization sub-problem involves extremely sparse matrices, thanks to the network structure. One key reason for this is the nature of the platoon dispersion model which is similar to a multiperiod inventory ¯ow model. Note that the Robertson's platoon dispersion equation means that the trac ¯ow (qsj …k†), which arrives during a given time step at the downstream end of a link, is a weighted combination of the arrival pattern at the downstream end of the link during the previous time step (qsj …k ÿ 1†) and the departure pattern from the upstream trac signal  seconds ago (qinj …k ÿ †). However, the recurrence platoon dispersion Eq. (4) can be transformed as the following form: qsj …k† ˆ

1 X iˆ

F…1 ÿ F†iÿ qinj …k ÿ i†;

…18†

where F is a dispersion parameter and  is a travel time coecient between the upstream and downstream intersection. The summation can be truncated after a reasonable number of terms (say 10±15). The transformed platoon dispersion equations directly translate to links on a time-expanded network, as in Fig. 2. A complete graphical representation of the network structure including the constrained turning movements at the intersection for multiple time steps is too complicated to show here; however, the network inventory-¯ow nature of the linear platoon dispersion equations should be clear from Fig. 2. The network size of the whole trac signal control problem can be reduced after each iteration of computation to a simpler and smaller network. This reduction process simpli®es the diculty of incorporating directly network simplex algorithm to platoon dispersion-based network trac signal control problem. The optimization algorithms employ the two standard principles of implicit enumeration: network simplex algorithm and branch and bound. Branch and bound is mainly used to obtain integer solutions when combined with some other relaxation method. Alternatively, the network simplex algorithm is mainly applied to solve minimum cost ¯ow problems (i.e. delay minimization); so it appears as subroutines in branch and bound methods. The elements of a network simplex algorithm are

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Fig. 2. Graphical representation of network programming formulation.

demands, nodes, arcs, costs, and capacities which determine the value of the objective function at each iteration. 4.2.2. Non-standard network ¯ow problem and special-case simplex The network problem presented above poses one diculty, however, and this is the ®xed split required from the upstream node to the platoon dispersion links. The standard network simplex algorithm does not involve such `®xed nodal splits fractions' [F, F(1ÿF), F(1ÿF)2, etc., as in Fig. 2, are these fractions]. This poses a diculty during the basis update step of network simplex. The reason is that when we attempt to change the ¯ow on any network loop (cycle) involving any of the platoon dispersion arcs, the ratio of ¯ows among the platoon dispersion arcs change. This has to be avoided to retain the split ratios, which means the simplex basis update schemes no longer apply. Thus, the class of network problems with such ®xed nodal splits are considered to be a class of problems with much higher diculty which do not have easy solutions. We develop a technique to handle this problem and the basic idea is straight forward. We can see that the arcs with costs in the network are only the inventory (queuing) arcs shown vertically in Fig. 2. This means that a restricted version of basis update for such a graph that is not based on normal augmenting cycles (loops) can be developed. This essentially involves the selection of a few inventory arcs (i.e. upstream intersections' arcs) and updating the current solution of split fraction arcs in the whole network. Once the inventory arc's ¯ow (note that this is not actual trac ¯ow, but rather queue length) is updated, the exit ¯ows and the platoon dispersion links can be updated. The algorithm operates in a decomposed fashion,

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starting from peripheral nodes and moving inwards, updating platoon arcs and in¯ows to other nodes. Note that the platoon dispersion arc ¯ows (which in turn depend on the signal settings) are adjusted after every iteration, and the network simplex operates essentially independently for each intersection. After about four or ®ve iterations, all the nodes are normally reached in reasonably sized networks, and the signal settings converge in a few more additional iterations (enough to adjust ¯ows along loops in the network). This approach of the solution process is implemented in iterations as described in the previous section by using network simplex algorithm as the subroutine of branch and bound method. In every iteration, ®rst of all, the branch and bound is performed to prevent con¯icting movements. As Eq. (18) shows, we can easily ®nd the ¯ow rates of the initial platoon (i.e. qinj …k†) during time step k by ignoring the split phenomenon on the road link. This is followed by splitting these platoon ¯ow rates as the initial arriving ¯ows of downstream intersection. The algorithm continues by ®nding the second iteration's ¯ow rates as the initial platoon ¯ow rates of next iteration's under the implementation of branch and bound, and so on. A platoon dispersion-based problem with ¯ow split constraints is constructed with the introduction of an arrival ¯ow prediction/estimation concept. Based on the branch and bound-and-network-simplex algorithm, the solution of the ®rst iteration (usually the minimum cost ¯ow computation of upstream intersections in a network) facilitates an initial solution of the second iteration. The solution procedure for each of the iterations can be stated as the following system control strategy. The following steps describe the basic control strategy governing the proposed models for network trac signal control with platoon dispersion constraints. Given the aforementioned system and all functions of its key elements, the operational procedures are summarized below. . Step 0. All the external intersections included in the set N are initialized with external entry ¯ows and initial branch and bound signal settings. . Step 1. For intersection n (in current set N) for each phase i carry out the branch and bound steps. If all intersections are completed, go to step 7. . Step 2. Check the minimum and maximum green constraints for all time steps k:  Condition 1: If green time is less than the minimum green time (i.e. Ui …k†4Gimin ), then extend the green. For instance, if i(k)=0, then Ui(k) is updated.  Condition 2: If Ui(k), the green time already used up by phase i at time step k, has reached the maximum green time Gimax (i.e. Ui …k†5Gimin ), then the green is terminated immediately, i.e., i(k) is changed to 1. If the integer variables for all time step k remain the same as in the previous iteration, update n to the next intersection and go to step 1. Else go to step 3. . Step 3. Examine and prevent the con¯icting movements which happen at intersections using the branch and bound method by setting the constraints ui …k† ‡ uj …k†51; where i, j  In and ui and uj are binary integers which are either 0 or 1.

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. Step 4. Using the network simplex-based algorithm to get the minimum cost ¯ow solution (i.e. delayed vehicles, TD) of the network for the green extension option evaluated by the branch and bound. (Here network simplex is operated at the given intersection and could even conceptually be replaced with vehicle clearance rules if they can guarantee optimality.) . Step 5. If the performance index is the bene®t of giving a green which is compared with that of terminating it by computing the tradeo€s incurred in vehicle delays and it is negative, then the optimal decision is not favorable to the intersection and a switchover decision is possible. Otherwise, the current green can be extended for another
T seconds, i.e. Ui …k ‡ 1† ˆ Ui …k† ‡
T: . Step 6. Go to step 2 to ensure of switchover being within the min±max constraints. . Step 7. Update platoon dispersion arc ¯ows and the corresponding in¯ows to existing intersections as well as new intersections joining the set N. If the platoon dispersion arc ¯ows are same as in the previous outer iteration (over the last set of intersections N), i.e. no signal settings changed, then STOP; else, go to step 1.

In the proposed models, the control decision is made every 5 s (i.e. duration of a time step) depending on the comparison of bene®ts between extending green and terminating it. The control strategy makes use of real-time trac state conditions, instead of pre-stipulated strategies. 4.2.3. Real-time aspects Ð implementation Although this paper assumes full information on external inputs, the formulation can be applied in a rolling horizon control scheme in a sensor-based environment. The ¯ow detectors can be located at the upstream junction output and at about 120 feet from the stop line an online updating scheme for the turning fractions as well as the platoon dispersion factors is possible, in addition to updating the side entry/exit ¯ows and external entry ¯ows. One thing needs to be noticed here is that vehicles are presumed to travel undelayed to the stop line before joining a vertical queue (if present). As a consequence of this assumption, detectors upstream of the intersection give advance information on vehicle arrivals at the stopline, the degree of forewarning depending on the undelayed travel time from the detector to the stop line. In addition, the more we go away from the present time, the more the demand forecasts and the model calculations become inaccurate. In practice, the extend of forewarning is limited and is unlikely to exceed 15 seconds. Any such implementation could present unforeseen challenges and sensitivity analyses are necessary to address questions regarding the choice of parameter values (e.g. roll period, stage length) and their in¯uence on the e€ectiveness of the solution procedure. Also of signi®cance is the deployment of detectors and issues such as whether the upstream output detectors are needed at all. These issues and issues pertaining to the computational eciency of the solution procedure are the focus of current research e€orts on this topic.

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5. Comparison of solution eciency via numerical tests The preliminary studies on the signal timings from this optimization have shown that the results are sound. Since other such network-wide optimization formulations are currently unavailable for implementation, no comparisons with other formulations are possible. We also point out that since our algorithm is not heuristic, the solutions will be optimal within the assumptions of the model. Thus, such comparisons will only be contrasting the assumptions of the models, not how well the assumptions re¯ect reality. In this section, however, we present results which are most important in determining the practicability of the algorithm, namely, its ability to be computationally solved in a fast manner. 5.1. Conventional solution approach via commercial programs Some restrictive versions of the network signal control mixed integer model with a limited number of intersections and time steps were solved using LINDO program. The optimal solution of simple cases were obtained by the LINDO. Even in the simple case of only three time intervals, the number of constraints exceed 10,000 and the number of variables exceed 2000. This huge dimensionality renders the solution process dicult, especially in the execution of simplex tableau process and the branch and bound techniques. The computation time needed is as many as 30 min on a personal computer (120 Mhz Pentium with 16MB RAM), and can even extend beyond a day for an increase in any such related components as number of intersections, time intervals, or number of phases. While providing an initial check on the correctness of the formulation, the computation results for the simple cases clearly demonstrate that standard treatment by mathematical programming are not practically feasible with such a complex formulation. 5.2. Comparison of network simplex programming algorithm with LINDO program As indicated in the previous section, the LINDO program was used as a temporary tool to ®nd the solution to a series of test cases to ensure validity of the model formulation. The characteristics and performance of the ecient network simplex programming algorithm is introduced and applied to solve such problems below. For illustration, we provide an example of an urban network. A simple network composed of one controlled four-approach junction surrounded by controlled junctions is selected as an application example. The network characteristics are shown in Fig. 3. The link lengths were all 1000 feet each and the average vehicle speed was 30 mph, respectively. This structure allows a ¯uctuated demand on the controlled links according to the varied green and red durations of the neighbor junctions. The entering network demand is assumed to be known along the time horizon. This is, of course, not a realistic con®guration from a trac point of view but this allows us to study the principal algorithm's behavior. Trac ¯ow generation is according to a Poisson distribution. Also, the percentage of turning movements and exit rates

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Fig. 3. Link-node diagram of urban network with ®ve intersections.

occurring in internal links are assumed to be zeros here in order to simplify this test example and ensure that a LINDO solution for comparison was possible. Two-phase operations are used in the evaluation and comparison. The saturation ¯ow rate of each of the phases is time-invariant and set to be 1800 vph. Five demand intervals of 5 s each and zero initial queues are used in the comparisons. A minimum green time of duration 10 s (including yellow and all-red time) and a maximum green time of 60 s are used in the control simulation. With the optimization control for this test network, the trac arrival ¯ows of the external approaches on each of the phases are using three cases of average arrival ¯ows (800 vphpl, 1000 vphpl and 1200 vphpl) to test our control model. The delay (veh-s) and computation times (s) are provided in Table 1. The percent change column gives the amount of delay or computation time reduction achieved by the network simplex programming algorithm as a percentage of the delay and computation time due to LINDO program. As shown by Table 1, for the ®ve-node network, 1200 vphpl case, the algorithm reaches the objective value of 937.84 veh-s utilizing 17 s on a personal computer (120 Mhz Pentium with 16MB RAM) by using network simplex programming algorithm. Compared with the same delay value and computation time 38 min (2280 s) obtained by LINDO program, the computation time reduction is signi®cantly achieved to 99.25% in this case. Note that the network simplex-based algorithm is applied here to a problem without rolling horizons. In a rolling horizon implementation, better initial solutions will cause each problem to be solved much faster than the times implied by the 17 s here. The signi®cant result here is how the solution technique is orders of magnitude faster than normal linear programming solutions. These results show that the

800 1000 1200

Trac volume (vphpl)

310.24 742.56 937.84

2250 2280 2280

310.24 742.56 937.84

15 15 17

Computation time (s)

Delay (veh/s)

Delay (veh/s)

Computation time (s)

Network simplex programming algorithm

LINDO program

Table 1 Comparisons of solution eciency for three simulated arrival ¯ows

0 0 0

Delay (veh/s)

Percent change

99.33 99.34 99.25

Computation time (s)

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formulation, though it is large, can be solved in a manner fast enough for real-time operation. In fact, we could say that the solution technique is very ecient as we solved a problem that has never been solved before, and showed that our fundamental assumption that the hypothesized network structure can be exploited for a model that runs hundreds of time faster than a non-network model. The solution times are expected to improve signi®cantly after we remove certain computational overhead as well. We expect real-time solution speeds for networks of up to a few tens of nodes. 6. Conclusions and future research This research presents a systematic approach to network trac signal control problem. The approach involves formulation and solution of a linear optimization problem for multiple intersections in a network, connected by explicit constraints capturing trac movement on the links connecting the intersections. Our model formulation is distinct from other models in that the trac arrival platoon is explicitly incorporated into the network signal control model. The platoon dispersion constraints directly translate to links of a time-expanded network, and thus the problem has the form of a linear multi-commodity network ¯ow problem. Even though, the ®xed nodal splits for the platoon constraints render it a nonstandard problem, we develop ecient special-purpose version of network simplex to solve the problem, in conjunction with a branch and bound scheme for the integer signal constraints. The results show extremely fast solutions from this algorithm. Several operational aspects of the algorithm and its potential application remain to be studied. One option of particular interest is a hierarchical operation where the algorithm provides strategic control for an extended trac network with an updating period of several seconds and complemented by an inferior, short-term reacting, possibly decentralized direct-control layer, and a superior adaptation layer that provides updated demand forecastings, estimation, and detection information (Papageorgiou, 1984; Stephanedes & Chang, 1993). Another possibility (and perhaps necessity) is to use the algorithm for smaller subnetworks based on real-time computational concerns, and to coordinate multiple subnetworks using a supervisory program. Real-world numerical results and evaluation is to follow in the future. The current and next steps related to this research include the following issues: (1) integration of the available demand-forecasting algorithms; (2) use of observed ¯ows and queues with stochastic feedback and ®ltering schemes for detected data; and (3) application of the overall methodology to real trac signal networks. References Ahuja, R. K., Magnanti, T. L., & Orlin, J. B. (1993). Network ¯ows theory, algorithms, and applications. New Jersey: Prentice Hall.

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