A Benchmark Problem in Transportation Networks
aa r X i v : . [ c s . S Y ] M a r A BENCHMARK PROBLEM IN TRANSPORTATIONNETWORKS
SAMUEL COOGAN AND MURAT ARCAK
Abstract.
In this note, we propose a case study of freeway traffic flow mod-eled as a hybrid system. We describe two general classes of networks that modelflow along a freeway with merging onramps. The admission rate of traffic flowfrom each onramp is metered via a control input. Both classes of networksare easily scaled to accommodate arbitrary state dimension. The model isdiscrete-time and possesses piecewise-affine dynamics. Moreover, we presentseveral control objectives that are especially relevant for traffic flow manage-ment. The proposed model is flexible and extensible and offers a benchmarkfor evaluating tools and techniques developed for hybrid systems. Introduction
Traffic flow theory has its foundations in the
Lighthill-Whitham-Richards (LWR)model , a first-order partial differential equation in the form of a conservation lawthat models traffic flow on a single road [1, 2]. Originally motivated by the need toefficiently simulate traffic flow, the cell transmission model was proposed as a finite-dimensional approximation to the LWR model [3, 4]. However, the cell transmissionmodel has since been established as an appropriate model of traffic flow in its ownright [5, 6, 7], and recent research has focused on studying the dynamical propertiesof this model [8, 9, 10].The cell transmission model considers traffic networks as interconnected links orcompartments with finite capacity to store vehicles. Vehicles flow from link to linkover time, and thus the occupancy of a link is time-varying. The cell transmissionmodel adopts a fluid approximation of traffic flow so that occupancy is not restrictedto integer values. Flow of vehicles from an upstream link to a downstream link isrestricted by the demand of vehicles on the upstream link to flow downstream, andthe supply of capacity available on the downstream link to accept incoming flow.These restrictions give rise to a hybrid model as the behavior is different in thesupply-restricted regime. For junctions with multiple incoming links or multipleoutgoing links, demand is divided among the outgoing links and supply is dividedamong the incoming links, and a variety of specific models for this division has beenproposed in the literature; see the above-cited references.In this note, we present a particular instantiation of the cell transmission modelespecially amenable to analysis and control as a hybrid system. We limit ourattention to models where each junction has a single incoming link and at mosttwo outgoing links (diverging junction), or at most two incoming links and a single
S. Coogan is with the School of Electrical and Computer Engineering and the School ofCivil and Environmental Engineering at the Georgia Institute of Technology, Atlanta, GA. [email protected] . M. Arcak is with the Department of Electrical Engineering and Com-puter Sciences, University of California, Berkeley, Berkeley, CA. [email protected] . outgoing link (merging junction), and the result is a piecewise-affine traffic flowmodel.The purpose of this note is to distill existing traffic models into a simple butextensible model of traffic flow and offer it as a practically motivated case studyin need of computationally efficient and scalable tools for control verification andsynthesis. The model presented here agrees with the various models presented inthe literature and cited above and illuminates the fundamental challenges.The note is organized as follows. In Section 2, we present models for traffic flowat merging and diverging junctions. These simple models can be interconnectedto create traffic networks with arbitrary topology. In Section 3, we suggest twoparticular classes of network topologies and provide detailed models for both. InSection 4, we present several control objectives for these two classes of networkswhich are especially relevant for traffic networks. In Section 5, we identify severalproperties of the traffic flow model and make connections to existing results in theliterature. Section 6 contains concluding remarks.2. Traffic Flow at Junctions
We begin by discussing the two elemental traffic flow models: the merging junc-tion where two traffic links merge to one downstream link as in Figure 1(a), andthe diverging junction where one link diverges to two downstream links as in Figure1(b). These two junction models capture the essential dynamics exhibited by trafficflow networks. In Section 3, we combine these models and propose two benchmarknetwork topologies. While this note considers networks with only merging nodesand diverging nodes, the model is easily extended to accommodate nodes withmultiple incoming and outgoing links [11].Let x i [ t ] denote the number of vehicles on link i at time t , that is, x i is the occupancy of link i . We adopt a macroscopic modeling approach and assume that x i [ t ] takes continuous values. The state of a traffic flow network is the collection oflink occupancies in the network so that, for the merging and the diverging junctions, x [ t ] = (cid:0) x [ t ] x [ t ] x [ t ] (cid:1) T is the system state.The flow of traffic from link to link through merging and diverging junctions isa function of the number of vehicles on a link wishing to flow downstream, which iscalled the demand of the link, the available downstream road space to accommodatethese vehicles, which is called the supply of the link, and possibly a control input,which is discussed below. Link demand is an increasing function in the numberof vehicles on the link, and the link supply is a decreasing function in the numberof vehicles on the link. In the transportation literature, the demand and supplyfunctions are referred to as the fundamental diagram relating link occupancy toflow.A common approach is to adopt a triangular fundamental diagram so that thedemand and supply functions are affine in x i with the additional restriction that thedemand saturates at some maximum value. Thus we model the demand function D ( x ) and the supply function S ( x ) as D ( x ) = min { c, vx } (1) S ( x ) = w (¯ x − x ) (2)where c is the capacity , v is the free-flow speed , w is the congestion-wave speed , and¯ x is the jam occupancy of a link [6]. Standard values for these parameters are given BENCHMARK PROBLEM IN TRANSPORTATION NETWORKS 3
Paramter Value UnitsLink length 1 milePeriod 0.5 min c
40 veh / period v / period w . / / period¯ x
320 vehicles β α α Table 1.
Parameter valuesin Table 1, which are appropriate for links modeling two lanes of traffic, with eachlink one mile long, and a time step of 30 seconds [6].We remark that it is common to consider different parameters for different linksto accommodate, e.g. , varying road geometry, varying link lengths, etc.
For nota-tional convenience and to establish a consistent model, we assume all parameterstake the values shown in Table 1 for all links.In traffic networks, link flow may be artificially restricted, or metered , via trafficsignaling devices, and, in this note, metering is considered to be the only availablecontrol input. For example, metering is commonly encountered on freeway onrampsto control admittance to the freeway. In particular, if a link is metered via controlinput u , then the traffic flow that exits the link cannot exceed u [ t ] at time t .The fundamental rule governing the dynamics of traffic networks is that the flowexiting a link should not exceed the link’s demand, nor the link’s metering rate ifthe link is controlled, and it should also not exceed downstream supply. Thus, fora junction consisting of one incoming link, labeled link 1, and one outgoing link,labeled link 2, the state equations are x [ t + 1] = x [ t ] − min { D ( x [ t ]) , S ( x [ t ]) } + d [ t ] , (3) x [ t + 1] = x [ t ] + min { D ( x [ t ]) , S ( x [ t ]) } − D ( x [ t ]) , (4)where x i [ t ] is the state, i.e. , occupancy, of link i ∈ { , } at time t , d [ t ] is thenumber of vehicles arriving on link 1 from upstream. We assume there are no linksdownstream of link 2 so that the flow exiting link 2 is equal to demand.We now extend this rule in a natural way for the merging junction and thediverging junction shown in Figure 1.2.1. Merge Junction.
At the merging junction show in Figure 1(a), links 1 and2 flow downstream to link 3. In this note, merging junctions are interpreted asmodeling a freeway entrance ramp that joins a freeway, and thus links 1 and 3are the freeway and link 2 is the onramp. Link 2 is metered via control input u .Let d [ t ] and d [ t ] be the exogenous, uncontrolled arrival of vehicles to links 1 and2, respectively. A fraction β ≤ − β fraction are assumed to exit the network via an unmodeled exitlink. All vehicles flowing from the onramp link 2 join link 3.Moreover, it is assumed that the supply of link 3 available to link 1 (respectively,link 2) is determined by a fixed weight α > α > β , α , and ¯ α used in this note are reported in Table 1. SAMUEL COOGAN AND MURAT ARCAK
12 3 1 23(a) (b)
Figure 1. (a) At a merging junction , traffic flow from two in-coming links merge and flow downstream to an outgoing link. Itis common for one of the incoming links (dashed) to model an onramp for which the rate of traffic flow can be controlled usingsignaling devices, thus providing a control input to the system.(b) At a diverging junction , traffic flows from one incoming link totwo outgoing links. Diverging junctions exhibit the first-in-first-out property whereby congestion (lack of supply) on one outgoinglink reduces flow to the other outgoing link.The state-update equations for the merge junction are given by x [ t + 1] = x [ t ] − min (cid:26) D ( x [ t ]) , αβ S ( x [ t ]) (cid:27) + d [ t ] , (5) x [ t + 1] = x [ t ] − min { D ( x [ t ]) , ¯ αS ( x [ t ]) , u [ t ] } + d [ t ] , (6) x [ t + 1] = x [ t ] − D ( x [ t ]) + min { βD ( x [ t ]) , αS ( x [ t ]) } + min { D ( x [ t ]) , ¯ αS ( x [ t ]) , u [ t ] } . (7) Remark 1.
Note that w S ( x i [ t ]) = (¯ x − x i [ t ]) is the available space on link i at time t so that we interpret wα (respectively, w ¯ α ) as the fraction of this spaceavailable to link (respectively, link ). Thus, occupancy will not exceed ¯ x so longas wα + w ¯ α ≤ , which holds for the values in Table 1. The model is referred to asan asymmetrical cell transmission model since ¯ α > and therefore flow from link may exceed supply [5, 6] . Diverge Junction.
A diverge junction models the division of flow from oneincoming link to two outgoing links. We assume that the outgoing flow of link 1 inFigure 1(b) divides evenly among the outgoing links 2 and 3. We do not assumeany of the links in the diverging junction are metered.Diverging junctions in traffic flow networks have been empirically observed toobey a first-in-first-out (FIFO) property whereby congestion on one outgoing linkrestricts flow to the other outgoing link, that is, lack of supply on link 2 restricts flowto link 3 and vice-versa. The intuition for this phenomenon is that traffic waiting tomove downstream to link 2 blocks traffic destined for link 3, even though link 3 hasadequate supply. This intuition has been empirically observed to extend to trafficflow on multi-lane freeways [12]. In the present model, we assume a full FIFO property whereby complete congestion on one outgoing link completely restrictsflow to other outgoing links. This is a common assumption in transportation flowmodels and can be relaxed to obtain partial FIFO models [13].
BENCHMARK PROBLEM IN TRANSPORTATION NETWORKS 5 x [ t + 1] = x [ t ] − min (cid:26) D ( x [ t ]) , αβ S ( x [ t ]) (cid:27) + d [ t ] , (11) x N [ t + 1] = x N [ t ] − D ( x N [ t ]) + min { βD ( x N − [ t ]) , αS ( x N [ t ]) } + min (cid:8) D ( x ( N − ′ [ t ]) , ¯ αS ( x N [ t ]) , u ( N − ′ [ t ] (cid:9) , (12) x i [ t + 1] = x i [ t ] − min (cid:26) D ( x i [ t ]) , αβ S ( x i +1 [ t ]) (cid:27) + min { βD ( x i − [ t ]) , αS ( x i [ t ]) } + min (cid:8) D ( x ( i − ′ [ t ]) , ¯ αS ( x i [ t ]) , u ( i − ′ [ t ] (cid:9) for i = 2 , . . . , N − , (13) x i ′ [ t + 1] = x i ′ [ t ] − min { D ( x i ′ [ t ]) , ¯ αS ( x i +1 [ t ]) , u i ′ [ t ] } + d i ′ [ t ]for i ′ = 1 ′ , . . . , ( N − ′ . (14) Figure 2.
State-update equations for the simple freeway. · · · N ′ ′ ′ ′ ( N − ′ Figure 3. A simple freeway network consists of only mergingjunctions and models a length of freeway with onramps. Thelength- N simple freeway consists of (2 N −
1) links, ( N −
1) ofwhich model onramps and thus possess control inputs. Therefore,the state dimension of the network is (2 N − N − x [ t + 1] = x [ t ] − min { D ( x [ t ]) , S ( x [ t ]) , S ( x [ t ]) } + d [ t ] , (8) x [ t + 1] = x [ t ] + min { . D ( x [ t ]) , S ( x [ t ]) , S ( x [ t ]) } − D ( x [ t ]) , (9) x [ t + 1] = x [ t ] + min { . D ( x [ t ]) , S ( x [ t ]) , S ( x [ t ]) } − D ( x [ t ]) . (10)3. Two Network Topologies
We now use the merge and diverge junction models presented in Section 2 asbuilding blocks to construct two classes of traffic networks. These two classesprovide an extensible benchmark problem for analysis and control of hybrid systems.3.1.
Simple Freeway Network.
The first class of benchmark networks consistsof only merging junctions and models a length of freeway with onramps. This classof network topologies is commonly encountered in studies of ramp metering forfreeways [5, 6, 14]. Given parameter N , the length- N simple freeway is composedof 2 N − N − onramps and thus possess controlinputs, as shown in Figure 3. The state dimension of the network is (2 N −
1) andthe input dimension is ( N − N simple freeway are given in Figure2. SAMUEL COOGAN AND MURAT ARCAK x ( − M ) [ t + 1] = x ( − M ) [ t ] − min (cid:26) D ( x ( − M ) [ t ]) , αβ S ( x ( − M +1) [ t ]) (cid:27) + d ( − M ) [ t ] , (15) x [ t + 1] = x [ t ] − min { D ( x [ t ]) , S ( x [ t ]) , S ( x N +1 [ t ]) } + min { βD ( x − [ t ]) , αS ( x [ t ]) } + min { D ( x − ′ [ t ]) , ¯ αS ( x [ t ]) , u − ′ [ t ] } , (16) x i [ t + 1] = x i [ t ] − min (cid:26) D ( x i [ t ]) , αβ S ( x i +1 [ t ]) (cid:27) + min { . D ( x [ t ]) , S ( x [ t ]) , S ( x N +1 [ t ]) } for i = 1 , N + 1 , (17) x i [ t + 1] = x i [ t ] − min (cid:26) D ( x i [ t ]) , αβ S ( x i +1 [ t ]) (cid:27) + min { βD ( x i − [ t ]) , αS ( x i [ t ]) } + min (cid:8) D ( x ( i − ′ [ t ]) , ¯ αS ( x i [ t ]) , u ( i − ′ [ t ] (cid:9) for i = − M + 1 , . . . , − , , . . . , N − , N + 2 , . . . , N − , (18) x i [ t + 1] = x i [ t + 1] + min { βD ( x i − [ t ]) , αS ( x i [ t ]) } + min (cid:8) D ( x ( i − ′ [ t ]) , ¯ αS ( x i [ t ]) , u ( i − ′ [ t ] (cid:9) − D ( x i [ t ])for i = N, N (19) x i ′ [ t + 1] = x i ′ [ t ] − min { D ( x i ′ [ t ]) , ¯ αS ( x i +1 [ t ]) , u i ′ [ t ] } + d i ′ [ t ]for i ′ = − M ′ , . . . , − ′ , ′ , ′ , . . . , ( N − ′ , ( N + 1) ′ , . . . , (2 N − ′ . (20) Figure 4.
State-update equations for the length-(
M, N ) diverging freeway.
Remark 2.
The simple freeway model may be modified to arrive at various alter-native configurations. For example, for fixed N , the number of states and inputsmay be reduced by eliminating some onramps from the model. Diverging Freeway Network.
The second class of benchmark networksmodels a freeway that diverges into two freeways. Given parameters M and N , the length- ( M, N ) diverging freeway is composed of 2 M + 4 N − M + 2( N − onramps and thus possess control inputs, as shown in Figure 5.The state-update equations for the length- N diverging freeway are given in Fig-ure 4. Remark 3.
As with the previous benchmark network, the diverging freeway modelmay be easily modified by eliminating onramps, which reduces the number of statesand inputs of the model.
Remark 4.
The incoming flows to the onramps and to the first link of the network( i.e. , link for the simple freeway and link − M for the diverging freeway) are notsubject to supply restrictions. The motivation for doing so is that the model is ableto accommodate arbitrary exogenous flows as discussed in Section 3.3. Exogenous Inputs.
For the simple freeway model, let d [ t ] = (cid:0) d [ t ] , d ′ [ t ] , d ′ [ t ] , . . . , d ( N − ′ [ t ] (cid:1) ∈ R N , (21) BENCHMARK PROBLEM IN TRANSPORTATION NETWORKS 7 − M − ′ − M ′ · · · N + 11 ′ ( N +1) ′ ′ ( N +2) ′ N + 2 ( N − ′ (2 N − ′ · · ·· · · N N Figure 5. A diverging freeway network models a freeway thatdiverges to two freeways. The length- ( M, N ) diverging freeway iscomposed of 2 M + 4 N − M + 2( N −
1) of which model onramps and thus possess control inputs.and, for the diverging freeway model, let d [ t ] = ( d ( − M ) [ t ] , d ( − M ) ′ [ t ] , d ( − M +1) ′ , . . . , d ( − ′ ,d ′ [ t ] , . . . , d ( N − ′ [ t ] , d ( N +1) ′ [ t ] , . . . , d (2 N − ′ [ t ]) ∈ R M +2 N − (22)so that d [ t ] is the vector of exogenous flows into the network at time t . Similarly,for the simple freeway, let u [ t ] = (cid:0) u ′ [ t ] , u ′ [ t ] , . . . , u ( N − ′ [ t ] (cid:1) ∈ R N − (23)and, for the diverging freeway, let u [ t ] = ( u ( − M ′ ) [ t ] , . . . , u ( − ′ , u ′ [ t ] , . . . , u ( N − ′ [ t ] ,u ( N +1) ′ [ t ] , . . . , u (2 N − ′ [ t ]) ∈ R M +2( N − (24)so that u [ t ] is the vector of inputs for the network at time t . Furthermore, let n = ( N − M + 4 N − m = ( N − M + 2( N −
1) for the diverging freeway, (26) q = ( N for the simple freeway, M + 2 N − d [ t ] for t ≥ feasible if there exists a sequence u [ t ]for t ≥ C > x i [ t ] < C for all t ≥ i ranging over the indices of links in the network. The sequence d [ t ] is saidto be infeasible otherwise. In other words, the sequence d [ t ] is feasible if thereexists a control sequence that fully accommodates the exogenous incoming flow.It is straightforward to show that if x i [0] ≤ ¯ x , then x i [ t ] ≤ ¯ x for all t ≥ i.e. , i ∈ { , , . . . , N } for the simple freeway or i ∈ {− M + 1 , − M + 2 , . . . , N } for the diverging freeway). Thus traffic may onlyaccumulate at onramps or the first link of the network ( i.e. , link 1 for the simple SAMUEL COOGAN AND MURAT ARCAK freeway and link − M for the diverging freeway), and therefore these are the onlylinks for which it may not be possible to find C satisfying x i [ t ] < C for all t ≥ e.g. , equilibriumconditions that may arise for extended periods of time [6].For example, [6] suggests exogenous flows that are infeasible but on the cusp offeasibility. Specifically, for the simple freeway network, such a choice is d [ t ] = 40 ∀ t (28) d i [ t ] = 10 + ǫ i ∀ t, i ∈ { ′ , ′ , . . . , ( N − ′ } . (29)When ǫ i = 0 for all i , the demand is feasible and the outgoing flow for link i ∈{ , , . . . , N } is 40 vehicles per period at equilibrium. Of these, 30 = β
40 advancedownstream to link i + 1, joined by 10 additional vehicles from link i ′ . However, theflow is infeasible when ǫ i > i since the outgoing flow of link i + 1 cannotexceed c = 40 vehicles per period. In the case of infeasible flows, it has been shownthat ramp metering can increase throughput of the network [6]. Similar choicesfor exogenous flow can be made for the diverging freeway network. Moreover,it is suggested in [15] to model the exogenous flow as belonging to a set, e.g. , d i [ t ] ∈ [10 − δ i ,
10 + δ i ] for some δ i > i ∈ { ′ , ′ , . . . , ( N − ′ } .4. Performance Specifications
We now suggest several performance metrics that are natural for traffic networks.
Total Travel Time
The total travel time (
TTT ) of the network up to time T is defined as TTT [ T ] = T X t =0 X i x i [ t ] (30)where i is assumed to vary over all state indices of the network. Total travel time isa useful metric when considering finite time horizons for which we seek to minimizethe total travel time at the end of the horizon. Throughput
Another common performance metric is throughput at time t , denoted W [ t ], anddefined as W [ t ] = X i ∈F (1 − β ) min (cid:26) D ( x i [ t ]) , αβ S ( x i +1 [ t ]) (cid:27) + X i ∈E D ( x i [ t ]) (31)where F = { , , . . . , N − } for the simple freeway, {− M, − M + 1 , . . . , − , , , N − , N + 1 , . . . , N − } for the diverging freeway , (32) E = ( { N } for the simple freeway, { N, N } for the diverging freeway. (33) BENCHMARK PROBLEM IN TRANSPORTATION NETWORKS 9
Throughput is a measure of the number of vehicles that exit the network in a giventime step, and total throughput over a horizon T is J [ T ] = T X t =0 W [ t ] . (34)Throughput is a useful metric in cases when the exogenous flow is infeasible or thetime horizon is infinite, in which case a discounted or average reward function maybe defined as J = ∞ X t =0 γ t W [ t ] or J = lim sup T →∞ T T X t =0 W [ t ] , (35)where γ < Congestion
Notice that D ( x crit ) = S ( x crit ) for x crit = max (cid:26) ¯ x − cw , w ¯ xv + w (cid:27) = 80 (36)where the second equality is valid for the values reported in Table 1. Link i is saidto be congested if x i > x crit .A third possible performance specification for the system is that x i ≤ x crit forall i . Motivated by specifications expressible in temporal logic [16, 17], we mayrequire that this conditions holds for all time, or eventually at some time in thefuture and forever thereafter, or at infinitely many time instants in the future, etc. For example, it is possible to characterize a degree of robustness for the system byconsidering these various possibilities [18].5.
Discussion
We now point to some important properties of the traffic model proposed inthis note. First, we note that the arguments of the minimization functions thatappear in (1), (5)–(7), and (8)–(10) are all affine so that the model is piecewiseaffine (PWA), that is, there exists a partition P , . . . , P M of R n and collections ofmatrices { A i } Mi =1 ⊂ R n × n and { B i } Mi =1 ⊂ R n × m such that x [ t + 1] = A i x [ t ] + B i u [ t ] + Ed [ t ] (37)whenever x [ t ] ∈ P i , that is, the system dynamics are affine in each region of thepartition. The matrix E ∈ { , } n × q is a binary matrix with no more than onenonzero entry per row whose role is to ensure that the disturbance vector conformswith the state vector appropriately.Since the model is piecewise affine, it is, in principle, amenable to tools suchas model predictive control [19] and synthesis for linear temporal logic (LTL) ob-jectives [20] which have been specialized to PWA systems. The latter approach isconsidered in [11] to meet LTL specifications.Next, it has been shown in [6, 8, 9] that traffic networks with only merging junc-tions are monotone dynamical systems for which trajectories maintain a partialorder on states [21, 22]. Moreover, general traffic networks with diverging junctionspossess dynamics that are mixed monotone [13], a generalization of monotone sys-tems [23]. For such systems, reachable sets are over-approximated by evaluating a certain decomposition function, derived from the dynamics, at only two extremalpoints, regardless of the state-space dimension. Thus, such systems are amenableto efficient finite abstraction [24], a typical requisite for applying formal methodstechniques to control systems [25]. This efficiency is exploited in [15] to synthesizecontrol strategies for traffic networks. Additionally, it is shown in [26] that mono-tone systems are especially amenable to classes of temporal logic specifications thatencourage lower occupancies.Moreover, the sparse interconnection structure of many traffic networks suggestcompositional techniques for further aiding scalability [27]. Finally, some workshave suggested that scalability can be improved by avoiding discretization of thestate space; in particular, [28] proposes computing abstraction based on sequencesof applied control inputs, and this approach is applied to a traffic flow model similarto that presented here. Nonetheless, these formal approaches are only applicableto relatively small networks (up to approximately 10 state dimensions).6. Conclusions
We have presented a general, hybrid model of traffic flow in vehicular trans-portation networks and suggested two simple classes of networks that encompassrealistic scenarios. Both classes are easily scaled to allow networks of arbitrary size.We have further characterized several performance metrics that are practically mo-tivated and fit well into optimization or formal methods frameworks. Traffic flowmodels have already proven to be effective case studies for new tools in hybrid sys-tems, and it is our hope that the transportation network model detailed in this notemay offer a practical and extensible benchmark problem for applying new tools andtechniques for hybrid systems.7.
Acknowledgments
Research funded in part by the National Science Foundation under grant 1446145.
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