A Physics-Based Finite-State Abstraction for Traffic Congestion Control
11 A Physics-Based Finite-State Abstraction forTraffic Congestion Control
Hossein Rastgoftar and Jean-Baptiste Jeannin
Abstract —This paper offers a finite-state abstraction of trafficcoordination and congestion in a network of interconnected roads(NOIR). By applying mass conservation, we model traffic coordi-nation as a Markov process. Model Predictive Control (MPC) isapplied to control traffic congestion through the boundary of thetraffic network. The optimal boundary inflow is assigned as thesolution of a constrained quadratic programming problem. Addi-tionally, the movement phases commanded by traffic signals aredetermined using receding horizon optimization. In simulation,we show how traffic congestion can be successfully controlledthrough optimizing boundary inflow and movement phases attraffic network junctions.
I. I
NTRODUCTION
Urban traffic congestion management is an active researcharea, and physics-based modeling of traffic coordination hasbeen extensively studied by researchers over the past threedecades. It is common to spatially discretize a network ofinterconnected roads (NOIR) using the Cell TransmissionModel (CTM) which applies mass conservation to model trafficcoordination [1], [2]. To control and analyze traffic congestion,the Fundamental Diagram is commonly used to assign a flow-density relation at every traffic cell. While the FundamentalDiagram can successfully determine the traffic state for small-scale urban road networks, it may not properly functionfor congestion analysis and control in large traffic networks.Modeling of backward propagation, spill-back congestion, andshock-wave propagation is quite challenging. The objective ofthis paper is to deal with these traffic congestion modelingand control challenges. In particular, this paper contributes anovel integrative data-driven physics-inspired approach to ob-tain a microscopic data-driven traffic coordination model and resiliently control congestion in large-scale traffic networks .Researchers have proposed light-based and physics-basedcontrol approaches to address traffic coordination challenges.Fixed-cycle control is the traditional approach for the oper-ation of traffic signals at intersections. The traffic networkstudy tool [3], [4] is a standard fixed-cycle control tool foroptimization of the signal timing. Balaji and Srinivasan [5]and Chiu [6] offer fuzzy-based signal control approachesto optimize the green time interval at junctions. Physics-based traffic coordination approaches commonly use the Fun-damental Diagram to determine traffic state (flow-densityrelation) [7], [8], model dynamic traffic coordination [9],incorporate spillback congestion [10], [11], infuse backwardpropagation [12], [13] effects into traffic simulation, or specify
The authors are with the Department of Aerospace Engineering, Univer-sity of Michigan, Ann Arbor, MI, 48109 USA e-mail: [email protected]. the feasibility conditions for traffic congestion control. Jafariand Savla [14] propose first order traffic dynamics inspiredby mass flow conservation, dynamic traffic assignment [15],[16], and a cell transmission model [1], [17] to model andcontrol freeway traffic coordination. Model predictive control(MPC) is an increasingly popular approach for model-basedtraffic coordination optimization [18]–[20]. Baskar et al. [21]apply MPC to determine the optimal platooning speed forautomated highway systems (AHS). Furthermore, researchershave applied fuzzy logic [22]–[25], neural networks [26]–[29],Markov Decision Process (MDP) [30], [31], formal methods[32], [33] and mixed nonlinear programming (MNLP) [34] formodel-based traffic management. Optimal control [14], [35]approaches have also been proposed. Rastgoftar et al. [36]model traffic coordination as a probabilistic process wheretraffic coordination is controlled only through boundary inletnodes.This paper studies the problem of traffic coordinationand congestion control in a network of interconnected roads(NOIR). We model traffic coordination as a mass conserva-tion problem governed by the continuity partial differentialequation (PDE). Through spatial and temporal discretization oftraffic coordination, this paper advances our previous work [36]by modeling traffic as a Markov process controlled throughramp meters (at boundary road elements) and traffic signals(at NOIR junctions). Given traffic feasibility conditions, MPCis applied to assign optimal boundary inflow such that trafficover-saturation is avoided at every NOIR road element. Asthe result, the optimal boundary inflow is continuously as-signed as the solution of a constrained quadratic programmingproblem, and incorporated into traffic congestion planning.Given optimal boundary inflow, movement phase optimizationis formulated as a receding horizon problem where discreteactions commanded by the traffic signals are assigned byminimization of coordination costs over a finite time horizon.Our proposed model ensures that traffic density is non-negativeeverywhere in the NOIR, if the traffic inflow is positive at everyinlet boundary roads. Therefore, traffic coordination controlcan be commanded by a low computation cost.This paper is organized as follows. Preliminary notions ofgraph theory presented in Section II are followed by trafficcoordination modeling presented in Section III. Finite stateabstraction of traffic coordination is presented in Section IV.Ramp-based and signal-based traffic congestion control ispresented in Section V. Simulation results are presented inSection VI followed by concluding remarks in Section VII. a r X i v : . [ ee ss . S Y ] J a n II. G
RAPH T HEORY N OTIONS
Consider a NOIR with 𝑚 junctions defined by set W = { , · · · , 𝑚 } . An example of such a NOIR is shown in Fig. 1 (a).NOIR roads are identified by set V 𝑅 where 𝑖 ∈ V 𝑅 is the indexnumber of a road directed from an upstream junction to adownstream junction. Set V 𝑅 can be partitioned into a set ofinlet boundary roads V 𝑖𝑛 and a set of non-inlet roads V 𝐼 suchthat V 𝑅 = V 𝑖𝑛 (cid:216) V 𝐼 . (1)We also define a single “Exit” road defined by singleton V 𝐸 .Note that the “Exit” road does not represent a real road element(See Fig. 1 (a)); it is defined to model traffic coordination by afinite-state Markov process. We spatially discretize the NOIRusing graph G (V , E) with node set V = V 𝑅 (cid:208) V 𝐸 and edgeset E ⊂ V × V . Note that the nodes of graph G are the roadsof our NOIR, and subsequently we use “road” and “node”interchangeably. Graph G is directed and the edge set E holdthe following properties:1) Traffic flow is directed from road 𝑖 , if ( 𝑖, 𝑗 ) ∈ E .2) Real roads defined by set V 𝑅 are all unidirectional.Therefore, ( 𝑗, 𝑖 ) ∉ E , if ( 𝑖, 𝑗 ) ∈ E .Given graph G (V , E) , global in-neighbor, global out-neighbor, inlet boundary nodes, non-inlet nodes, and “Exit”node are formally defined as follows: Definition 1.
Given edge set E , the global in-neighbors ofroad 𝑖 are defined by set I 𝑖 = { 𝑗 ∈ V 𝑅 : ( 𝑗, 𝑖 ) ∈ E} . (2) Definition 2.
Given edge set E , the global out-neighbors ofroad 𝑖 are defined by set O 𝑖 = { 𝑗 ∈ V : ( 𝑖, 𝑗 ) ∈ E} . (3) Definition 3.
Inlet boundary roads have no in-neighbors at anytime, and they are formally defined by set V 𝑖𝑛 = { 𝑖 ∈ V 𝑅 : I 𝑖 = ∅ ∧ O 𝑖 ≠ ∅} . (4) Definition 4.
Non-inlet roads have at least one in-neighbor andone out-neighbor at any time, and they are formally definedby set V 𝐼 = V 𝑅 \ V 𝑖𝑛 . (5) Definition 5.
The “Exit” node is formally defined as follows: V 𝐸 = { 𝑖 ∈ V : I 𝑖 ≠ ∅ ∧ O 𝑖 = ∅} (6)where we assume that V 𝐸 is a singleton.Without loss of generality, inlet boundary nodes are indexedfrom through 𝑁 𝑖𝑛 , non-inlet roads are indexed from 𝑁 𝑖𝑛 + through 𝑁 , and the “Exit” node is indexed by 𝑁 + . Therefore V 𝑖𝑛 = { , · · · , 𝑁 𝑖𝑛 } , V 𝐼 = { 𝑁 𝑖𝑛 + , · · · , 𝑁 } , and V 𝐸 = { 𝑁 + } define the inlet, non-inlet, and “Exit” nodes, respectively. Weuse graph G (V , E) to define interconnections between theNOIR roads, V = V 𝑅 (cid:208) V 𝐸 and E ⊂ V × V define nodes andedges of graph G .The NOIR shown in Fig. 1 contains unidirectional “real”roads identified by set V 𝑅 = { , · · · , } and a virtual “Exit” node identified by set V 𝐸 = { } , i.e. V = V 𝑅 (cid:208) V 𝐸 . Note thatroads , · · · , ∈ V 𝐼 ⊂ V 𝑅 are in-neighbors to the “Exit” node ∈ V 𝐸 , as represented by the dotted lines. Thus I = { , · · · , } . Inlet nodes are identified by V 𝑖𝑛 = { , · · · , } and V 𝐼 = { , · · · , } defines all non-inlet roads. Movement Phase Rotation:
At each intersection, we define movement phases representing the different possible configura-tions of traffic light states at that intersection or, equivalently,the different possible paths that are allowed at that intersection.For instance, in the example of Fig. 1, intersection number 10has three lights – at the ends of roads 33, 35 and 50 – andthree different movement phases: • the first movement phase 𝜆 , corresponds to a greenlight at the end of road 50, and red lights at the endsof roads 33 and 35; equivalently, cars are allowed tocirculate from road 50 to roads 34, 13 or 36, and noother circulation is allowed; • the second movement phase 𝜆 , corresponds to a greenlight at the end of road 35, and red lights at the end ofroads 33 and 50; cars are only allowed to circulate fromroad 35 to either road 13 or 36; • the third movement phase 𝜆 , corresponds to a greenlight at the end of road 33, and red lights at the endof roads 35 and 50 to be red; cars are only allowed tocirculate from road 33 to either road 13 or 34.Those three movement phases define the three possible con-figurations of the lights at intersection number 10, and overtime the lights of intersection 10 alternatively go over thosemovement phases.Formally, let M 𝑖𝑛, 𝑗 ⊂ V 𝑅 define incoming roads and M 𝑜𝑢𝑡, 𝑗 ⊂ V 𝑅 define outcoming roads at junction 𝑗 ∈ W .Every junction 𝑗 is associated with 𝜇 𝑗 movement phasesthat can be commended by the traffic signals. The set ofedges 𝜆 𝑗,𝑘 ⊂ M 𝑖𝑛, 𝑗 × M 𝑜𝑢𝑡, 𝑗 ⊂ E is the 𝑘 -th movement phasecommanded at junction 𝑗 ∈ W where 𝑘 = , · · · , 𝜇 𝑗 . Movementphases at junction 𝑗 ∈ W are defined by finite set 𝚲 𝑗 asfollows: 𝚲 𝑗 = 𝜇 𝑗 (cid:216) 𝑘 = { 𝜆 𝑗,𝑘 } = { 𝜆 𝑗, , ..., 𝜆 𝑗,𝜇 𝑗 } (7)where 𝑗 ∈ W and 𝑘 = , · · · , 𝜇 𝑗 . Note that 𝚲 𝑗 is a set of subsetsof edge set E , i.e., is contained in the powerset of E . We candefine 𝚲 = 𝚲 × · · · × 𝚲 𝑚 (8)as the set of all possible movement phases across the NOIR.Transitions of movement phases are cyclic at every junction 𝑗 ∈ W , and defined by cycle graph C 𝑗 (cid:0) 𝚲 𝑗 , 𝚵 𝑗 (cid:1) with node set 𝚲 𝑗 and edge set 𝚵 𝑗 = (cid:110)(cid:0) 𝜆 𝑗, , 𝜆 𝑗, (cid:1) , · · · , (cid:16) 𝜆 𝑗,𝜇 𝑗 − , 𝜆 𝑗,𝜇 𝑗 (cid:17) , (cid:16) 𝜆 𝑗,𝜇 𝑗 , 𝜆 𝑗, (cid:17)(cid:111) (9)Intuitively, first 𝜆 𝑗, is the active movement phase defining thecurrent traffic light states and equivalent authorized paths atjunction 𝑗 ∈ W ; then the active movement phase is switched (a) (b) Fig. 1: (a) Example NOIR with unidirectional roads. (b) Three possible movement phases at junction ∈ W .to 𝜆 𝑗, , then to 𝜆 𝑗, ,..., then to 𝜆 𝑗,𝜇 𝑗 , then back to 𝜆 𝑗, to restartthe cycle.Fig. 1 (b) shows all possible movement phases at junc-tion ∈ W of the NOIR shown in Fig. 1 (a), where W = { , · · · , } defines the junctions. The incoming andoutcoming roads are defined by set M 𝑖𝑛, = { , , } and M 𝑜𝑢𝑡, = { , , } , respectively. There are three move-ment phases 𝜆 , = {( , ) , ( , ) , ( , )} ⊂ E , 𝜆 , = {( , ) , ( , )} ⊂ E , and 𝜆 , = {( , ) , ( , )} ⊂ E .Note that U-turns are disallowed at every junction of theExample NOIR shown in Fig. 1. Movement Phase Activation Time:
It is assumed thatmovement phase 𝜆 𝑗,𝑘 ∈ 𝚲 𝑗 ( 𝑘 = , · · · , 𝜇 𝑗 ) cannot be activemore that 𝑇 𝐿, 𝑗 time steps, where 𝑇 𝐿, 𝑗 ∈ N is equivalentto 𝑇 𝐿, 𝑗 Δ 𝑇 seconds, and Δ 𝑇 is a known constant time stepinterval. Because movement rotation is cyclic at every junction 𝑗 ∈ W , we define the maximum activation time 𝑇 𝐿, 𝑗 for everymovement phase at NOIR junction 𝑗 ∈ W . Define 𝑇 𝑗 as theactivation time of a movement phase at junction 𝑗 ∈ W , where 𝑇 𝑗 ≤ 𝑇 𝐿, 𝑗 . Note that 𝑇 𝑗 is independent of index 𝑘 ∈ { , · · · , 𝜇 𝑗 } and is counted from the start time of a movement phase 𝜆 𝑗,𝑘 at junction 𝑗 ∈ W . Given 𝑇 𝑗 and 𝑇 𝐿, 𝑗 , we define activationindex 𝑗 ∈ W , 𝜏 𝑗 = (cid:22) 𝑇 𝑗 𝑇 𝐿, 𝑗 (cid:23) ∈ { , } at every intersection 𝑗 ∈ W , where (cid:98)·(cid:99) denotes the floor func-tion. Because 𝑇 𝑗 ≤ 𝑇 𝐿, 𝑗 , 𝜏 𝑗 ∈ { , } is a binary variable assign-ing whether the active movement phase must be overridden ornot. If 𝜏 𝑗 = , the current movement 𝜆 𝑗,𝑘 ( 𝑘 = , · · · , 𝜇 𝑗 , 𝑗 ∈ W )can still remain active. Otherwise, the active movement phaseis overridden and the next movement phase must be selected. The network movement phase is denoted by 𝜆 = ( 𝜆 , · · · , 𝜆 𝑚 ) ∈ 𝚲 where 𝜆 𝑗 ∈ 𝚲 𝑗 and 𝑗 ∈ W . We define theswitching communication graph G 𝜆 (V , E 𝜆 ) to specify theinter-road connection under movement phase 𝜆 ∈ 𝚲 , where E 𝜆 ⊂ E defines the edges of graph G . Per movement phasedefinition given in (7), E 𝜆 = ∪ 𝑚𝑘 = 𝜆 𝑘 . In-neighbors and out-neighbors of road (or Exit node) 𝑖 ∈ V is defined by thefollowing sets: 𝑖 ∈ V , 𝜆 ∈ 𝚲 , I 𝑖,𝜆 = { 𝑗 ∈ V 𝑅 : ( 𝑗, 𝑖 ) ∈ E 𝜆 } , (10a) 𝑖 ∈ V , 𝜆 ∈ 𝚲 , O 𝑖,𝜆 = { 𝑗 ∈ V : ( 𝑖, 𝑗 ) ∈ E 𝜆 } . (10b)Given the above definitions, for any 𝜆 ∈ 𝚲 , I 𝑖,𝜆 ⊂ I 𝑖 and O 𝑖,𝜆 ⊂O 𝑖 , thus:1) for every 𝜆 ∈ 𝚲 , in-neighbor set I 𝑖,𝜆 = ∅ if 𝑖 ∈ V 𝑖𝑛 ;2) for every 𝜆 ∈ 𝚲 , out-neighbor set O 𝑖,𝜆 = ∅ if 𝑖 ∈ V 𝐸 .III.T RAFFIC C OORDINATION M ODEL
We use the mass conservation law to model traffic at everyNOIR road element 𝑖 ∈ V . Let 𝜌 𝑖 , 𝑦 𝑖 , and 𝑧 𝑖 denote trafficdensity, traffic inflow, and traffic outflow at every road element 𝑖 ∈ V . Traffic dynamics governed by mass conservation is: 𝜌 𝑖 ( 𝑘 + ) = 𝜌 𝑖 ( 𝑘 ) + 𝑦 𝑖 ( 𝑘 ) − 𝑧 𝑖 ( 𝑘 ) , (11)where 𝑧 𝑖 ( 𝑘 ) = (cid:26) 𝑝 𝑖 ( 𝜆 ) 𝜌 𝑖 ( 𝑘 ) 𝑖 ∈ V 𝑅 , ∀ 𝜆 ∈ 𝚲 𝜌 𝑖 ( 𝑘 ) + 𝑦 𝑖 ( 𝑘 ) 𝑖 ∈ V 𝐸 , ∀ 𝜆 ∈ 𝚲 (12a) 𝑦 𝑖 ( 𝑘 ) = (cid:26) 𝑢 𝑖 ( 𝑘 ) 𝑖 ∈ V 𝑖𝑛 , ∀ 𝜆 ∈ 𝚲 (cid:205) 𝑗 ∈I 𝑖,𝜆 𝑞 𝑖, 𝑗 ( 𝜆 ) 𝑧 𝑗 ( 𝑘 ) + 𝑑 𝑖 𝑖 ∈ V \ V 𝑖𝑛 , ∀ 𝜆 ∈ 𝚲 (12b) and inflow 𝑦 𝑖 ≥ at road element 𝑖 ∈ V 𝑖𝑛 has the followingproperties:1) If 𝑖 ∈ V 𝑖𝑛 , 𝑦 𝑖 = 𝑢 𝑖 can be controlled by a ramp meter.2) If 𝑖 ∈ V 𝐼 , 𝑑 𝑖 ≥ is given as a non-zero-mean Gaussianprocess.Note that 𝑑 𝑖 is uncontrolled at road element 𝑖 ∈ V 𝑅 \ V 𝑖𝑛 .Variable 𝑝 𝑖 ( 𝜆 ) ∈ [ , ] is the traffic outflow probability, and 𝑞 𝑖, 𝑗 ( 𝜆 ) is the outflow fraction of road element 𝑗 directedtowards 𝑖 ∈ O 𝑗,𝜆 when 𝜆 ∈ 𝚲 is the active movement phaseover time interval [ 𝑡 𝑘 , 𝑡 𝑘 + ] . Note that ∑︁ 𝑗 ∈O 𝑖,𝜆 𝑞 𝑗,𝑖 ( 𝜆 ) = (13)for every 𝜆 ∈ 𝚲 . We define P ( 𝜆 ) = diag ( 𝑝 ( 𝜆 ) , · · · , 𝑝 𝑁 ( 𝜆 ) , 𝑝 𝑁 + ( 𝜆 )) , where 𝑝 𝑁 + ( 𝜆 ) = ∀ 𝜆 ∈ Λ . This implies that the outflow of the exit node is zero.Also, matrix Q ( 𝜆 ) = (cid:2) 𝑞 𝑖, 𝑗 ( 𝜆 ) (cid:3) ∈ R ( 𝑁 + )×( 𝑁 + ) is non-negative,and 𝑞 𝑁 + , 𝑗 ( 𝜆 ) = (cid:26) 𝑗 = 𝑁 + ∈ V 𝐸 . (14)Eq. (14) implies that traffic does not flow from the exit node 𝑁 + ∈ V 𝐸 to any other element 𝑗 ∈ V 𝑅 \ V 𝐸 . The trafficnetwork dynamics is given by x ( 𝑘 + ) = A ( 𝜆 ) x ( 𝑘 ) + g ( 𝑘 ) (15)where x ( 𝑘 ) = (cid:2) 𝜌 ( 𝑘 ) · · · 𝜌 𝑁 + ( 𝑘 ) (cid:3) 𝑇 and g = (cid:2) g 𝑇𝑅 𝑔 𝑁 + (cid:3) 𝑇 = [ 𝑔 𝑖 ] ∈ R ( 𝑁 + )× is defined as follows: 𝑔 𝑖 ( 𝑘 ) = 𝑢 𝑖 ( 𝑘 ) 𝑖 ∈ V 𝑖𝑛 𝑑 𝑖 ( 𝑘 ) 𝑖 ∈ V 𝑅 \ V 𝑖𝑛 𝑖 ∈ V 𝐸 . (16)Also, A ( 𝜆 ) = I − P ( 𝜆 ) + Q ( 𝜆 ) P ( 𝜆 ) = (cid:20) C ( 𝜆 ) ( 𝜆 ) (cid:21) , where every column of non-negative matrix A : 𝚲 → R ( 𝑁 + )×( 𝑁 + ) sums to for every movement phase 𝜆 ∈ Λ , C : 𝚲 → R 𝑁 × 𝑁 , and D ( 𝜆 ) ∈ R × 𝑁 . Eigenvalues of matrix C ( 𝜆 ) are all placed inside a disk of radius 𝑟 𝜆 < with center atthe origin. Note that the 𝑖 -th entry of matrix D : 𝚲 → R × 𝑁 specifies the fraction of traffic flow exiting the NOIR fromnode 𝑖 ∈ V 𝑅 . Traffic dynamics at non-exit nodes is given by x 𝑅 ( 𝑘 + ) = C ( 𝜆 ) x 𝑅 ( 𝑘 ) + g 𝑅 ( 𝑘 ) , (17)where x 𝑅 ( 𝑘 ) = (cid:2) 𝜌 ( 𝑘 ) · · · 𝜌 𝑁 ( 𝑘 ) (cid:3) 𝑇 .IV. P ROBLEM S PECIFICATION
Linear Temporal Logic (LTL) is used to specify theconservation-based traffic coordination dynamics [37] andpresent the feasibility conditions. Every LTL formula con-sists of a set of atomic propositions, logical operators, andtemporal operators. Logical operators include ¬ (“negation”), ∨ (“disjunction”), ∧ (“conjunction”), and ⇒ (“implication”). LTL formulae also use temporal operators (cid:3) (“always”), (cid:13) (“next”), ♦ (“eventually”), and U (“until”).We extend discrete-time LTL with the syntactic sugar (cid:3) { ,...,𝑁 𝜏 } to specify satisfaction of a certain property inthe next 𝑁 𝜏 + time steps. More specifically, (cid:3) { ,...,𝑁 𝜏 } 𝜑 atdiscrete time 𝑘 if and only if 𝜑 is satisfied at discrete times 𝑘 to time 𝑘 + 𝑁 𝜏 [36].The problem of traffic coordination can be formally speci-fied by a finite-state abstraction defined by tuple M = (S , A , H , C ) , where S is the state set, A is the discrete action set, H : S × A → S is the state transition relation, and
C :
S × A → R + is the immediate cost function. A. State set S Set S is mathematically defined by S = { 𝑠 = ( x , g , 𝜆, 𝜏 ) (cid:12)(cid:12) x ∈ X , g ∈ G , 𝜆 ∈ 𝚲 , 𝜏 ∈ { , } 𝑚 } , (18)where the traffic density vector x = (cid:2) 𝜌 · · · 𝜌 𝑁 (cid:3) 𝑇 ∈ R 𝑁 + and input vector g ∈ G ∈ R 𝑁 𝑖𝑛 × were introduced in Section III,and X and G are compact sets. Also, 𝜆 = (cid:16) 𝜆 ,𝜁 , · · · , 𝜆 𝑚,𝜁𝑚 (cid:17) ∈ 𝚲 is a movement phase, and 𝜏 = ( 𝜏 , · · · , 𝜏 𝑚 ) ∈ { , } 𝑚 where 𝜏 𝑖 ∈{ , } is the activation index at junction 𝑖 ∈ W . An executionof the proposed system is expressed by 𝑠 = 𝑠 𝑠 𝑠 , · · · where 𝑠 𝑘 = ( x [ 𝑘 ] , g 𝑘 , 𝜆 [ 𝑘 ] , 𝜏 [ 𝑘 ]) is the state of the system at time 𝑘 . Feasibility Condition 1:
Traffic density, defined as thenumber of cars at a road element, is a positive quantityeverywhere in the NOIR. It is also assumed that every roadelement has maximum capacity 𝜌 max . Therefore, the numberof cars cannot exceed 𝜌 max in any road element 𝑖 ∈ V . Thesetwo requirements can be formally specified as follows: (cid:219) 𝑖 ∈V (cid:3) { ,...,𝑁 𝜏 } ( 𝜌 𝑖 ≥ ∧ 𝜌 𝑖 ≤ 𝜌 max ) . ( Φ )If feasibility condition Φ is satisfied at every road element,then traffic over-saturation is avoided everywhere in the NOIR,at every discrete time 𝑘 . Optional Condition 2:
Boundary inflow should satisfy thefollowing feasibility condition at every discrete time 𝑘 : (cid:3) { ,...,𝑁 𝜏 } (cid:32) ∑︁ 𝑖 ∈V 𝑖𝑛 𝑢 𝑖 = 𝑢 (cid:33) . ( Φ )Boundary condition ( Φ ) constrains the number of vehiclesentering the NOIR to be exactly 𝑢 at any time 𝑘 . Note that 𝑢 is an upper bound on vehicles entering the NOIR. However,in the simulation results presented, traffic demand is significantsuch that the NOIR is maximally utilized by as many vehiclesas possible. B. Action Set A Action set A : 𝚲 × T → 𝚲 assigns the next acceptablemovement at every junction 𝑖 ∈ W , given the current NOIRactivation index 𝜏 ∈ T = { , } 𝑚 and movement phase 𝜆 = ( 𝜆 , · · · , 𝜆 𝑚 ) , i.e. 𝜏 = ( 𝜏 , · · · , 𝜏 𝑚 ) , 𝜏 𝑖 ∈ { , } , 𝑖 ∈ W . We write 𝜆 + 𝑖 for the value of 𝜆 𝑖 in the next state, i.e. 𝜆 + 𝑖 ( 𝑘 ) = 𝜆 𝑖 ( 𝑘 + ) ,and similarly for other variables. Actions are constrained andmust satisfy one of the following LTL formula: ( 𝜏 𝑖 = ) ⇒ (cid:0)(cid:0) 𝜆 𝑖 , 𝜆 + 𝑖 (cid:1) ∈ Ξ 𝑖 ∨ (cid:0) 𝜆 + 𝑖 = 𝜆 𝑖 (cid:1)(cid:1) , ( Φ ,𝑖 ) ( 𝜏 𝑖 = ) ⇒ (cid:0) 𝜆 𝑖 , 𝜆 + 𝑖 (cid:1) ∈ Ξ 𝑖 , ( Φ ,𝑖 )Combining ( Φ ,𝑖 ) and ( Φ ,𝑖 ), the next movement phase mustsatisfy the following LTL formula: (cid:219) 𝑖 ∈W (cid:3) { ,...,𝑁 𝜏 } (cid:0)(cid:0)(cid:0)(cid:0) 𝜆 𝑖 , 𝜆 + 𝑖 (cid:1) ∈ Ξ 𝑖 (cid:1) U ( 𝜏 𝑖 = ) (cid:1) ∨ (cid:0)(cid:0) 𝜆 𝑖 , 𝜆 + 𝑖 (cid:1) ∈ Ξ 𝑖 (cid:1)(cid:1) . ( Φ ) Remark 1.
Set
A ( 𝜆, 𝜏 ) ⊂ 𝚲 is defined as follows: A ( 𝜆, 𝜏 ) = { 𝜆 + ∈ 𝚲 | ∀ 𝑖 ∈ W , ( 𝜆 𝑖 , 𝜆 + 𝑖 ) ∈ Ξ 𝑖 ∨ ( 𝜏 𝑖 = ∧ 𝜆 + 𝑖 = 𝜆 𝑖 )} . (20) C. State Transition Function
The state transition relation H defines transition from“current” state 𝑠 = ( x , g , 𝜆, 𝜏 ) ∈ S to “next” state 𝑠 + = ( x + , g + , 𝜆 + , 𝜏 + ) ∈ S given action 𝑎 ( 𝜆, 𝜏 ) ∈ A ( 𝜆, 𝜏 ) . Currentand next movement phases must satisfy condition ( Φ ) below.Transition of current activation index 𝜏 must satisfy thefollowing properties: (cid:219) 𝑖 ∈W (cid:0)(cid:0) 𝜏 + 𝑖 = (cid:1) U (cid:0) 𝑇 𝑖 = 𝑇 𝐿,𝑖 (cid:1)(cid:1) . ( Φ )Note that the 𝑇 𝑖 is reset every time movement phase is updatedat junction 𝑖 ∈ W . This requirement is formally specified asfollows: ∀ 𝑖 ∈ W , (cid:0) 𝜆 + 𝑖 ≠ 𝜆 𝑖 (cid:1) ⇒ (cid:0) 𝑇 + 𝑖 = (cid:1) (21)This paper assumes that 𝑔 𝑖 = 𝑑 𝑖 is a Gaussian process for 𝑖 ∈ V 𝐼 is an non-inlet road, i.e. 𝑑 𝑖 ∼ N (cid:0) ¯ 𝑑 𝑖 , 𝜎 𝑖 (cid:1) . Per Eq. (16), 𝑔 𝑖 = 𝑢 𝑖 for 𝑖 ∈ V 𝑖𝑛 where 𝑢 𝑖 is determined as the solution of areceding horizon optimization problem presented in SectionV. Therefore (cid:32) (cid:219) 𝑖 ∈V 𝑖𝑛 𝑔 + 𝑖 = 𝑢 + 𝑖 (cid:33) ∧ (cid:32) (cid:219) 𝑖 ∈V 𝐼 𝑔 + 𝑖 = 𝑦 𝑖 (cid:33) ∧ (cid:32) (cid:219) 𝑖 ∈V 𝐸 𝑔 + 𝑖 = (cid:33) . (22)Transition of x is governed by (15), thus x + = A ( 𝜆 ) x + g (23)where 𝜆 ∈ 𝚲 . D. Cost Function
Given Eq. (15), an 𝑁 𝜏 -step expected transition is given by x 𝑁 𝜏 + = 𝚯 ℎ ( 𝜆 ) x + 𝚪 𝑁 𝜏 g ... g 𝑁 𝜏 , (24)where g , · · · , g 𝑁 𝜏 ∈ G , x ∈ X , 𝜆 ∈ 𝚲 , 𝚯 𝑁 𝜏 ( 𝜆 ) = A 𝑁 𝜏 ( 𝜆 ) and 𝚪 𝑁 𝜏 = (cid:2) 𝚯 𝑁 𝜏 − · · · 𝚯 I (cid:3) ∈ R ( 𝑁 + )× 𝑁 𝜏 ( 𝑁 + ) . The cost function C is defined by C (cid:0) x , g , · · · , g 𝑁 𝜏 , 𝜆 (cid:1) = 𝑁 𝜏 ∑︁ ℎ = x 𝑇ℎ + F 𝑇 Fx ℎ + = (cid:2) x 𝑇 g 𝑇 · · · g 𝑇𝑁 𝜏 (cid:3) W x g ... g 𝑁 𝜏 (25)where F = (cid:20) I 𝑁 𝑁 × × 𝑁 (cid:21) , and W = (cid:34) (cid:205) 𝑁 𝜏 ℎ = 𝚯 𝑇ℎ F 𝑇 F 𝚯 ℎ (cid:205) 𝑁 𝜏 − ℎ = 𝚯 𝑇𝑁 𝜏 F 𝑇 F 𝚪 ℎ (cid:205) 𝑁 𝜏 ℎ = 𝚪 𝑇ℎ F 𝑇 F 𝚯 𝑁 𝜏 (cid:205) 𝑁 𝜏 ℎ = 𝚪 𝑇ℎ F 𝑇 F 𝚪 ℎ (cid:35) . V. T
RAFFIC C ONGESTION C ONTROL
The objective of the traffic congestion control is to determineoptimal inflow and movement phase such that cost function C , defined in Eq. (25), is minimized. Optimal traffic inflowis assigned with MPC while optimal movement phases areassigned as the solution of a RHO problem.The optimal boundary inflow g ∗ is assigned by solving thefollowing optimization problem: x ∈ X , 𝜆 ∈ 𝚲 , (cid:16) g ∗ , · · · , g ∗ 𝑁 𝜏 (cid:17) = argmin g , ··· , g 𝑁𝜏 ∈ G C (cid:0) x , g , · · · , g 𝑁 𝜏 , 𝜆 (cid:1) , (26)subject to the conditions ( Φ ) and ( Φ ).The optimal movement phase 𝜆 +∗ is assigned by solving thefollowing optimization problem: x ∈ X , g , · · · , g 𝑁 𝜏 ∈ G , 𝜆 ∈ 𝚲 ,𝜆 +∗ = argmin 𝜆 + ∈A( 𝜆,𝜏 ) C (cid:0) x , g , · · · , g 𝑁 𝜏 , 𝜆 (cid:1) , (27)subject to the following conditions (cid:211) 𝑖 ∈W Φ 𝑖, , (cid:211) 𝑖 ∈W Φ 𝑖, , and Φ . Fig. 2: Optimal boundary inflow rates 𝑢 through 𝑢 versusdiscrete time 𝑘 . VI.S IMULATION R ESULTS
Traffic coordination is investigated in simulation for theexample NOIR shown in Fig. 1 (a) consisting of 𝑁 = unidirectional roads. Traffic coordination is controlled throughthe NOIR inlet boundary nodes defined by V 𝑖𝑛 = { , · · · , } and traffic signals at junction nodes W = { , · · · , } .This paper assumes that the time interval between twoconsecutive discrete times 𝑘 and 𝑘 + is Δ 𝑡 = 𝑠 . It isassumed that the inflow 𝑦 𝑖 = ± . is randomly enteredthrough every road element 𝑖 ∈ V 𝐼 . For simulation 𝑢 = ischosen. Therefore, a total of vehicles are allowed to enterthe NOIR through the NOIR inlet boundary road elements atevery discrete time 𝑘 . Traffic coordination is controlled throughthe ramp meter at the NOIR boundary road elements and trafficsignals at NOIR intersections by solving the optimizationproblem developed in Section V.In Fig. 2, boundary inflow rates 𝑢 through 𝑢 are plottedversus time for 𝑘 = , · · · , . For the simulation, 𝜌 max = isconsidered. Fig. 4 plots traffic density 𝜌 𝑖 at every road element 𝑖 ∈ V versus discrete time 𝑘 . It is seen that 𝜌 ( 𝑘 ) < 𝜌 max = atevery discrete time 𝑘 . Thus, traffic oversaturation is ensured.Also, the total traffic density 𝑟 net ( 𝑘 ) = × 𝑁 x 𝑅 ( 𝑘 ) is plottedversus discrete time 𝑘 in Fig. 5. For simulation, we choose 𝑇 𝐿,𝑖 = . Therefore, a movement phase cannot be active morethan × Δ 𝑇 = 𝑠 . A movement phase at junction 𝑖 ∈ W isrepresented by a directed tree containing a root node andterminal nodes per the example movement phase shown in Fig.1 (b). The root node represents the active road with incomingtraffic flow, and terminal nodes represent the active outgoingroads. In Fig. 3, active incoming roads are shown at NOIRjunctions , · · · , ∈ W for 𝑘 = , · · · , .Fig. 5 plots the net traffic density of the NOIR versusdiscrete time 𝑘 for 𝑘 = , · · · , . It is seen that net trafficdensity reaches the steady-state value in about eight time stepswhile traffic consistently enters and leaves the NOIR.VII. C ONCLUSION
This paper offers a physics-inspired approach to model andcontrol traffic coordination in a network of interconnectedroads (NOIR). Traffic coordination modeled as a Markovprocess is obtained by spatial and temporal discretization of themass conservation continuity equation. We showed how trafficcongestion can be effectively controlled through ramp metersand traffic signals located at boundaries and junctions of theNOIR. In particular, MPC is applied to control the boundary Fig. 3: Optimal movement phases at NOIR junctions at 𝑘 = , · · · , .Fig. 4: Traffic density at every NOIR road for 𝑘 = , · · · , .inflow while a RHO planner optimizes movement phasescommanded by traffic signals at NOIR junctions. Simulationresults show that integration of boundary and signal controlscan effectively manage urban traffic congestion.VIII.A CKNOWLEDGEMENT
This work has been supported by the National ScienceFoundation under Award Nos. 1914581 and 1739525. Theauthors gratefully thank Professor Ella Atkins for the usefulcomments on this paper.R
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