Vehicular Demand Management in Traffic Networks Comprised by Macroscopically Homogeneous Regions under Inter-boundary Constraints

Vehicular Demand Management in Traffic Networks Comprised by Macroscopically Homogeneous Regions under Inter-boundary Constraints Michalis Rampa, Andreas Kasisa, Charalambos Menelaouaand Stelios Timotheoua Abstract— Vehicular demand management strategies emerge as a potential solution to traffic congestion. These strategies aim to regulate the inflow of vehicles across specific regions of a vehicular transportation network. This work investigates the stability and optimality of vehicular demand management schemes in regional traffic networks, considering inter-boundary flow constraints and triangular macroscopic fundamental diagram relationships between densities and flows. We first formulate an optimization problem that aims to maximize the total vehicular throughput at steady-state. Due to the triangular macroscopic fundamental diagram relationships and inter-boundary flow constraints, the optimization problem is nonconvex. We tackle this challenge by reformulating the problem as a Mixed Integer Linear Program that can be solved with standard mathematical programming solvers to determine the optimal operating set-points. Nonetheless, it has been demonstrated that operating at maximum throughput setpoints, particularly near local critical density points, may lead to instability and gridlock. To address this issue, we propose a decentralized proportional vehicular demand management controller, accompanied by proper local design conditions, such that stability is guaranteed. The effectiveness and practicality of the proposed approach are demonstrated through numerical simulations in a six-region traffic network system, that showcase the impact, in terms of maximum throughput, of incorporating inter-boundary constraints. I. INTRODUCTION Traffic congestion remains one of the most pressing challenges for modern vehicular transportation networks (VTN). Congestion arises from high traffic demand, with many vehicles using the same road, especially during peak hours. When demand surpasses the infrastructure’s capacity, traffic flow deteriorates, leading to congestion [1]. In extreme cases, this can escalate to gridlock, a state where vehicles are at a complete standstill, and the VTN is unable to admit any more traffic, causing a total system breakdown. To reduce congestion, various traffic and demand management strategies have been proposed. Given the large scale of VTN, most contemporary approaches rely on macroscopic This work is supported by the European Union (i. ERC, URANUS, No. 101088124, and ii. Horizon 2020 Teaming, KIOS CoE, No. 739551), and the Government of the Republic of Cyprus through the Deputy Ministry of Research, Innovation, and Digital Strategy. Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them. aAll the authors are with KIOS Research and Innovation Center of Excellence, 1 Panepistimiou Avenue, 2109 Aglantzia, Nicosia, Cyprus. Emails: {ramp.michalis,kasis.andreas,menelaou.charalampos, timotheou.stelios}@ucy.ac.cy traffic dynamics, which offer a simplified, aggregated view of the system by dividing the network into homogeneous regions [2]. Within each region, traffic dynamics are modelled using key parameters such as flow, density, and speed [3]. Macroscopic approaches reduce computational complexity by focusing on the collective behaviour of traffic rather than tracking individual vehicles, which would be computationally infeasible in large-scale VTNs. The primary goal of both traffic and demand management strategies is to sustain network operation at maximum capacity, which is achieved near the critical density point—a threshold that separates the free-flow regime from the congested regime [3]. Traffic management strategies focus on optimizing the flow of vehicles across the network using methods such as route guidance, gating strategies, and perimeter control [4]–[11]. These approaches aim to redistribute traffic or restrict access to critical areas, such as city centers, to avoid congestion. However, they often lead to congestion in unprotected regions, as traffic is merely shifted rather than reduced [12]. In contrast, vehicular demand management (VDM) strategies have the potential to completely eliminate congestion by regulating the rate at which vehicles enter a VTN. Unlike traffic management, VDM controls both the spatial and temporal distribution of traffic flow, making it possible to optimize overall system performance. VDM strategies can queue vehicles outside the VTN, schedule departure times, or encourage the use of alternative transportation modes to avoid overloading the VTN [13]. By regulating vehicle inflow, VDM ensures that the VTN operates near its critical density, thereby preventing excessive vehicle accumulation and maintaining smooth traffic flow [14]. Despite the success of VDM strategies in optimizing traffic flows, a crucial aspect of traffic management is often overlooked: stability. Stability is vital, particularly when operating near the critical density point, where small disturbances, such as slight increases in demand or irregularities in flow, can lead to rapid destabilization of the VTN, severely degrading performance and potentially leading to gridlock [15]. Although many existing traffic and demand management strategies effectively mitigate congestion, few explicitly account for stability, despite its importance in ensuring the long-term success of congestion management. An additional factor that contributes to congestion is the presence of inherent inter-boundary flow constraints (IBCs) between regions that are innate characteristics of VTNs. IBCs limit the amount of vehicles a region can admit from neighbouring areas based on its current occupancy, becoming tighter as local vehicle numbers increase. This behaviour significantly complicates the task of congestion mitigation as it introduces additional non-linearities and challenges in the VTN optimization and control design, making it essential to account for. Moreover, to the author’s best knowledge, the stability and optimality properties of VDM schemes in VTNs in the presence of IBCs have not been explored. Contribution: This work addresses this critical gap, by focusing on the stability properties of VDM schemes and investigating optimality conditions that achieve maximum throughput response in VTNs, while explicitly accounting for regional IBCs. These constraints introduce non-linearities into traffic network dynamics and additional challenges to the optimization and control design. Specifically, we consider a class of VTNs with density-flow relationships described by triangular macroscopic fundamental diagrams (MFD) and investigate how macroscopic VDM schemes can be designed to maintain network stability. First, an optimization problem aiming to maximize vehicle throughput with respect to a VTN under steady-state conditions is formulated. Regional IBCs render the formulated optimization problem nonlinear and non-convex, making standard solving tools not directly applicable. We resolve this by transforming it to a Mixed Integer Linear Program (MILP). However, as demonstrated in [15], maximum-throughput solutions, which often coincide with local critical density values, might result to unstable behaviour. To address this, a decentralized proportional feedback VDM scheme is developed and supplemented with local conditions on its parameters to analytically guarantee VTN stability. The developed stability conditions explicitly account for IBCs. The developed analytic results are validated through numerical simulations on a 6-region VTN, which demonstrate the effectiveness of the proposed strategy, and highlight the importance of IBCs, showing a significant improvement in throughput when those are taken into account. By simultaneously enabling optimal throughput and stability guarantees, this work provides a robust and practical solution to congestion in modern VTNs. The main contributions of this work are summarized below. (i) We formulate an optimization problem aimed at maximizing vehicle throughput in large-scale networks, explicitly considering regional IBCs. The optimization problem is transformed into a MILP to account for the non-linearities present in the system, and enable the use of standard solvers. (ii) We propose a decentralized proportional VDM controller, and provide locally verifiable conditions that take into account regional IBCs, such that VTN stability is analytically guaranteed. Paper structure: The paper is organized as follows. The problem formulation is given in Section II. It entails the dynamics of the VTN, the formulation of an optimization problem that aims to maximize vehicular throughput and the problem statement. Our approach in solving the maximum vehicular throughput optimization problem is given in Section III. Section IV presents a decentralized proportional VDM scheme with enhanced stability properties and the main stability results of this work. The effectiveness of the developed analytic results is showcased in Section V through simulations on a 6-region VTN. Section VI contains the conclusions of this work. Notation: We denote real numbers by R. The set of real non-negative numbers is denoted by R+. Small bold letters denote vectors with the ith component of a vector xdenoted by xi. We use Rnto denote the set of ndimensional vectors with real entries with the non-negative orthant of Rngiven by Rn +. The min-max operator is given by [x]b a= max(min(x, b), a), where a, b ∈Rand a≤b. Capital calligraphic letters denote sets. The n×1vector with all elements equal to 0 is denoted by 0n∈Rn. A function f:Rn→Ris positive (negative) definite if f(0n) = 0 and f(x)>0(f(x)<0) for every x=0n. We write N \{i}to denote the set Nexcluding the element i. The right superscript x∗denotes the equilibrium value of xand the hat designation, ˆx, denotes that xis an optimal element with respect to a given optimization problem. Units of considered variables are given only at the first instance for compactness. II. PROBLEM FORMULATION A. Model description The traffic area under interest is modelled as a VTN of n nodes (n≥2) connected by edges between nodes. Each node corresponds to a specific region of the VTN and each edge to a specific road segment connecting two nodes. A directed graph G={N,E} describes the VTN structure with the sets of regions1and roads denoted by N={1,2, ..., n}, and E ⊆ N × N , respectively. We use ϵi,j = (i, j)∈ E to denote the road allowing region jto receive vehicles from region i. Predecessor regions that can directly send vehicles to a region ibelong to the set Pi={j∈ N :ϵj,i ∈ E}. Successor regions that can directly receive vehicles from a region ibelong to the set Si={l∈ N :ϵi,l ∈ E}. Vehicles enter the multi-region VTN via origin regions and every vehicle that enters the network, does so to reach a destination region that belongs to the VTN. In this work, all regions are considered to be both origin and destination regions. Since regions are treated as homogeneous, the collective inter-regional traffic flow behaviour of a region i, gi(ρi(t)) [veh/h], where ρi(t) [veh/km]is the region’s vehicle density, is described by a triangular (piecewiselinear) MFD [3], [16], [17], see Fig. 1, i.e. a triangular function of the region’s vehicle density, given by gi(ρi) =rifi(ρi), i ∈ N,(1a) fi(ρi) = min(vf iρi, bC i−vC iρi), i ∈ N,(1b) ri=Lil−1 i, i ∈ N,(1c) 1In this work the terms node and region and edge and road are used interchangeably. where, ri∈R+is the region itrip completion ratio, Li[km] is the length of region i,li[km]is the average trip length of a vehicle in the same region, and vf i[km/h]is the free-flow speed of region igiven by vf i=qC i(ρC i)−1, i ∈ N,(2) with qC i[veh/h]being the capacity flow2of region i, and ρC i[veh/km], being the critical density threshold of region i. The backward congestion propagation speed, vC i[km/h], and the constant bC i[veh/h], for region iin (1b) are given by vC i=qC i(ρJ i−ρC i)−1, i ∈ N,(3a) bC i=ρJ iqC i(ρJ i−ρC i)−1, i ∈ N,(3b) where ρJ i[veh/km]is the jam density threshold of region i. Two traffic modes of operation are identified; a free-flow mode, [0, ρC i), and a congested mode, [ρC i, ρJ i], see Fig. 1. The MFDs total inter-regional flow, gi(ρi(t)), is needed in the calculation of the flows that exit the network through destination regions and the transfer flows between regions. Transfer flows between regions are limited by successor regions inter-boundary capacities via inter-boundary region ito region jcapacity functions, cij :R+→R+, given by cij(ρj) = min(cmax ij , cmax ij ρJ j−ρj ρJ j−¯ρij ), i ∈ N, j ∈ Si,(4) where cmax ij ∈R+is the critical inter-boundary capacity flow threshold between region iand region j, and ¯ρij ∈(0, ρJ j)is the critical inter-boundary density threshold between region iand region j. Equations (1a) and (4) yield the transfer flow from region ito region j, given by gij(ρi, ρj) = min (wij gi(ρi), cij (ρj)) ,(i, j)∈ E,(5) where wij ∈R+are the region ioutflow split constants satisfying wii +X j∈Si wij = 1, i ∈ N.(6) The term wii ∈R+in (6) is the rate that vehicles end their trip in region i. The VTN density state dynamics for each region i,ρi(t), is given by ˙ρi(t) = 1 Li−X l∈Si gil(ρi(t), ρl(t)) −wiigi(ρi(t)) +ui(t) + X j∈Pi gji(ρj(t), ρi(t)),∀i∈ N.(7) In the right hand side of (7), the first term is the flow towards successor nodes, the second term is the flow exiting the VTN through destination region i, the third term, ui(t) [veh/h], is the serviced demand admitted to the network through the origin region i(considered as a control variable), and the last term is the inflow from predecessor nodes. For ease of 2Capacity flow is the maximum flow that can be supported by region i; it is yielded by the triangular function fi(ρi)at the critical density point, ρC i. ρi gi gC i ρC iρJ i gi(ρi) cji(ρi) ¯ρji Fig. 1. Region itotal inter-regional traffic flow gi(ρi(t)), approximated by a triangular macroscopic fundamental diagram and region iinter-regional constraint function cji(ρi(t)). referencing the dynamics are defined in a compact form as follows: Traffic Model 1: Equations (1), (4), (5) and (7) capture the evolution of the traffic states of the considered n-region connected VTN. B. Optimization problem The design of efficient VDM schemes requires the formulation of an optimization problem satisfying key metrics for demand allocation. Here, by explicitly taking into account regional IBCs, an optimization problem yielding throughputoptimal equilibria under steady-state conditions is formulated, i.e. a state of maximum vehicle throughput relative to the VTN boundaries. Optimization Problem 1: For a VTN with dynamics described by Traffic Model 1, solve: max (ρ,u)X i∈N ui(8a) s.t. : 0 ≤ρi≤ρC i, i ∈ N,(8b) umin i≤ui≤umax i, i ∈ N,(8c) X j∈Pi gji(ρj, ρi) + ui −wiigi(ρi)−X l∈Si gil(ρi, ρl)=0, i ∈ N,(8d) where umin i∈R+, umax i∈R+are constants associated with the ability of each region to accommodate demand. Remark 1: Constraint (8b) stems from the selection of an equilibrium point to be characterized by free-flow conditions. In (8c), constraint umin iis employed to ensure that the solution of Optimization Problem 1 simultaneously serves the vehicle demand goals and ensures sufficient control authority since no negative control values are allowed, i.e. the controller cannot take vehicles out from a region or the network. An admitted demand upper bound constraint, umax i, is also placed since in practice the traffic network parameters and the network structure constrain the range of values of ui,∀i∈ N. Also, constraint (8d) follows from (7) at equilibrium. Finally, it is assumed that a feasible solution to this problem exists. C. Problem Statement Since vehicular congestion continues to be a towering issue of modern VTNs, this work intends to address the following problem: Problem 1: For the Traffic Model 1: (i) obtain an optimal VDM solution, by solving Optimization Problem 1, (ii) develop a decentralized VDM control scheme that is based on locally available information and conditions that enable convergence guarantees and are applicable to arbitrary connected VTN configurations. The first objective aims for the VDM solution to enable maximum throughput. The second objective aims to develop a decentralized VDM controller, accompanied by suitable stability guarantees, that stabilizes the system states to a reference equilibrium and is applicable to arbitrary connected VTN configurations enabling scalable designs. III. OPTIMAL VEHICULAR DEMAND MANAGEMENT SOLUTION Optimization Problem 1 does not follow any standard form, as it is nonlinear due to the presence of the nonlinear functions gij and the terms cij nested in gij, within constraint (8d). The cij and gij functions employ min operators (see (4), (5)) which significantly complicates solving (8). To facilitate obtaining a solution, Optimization Problem 1 is transformed into a MILP by following the methodology described in [18, Sec. 2.2]. Pointedly in [0, ρC i], i ∈ N , the functions gij in (8d) satisfy gij(ρi, ρj) = min wij rivf iρi, cmax ij , cmax ij ρJ j−ρj ρJ j−¯ρij , j ∈ Si. (9) According to [18, Sec. 2.2], in any optimization problem, for continuous variables x1, x2, x3, the term min{x1, x2, x3} can be replaced in the optimization problem with a new variable y, by introducing the following additional constraints xmin i≤xi≤xmax i,∀i∈ {1,2,3},(10a) y≤xi,∀i∈ {1,2,3},(10b) y≥xi−(xmax i−min{xmin 1, xmin 2 , xmin 3})(1 −di),∀i∈ {1,2,3},(10c) 3 X i=1 di=1,(10d) di=(1,if,min{x1, x2, x3}=xi, 0,otherwise, ∀i∈ {1,2,3}. (10e) This substitution generates a standard MILP form, presented below, that can be solved by conventional solvers. Thus the terms gil(ρi, ρl)in (8d) are replaced with intermediate variables yil, and appropriate MILP constraints by using binary variables, dilk ∈ {0,1},(i, l)∈ E,k∈ {1,2,3}. For compactness we define the inter-boundary flow vector, y∈R|E| +comprised by all the elements yil such that (i, l)∈ E. We also define the binary vectors d1∈ B|E|,d2∈ B|E|,d3∈ B|E|, where B={0,1}, comprised respectively by all the elements dil1,dil2, and dil3, such that (i, l)∈ E. These actions result in the following MILP formulation. Optimization Problem 2: For a VTN with traffic dynamics described by Traffic Model 1, solve: max (ρ,u,y, d1,d2,d3)X i∈N ui(11a) s.t. :Constraints (8b),(8c),(11b) X j∈Pi yji +ui−wiirivf iρi−X k∈Si yik = 0, i ∈ N, (11c) yνl −wνlrνvf νρν≤0,(ν, l)∈ E,(11d) yνl −cmax νl ≤0,(ν, l)∈ E,(11e) yνl +cmax νl ρl ρJ l−¯ρνl ≤cmax νl ρJ l ρJ l−¯ρνl ,(ν, l)∈ E,(11f) wνlrνvf ν(ρν+ρC νdνl1) −yνl ≤wνlrνvf νρC ν,(ν, l)∈ E,(11g) cmax νl dνl2−yνl ≤0,(ν, l)∈ E,(11h) cmax νl (ρJ ldνl3−ρl) ρJ l−¯ρνl −yνl ≤0,(ν, l)∈ E,(11i) dνl1+dνl2+dνl3= 1,(ν, l)∈ E,(11j) where for the binary variables, dνlk,(ν, l)∈ E,k∈ {1,2,3}, the following hold dνl1=(1,if gνl(ρν, ρl) = wνlrνvf νρν 0,otherwise ,(12a) dνl2=(1,if gνl(ρν, ρl) = cmax νl 0,otherwise ,(12b) dνl3=(1,if gνl(ρν, ρl) = cmax νl ρJ l−ρl ρJ l−¯ρνl 0,otherwise .(12c) It is signified that Optimization Problem 2 is equivalent to Optimization Problem 1 and as already mentioned Optimization Problem 2 has mixed-integer linear constraints and can be solved via standard commercial solvers to obtain optimal network states/equilibria, (ˆ ρ,ˆ u), characterized by maximum serviced vehicle demand and free-flow operating conditions. This serves the VTN demand allocation task as defined in Problem 1, Item i. It should be noted that optimal maximum-throughput solutions often entail local equilibrium densities at critical values. Operation at these values might result to instability and gridlock [15]. Below, we develop a VDM stabilizing controller that allows stability guarantees to be deduced in the presence of IBCs, even when equilibrium points coincide with critical densities. Such scheme enables stability and maximum throughput to be simultaneously attained, while explicitly accounting for IBCs. IV. TRAFFIC NETWORK STABILITY By explicitly taking into account regional IBCs, this section investigates the stability properties of a VTN under a developed proportional feedback VDM scheme. A formal definition of the equilibrium points of Traffic Model 1, to be referenced during the analysis, is given next. Definition 1: An equilibrium point, ρ∗∈Rn +, of Traffic Model 1, satisfies −wiigi(ρ∗ i)−X l∈Si gil(ρ∗ i, ρ∗ l) +X j∈Pi gji(ρ∗ j, ρ∗ i) + u∗ i= 0,∀i∈ N.(13) where u∗ iis the steady-state value of ui. An investigation on the response of Traffic Model 1 when the equilibrium points are within an open set of the domain, i.e. ∀i∈ N, ρ∗ i∈(0, ρJ i), is provided next. In addition, sufficient conditions for stability are derived, in the presence of IBCs, via Lyapunov analysis. The inherent compromise between optimality and stability identified in [15] (instability-prone behaviour) drives the development of a stabilizing controller that accounts for IBCs and ensures system state convergence guarantees. It is given by ui(t) = [¯ui−kpiρi(t)]umax i 0, kpi∈R+\ {0}, i ∈ N,(14) where the admitted flow constant, ¯ui, satisfies by design the following condition 0<¯ui−kpiρ∗ i<¯umax i.(15) and can be selected by taking into account optimality goals. Knowledge of the local equilibrium density, ρ∗ i, i ∈ N, is required for proper selection of the constants ¯uiand kpi. This value can be extracted by historical data, or when feasible, by solving Optimization Problem 2. If the local equilibrium is not available, ¯uiand kpican be selected empirically to satisfy (15), thus yielding a range of local equilibrium values. Finally, it is noted that within the theorem we will make use of the following Lipschitz constants characterizing (1b)3and (4)4 vL i= max(vf i, vC i), i ∈ N,(16a) vL ij =cmax ij /(ρJ j−¯ρij),(i, j)∈ E.(16b) Theorem 1: Consider Traffic Model 1 under the action of (14), (15), and consider an equilibrium point ρ∗ i∈ (0, ρJ i),∀i∈ N. Then, if there exist γil ∈R,(i, l)∈ E, such that kpi>max{X l∈Si αil,X l∈Pi vL li 2γli +X l∈Si γilvL il 2} + max{X j∈Pi vL ji,X j∈Si αij 2γij +X j∈Pi γjiαji 2}+wiirivL i, (17a) αji =wjirjvL j, j ∈ Pi,(17b) 3Equation (1b) implies that fi:R+→R+satisfies the Lipschitz condition globally with the Lipschitz constant vL igiven by (16a). 4Similarly, (4) implies that each IBC function, cij :R+→R+also satisfies the Lipschitz condition globally, with Lipschitz constant vL ij given by (16b). for all i∈ N then, the solutions of the Traffic Model 1 locally converge to the equilibrium point ρ∗. Additionally, if u∗=ˆ u, where ˆ uis an element of the solution to Optimization Problem 1, then the equilibrium point globally solves Optimization Problem 1. Proof: Onward, time dependence is dropped for compactness. We use e= [ρ1−ρ∗ 1, .., ρn−ρ∗ n]T∈Rnto denote the vehicular density error state. A Lyapunov candidate V:Rn→Ris selected as V(e) = X i∈N Li 2e2 i.(18) The considered solutions to Traffic Model 1 lie within a connected local subset K ⊂ Rn, with Kgiven by K=e:ρ∗ i∈(0, ρJ i), i ∈ N, V (e)≤ε.(19) Due to (15), a proper choice of ε∈R+\{0}, results in a VDM controller response that does not violate its bounds. Via (7), (14), the derivative of Vis ˙ V=X i∈N ei−wiigi(ρi)−X l∈Si gil(ρi, ρl) +X j∈Pi gji(ρj, ρi) + ¯ui−kpiρi.(20) Subtracting (13) from (20), and noting that due to (15) there exists εsuch that u∗ i= ¯ui−kpiρ∗ i, yields ˙ V=X i∈N ei−kpiei−wii(gi(ρi)−gi(ρ∗ i)) −X l∈Si (gil(ρi, ρl)−gil(ρ∗ i, ρ∗ l)) +X j∈Pi (gji(ρj, ρi(t)) −gji(ρ∗ j, ρ∗ i)).(21) Employing (16a), (16b), (17b), in (21) results in ˙ V≤X i∈N (wiirivL i−kpi)e2 i+X l∈Si max{αile2 i, vL il|el||ei|} +X j∈Pi max{αji|ej||ei|, vL jie2 i}.(22) Using the following perfect squares identity βxy =β 2γx2+γβ 2y2− sβ 2γx−rγβ 2y!2 (23) on (22) leads to ˙ V≤X i∈N (wiirivL i−kpi)e2 i+ max{X l∈Pi vL li 2γli e2 i +X l∈Si γilvL il 2e2 i−svL il 2γil el−rγilvL il 2ei2,X l∈Si αile2 i} + max{X j∈Si αij 2γij e2 i+X j∈Pi γjiαji 2e2 i −rαji 2γji ej−rγjiαji 2ei2,X j∈Pi vL jie2 i}.(24) From (24) and (17a), it follows that ˙ V(e)≤ − X i∈N ξie2 i<0,e∈ K \ {0},(25a) ξi=kpi−wiirivL i−max{X l∈Si αil,X l∈Pi vL li 2γli +X l∈Si γilvL il 2} − max{X j∈Pi vL ji,X j∈Si αij 2γij +X j∈Pi γjiαji 2},(25b) ˙ V(0) =0,(25c) and via [19, Theorem 4.1], asymptotic stability of e=0 follows suit; i.e. limt→∞ ρ(t) = ρ∗. It is signified that, if u∗ i= ˆui,∀i∈ N then by (13), and the analysis above, the VTN dynamics converge to ˆ ρ, thus globally solving Optimization Problem 1. Theorem 1 provides stability guarantees for the VDM controller (14) and also yields a sufficient condition for gain selection, i.e. (17a), that ensures VTN stability. It is noted that, when ¯ui= ˆui+kpiˆρi, then the obtained equilibrium point solves Optimization Problem 1, enabling globally optimal throughput behaviour. It is also noted that the obtained results apply to any connected VTN configuration. Concluding, all objectives of Problem 1 are achieved. Next, the results presented in this work are verified with numerical simulations. V. NUMERICAL RESULTS A representative simulation of a 6-region VTN is conducted to validate the analytical results of this study. The simulations aim to demonstrate the stability properties of the developed controller in the presence of disturbances, the optimality properties of the considered setup, and also highlight the importance of including IBCs, by showcasing the impact, in terms of throughput, when IBCs are neglected. The VTN parameters are given in Table I. The VTN topology can be inferred from the split-ratio matrix wgiven in Table I. Two simulations are conducted, to be referred to as Cases 1 and 2, as described below. (i) Case 1 (black line) employs the developed VDM control law (14) with ¯ui= ˆui+kpiˆρi, i ∈ N (see Table II for ˆui,ˆρi) where ˆui,ˆρiwere obtained by solving Optimization Problem 1 without taking into account the existence of the IBCs5. (ii) Case 2 (blue line) also uses the developed VDM control law (14) with ¯ui= ˆui+kpiˆρi, i ∈ N but the optimal solution ˆui,ˆρitakes into account the IBCs as it is obtained by solving Optimization Problem 2 (see Table III for ˆui,ˆρi). The gain values kpifor both controllers are the same and are given in Table I; they were selected to satisfy the gain selection condition (17) of Theorem 1. 5When the IBCs are disregarded, Optimization Problem 1 becomes linear and can be easily solved with a standard solver. TABLE I 6-REGION NETWORK PARAMETERS AND CONTROL GAIN VALUES Parameter & Value L= [1.2,1,0.85,0.9,1.02,0.88] l= [0.6,0.45,0.35,0.4,0.48,0.34] vf= [30,35,32,34,35,31] ρJ= [118,125,98,115,120,106] ρC= [26.3,28.2,24.4,25.3,23.8,21.9] w=        0.25 0.25 0 0 0.25 0.25 0.15 0.35 0.3 0 0 0.2 0 0.1 0.3 0.4 0 0.2 0 0 0.24 0.16 0.3 0.3 0.05 0.1 0 0.25 0.3 0.3 0.32 0.03 0.23 0.17 0.1 0.15        cmax =        0 117.6 0 0 106.1 124.3 151.3 0 140.9 0 0 155.5 0 116.4 0 134.5 0 123 0 0 122.8 0 115.6 135.5 127.7 124.2 0 143.4 0 131.2 104.1 101.2 97 116.9 91.2 0        ¯ ρ=        0 19.2 0 0 13.2 21.1 16.9 0 21.2 0 0 22.6 0 17.8 0 17.1 0 19.5 0 0 19 0 12.7 20.2 14.3 17.4 0 16.7 0 19.1 13.1 16 16.4 15.3 11 0        kp= [103.1,115.8,135.1,140.4,123.5,163.9] γil = 1,(i, l)∈ E qC= [789,987,780.8,860.2,833,678.9] umin = 0.01qC umax =qC TABLE II THROUGHPUT-OPTIMAL SOLUTION (WITHOUT IBCS) Parameter & Value ˆ ρ= [24.2,21.7,24.4,17,12.6,21.9] ˆ u= [589.4,987,674.3,8.6,8.3,6.8] A1[h] simulation scenario is carried out where, by utilizing the corresponding optimal solutions for Cases 1 and 2, the VDM controllers aim to achieve maximum throughput operation. For both cases, at t= 40 [min] a disturbance is introduced in region 2 for a duration of 1[min], by increasing u2by 0.5ˆu2. The simulation results are discussed next. A key point to note is the significant difference in throughput response shown in Fig. 2, between the optimal solutions (ˆui,ˆρi) obtained for Case-1 (see Table II) and Case-2 (see Table III). For this example, when IBCs are taken into account in the optimization problem, the throughput is improved by 24.4%. Moreover, for Case-1, a steady state system-wide equilibrium is achieved. However, i) the system stabilizes at a sub-optimal operation point (see Fig. 2), and ii) it is unable to converge to the optimal solution given in Table II (see Fig. 3). This is not due to the inability of the controller to steer the system to a desired equilibrium but due to the fact that Case-1 doesn’t account for IBCs despite their active presence in the actual network. Thus the obtained ”optimal” solution when IBCs are excluded yields a suboptimal systemwide equilibrium that also corresponds to region 3 being congested (see Fig. 5). The fact that region 3 is congested is inferred by observing that the region flow velocity in Fig. 5 TABLE III THROUGHPUT-OPTIMAL SOLUTION (WITH IBCS) Parameter & Value ˆ ρ= [26.3,28.2,24.4,25.3,23.8,21.9] ˆ u= [403.8,818.2,579.9,301.3,709.4,22.7] 0 10 20 30 40 50 60 0 1000 2000 3000 4000 24.4% improvement in throughput for Case-2 Fig. 2. Total network outflow [veh/h]for Case-1 (black line) and Case-2 (blue line). Due to the fact that the Case-1 optimal solution disregards IBCs the network operates sub-optimally with 24.4% reduced throughput. Case-2 operates optimally with maximum throughput response. is lower than the free-flow value. In contrast, for Case-2, a system-wide equilibrium is rapidly achieved as the solutions are unhindered to converge to the desired optimal solution given in Table III (see Fig. 4). The controller easily steers the system to the desired equilibrium since Case-2 solution accounts for IBCs. Moreover all regions operate in free-flow mode as it is evident by the flow speed (see Fig. 5) and at maximum throughput (see Fig. 2). Despite the demand variation at t= 40 [min] the derived controller recovers the states to an equilibrium very fast for both cases thus demonstrating its effectiveness. Additionally, the ability of the developed controller to stabilize the system even from congestion is highlighted from the fact that it recovers the system to equilibrium for Case-2. Pointedly, the optimal solution for region 2 coincides with the critical density (see ˆρ2= 28.2in Table III and ρC 2= 28.2in Table I) and due to the disturbance, region 2 gets congested momentarily. This is evident by the speed drop from the freeflow nominal value observed in Fig. 5 (blue line). Despite this, the controller is able to recover and steer the solutions back to the desired operating equilibrium, ensuring fast recovery to maximum throughput operation. The simulation scenarios clearly validate the derived theoretic results and highlight the importance of taking IBCs into consideration in the design of system operation. VI. CONCLUSIONS This work considered the stability and optimality properties of vehicular demand management schemes in the presence of inter-boundary region to region flow constraints. An optimization problem that aimed to maximize vehicle throughput was formulated, taking into account vehicular 0 50 0 20 0 50 0 20 0 50 0 20 0 50 0 20 0 50 0 10 20 0 50 0 10 20 Fig. 3. Case-1 network density [veh/km]. Steady state is achieved in both cases. However, the system stabilizes at a sub-optimal operation point due to the action of IBCs. The solutions are unable to attain the optimal solution yielded by Optimization Problem 1 (IBCs not taken into account in this solution). Also region 3 is congested since the flow velocity in region 3 (see Fig. 5) is lower than the free-flow value. Despite the demand variation at t= 40 [min] the derived controller recovers the states to an equilibrium very fast. 0 50 0 20 0 50 0 20 0 50 0 20 0 50 0 20 0 50 0 10 20 0 50 0 10 20 Fig. 4. Case-2 network density [veh/km]. Steady state is rapidly achieved and at the intended optimal operation point as the solutions converge to the optimal solution yielded by Optimization Problem 2 (IBCs are taken into account in this solution). All regions operate in free-flow mode. Despite the demand variation at t= 40 [min] the derived controller recovers the states to an equilibrium very fast. traffic network dynamics and inter-boundary constraints. The problem was non-convex and non-linear and was transformed to a Mixed Integer Linear Program to enable the use of standard solvers. A decentralized proportional vehicular demand management controller was developed, accompanied by local design conditions that explicitly account for inter-boundary constraints, such that stability is analytically guaranteed. The effectiveness and practicality of the derived analytic results was showcased with numerical simulations in a 6region vehicular traffic network system, which highlight the importance of inter-boundary constraints, demonstrating a significant improvement in vehicle throughput when those are taken into account. 0 50 0 20 0 50 0 20 40 0 50 0 20 0 50 0 20 40 0 50 0 20 40 0 50 0 20 Fig. 5. Network speed for Case-1 (black line) and Case-2 (blue-line) [km/h]. Case-1 operates suboptimally since the flow velocity in region 3 is lower than the free-flow value. Hence region 3 is congested. 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