Flow-Maximizing Equilibria of the Cell Transmission Model*

Transcription

1 Flow-Maximizing Equilibria of the Cell Transmission Model* Marius Schmitt, Paul Goulart, Angelos Georghiou and John Lygeros Abstract We consider the freeway ramp metering problem, based on the Cell Transmission Model and address the question of how well decentralized control strategies, e.g. local feedbac controllers at every onramp, can maximize the traffic flow asymptotically under time-invariant boundary conditions. We extend previous results on the structure of steady-state solutions of the Cell Transmission Model and use them to optimize over the set of equilibria. By using duality arguments, we derive optimality conditions and show that closed-loop equilibria of certain decentralized feedbac controllers, in particular the practically successful ALINEA method, are in fact globally optimal. I. INTRODUCTION Active traffic control schemes have been established as an effective and practically useful tool to improve traffic flows on congestion-prone road networs [6]. This wor concentrates on the freeway ramp metering problem, where we can actively control the number of cars that enter the freeway using a specific onramp. A survey of ramp metering strategies can be found in [8]. To model the freeway traffic dynamics, we mae use of the Cell Transmission Model CTM, which was originally derived as a firstorder Godunov approximation of the inematic wave partial differential equation [], [3]. More precisely, we adopt the asymmetric CTM [4], [5], which simplifies the model of onramp-mainline merges in comparison to the originally proposed formulation. Its popularity for model-based control stems from the simplicity of the model equations, allowing for computationally efficient solutions methods for optimal control problems [4], [9]. A variety of local feedbac strategies, i.e., ramp metering controllers that only receive measurements from sensors in close vicinity of the onramp location, have been described in literature [7], [4], [7]. These strategies have been shown to come close to the performance of optimal control strategies in practical applications, even though they only aim to maximize bottlenec flows locally [3], [5]. While it is apparent that such local feedbac controllers are far easier to implement and to configure than centralised, model-based optimal control strategies, it is not obvious why the former tend to come close in performance to the latter in practice [9]. The optimal control strategy for a special case no internal freeway queues is given in [8] and the structure of the explicit solution explains why some local metering *The research leading to these results has received funding from the EU project SPEEDD FP7-ICT Automatic Control Laboratory, ETH Zurich, Switzerland, <schmittm,angelosg,lygeros>@control.ee.ethz.ch Department of Engineering Science, University of Oxford, UK, paul.goulart@eng.ox.ac.u algorithms [...] are successful they are really close to the most-efficient logic. In this wor, we address the question of how decentralized control strategies, such as local feedbac controllers at every onramp, compare to optimal control strategies asymptotically under time-invariant boundary conditions. The idea to focus on traffic controllers which are only required to achieve convergence to an optimal equilibrium instead of solving the far more challenging problem of optimizing the transient behavior has recently received attention in a series of papers on traffic density balancing [], [], []. It is well nown that the problem of maximizing the flow over the set of equilibria of the CTM can be posed as a linear program LP. Using duality arguments we derive simple optimality conditions for flow-maximizing equilibria. We show that all closed-loop equilibria of certain local feedbac controllers, in particular the practically successful ALINEA method [7], are in fact globally optimal in the CTM. The paper is organized as follows: Section II introduces the CTM. In Section III, we extend previous results on the structure of equilibria of the CTM. In Section IV, we optimize over the set of equilibria of the CTM and derive optimality conditions. Section V demonstrates the results on a simple example and Section VI comments on the main results and discusses practical implications. II. PROBLEM FORMULATION Throughout this wor, we consider a freeway as depicted in Figure. In this section, we introduce the asymmetric CTM [4], [5] that we use to model such a freeway. All parameters and variables of the CTM are summarized in Table II. The discrete-time CTM admits the following intuitive explanation: The freeway is partitioned into n sections or cells of length l, {,...,n. The state of the freeway is described by the traffic density ρ t in each cell at sampling time t. The CTM is a first order model, so the velocity is not part of the state. The evolution of the traffic is described by the traffic flows φ t, i.e., the number of cars that move from cell to cell + in one sampling interval of length t. We model the off-ramp flows r t = β β φ t as a proportion of the mainline flow φ t with the constant split ratios β [, and β := β. The onramp demands u t are the flows entering the freeway via an onramp at cell. The flow of cars that aim to enter the freeway via the mainline is the called the mainline demand u t. We can The sampling period t is restricted to t min l to ensure convergence. v

2 Cell - Cell u r u r u + Cell + + Fig.. Setch of the cell transmission model r + thus formulate the conservation law for each cell as: ρ t + = ρ t + t φ t + u t φ t. l β The road conditions are described by the so-called free-flow velocity v, the congestion-wave speed w and the jam density ρ. The flows φ t are limited by the number of cars in the origin cell that want to travel downstream β v ρ t, the mainline capacity F and the available free space ρ + ρ + tw + in the receiving cell: φ t = min { β v ρ t,f, ρ + ρ + tw +. This relationship can be visualized with the so-called fundamental diagram as depicted in Figure. The critical density ρ c = w v +w ρ is the density value at which the fundamental diagram is maximized. The pea value of both the sending as well as the receiving cell determine the values of the capacities F, which are defined as F := min { β v ρ c, ρ + ρ+ c w +. This equation is slightly adapted for the first and the last cell, where we define F = min{u, ρ ρ cw and F n = min { βn v n ρn, c φ n. Here, φ n is some arbitrary, constant bound on the outflow from the freeway. For a given initial state ρ and ramp metering rates u t for n and t N, the evolution of the freeway is described by the following equations: ρ t + = ρ t + t φ t + u t φ t, l β φ t = min{f, ρ ρ tw, φ t = min { β v ρ t,f, ρ + ρ + tw +, φ n t = min { βn v n ρ n t,f n. Note that this model includes the implicit assumption that congestion does not spill bac onto the onramps. While this assumption might not be satisfied for an uncontrolled freeway, it was shown to be satisfied by a large margin for a freeway controlled by ramp metering in a field study [4]. In the following, we will also use the vector notation u := u,...,u n, φ := φ,...,φ n, ρ := ρ,...,ρ n etc. Furthermore, R + denotes the set of nonnegative real numbers. Note that some authors also consider mainline capacities < F min { β v ρ c, ρ + ρ c + w + which leads to fundamental diagrams of trapezoidal shape. + Parameters Variables Symbol Range Name/Definition Unit φ [, flow /h ρ [, density /m u [, onramp flow /h β [, split ratio β,] β l, cell length m v, free-flow velocity m/h w, congestion-wave speed m/h ρ, jam density /m l t,min v sampling time interval h F TABLE I SUMMARY OF SYMBOLS c v + + w + III. F EQUILIBRIAs + OF THE + CTM We start by deriving some general properties of equilibria of the CTM, which will d be important tools in the analysis of flow-optimality in the following sections. In the following, if the time index is omitted c for some variable, e.g. φ instead of φ t, the variable is to be understood as a steady-state value. The equations describing steady-state solutions or equilibria of the CTM can be derived by imposing ρ t + = ρ t =: ρ and removing the time index t from all variables, which yields: v Fig.. Setch of the fundamental diagram for uniform freeway conditions, i.e., parameters and densities that do not differ between consecutive cells. φ = φ + u β, n, φ = min{f, ρ ρ w, φ = min { β v ρ,f, ρ + ρ + w +, < n, φ n = min { βn v n ρ n,f n. For ease of notation, we introduce the set of equilibria E := {u,φ,ρ : u,φ,ρ R n + R n+ + R n + satisfy. For the special case of feasible demands, i.e. the case when both onramp and mainline demand can be fully served, the equilibrium sets have been computed explicitly in [5]. We call a section a bottlenec if φ = F. The locations of the bottlenecs are important in the analysis of the CTM equilibria [5]. Assuming that there are m bottlenecs b,b,...,b m, these bottlenecs partition the freeway into m segments S = {,...,b, S = {b +,...,b,..., S m = {b m +,...,n, as depicted in Figure 3. Note that the first and the last segment may be empty if φ or φ n are bottlenec flows. We say that a cell is congested if the density exceeds the critical density ρ > ρ c. Conversely, a cell operates in free-flow if ρ < ρ c. Obviously, it is also possible that a cell operates exactly at the critical density ρ = ρ. In an arbitrary equilibrium, some cells in a segment

3 b j = c < c q j Segement S j b j > c b = F b b j = F bj b j = F bj b j+ = F bj+ b b j b j b j+ Segment S Segment S j Segment S j+ Segment S m Fig. 3. Segments, defined according to the bottlenec locations, in a CTM representation of a freeway. might be congested while other cells operate in free-flow or at the critical density. The order of congestion and free-flow conditions is not arbitrary, however, but follows a specific pattern instead: Lemma : Let u,φ,ρ E be an equilibrium of the CTM. For every nonempty segment S j = {b j +,...,b j, j {,...,m, there exists an index q j S j such that all upstream cells within the segment operate in free-flow: ρ < ρc, S j, < q j and all downstream cells within the segment are congested: ρ > ρc, S j, > q j. Proof: If all cells within segment S j = {b j +,...,b j operate in free-flow, choose q j := b j and the statement holds by definiton. If this is not the case, there exists at least one cell within segment S j that does not operate in free-flow. Then define q j to be the index of the first cell in downstream direction that does not operate in free-flow: q j := argmin { S j : ρ ρc. For this choice, the first assertion holds by definition of q j. The second assertion can be proven by contradiction: Assume some cells downstream of cell q j, but within the same segment, are not congested. Let n j := argmin { S j, > q j : ρ ρc be the most upstream one of those cells. By definition, cell n j is not in free-flow, and cell n j is not congested. Therefore we have { φ n j = min βn j v n j ρn j,f n j, ρ n j ρn w j n j { min βn j v n j ρn c j,f n j, ρ n j ρn c w j n j = F n j. It follows that φ n j = F n j, i.e. cell n j is a bottlenec. This is a contradiction to the definition of a segment, in particular, there are no bottlenecs within a segment. Intuitively, this means that there exists only one congestion queue per segment 3, which starts in cell q j. Furthermore, there can at most be one cell within a segment cell q j in which the density actually equals the critical density. The resulting congestion/ free-flow pattern is visualized in Figure 4. IV. OPTIMAL EQUILIBRIA In this section, we will address the problem of optimizing over steady-state equilibria of the CTM. In particular, we will derive optimality conditions tailored for the analysis of decentralized control approaches. Intuitively, such control strategies are based on the following consideration: Assume a decentralized controller with the objective of moving 3 A similar result is stated in [5, Lemma 4.3]. The cases ρ < ρ c and ρ = ρ c are not distinguished, however, but this distinction will be important in our further analysis. bottlenec: < c Congestion queue = c Segment S j bottlenec: > c Fig. 4. Pattern of the freeway state within one segment. Note that there is not necessarily a cell in which the critical density is exactly achieved = ρ F = F bj bj = ρ c. In this case, cell q i is the last cell in free flow: ρ < ρ bj+ c. bj+ b j b j b j+ the local density at every controlled onramp as close as possible Segement to thescritical j density. ItSegement is easy to S j+ chec that each individual flow will be maximal if ρ = ρ c is achieved in every cell. However, this will in general not be the case due to saturation of the available control inputs the ramp metering rates, in particular if multiple local controllers interact while controlling a freeway. Therefore, the performance of such closed-loop equilibria in terms of flow maximization is not clear a priori. We will first state the nonconvex main problem and show the equivalence of this problem with a suitable linear relaxation. Duality arguments then yield sufficient, decentralized conditions for optimality that only relate each local ramp metering rate to the density in the came cell. Consider a freeway modeled by the CTM and controlled by ramp metering. We want to find the equilibrium that maximizes a positive combination of all flows c φ c R n+ +, : c > by choosing appropriate steady-state ramp metering rates u. The ramp metering rates are assumed to be constrained by box constraints u u ū. In the simplest case, the lower limit might be equal to zero to prevent negative flows and the upper bounds will model the maximal number of cars that want to enter the freeway at a certain onramp per time period, i.e. a constant onramp demand. Thus we consider the following optimization problem over equilibria of the CTM: maximize u,φ,ρ c φ subject to u u ū, n, u,φ,ρ E. Note that Problem includes the maximization of the Total Travel Distance TTD := φ + n φ = β or the total discharge flows r tot := φ n + n β = β φ as special cases. Also note that Problem is nonconvex, due to the nonconvex fundamental diagram that maes the set of equilibria E nonconvex. Even though the flow-constraints are nonconvex, it is wellnown that there exists an uncongested, flow-maximizing equilibrium for freeways under fairly general conditions [6], []. One can find this solution by solving a suitable Linear Program, for example by introducing a condition that restricts

4 the freeway to free-flow conditions We want to avoid the restriction to free-flow conditions and consider the following relaxation: maximize u,φ c φ 3a subject to u u ū, n, 3b φ = φ + u β, n, 3c φ F, n, 3d in which the nonconvex flow-constraints have been relaxed, thus removing the densities from the problem. We also introduce the dual of problem 3 which is given as: minimize ν u ξ ū λ F µ.ν,ξ,λ subject to µ + ν ξ = n c + µ β µ + + λ = µ = µ n+ = n µ R n+,ν R n +,ξ R n +, λ R n+ + To simplify notation, we have introduced constants µ and µ n+. The dual variables ν and ξ correspond to the lower and upper bounds in 3b, µ belong to the equality constraints 3c and λ belong to the inequality constraints 3d. Lemma : Let c R n+ +, : c >. Then problem is equivalent to the relaxation 3, in the sense that the objective values are equal. Furthermore, given a maximizer u,φ of the relaxed problem 3, we can compute densities ρ such that u,φ,ρ is a maximizer of. Proof: Since 3 is a relaxation of, it is sufficient to show that for any optimizer φ,u of 3, we can find ρ such that φ,u,ρ is feasible for. Let u,φ,ν,ξ, µ,λ be a primal-dual solution to 3, 4. First, we show by contradiction that there exists at least one bottlenec: Assume there does not exist a bottlenec, i.e. φ < F, {,...,n. Because of primal-dual complementarity, this implies that the corresponding multipliers λ =, {,...,n. Using the dual problem, we can then compute µ iteratively: µ = c, µ + = c + µ β Since c and β >, it follows that µ, {,...,n+. Since there exists c l >, it follows that µ <, {l +,...,n +. This contradicts µ n+ =. Recall that b is the index of the first bottlenec. Feasible densities can be constructed by chosing ρ := ρ φ w F b ρ := φ β v F For b, it follows that β v ρ = β v ρ φ β v ρ w β v v + w ρ β v b < n ρ F w = β v w v + w ρ = F 4 and therefore we have indeed: φ = min { β v ρ,f, ρ + ρ+ w + = ρ + ρ + w + = φ Conversely, for b < n, it follows that ρ ρ w = ρ φ w β v ρ w v + w ρ ρ F β v w w β v w v + w ρ = F and liewise: φ = min { β v ρ,f, ρ + ρ+ w + = β v ρ = φ Thus the proposed densities ρ together with u and φ indeed satisfy the fundamental diagram, meaning ρ,u,φ E, which completes the proof. Remar : If the complete traffic demand can be served by the freeway, we have that φ = u = F and therefore a bottlenec at =. This somewhat counterintuitive definition, which interprets the depletion of the mainline demand as an artificial bottlenec, helps to avoid cumbersome case distinctions in the analysis of the equilibria. Remar : Note that the solution includes equilibria in which some cells are congested, in general. While optimizing over free-flow conditions will yield a global optimum, the CTM also allows for partly congested equilibria, which achieve the same objective value. We are now ready to state the following optimality conditions: Theorem : Let u,φ,ρ E be an equilibrium of the CTM and assume that the equilibrium ramp metering u rates satisfy: u = ū : ρ < ρc free-flow, u u ū : ρ = ρc critical density, u = u : ρ > ρc congestion. Then u,φ,ρ solves the maximal flow problem. Proof: Feasibility of the primal problem holds by assumption. To verify optimality, it is sufficient to show that the equilibrium flows solve the LP-relaxation 3, according to Lemma. We will show that for any solution to the primal problem which satisfies 5, we can construct a complementary dual solution, thus proving that the primal solution is indeed optimal. The complementarity conditions are given as u u ν, u ū ξ, φ F λ. For ease of notation, we define the following index sets: The set of all bottlenecs B := { : F = φ, the set of all congested cells C := { : ρ > ρc, the set of all cells in free flow F := { : ρ < ρc and the set of all cells in which the 5

5 b t S = F b µ µ qj = bj µ apple bj b j > c < c qj = c qj q j Segment S j Fig. 5. Dual variables corresponding to the congestion/ free-flow pattern of the freeway within one segment. = F = F = F bj bj bj bj bj+ bj+ b j critical density is achieved b j D := { : ρ = ρc. By bcombining j+ the constraints for dual feasibility, the complementarity constraints and Assumption Segment S j 5 on the dependence Segment S j+ of the ramp metering rates on the densities, we arrive at the following optimality conditions: u,φ,ρ E u, u,ρ satisfy 5, 6a c + µ β µ + + λ =, b j 6b µ = if D, λ = if / B, 6c µ if F, µ if C, 6d λ if B. 6e We will now find a complementary dual solution explicitly, by constructing such a solution for every segment S j. Define for every nonempty segment S j the index q j S j as in Lemma and choose: µ q j :=. Then feasible multipliers µ for all other cells in segment S j can be computed by iterating in upstream/downstream direction, starting from the cell q j : µ := β µ + + c, b j < < q j, µ := µ β c, q j < b j, again be verified by induction: µ q j = by definition. Assume now that µ. Then µ + = µ β c for all cells with indices q j < b j. Recall now that according to Lemma, the congestion pattern within a segment is ordered. In particular, all cells b j < < q j operate in free-flow and all cells q j < b j are congested. Satisfaction of conditions 6d immediately follows. We can also conclude from the previous analysis that for Segment S m every bottlenec b j, it holds that µ b j since cell b j is located downstream of cell q j and also µ b j + since cell b j + is located upstream of cell q j +. Therefore λb j := c b j µ b j + µ β b b j + and conditions 6e follow. j It is interesting to note that the result does not depend on the choice of the weights for the individual mainline flows c as long as all are non-negative, so all such objectives will be jointly maximized by an equilibrium satisfying the optimality condition 5. This is a consequence of the assumption of constant split rations β, under which it is impossible to trade off the mainline flow at some part of the freeway with the mainline flow at another place. V. APPLICATION To demonstrate the practical relevance of the previously derived optimality conditions, we use them to analyze the well-nown decentralized ramp-metering strategy ALINEA [7]. ALINEA is in essence an integral controller that aims at stabilizing the local densities at the critical density. It is easy to verify that every closed-loop equilibrium of a freeway controlled by local ALINEA controllers satisfies the optimality conditions 5. We implement the standard ALINEA controller and the multipliers λ are computed as λ := c µ β + µ +, B, λ :=, B. The resulting pattern within one segment is depicted in Figure 5. We will now verify that the proposed solution indeed solves the optimality conditions 6. Note that condition 6a is satisfied by assumption. Conditions 6b and 6c are satisfied by construction. Conditions 6d and 6e remain to be checed. We will first show that conditions 6d hold for a generic segment S j. For this we need some intermediate steps: Recall that the parameters β and c are nonnegative. We claim that the multipliers of all cells upstream of cell q j, but within segment S j, are nonnegative. This can be verified by induction: µ q j = by definition. Assume now that µ. Then µ = β µ + c for all cells with indices b j < < q j. Conversely, the multipliers of all cells downstream of cell q j, but within segment S j, are nonpositive. This can u t := [u t + K I ρ c ρ t]ū u in which [...] x x denotes saturation at x and x, so [x] x x = min{max{x,x, x and K I := 7/ρ c is the integral gain chosen as recommended in [7]. Closed-loop simulation results of this controller operating on simple freeways represented by the CTM are depicted in Figure 6. We compute the optimal equilibrium flows separately by solving for equal weights c = c = c = c 3 numerically and normalize the flows with the optimal equilibrium flows. In both examples, the closed-loop system converges to the optimal closed-loop equilibrium, as predicted. In the first example, the densities converge to the respective critical densities and it is not surprising that the resulting flows are optimal. In the second example, however, the controller of cell 3 saturates, i.e. u 3 t = u 3 for t 3min. Effectively, the bottlenec flow is now controlled by the controller in cell, which reduces its inflow only once the congestion spills bac. We observe convergence to a partly congested equilibrium: ρ t > ρ c for t 8min and ρ 3 t ρ3 c for t > 3min. Also, the convergence is slow and heavy oscillations occur, effects that have already been described [5]. Nevertheless, even this closed-loop equilibrium turns out to be flow-optimal.

6 flow density onramp flow time min time min = u ū u u 3 u u 3 u = flow density onramp flow time min 3 u = time min 3 3 ū 5 5 time min time min 3 3 = F = F 3 = F 3 3 = F 3 Example. Example. Fig. 6. Simulation results using the CTM. Note that the flows are normalized such that a flow of means that the optimal steady-state flow computed separately is achieved. Similarly, the densities are normalized by the respective critical densities. Note the different timescales. VI. CONCLUSIONS This wor analyzes flow-maximizing equilibria of the CTM. We find simple optimality conditions that are tailored to decentralized control approaches, since only constraints between local ramp metering rates and the densities in the adjacent cells are imposed. In particular, the optimality conditions are satisfied for all closed-loop equilibria of a freeway controlled by independent local ALINEA controllers, thus proving that all such equilibria are globally optimal w.r.t. flow maximization. It is important to eep in mind that this result holds only under the assumptions made in the CTM. In particular, the equilibrium may contain congested bottlenecs, which lead in practice to an empirically observed reduction of the bottlenec capacity. Such effects are not described by the simplistic CTM. In this sense, the simulations described in the previous section are not meant to be a realistic assessment of the real-world traffic behavior, but rather a way to demonstrate the properties of the idealized CTM. Note however, that the equilibrium densities are not unique and techniques described in [5] or [5] can be employed to steer the system to the preferred equilibrium usually the unique, least congested equilibrium in order to avoid the aforementioned capacity drop. REFERENCES [] Cheng-I Chen, José B. Cruz Jr, and Jean G. Paquet. Entrance ramp control for travel-rate maximization in expressways. Transportation Research, 86:53 58, 974. [] Carlos F. Daganzo. 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