Queue length estimation on urban corridors Guillaume Costeseque with Edward S. Canepa (KAUST) and Chris G. Claudel (UT, Austin) Inria Sophia-Antipolis Méditerranée VIII Workshop on the Mathematical Foundations of Traffic March 08, 2017 G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 1 / 25
Motivation Traffic control strategies Main control schemes: Highways Variable speed limits Ramp metering Dynamic lane management Arterial streets Adaptative traffic signal timings [Source: TRI Old Dominion University website] G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 2 / 25
Motivation Traffic control strategies Main control schemes: Highways Variable speed limits Ramp metering Dynamic lane management Arterial streets Adaptative traffic signal timings [Source: TRI Old Dominion University website] G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 2 / 25
Motivation Why introducing bounded acceleration? Traffic light: What scalar conservation laws theory teaches us t k + x Q(k) =0, Q(k) =min{v f k, w (k κ)} x Q (B) (A) (B) (A) (A) w (C) t vf (A) (A) 0 κ (C) k G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 3 / 25
Motivation Why introducing bounded acceleration? Car trajectories (Assuming no Italian taxi drivers...) x t G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 3 / 25
Motivation Why introducing bounded acceleration? Bounded acceleration phase [Lebacque, 2003, Leclercq, 2007] x Q (B) (A) (B) (A) (A) w (C) t vf (A) (A) 0 κ (C) k G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 3 / 25
Motivation Why introducing bounded acceleration? Car trajectories with bounded acceleration phase x t G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 3 / 25
Motivation Outline 1 Introduction 2 Optimization problem 3 Model and data constraints 4 Application to Lankershim Bvd, LA G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 4 / 25
Introduction Outline 1 Introduction 2 Optimization problem 3 Model and data constraints 4 Application to Lankershim Bvd, LA G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 5 / 25
Introduction Quick review of queue length estimation methods Queue length estimation at signalized intersections: [data-driven] input-output techniques (-) Need good estimate of the initial queue length G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 6 / 25
Introduction Quick review of queue length estimation methods Queue length estimation at signalized intersections: [data-driven] input-output techniques (-) Need good estimate of the initial queue length [data-driven] statistical/probabilistic approaches (-) Strongly depend on realistic vehicles arrival patterns VS sparsely available GPS data G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 6 / 25
Introduction Quick review of queue length estimation methods Queue length estimation at signalized intersections: [data-driven] input-output techniques (-) Need good estimate of the initial queue length [data-driven] statistical/probabilistic approaches (-) Strongly depend on realistic vehicles arrival patterns VS sparsely available GPS data [model based] shockwaves-based approach (-) Previous works do not account for bounded acceleration G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 6 / 25
Introduction Our approach Our focus Shockwaves-based approach: optimization-based framework [Anderson et al., 2013] G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 7 / 25
Introduction Our approach Our focus Shockwaves-based approach: optimization-based framework [Anderson et al., 2013] + explicit solutions for the macroscopic traffic flow models G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 7 / 25
Introduction Our approach Our focus Shockwaves-based approach: optimization-based framework [Anderson et al., 2013] +explicitsolutionsforthemacroscopictrafficflowmodels Basic assumptions: triangular fundamental diagram (FD) piecewise affine conditions Q(k) =min{v f k, w(k κ)} G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 7 / 25
Introduction LWR and LWR-BA models LWR model [Lighthill and Whitham, 1955, Richards, 1956]: scalar conservation law t k + x Q(k) =0, on (0, + ) R, (1) G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 8 / 25
Introduction LWR and LWR-BA models LWR model [Lighthill and Whitham, 1955, Richards, 1956]: scalar conservation law t k + x Q(k) =0, on (0, + ) R, (1) LWR model with bounded acceleration [Lebacque, 2002, Lebacque, 2003, Leclercq, 2002, Leclercq, 2007] { t k + x Q(k) =0, if v = V e (k), t k + x (kv) =0 if v < V e (k), t v + v x v = a (2) a is the maximal acceleration rate V e : k V e (k) equilibriumspeedsuchthatq(k) =kv e (k) G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 8 / 25
Introduction Hamilton-Jacobi setting Consider the Moskowitz function M(t, x) = + x k(t, y)dy (3) such that x M = k and t M = kv Then the LWR with bounded acceleration can be recast as t M Q ( x M)=0, if v = V e ( x M), { (4) t M + v x M =0, if v < V e ( x M) t v + v x v = a, G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 9 / 25
Introduction Hamilton-Jacobi setting Explicit solutions Viability theory + Lax-Hopf formula [Claudel and Bayen, 2010a, Claudel and Bayen, 2010b] = explicit solutions LWR model LWR model with bounded acceleration [Mazaré et al., 2011] [Qiu et al., 2013] G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 10 / 25
Optimization problem Outline 1 Introduction 2 Optimization problem 3 Model and data constraints 4 Application to Lankershim Bvd, LA G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 11 / 25
Optimization problem Initial and boundary conditions Piecewise affine conditions x x n c (j) down c (i) ini c (l) intern x 0 t 0 c (j) up t max t G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 12 / 25
Optimization problem Initial and boundary conditions Piecewise affine conditions Initial conditions c (i) ini (x) = { k i x + b i, if x [x i, x i+1 ], +, Upstream boundary conditions c (j) up (t) = else, { q j t + d j, if t [t j, t j+1 ], +, Downstream boundary conditions c (j) down (t) = Internal boundary condition c (l) intern (t, x) = else, { p j t + b j, if t [t j, t j+1 ], +, else, { M (l) + q (l) intern (t t(l) +, min ), if (t, x) D(l), else G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 13 / 25
Optimization problem Setting of the MILP Decision variable ( ) y :=...,k i,...,...,q }{{} j,...,...,p j,...,...,m (l), q (l) intern }{{}}{{}}{{,... } initial densities upstream flows downstream flows internal conditions G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 14 / 25
Optimization problem Setting of the MILP Decision variable ( ) y :=...,k i,...,...,q }{{} j,...,...,p j,...,...,m (l), q (l) intern }{{}}{{}}{{,... } initial densities upstream flows downstream flows internal conditions Optimization problem as a Mixed Integer Linear Programming (MILP) Maximize g(y) { A model y b model, subject to C data y d data, (model constraints), (data constraints). G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 14 / 25
Optimization problem Setting of the MILP Decision variable ( ) y :=...,k i,...,...,q }{{} j,...,...,p j,...,...,m (l), q (l) intern }{{}}{{}}{{,... } initial densities upstream flows downstream flows internal conditions Optimization problem as a Mixed Integer Linear Programming (MILP) Maximize g(y) { A model y b model, subject to C data y d data, (model constraints), (data constraints). Objective function: maximize the downstream outflows g(y) =(0 R n, 0 R m, 1 R m, 0 R o R o ) y T = m 1 j=0 p j G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 14 / 25
Optimization problem Queue estimation Algorithm 1 Compute the optimal solution to the MILP ( y :=...,ki,...,...,qj,...,...,pj,... }{{}}{{}}{{} initial densities upstream flows downstream flows =argmax y g(y),..., (M (l)) (, q intern) (l),... } {{ } internal conditions 2 Compute the traffic states M and k = x M thanks to the explicit solutions [Qiu et al., 2013] 3 Deduce queue lengths by computing for any time step the extremal points of { } Q ε (t) := (α, β) ξ α<β χ, k(t, z) κ ε, z [α, β] where ε>0 is a prescribed sensitivity parameter ) G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 15 / 25
Model and data constraints Outline 1 Introduction 2 Optimization problem 3 Model and data constraints 4 Application to Lankershim Bvd, LA G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 16 / 25
Model and data constraints Model constraints Compatibility conditions Proposition (Compatibility conditions [Claudel and Bayen, 2011]) Consider a family of value conditions c j and define their minimum c(t, x) :=min j J c j(t, x). Then, the solution M of the LWR-BA PDE verifies if and only if M(t, x) =c(t, x), for any (t, x) Dom (c), M ci (t, x) c j (t, x), for all i, j J, and (t, x) Dom(c j ). G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 17 / 25
Model and data constraints Model constraints x x n (iv) v f (iii) x i+1 c (i) ini x i x 0 x n t 0 x (i) w w (ii) w v f (iv) v f v f t max t Check and M c (i) ini M c (j) up c (j) up c (i) ini only for crossing points of domains of influence (v) (ii) x 0 c (j) up (iii) t 0 tmax t t j t j+1 (i) w G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 18 / 25
Model and data constraints Data constraints Data constraints Assume that the data constraints are linear w.r.t. the decision variable y C data y d data. 1 Downstream outflow constraint (red light) p j =0, j s.t. Ω red [t j, t j+1 ], 2 [Loops] Upstream flow data q meas with errors e meas flow (1 e meas flow )qmeas (t) q j (1 + e meas flow )qmeas (t), t [t j, t j+1 ] 3 [GPS] Travel times data dtravel meas with errors etime meas M (t meas exit d meas travel e meas time,ξ) M(texit meas,χ) M (t meas exit d meas travel + e meas time,ξ). G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 19 / 25
Application to Lankershim Bvd, LA Outline 1 Introduction 2 Optimization problem 3 Model and data constraints 4 Application to Lankershim Bvd, LA G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 20 / 25
Application to Lankershim Bvd, LA NGSIM dataset (2006) G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 21 / 25
Application to Lankershim Bvd, LA NGSIM dataset (2006) monitored section = 5 blocks and 4 signalized intersections individual trajectories for each vehicle (+2,400) over 30 min G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 21 / 25
Queue Estimation on Networks Link 1 24
Queue Estimation on Networks Link 2 25
End of the talk Thanks for your attention Any question? guillaume.costeseque@inria.fr G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 25 / 25
References Some references I Anderson, L. A., Canepa, E. S., Horowitz, R., Claudel, C. G., and Bayen, A. M. (2013). Optimization-based queue estimation on an arterial traffic link with measurement uncertainties. Transportation Research Board 93rd Annual Meeting. Paper 14-4570. Claudel, C. G. and Bayen, A. M. (2010a). Lax Hopf based incorporation of internal boundary conditions into Hamilton Jacobi equation. Part I: Theory. Automatic Control, IEEE Transactions on, 55(5):1142 1157. Claudel, C. G. and Bayen, A. M. (2010b). Lax Hopf based incorporation of internal boundary conditions into Hamilton Jacobi equation. Part II: Computational methods. Automatic Control, IEEE Transactions on, 55(5):1158 1174. G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 26 / 25
References Some references II Claudel, C. G. and Bayen, A. M. (2011). Convex formulations of data assimilation problems for a class of Hamilton Jacobi equations. SIAM Journal on Control and Optimization, 49(2):383 402. Lebacque, J.-P. (2002). AtwophaseextensionoftheLWRmodelbasedontheboundednessoftraffic acceleration. In Transportation and Traffic Theory in the 21st Century. Proceedings of the 15th International Symposium on Transportation and Traffic Theory. Lebacque, J.-P. (2003). Two-phase bounded-acceleration traffic flow model: analytical solutions and applications. Transportation Research Record: Journal of the Transportation Research Board, 1852(1):220 230. G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 27 / 25
References Some references III Leclercq, L. (2002). Modélisation dynamique du trafic et applications à l estimation du bruit routier. PhD thesis, Villeurbanne, INSA. Leclercq, L. (2007). Bounded acceleration close to fixed and moving bottlenecks. Transportation Research Part B: Methodological,41(3):309 319. Lighthill, M. J. and Whitham, G. B. (1955). On kinematic waves II. A theory of traffic flow on long crowded roads. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 229(1178):317 345. Mazaré, P.-E., Dehwah, A. H., Claudel, C. G., and Bayen, A. M. (2011). Analytical and grid-free solutions to the Lighthill Whitham Richards traffic flow model. Transportation Research Part B: Methodological,45(10):1727 1748. G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 28 / 25
References Some references IV Qiu, S., Abdelaziz, M., Abdellatif, F., and Claudel, C. G. (2013). Exact and grid-free solutions to the Lighthill Whitham Richards traffic flow model with bounded acceleration for a class of fundamental diagrams. Transportation Research Part B: Methodological,55:282 306. Richards, P. I. (1956). Shock waves on the highway. Operations research, 4(1):42 51. G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 29 / 25
Appendices Outline 5 References 6 Appendices Initial condition: free-flow case Initial condition: congested case Upstream condition: free-flow case Upstream condition: congested case Downstream condition: free-flow case Downstream condition: congested case Junction setting G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 30 / 25
Appendices Initial condition: free-flow case x x n (iii) v f (ii) v f x i+1 c (i) ini x i (i) w x 0 t 0 t max t G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 31 / 25
Appendices Initial condition: congested case x x n (iv) v f (iii) x i+1 c (i) ini x i (i) w w (ii) w x 0 t 0 t max t G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 32 / 25
Appendices Upstream condition: free-flow case x x n (iii) v f vf (ii) x 0 c (j) up t 0 t max t t j t j+1 (i) G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 33 / 25
Appendices Upstream condition: congested case x x n v f (iv) v f v f (v) (ii) x 0 c (j) up (iii) t 0 t max t t j t j+1 (i) w G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 34 / 25
Appendices Downstream condition: free-flow case x n x c (j) down (iv) (i) (ii) (iii) w w w x 0 t t 0 t j t j+1 t max G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 35 / 25
Appendices Downstream condition: congested case x x n x (l) min v f (vi) c (l) intern (iii) v f w (iv) (ii) (i) v f w w (v) x 0 t 0 t (l) min t (l) max t max t G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 36 / 25
Appendices Junction setting f out f rampout f rampin f in G. Costeseque Queue length estimation on arterials Roma, March 08th 2017 37 / 25