Motivation Traffic control strategies Main control schemes: Highways Variable speed limits Ramp metering Dynamic lane management Arterial streets Adap

Transcription

1 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 / 25

2 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 / 25

3 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 / 25

4 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 / 25

5 Motivation Why introducing bounded acceleration? Car trajectories (Assuming no Italian taxi drivers...) x t G. Costeseque Queue length estimation on arterials Roma, March 08th / 25

6 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 / 25

7 Motivation Why introducing bounded acceleration? Car trajectories with bounded acceleration phase x t G. Costeseque Queue length estimation on arterials Roma, March 08th / 25

8 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 / 25

9 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 / 25

10 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 / 25

11 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 / 25

12 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 / 25

13 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 / 25

14 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 / 25

15 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 / 25

16 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 / 25

17 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 / 25

18 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 / 25

19 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 / 25

20 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 / 25

21 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 / 25

22 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 / 25

23 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 / 25

24 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 / 25

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 / 25

26 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 / 25

27 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 / 25

28 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 / 25

29 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 / 25

30 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 / 25

31 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 / 25

32 Application to Lankershim Bvd, LA NGSIM dataset (2006) G. Costeseque Queue length estimation on arterials Roma, March 08th / 25

33 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 / 25

34 Queue Estimation on Networks Link 1 24

35 Queue Estimation on Networks Link 2 25

36 End of the talk Thanks for your attention Any question? G. Costeseque Queue length estimation on arterials Roma, March 08th / 25

37 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 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): 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): G. Costeseque Queue length estimation on arterials Roma, March 08th / 25

38 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): 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): G. Costeseque Queue length estimation on arterials Roma, March 08th / 25

39 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): 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): 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): G. Costeseque Queue length estimation on arterials Roma, March 08th / 25

40 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: Richards, P. I. (1956). Shock waves on the highway. Operations research, 4(1): G. Costeseque Queue length estimation on arterials Roma, March 08th / 25

41 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 / 25

42 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 / 25

43 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 / 25

44 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 / 25

45 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 / 25

46 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 / 25

47 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 / 25

48 Appendices Junction setting f out f rampout f rampin f in G. Costeseque Queue length estimation on arterials Roma, March 08th / 25