MINLP in Air Traffic Management: aircraft conflict avoidance

Transcription

1 MINLP in Air Traffic Management: aircraft conflict avoidance Sonia Cafieri ENAC - École Nationale de l Aviation Civile University of Toulouse France thanks to: Riadh Omheni (ENAC, Toulouse) David Rey (New South Wales, Australia) Second Sevilla Workshop Mixed-Integer Nonlinear Programming Seville, March 30-April 1, 2015 Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

2 Some question... Did you fly to Seville? Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

3 Some question... Did you fly to Seville? Did you think to in-flight safety? How in-flight safety is guaranteed? What does impact it most? Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

4 Air Traffic Management (ATM) & Control ATM : making sure that aicraft are safely guided in the skies and on the ground Air Traffic growing on the world scale Eurocontrol forecast needs increasing automation BOEING long-term market forecast Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

5 Aircraft Conflict Avoidance Aircraft i and j are in conflict if their horizontal distance is less than d: x i (t) x j (t) d t (d = 5NM) their altitude difference is less than h: h i (t) h j (t) h t (h = 1000ft) NM (nautical mile)= 1852 m 1 ft (feet) = m Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

6 Outline 1 Conflict Avoidance in ATM: background 2 MINLP models Aircraft separation modeling Maximizing the number of solved conflicts by speed regulation Solving conflicts by heading angle changes 3 Numerical results 4 Conclusions and Perspectives Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

7 Outline 1 Conflict Avoidance in ATM: background 2 MINLP models Aircraft separation modeling Maximizing the number of solved conflicts by speed regulation Solving conflicts by heading angle changes 3 Numerical results 4 Conclusions and Perspectives Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

8 Aircraft Conflict Avoidance: operational viewpoint Resolution of conflicts currently still largely performed manually by air traffic controllers SESAR & NextGen promote automation projects: Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

9 Aircraft Conflict Avoidance: operational viewpoint Resolution of conflicts currently still largely performed manually by air traffic controllers SESAR & NextGen promote automation projects: Aircraft separation strategies Heading angle deviation Altitude modification Speed adjustements suggested by ERASMUS (En-Route Air Traffic Soft Management Ultimate System) project ( ), allows one to perform a subliminal control Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

10 Aircraft Conflict Avoidance: mathematical/or viewpoint Optimal Control (Tomlin et al. 2004, Cellier et al. 2012) Evolutionary computation (Durand&Alliot 1995, 1998, Delahaye et al. 1996) Mathematical Programming based approaches: Mixed-Integer Linear and Nonlinear Optimization Richards & How, 2002 (MILP) Pallottino, Feron, Bicchi, 2004 (MILP) Christodoulou & Costoulakis, 2004 (MINLP) Vela et al., 2010 (MILP) Alonso-Ayuso, Escudero, Martín-Campo, 2011, 2012, 2013 (MILP, MINLP) Rey et al., 2012 (MILP), 2013 (MINLP) Cafieri & Durand, 2014 (MINLP) Cafieri, 2014 (MINLP) in general, subject to some simplifying hypotesis Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

11 Aircraft Conflict Avoidance: mathematical/or viewpoint Challenge Propose a model corresponding as much as possible to a realistic situation computationally affordable Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

12 Outline 1 Conflict Avoidance in ATM: background 2 MINLP models Aircraft separation modeling Maximizing the number of solved conflicts by speed regulation Solving conflicts by heading angle changes 3 Numerical results 4 Conclusions and Perspectives Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

13 Modeling and optimizing aircraft separation Give a computationally-treatable expression for aircraft separation Choose a separation maneuver decision variables Give a Mathematical Programming formulation for aircraft conflict avoidance Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

14 Modeling and optimizing aircraft separation Give a computationally-treatable expression for aircraft separation Choose a separation maneuver decision variables Give a Mathematical Programming formulation for aircraft conflict avoidance Question: once a separation maneuver (e.g. speed change) is chosen, is the corresponding optimization problem always feasible? combine different separation maneuvers complex models resort to feasibility seeking methods no optimal solution accept that not all conflicts are solved with the selected maneuver and devise suitable pre-processing procedures Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

15 Outline 1 Conflict Avoidance in ATM: background 2 MINLP models Aircraft separation modeling Maximizing the number of solved conflicts by speed regulation Solving conflicts by heading angle changes 3 Numerical results 4 Conclusions and Perspectives Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

16 Aircraft separation Separation condition for aircraft i and j: x r (t) = x i(t) x j (t) d t (0, T) * d = minimum required separation distance * x r (t) = x i(t) x j (t) = x r0 + v r t x r0 v r = relative initial position of aircraft i and j = relative speed of aircraft i and j Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

17 Aircraft separation Separation condition for aircraft i and j: x r (t) = x i(t) x j (t) d t (0, T) depends on t * d = minimum required separation distance * x r (t) = x i(t) x j (t) = x r0 + v r t x r0 v r = relative initial position of aircraft i and j = relative speed of aircraft i and j i.e. x r0 + v r t 2 d 2 t (0, T) v r 2 t 2 + 2(x r0 vr ) t + ( xr0 2 d 2 ) 0 Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

18 Aircraft separation 1) By differentiation, compute the minimizer t m = ( ) x r0 v r / v r 2 and substitute: x r0 2 (xr0 v r )2 v r d Separation: if t m > 0 then condition x r0 2 (xr0 v r )2 v r d Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

19 Aircraft separation 1) By differentiation, compute the minimizer t m = ( ) x r0 v r / v r 2 and substitute: x r0 2 (xr0 v r )2 v r d 2 0 does not depend on t 2 Separation: if t m > 0 then condition x r0 2 (xr0 v r )2 v r d ) Compute the discriminant: = (x r0 v r )2 v r 2 ( x r0 2 d 2 ) does not depend on t Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

20 Aircraft separation 1) By differentiation, compute the minimizer t m = ( ) x r0 v r / v r 2 and substitute: x r0 2 (xr0 v r )2 v r d Separation: if t m > 0 then condition x r0 2 (xr0 v r )2 v r d ) Compute the discriminant: = (x r0 v r )2 v r 2 ( x r0 2 d 2 ) Separation: 2.1) 0 no solutions, aircraft separated 2.2) > 0 and both roots t, t negative t = 2(xr0 v r ) 2 v r and t = 2(xr0 v r ) v r 2 Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

21 Outline 1 Conflict Avoidance in ATM: background 2 MINLP models Aircraft separation modeling Maximizing the number of solved conflicts by speed regulation Solving conflicts by heading angle changes 3 Numerical results 4 Conclusions and Perspectives Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

22 Maximizing solved conflicts by speed regulation separation maneuver selection = Separation based on speed regulation using subliminal control How many conflicts can be solved using the selected maneuver? Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

23 Maximizing solved conflicts by speed regulation separation maneuver selection = Separation based on speed regulation using subliminal control How many conflicts can be solved using the selected maneuver? Maximize the number of conflits that are solved by speed regulation discriminate between conflicts that can be solved by speed changes and those needing other maneuvers may be used as a pre-processing for another model Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

24 MINLP formulation: Max VC (1/4) A = set of n aircraft Variables: { 1 if i and j are separated (no conflict) i, j A, i < j z = 0 otherwise v i, v min, v max i A, aircraft speeds (continuous) speed change between -6% and +3% of the original speed subliminal control i, j A, auxiliary var. for the inner product x r0 v r (continuous) Objective: maximize the number of solved conflicts: max i,j A, i<j z Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

25 MINLP formulation: Max VC (2/4) Constraints: integrality constraints on z variables bounds on v variables separation constraints on pairs of aircraft Separation constraints: count the number of aircraft pairs for which 0 (condition 2.1) ( (x r0 v r )2 v r 2 ( x r0 2 d 2 ) ) (2 z 1) 0 i, j A, i < j Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

26 MINLP formulation: Max VC (3/4) Pairs of separated aircraft may potentially be not counted couple the -based condition with a simple geometric condition - Assume that trajectories are straight lines intersecting in one point - look at the sign of the scalar product x r0 v r Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

27 MINLP formulation: Max VC (3/4) Pairs of separated aircraft may potentially be not counted couple the -based condition with a simple geometric condition If x r0 v r 0 aircraft diverging separated if separated at their initial positions 0 OR x r0 v r 0 aircraft separated Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

28 MINLP formulation: Max VC (3/4) Pairs of separated aircraft may potentially be not counted couple the -based condition with a simple geometric condition If x r0 v r 0 aircraft diverging separated if separated at their initial positions 0 OR x r0 v r 0 aircraft separated Separation constraints: (x r0 v r )2 (2 z 1) v r 2 ( x r0 2 d 2 ) (2 z 1) OR (x r0 v r ) (2 y 1) 0 i, j A, i < j i, j A, i < j Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

29 MINLP formulation: Max VC (4/4) Modeling the OR condition: use an additional variable 0 w 1, i, j, i < j Separation constraints: (x r0 v r )2 (2 z 1) v r 2 ( x r0 2 d 2 ) (2 z 1) (x r0 v r ) (2 y 1) 0 w z w y w z + y i, j A, i < j i, j A, i < j i, j A, i < j i, j A, i < j i, j A, i < j and maximize max i,j A, i<j w Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

30 Numerical results Test problems Solution approach Global solution spatial Branch-and-Bound COUENNE 0.4 (Belotti et al., 2008) AMPL model conflict zone n aircraft n {1,..., 6} aircraft n(n 1)/2 conflicts d = 5 NM, v = 400 NM/h Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

31 Numerical solution Max VC n radius n confl obj time (s) pb_n pb_n pb_n pb_n pb_n CP_n CP_n Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

32 Numerical solution Max VC n radius n confl obj time (s) pb_n pb_n pb_n pb_n pb_n CP_n CP_n promising results; not guaranteed solving all conflicts Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

33 Numerical solution Max VC n radius n confl obj time (s) pb_n pb_n pb_n pb_n pb_n CP_n CP_n Example pb_n5, n = 5 (10 conflicts) aircraft v ratio = v/ v aircraft accelerated, 3 decelerated speed variations in [-6%, +3%] around the original velocity (subliminal control) Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

34 Outline 1 Conflict Avoidance in ATM: background 2 MINLP models Aircraft separation modeling Maximizing the number of solved conflicts by speed regulation Solving conflicts by heading angle changes 3 Numerical results 4 Conclusions and Perspectives Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

35 Solving all conflicts with Riadh Omheni (ENAC, Toulouse) If not all conflicts are solved by speed changes Solve conflicts by heading angle changes = new model using the first one as a pre-processing step speed changes + heading angle changes Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

36 Solving all conflicts with Riadh Omheni (ENAC, Toulouse) If not all conflicts are solved by speed changes Solve conflicts by heading angle changes = new model using the first one as a pre-processing step speed changes + heading angle changes Algorithm Input: n, A maximize the number of solved conflicts by speed regulation if (there are still unsolved conflicts) then solve conflicts by heading angle changes minimizing angle deviations endif Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

37 Solving conflicts by heading angle changes (HAC) Recall separation (condition 1): if t m > 0 then x r0 2 (xr0 v r )2 v r d Relative velocity vector: v r = cos(φ i )v i cos(φ j )v j sin(φ i )v i sin(φ j )v j Changing heading angle changing Φ = φ + θ v r = cos(φ i + θ i )v i cos(φ j + θ j )v j sin(φ i + θ i )v i sin(φ j + θ j )v j with θ bounded and v i, v j kept unchanged. Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

38 MINLP formulation: HAC (1/3) Variables: θ i, ( θ min, θ max ), i A angle variation of aircraft i (continuous) v 2r, (i, j) A, i < j square of the relative velocity of i and j (continuous) p, (i, j) A, i < j t m, (i, j) A, i < j (continuous) inner product v r xr0 (continuous) y, (i, j) A, i < j, used to check if t m > 0 (binary) Objective: min i A θ 2 i minimizing angle deviations Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

39 MINLP formulation: HAC (2/3) Constraints: definition of v 2r (quadratic, trigonometric functions) v 2r = ( v i cos(φ i + θ i ) v j cos(φ j + θ j ) ) 2 + ( v i sin(φ i + θ i ) v j sin(φ j + θ j ) ) 2 i, j A, i < j inner product in the separation condition (trigonometric functions) p = (x 0 i,1 x0 j,1 ) ( v i cos(φ i + θ i ) v j cos(φ j + θ j ) ) + (x 0 i,2 x0 j,2 ) ( v i sin(φ i + θ i ) v j sin(φ j + θ j ) ) i, j A, i < j definition of t m check sign of t m (bilinear) t m v2r + p = 0 (i, j) A, i < j (bilinear with binary var.) t m (2y 1) 0 (i, j) A, i < j separation (quadratic + linear term, product with binary var.) y ( (x 0r v2r ) (p ) 2 ((d) 2 v 2r )) 0 (i, j) A, i < j Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

40 MINLP: HAC reformulation (3/3) Compute bounds on variables v r, p and t m ( obtain (t m ) min, (t m ) max) Reformulate products of binary variables y reformulate t m (2y 1) 0 and continuous variables (i, j) A, i < j to t m i,j (t m ) min (1 y ) (i, j) A, i < j t m i,j (t m ) max y reformulate y ( (x 0r v2r ) (p ) 2 ((d) 2 v 2r )) 0 (i, j) A, i < j to (x 0r d 2 ) v 2r (p ) 2 bigm (1 y ) (i, j) A, i < j Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

41 Outline 1 Conflict Avoidance in ATM: background 2 MINLP models Aircraft separation modeling Maximizing the number of solved conflicts by speed regulation Solving conflicts by heading angle changes 3 Numerical results 4 Conclusions and Perspectives Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

42 Numerical results: HAC HAC n n c nrc time (s) obj pb-hth_n pb-hth_n pb-hth_n pb-hth_n CP_n CP_n CP full _n CP full _n CP half _n CP half _n RCP full _n RCP full _n promising computing time - exact solution Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

43 Numerical results: HAC Example: n = 3 aircraft, 1 head-to-head Φ j Φ i Φ k Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

44 Numerical results: HAC Example: n = 3 aircraft, 1 head-to-head Φ j Φ j Θ j Θ i Φ i Φ k Φ i Θ k = 0 Φ k Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

45 Numerical results: Max VC + HAC Max VC n n c nrc time (s) pb-hth_n pb-hth_n pb-hth_n pb-hth_n CP_n CP_n CP full _n CP full _n CP half _n CP half _n RCP full _n RCP full _n Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

46 Numerical results: Max VC + HAC Max VC HAC n n c (pre-processing) (with pre-processing) n rc time (s) n rc time (s) obj pb-hth_n pb-hth_n pb-hth_n pb-hth_n CP_n CP_n CP full _n CP full _n CP half _n CP half _n RCP full _n RCP full _n Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

47 Numerical results: Max VC + HAC Max VC HAC HAC n n c (pre-processing) (with pre-processing) n rc time (s) n rc time (s) obj n rc time (s) obj pb-hth_n pb-hth_n pb-hth_n pb-hth_n CP_n CP_n CP full _n CP full _n CP half _n CP half _n RCP full _n RCP full _n pre-processing reduces, sometimes significantly, computing time Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

48 Numerical results: Max VC + HAC Max VC HAC HAC n n c (pre-processing) (with pre-processing) n rc time (s) n rc time (s) obj n rc time (s) obj pb-hth_n pb-hth_n pb-hth_n pb-hth_n CP_n CP_n CP full _n CP full _n CP half _n CP half _n RCP full _n RCP full _n pre-processing reduces, sometimes significantly, computing time Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

49 Outline 1 Conflict Avoidance in ATM: background 2 MINLP models Aircraft separation modeling Maximizing the number of solved conflicts by speed regulation Solving conflicts by heading angle changes 3 Numerical results 4 Conclusions and Perspectives Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

50 Conclusions and Perspectives Promising results combining two MINLP models based on different maneuvers to separate aircraft MINLP model for maximizing the number of solved aircraft conflicts: helps filtering conflicts with respect to the selected separation maneuver Future work: decomposition-based approaches with D. Rey (New South Wales, Australia) solution by Interval Branch&Bound techniques with F. Messine (Toulouse, France) ATOMIC = Air Traffic Optimization by Mixed-Integer Computation project funded by French National Agency of Research Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36

51 The end Thank you! Sonia Cafieri (ENAC) MINLP in Air Traffic Management Seville, / 36