Wind-Based Robust Trajectory Optimization using Meteorological Ensemble Probabilistic Forecasts

Transcription

1 Wind-Based Robust Trajectory Optimization using Meteorological Ensemble Probabilistic Forecasts Daniel González Arribas, Manuel Soler, Manuel Sanjurjo Rivo Area of Aerospace Engineering Universidad Carlos III de Madrid SESAR Innovation Days TU Delft, 8th October 2016 González Arribas, Soler and Sanjurjo SESAR Innovation Days / 42

2 Table of contents 1 Introduction Motivation 2 Methodology Ensemble Prediction Systems Dynamic system modelling Robust nonlinear optimal control 3 Results Case study Constant airspeed Variable airspeed 4 Discussion Conclusions Future work González Arribas, Soler and Sanjurjo SESAR Innovation Days / 42

3 Motivation Uncertainty and 4D trajectories t + 0 t 1 t 2 Weather? Mass? p( x) Performance?... = t 0 t 1 t 2 González Arribas, Soler and Sanjurjo SESAR Innovation Days / 42

4 Introduction Methodology Results Discussion Motivation TBO-Met project Pre-Tactical Phase up to 3 Hours before departure ECMWF EPS forecasts Tactical Phase (during execution) Nowcasts Nowcasts available 3 to 1/2 hours in advance Robust Trajectory Planning Convection Probability fields Wind Uncertainty fields Time lead or lag (s) Storm avoidance Ground speed (m/s) probability of high-risk area Heading (rad) González Arribas, Soler and Sanjurjo Agreed RBT SBT T -6Hours Course True heading Distance (km) T -3Hours... T Executed RBT (departure) SESAR Innovation Days 2016 E- 1/2 to 3Hours Revised RBT E (event) / 42

5 Motivation Research scope González Arribas, Soler and Sanjurjo SESAR Innovation Days / 42

6 Motivation 4D Trajectory Planning González Arribas, Soler and Sanjurjo SESAR Innovation Days / 42

7 Motivation Contributions of this work Developing a methodology for 4D flight plan optimization under uncertainty Incorporate predictability into this methodology as an objective González Arribas, Soler and Sanjurjo SESAR Innovation Days / 42

8 Table of contents 1 Introduction Motivation 2 Methodology Ensemble Prediction Systems Dynamic system modelling Robust nonlinear optimal control 3 Results Case study Constant airspeed Variable airspeed 4 Discussion Conclusions Future work González Arribas, Soler and Sanjurjo SESAR Innovation Days / 42

9 Ensemble Prediction Systems Why use ensemble forecasting? Weather forecasts are uncertain, because nonlinear, complex, chaotic dynamics amplify: 1 Uncertainty in initial conditions 2 Physical modelling/parametrization errors 3 Numerical error and computational limitations Deterministic forecasts have limited usefulness González Arribas, Soler and Sanjurjo SESAR Innovation Days / 42

10 Ensemble Prediction Systems Why use ensemble forecasting? Weather forecasts are uncertain, because nonlinear, complex, chaotic dynamics amplify: 1 Uncertainty in initial conditions 2 Physical modelling/parametrization errors 3 Numerical error and computational limitations Deterministic forecasts have limited usefulness González Arribas, Soler and Sanjurjo SESAR Innovation Days / 42

11 Ensemble Prediction Systems Why use ensemble forecasting? Weather forecasts are uncertain, because nonlinear, complex, chaotic dynamics amplify: 1 Uncertainty in initial conditions 2 Physical modelling/parametrization errors 3 Numerical error and computational limitations Deterministic forecasts have limited usefulness González Arribas, Soler and Sanjurjo SESAR Innovation Days / 42

12 Ensemble Prediction Systems Ensemble Prediction Systems Deterministic forecast Initial condition uncertainty Ensemble members Figure: Uncertainty propagation González Arribas, Soler and Sanjurjo SESAR Innovation Days / 42

13 Introduction Methodology Results Discussion Ensemble Prediction Systems Ensemble Prediction Systems II Figure: A subset of the PEARP ensemble forecast at 250 hpa González Arribas, Soler and Sanjurjo SESAR Innovation Days / 42

14 Dynamic system modelling The 3-DoF Point-Mass Model 3 Position states: 3 Velocity states: Latitude ( ) Longitude ( ) Altitude (z) 1 Additional state: Mass (m) Airspeed (v) Heading ( ) Flight Path Angle ( ) 3 Controls: Throttle ( ) Lift Coeff. (C L) Bank Angle ( ) Heading N v W E Angle of Attack Flight Path Angle v Bank angle S González Arribas, Soler and Sanjurjo SESAR Innovation Days / 42

15 Dynamic system modelling Phases of a flight Cruise Initial climb Descent Approach & Landing Takeoff González Arribas, Soler and Sanjurjo SESAR Innovation Days / 42

16 Dynamic system modelling Dynamic model for cruise Cruise dynamical system φ = v cos(ψ) + w y(φ, λ, t) R M + z λ = v sin(ψ) + w x(φ, λ, t) (R M + z) cos(φ) v = m 1 (Thrust(π, v, ρ, T ) Drag(v, m, ρ, T )) ṁ = Fuel Consumption(π, v, ρ, T ) Forces, constraints & fuel consumption from BADA model González Arribas, Soler and Sanjurjo SESAR Innovation Days / 42

17 Dynamic system modelling Wind impact on the dynamics N True Airspeed (TAS) Wind speed Heading Ground speed W E S González Arribas, Soler and Sanjurjo SESAR Innovation Days / 42

18 Robust nonlinear optimal control The Optimal Control Problem OCP x R n, u R m, g R ng subject to: min J = ϕ(x 0, x f, t 0, t f ) + tf t 0 dx/dt = f (x, u, t) L(x, u, t)dt g L g(x, u, t) g U b L b(x 0, x f, t 0, t f ) b U Common cost functional in ATM: J = m f + CI t f González Arribas, Soler and Sanjurjo SESAR Innovation Days / 42

19 Robust nonlinear optimal control Direct collocation methods NLP M min J = ϕ(x 0, x M, t 0, t M ) + (t f t 0 ) w k L(x k, u k, t k ) k=0 subject to: D k (x 0,..., x N ) = f (x k, u k, t k ), k {0,..., M} g L g(x k, u k, t k ) g U, k {0,..., M} b L b(x 0, x M, t 0, t M ) b U González Arribas, Soler and Sanjurjo SESAR Innovation Days / 42

20 Robust nonlinear optimal control The Dynamical System with Uncertainty The dynamical system with uncertainty dx/dt = f (x, u, t, p) Uncertainty is represented by a random variable p R np Note that the uncertainty p is static (i.e. this is not a SDE) González Arribas, Soler and Sanjurjo SESAR Innovation Days / 42

21 Robust nonlinear optimal control The robust OCP Discretize p with a Polynomial Chaos rule Approximate expectation with E(h(p)) = N i=0 w ih(p i ) Create a copy of the state x i for each p i and average the cost functional Problem size is O(nN) Dynamical system for the open-loop ROCP dx i /dt = f (x i, u, t, p i ), i {0,..., N} x 0... x N u States t Controls t González Arribas, Soler and Sanjurjo SESAR Innovation Days / 42

22 Robust nonlinear optimal control The robust OCP II Declare q m states to be tracked Make up to q controls scenario-specific Dynamical system for the state-tracking ROCP dx i /dt = f (x i, u, t, p i ), i {0,..., N} T (x i x j ) = 0 where T R n q is the (full-rank) tracking matrix x 0... x N x 0... x N u Tracked states t Non-tracked states t Controls t González Arribas, Soler and Sanjurjo SESAR Innovation Days / 42

23 Robust nonlinear optimal control Applying the ST-ROCP formulation to our problem EPS already have a discretized p (with w i = N 1 ) Tracking of φ and λ is not practical because it forces to fully absorb uncertainty in v Not efficient Not operational Small feasible set (need to go slow to keep v margin for unfavourable scenarios) González Arribas, Soler and Sanjurjo SESAR Innovation Days / 42

24 Robust nonlinear optimal control Flight plans FL380 M0.83 FL340 TAS240 FL360 M0.81 González Arribas, Soler and Sanjurjo SESAR Innovation Days / 42

25 Robust nonlinear optimal control Dynamic system reformulation Use ground distance (r) flown as independent variable. We can do this if v G > 0 (true for realistic winds), as r(t) is strictly increasing. Groundspeed is v G = (v cos ψ + w y ) 2 + (v sin ψ + w x ) 2. Reformulated cruise dynamical system dx/dr = ẋ v G dt/dr = v G (r) 1 Cost functional: J = E[ m f + CI t f ] + DP(t f,max t f,min ) with t f,min t f,i t f,max González Arribas, Soler and Sanjurjo SESAR Innovation Days / 42

26 Table of contents 1 Introduction Motivation 2 Methodology Ensemble Prediction Systems Dynamic system modelling Robust nonlinear optimal control 3 Results Case study Constant airspeed Variable airspeed 4 Discussion Conclusions Future work González Arribas, Soler and Sanjurjo SESAR Innovation Days / 42

27 Case study Description A330 from NY to Lisbon Flying at M=.82 and FL380 20th of January, hpa level forecast from the PEARP ensemble (35 members, from Météo France) González Arribas, Soler and Sanjurjo SESAR Innovation Days / 42

28 Constant airspeed Routes González Arribas, Soler and Sanjurjo SESAR Innovation Days / 42

29 Constant airspeed Groundspeed Ground speed (m/s) Distance (km) González Arribas, Soler and Sanjurjo SESAR Innovation Days / 42

30 Constant airspeed Time lead or lag Time lead or lag (s) Distance (km) González Arribas, Soler and Sanjurjo SESAR Innovation Days / 42

31 Constant airspeed Time lead or lag Heading (rad) Course True heading Distance (km) González Arribas, Soler and Sanjurjo SESAR Innovation Days / 42

32 Constant airspeed Routes González Arribas, Soler and Sanjurjo SESAR Innovation Days / 42

33 Constant airspeed Groundspeed Ground speed (m/s) Distance (km) González Arribas, Soler and Sanjurjo SESAR Innovation Days / 42

34 Constant airspeed Time lead or lag Time lead or lag (s) Distance (km) González Arribas, Soler and Sanjurjo SESAR Innovation Days / 42

35 Constant airspeed Time lead or lag Heading (rad) Course True heading Distance (km) González Arribas, Soler and Sanjurjo SESAR Innovation Days / 42

36 Constant airspeed Predictability-efficiency trade-off p González Arribas, Soler and Sanjurjo SESAR Innovation Days / 42

37 Variable airspeed Routes (Variable velocity (CI=10) González Arribas, Soler and Sanjurjo SESAR Innovation Days / 42

38 Variable airspeed Airspeeds (CI=10) DP=0 DP=20 M Distance (km) González Arribas, Soler and Sanjurjo SESAR Innovation Days / 42

39 Variable airspeed Predictability-efficiency trade-off Arrival time range (minutes) Cost (Equivalent Kg) González Arribas, Soler and Sanjurjo SESAR Innovation Days / 42

40 Table of contents 1 Introduction Motivation 2 Methodology Ensemble Prediction Systems Dynamic system modelling Robust nonlinear optimal control 3 Results Case study Constant airspeed Variable airspeed 4 Discussion Conclusions Future work González Arribas, Soler and Sanjurjo SESAR Innovation Days / 42

41 Conclusions Main takeaways We have developed a methodology to do flight planning under uncertainty and include predictability as an objective Adapted to operational concepts Scaling of O(nN) is manageable (parallelization possibilities?) Same advantages and drawbacks of other direct methods (possibility of local optima but better handling of more complex, nonlinear and higher-dimensional systems compared to raw DP) Natural handling of EPS forecasts González Arribas, Soler and Sanjurjo SESAR Innovation Days / 42

42 Future work Future work Dynamic Weather, BADA 4 Extension to 4D (altitude changes) Assesment of potential benefits Combination with other uncertainty sources Extension to tactical avoidance of thunderstorms Conflict avoidance & resolution? González Arribas, Soler and Sanjurjo SESAR Innovation Days / 42

43 Future work Acknowledgements HALA! network PhD scholarship TBO-Met project (check out poster!) González Arribas, Soler and Sanjurjo SESAR Innovation Days / 42

44 Wind-Based Robust Trajectory Optimization using Meteorological Ensemble Probabilistic Forecasts Daniel González Arribas, Manuel Soler, Manuel Sanjurjo Rivo Area of Aerospace Engineering Universidad Carlos III de Madrid SESAR Innovation Days TU Delft, 8th October 2016 González Arribas, Soler and Sanjurjo SESAR Innovation Days / 42