Assimilation de Données, Observabilité et Quantification d Incertitudes appliqués à l hydraulique à surface libre

Size: px

Start display at page:

Download "Assimilation de Données, Observabilité et Quantification d Incertitudes appliqués à l hydraulique à surface libre"

Mitchell Hampton
5 years ago
Views:

1 Assimilation de Données, Observabilité et Quantification d Incertitudes appliqués à l hydraulique à surface libre Pierre-Olivier Malaterre, Igor Gejadze, Hind Oubanas UMR G-EAU, Irstea, Montpellier, France Pour mieux affirmer ses missions, le Cemagref devient Irstea Colloque National d Assimilation 2 décembre 206

2 Plan Systèmes étudiés : canaux d'irrigation, fleuves Questions : problèmes inverses & incertitudes Modèles : Saint-Venant D-.5D non linéaire (SIC) Modèle Linéaire Tangent, Adjoint Filtre de Kalman, pour un canal d irrigation (X, Q p, C d ) FK, Evolution des Erreurs, Observabilité 4D-Var, données SWOT sur la Garonne 4D-Var, quantification d'incertitudes Conclusions

3 Irrigation canal Water for irrigation, industries and domestic uses Automatic gates SCADA systems

4 Control objectives Distribute water to users (offtaes) Minimize water losses Satisfy constraints (levels, discharges, etc) By mean of control actions (manual or automatic) on cross devices (gates, weirs, pumps) Automatic control = type of inverse problem

5 Inverse problems for controlled Irrigation Canals Known: Bathymetry, boundary conditions (up, mid, down) Observations: Water levels close to cross devices Unnowns / Uncertainties (active & passive): Offtae withdrawal or inflow (Q p ) Cross device discharge coefficients (C d ) Hydraulic state (Q, Z) (e.g. state feedbac ctrl) Friction coefficients Objective: water loss or theft, fault detection, model update for operation & maintenance, etc.

Surface Water and Ocean Topography (SWOT) mission (202) Scientific requirements Observable river width > 00 m Height accuracy Slope accuracy Width accuracy Data collection 0 cm over area > m².

6 Surface Water and Ocean Topography (SWOT) mission (202) Scientific requirements Observable river width > 00 m Height accuracy Slope accuracy Width accuracy Data collection 0 cm over area > m².7 cm/m over area > m 5% of the evaluated river 90% of all ocean/continents within the orbit during 90% of the operational time. August 22, 206 hind.oubanas@irstea.fr 6

7 Inverse problems for rivers observed from space Known: River length, mean slope, mean annual discharge!? boundary condition (down)!? Observations: Water levels locally, or globally (SWOT swaths), local slope, width Unnowns / Uncertainties (active & passive): Boundary conditions (Q up ) Bathymetry, Friction (strong effect) Hydraulic state (Q, Z) outside observations windows (time and space) Tributaries inflows (Q p ) Objective: flood control, resource availability, navigation, water balance

8 Example (5-pool irrigation canal) y y 2 y 3 y4 y 5 y 6 y 7 y 8 y 9 y 0 Q p Q p2 Q p3 Q p4 Q p5 5 pools, 5 unnown withdrawals Q p, 0 water level measurements Y

9 SIC² : Simulation and Integration for Controls of Canals - D/.5D hydraulic model (Multiple beds + Storage areas) - Flow dynamics of rivers - Irrigation canals, drainage networ, etc. - Based on the full Saint-Venant equations - TLM & Adjoint (Tapenade) Gejadze & Malaterre 206 August 22, 206 hind.oubanas@irstea.fr 9

10 SIC² : Simulation and Integration for Controls of Canals Regular sections : t A + x Q = Q L t Q + x Q 2 /A + ga x Z = gas f + C Q L v, S f = - D/.5D hydraulic model (Multiple beds + Storage areas) - Flow dynamics of rivers - Irrigation canals, drainage networ, etc. - Based on the full Saint-Venant equations - TLM & Adjoint (Tapenade) Q 2 K 2 S A 2, t [0, T] R4/3 Q(x, t): Discharge Z(x, t): Water level A x, t : Wetted cross sectional area v x, t = Q/A: Mean velocity R: Hydraulic radius K S : Stricler coefficient C : Lateral discharge coefficient Gejadze & Malaterre 206 August 22, 206 hind.oubanas@irstea.fr 0

11 SIC² : Simulation and Integration for Controls of Canals Regular sections : t A + x Q = Q L t Q + x Q 2 /A + ga x Z = gas f + C Q L v, S f = Singular sections : Q S3,i Q S3,i+ = 0, Q S3,i = F(Z S3,i, Z S3,i+, C d S 3,i ) - D/.5D hydraulic model (Multiple beds + Storage areas) - Flow dynamics of rivers - Irrigation canals, drainage networ, etc. - Based on the full Saint-Venant equations - TLM & Adjoint (Tapenade) Q 2 K 2 S A 2, t [0, T] R4/3 Q(x, t): Discharge Z(x, t): Water level A x, t : Wetted cross sectional area v x, t = Q/A: Mean velocity R: Hydraulic radius K S : Stricler coefficient C : Lateral discharge coefficient C d : Cross device coefficient S i, : th cross section of the reach i Gejadze & Malaterre 206 August 22, 206 hind.oubanas@irstea.fr

12 SIC² : Simulation and Integration for Controls of Canals Regular sections : t A + x Q = Q L t Q + x Q 2 /A + ga x Z = gas f + C Q L v, S f = Singular sections : Q S3,i Q S3,i+ = 0, Q S3,i = F(Z S3,i, Z S3,i+, C d S 3,i ) Boundary conditions: - D/.5D hydraulic model (Multiple beds + Storage areas) - Flow dynamics of rivers - Irrigation canals, drainage networ, etc. - Based on the full Saint-Venant equations - TLM & Adjoint (Tapenade) Q 2 K 2 S A 2, t [0, T] R4/3 Upstream nodes : Q t or Z(t) Downstream nodes : Q t = f(z t, p rc ) Q(x, t): Discharge Z(x, t): Water level A x, t : Wetted cross sectional area v x, t = Q/A: Mean velocity R: Hydraulic radius K S : Stricler coefficient C : Lateral discharge coefficient C d : Cross device coefficient S i, : th cross section of the reach i p rc : rating curve parameters Gejadze & Malaterre 206 August 22, 206 hind.oubanas@irstea.fr 2

13 LQG (Linear Quadratic Gaussian) Perturbations (w) Q p Canal Sorties contrôlées (z) Z av Commandes (u) LQG Mesures (y)

14 Design du contrôleur LQG Minimisation d'un critère J : Contraintes dynamiques : J y y Q y y u R u T y T N 2 0 ( ( ) *( ))..( ( ) *( )) ( ).. ( ) ) (. ) ( ) (. ) (. ) (. ) (. ) ( C x y u B u B B u A x x c c p p u : vecteur des ouvertures des 5 vannes u p : vecteur des débits aux 5 prises u c : vecteur des coefficients de débit aux 5 ouvrages en travers

15 Solution optimale Solution optimale (retour d état) : u( ) K ( ). x( ) LQ? K LQ matrice gain du contrôleur LQ, solution d une équation de Riccati

16 Observateur d'état et de perturbation Cx y y y L u u y y L u u y y L u B u B B u A x x c c c p p p c c p p ˆ ˆ ˆ).( ˆ ˆ ˆ).( ˆ ˆ ˆ).( ˆ. ˆ.. ˆ. ˆ L, L p, L c : par placement de pole (A-LC) pour avoir convergence de l'erreur de reconstruction de l état x vers 0 => Observateur de Luenberger (théorie de l'observabilité, théorie de la détectabilité, principe de séparation) L, L p, L c : Minimisation de l'erreur (variance) de reconstruction => Filtre de Kalman

17 Kalman Filter equations Q A AP P BU AX X t ~ ~ ~ ˆ ~ ˆ t t P C K I P CX Y K X X R C CP C P K ~ ~ ˆ ˆ ˆ ~ ~ ~ c p c p c p c p v u u x C y w u B u u x B B A u u x Initial linear model + noise with given covariance matrices Q and R Augmented state for the reconstruction of x, u p (Q p ) and u c (C d ) No observability requirement!!!

18 Uncertainty reduction (on the 5 Q p and 5 C d ) y y 2 y 3 y4 y 5 y 6 y 7 y 8 y 9 y 0 Cd Cd 2 Q p Q p2 Cd 3 Cd 4 Cd 5 Q p3 Q p4 Q p5 Is it possible to identify Q p and C d from limited water level measurements Y?

19 Scenario tested with a Kalman Filter Time step for the simulations is 300 seconds. T end = 48h The 5 offtaes are operated at time 6 h (-0.00 m3/s) and again at time 20 h (+0.00 m3/s) returning to the initial discharge values of m3/s All cross regulators eep the same gate positions (open loop), but the discharge coefficient of their gates is changed from 0.82 to 0.66 at time 30 mn, then to 0.72 at time 6 h and returning to initial value 0.82 at 24 h The initial state X 0 is taen using random variables For the matrix Q, we choose σ Q = 0. m 3 /s and σ Z = 0. m for the normal hydraulic states and σ QC = for the augmented states (Q p and C d ). For the matrix R, we choose σ R = 0.0 m There are 0 water level measurements (y, y 2,, y 0 ), or less!

20 State reconstruction Twin exp X (Q & Z) 4 2 X Q 0 nx= Time (h) X Z Time (h)

21 State reconstruction Twin exp - C d Cd Reconstruction of discharges coefficients Cd at cross devices Cd Cd2 Cd3 Cd4 Cd5 Good and fast convergence Despite wrong initial state X 0 (No additional noise) Time (h)

22 State reconstruction Twin exp - Q p Qp Reconstruction of discharges Qp at offtaes Qp Qp2 Qp3 Qp4 Qp5 Good and fast convergence Despite wrong initial state X 0 (No additional noise) Time (h)

23 No convergence towards the correct values (case with 9 measurements only: y 8 removed) Reconstruction of discharges Qp at offtaes wrong! Qp Qp2 Qp3 Qp4 Qp Reconstruction of discharges coefficients Cd at cross devices wrong! Cd Cd2 Cd3 Cd4 Cd5 0 0 Qp Cd wrong! Time (h) Time (h) Timing is good but not the values Very bad for Q p3 and Q p4 (the 2 influenced a lot by y 8 ), surprisingly good for C d (except C d4 for the same reasons)

24 Variances of Pb on X Z Variances of Pb on Qp Increase of the local variance (close to y 8 ) during time iterations: we now that the values are not good State index Variance Measurement points State index Variances of Pb on X Q Variances of Pb on X Z State index Variances of Pb on Cd Variances of Pb on Qp Variance State index Variances of Pb on X Q Variances of Pb on Cd State index State index

25 Convergence study o a a o t a t a a t a o t a a t a o t a a t a o t a a t b a a t b b a t a a K C K A C K A K C K X X A C K X X A X A X A C C X A C K X A X A B U C X A C C B U C X A C K X A X A B U C X A C X C K X A B U X A X C Y K B U X A X X C Y K X X X ) ( 0 lim lim 0 0 et si? a a a a E K C A A A C K A A C K A E E E E A C K A E We want that the estimation error converges toward 0 Convergence condition : If for > n this term is Schur =0 or

26 With the 0 measurements: Convergence of the reconstruction error -> 0 Max eig. (A-KCA) = < Then the error between the estimated state X a and the true state X t is converging towards 0 (within the linear assumptions) Poles of the Kalman Filter during iterations /T /T /T 0.9/T 0.8/T 0.8/T 0.7/T 0.6/T Poles 0.5/T of the 0.4/T Kalman Filter. Max = /T /T /T /T 0./T 0.7/T 0.3/T /T 0.4/T 0.5/T

27 Non convergence if only 9 measurements or less (e.g.: remove y 8 ) Max eigenvalue = Poles of the Kalman Filter during iterations /T /T /T 0.9/T 0.8/T 0.8/T 0.7/T 0.6/T 0.5/T Poles 0.4/T of the Kalman Filter. Max = /T /T /T /T 0./T 0.7/T 0.3/T /T 0.4/T 0.5/T

28 Questions to solve Does it depend only on A, C? Is-it possible to choose or analyze (Q,R) so that the resulting gain matrix K, leads to a Schur (A-KCA) matrix?

29 Luenberger Observer Kalman Filter Observability Observability C CA 2 CA O n CA O full ran = n K such that p, K such that poles(a-kc)=p A-KC Schur Convergence of estimation? Condition on A, C??? (Q,R) K such that? (Q,R) K such that A-KCA Schur Convergence de of l estimation

30 Detectability Detectability non observables states of O are stables K such that A-KC Schur Convergence of estimation Rm: Same property for A-KCA

31 Convergence Sufficient condition for convergence : Let (A, C) pair be detectable Let Q matrix factorizable in Q=ΓΓ T, such that pair (A, Γ) be stabilizable Let R matrix be definite positive Then the Riccati algebraic equation for the Kalman Gain has and only positive solution and the corresponding matrix (A-KC) is Schur Rm: depends on A, C and Q Def: (A,B) controllable if ran(b AB A 2 B A n- B) = n Def: stabilizable if non controllable poles are stable

32 Influence de Q sur l'observabilité Non Oui "Optimal"

33 Conclusions (cadre linéaire FK). Les 0 mesures de niveau permettent d identifier les 5 Q p et 5 C d 2. Le problème inverse est bien posé, et l erreur de reconstruction de l état analysé tend bien vers 0 3. Dès qu on enlève mesure on perd ces résultats : l erreur ne tends pas vers 0, certaines composantes du vecteur de contrôle ne sont pas bien reconstituées 4. Un test simple a priori (eig value) ou a posteriori (variance) permet de vérifier cette propriété qu on souhaite 5. L observabilité (A,C) serait suffisante, mais n est pas nécessaire. Et difficile à vérifier en grande dimension (nx 0 2, 0 3 ) 6. La détectabilité (A,C) + qq autres propriétés sont suffisantes 7. Les covariances des erreurs de modèle Q sont importantes dans ces conditions 8. Le cadre linéaire précédent propose des outils puissants mais a des limites (linéarité, dimension, sensibilité au bruit, etc.) -> 4D-Var pour la suite

34 Surface Water and Ocean Topography (SWOT) mission Scientific requirements Observable river width > 00 m Height accuracy Slope accuracy Width accuracy Data collection 0 cm over area > m².7 cm/m over area > m 5% of the evaluated river 90% of all ocean/continents within the orbit during 90% of the operational time. August 22, 206 hind.oubanas@irstea.fr 34

35 Data assimilation method Variational approach Observations: Y = Y t T + ε o Y, ε o ~N 0, O, R = E ε o ε o Bacground: U b = U t T + ε b U, ε b ~N 0, B, B = E ε b ε b Control vector: U U Classical cost function: Tihonov cost function: Iterative regularization: J U = 2 R 2 G U Y J U, α = 2 R 2 G U Y B 2 U U b 2 + α2 2 B 2 U U b, α > 0, Tihonov regularization parameter J W = 2 R 2 G U b + B 2W, U 0 Y, J W ~χ 2 (M) 2 U = U b + B 2W, U U : Control vector U b U : Bacground Y Y : Observation vector M : Observation space dimension G U Y : Nonlinear mapping operator B : Bacground covariance matrix R: Observation covariance matrix August 22,

36 Data assimilation method Variational approach Limited-memory Broyden Fletcher Goldfarb Shanno (L-BFGS) method : W i+ = W i + β i H i J W i W, W 0 = 0 Gradient of the cost function Adjoint Model J W i W = B 2 G U b + B 2W, U 0 R G U b + B 2W, U 0 Y Automatic differentiation TAPENADE (INRIA) (Gejadze & Malaterre 206) August 22, 206 hind.oubanas@irstea.fr 36

37 Experimental framewor Study area / period Garonne River France Downstream reach : 50 m Mean width : 70m Mean slope : 28cm/m T c ~24h August 22, 206 hind.oubanas@irstea.fr 37

38 Experiment () : Estimation of Q, given K S and Z b Estimation of upstream discharge Q assuming nown the bed level Z b and the friction coefficient K S, investigating the influence of the SWOT temporal frequency. Identical twin experiments framewor Assimilation of water surface elevation Z observations Observations error σ = 0 cm Observations time period: from day to 5 days Observation spatial sampling: each 0 m The first guess on discharge is taen as the mean annual value Bathymetry and friction assumed nown Sequential version (DA sub-window: 75-day period) rrmse(q) = T T 0 Q estimate t Q true t 2 dt /2 August 22, 206 hind.oubanas@irstea.fr 38

39 Experiment () : Estimation of Q, given K S and Z b Discharge hydrograph at Tonneins from 0/0/200 to 3/05/200 (a) -day, (b) 2-day, (c) 4-day, (d) 5-day obervation period (a) (b) (c) (d) Q rrmse 2.% 9.5% 2.9% 8.2% August 22, 206 hind.oubanas@irstea.fr 39

40 Experiment () : Estimation of Q, given K S and Z b Nyquist sampling theorem The sampling frequency should be at least twice the highest frequency contained in the signal. Discharge hydrograph at Tonneins from 0/0/200 to 3/05/200 (a) -day, (b) 2-day, (c) 4-day, (d) 5-day obervation period (a) (b) (c) (d) Q rrmse 2.% 9.5% 2.9% 8.2% August 22, 206 hind.oubanas@irstea.fr 40

41 Experiment (2) : Simultaneous estimation of Q and K S, given exact Z b Estimation of upstream discharge Q under uncertainty in the friction coefficient K S, assuming nown the bed level Z b First guess on the friction coefficient is taen as a 20% error of the mean value Bathymetry assumed nown Sequential version (DA sub-window: 75-day period) August 22, 206 hind.oubanas@irstea.fr 4

42 Experiment (2) : Simultaneous estimation of Q and K S, given exact Z b A-I A-II Q rrmse 2.9% 2.6% K S rrmse 20.4% 3.4% A-I : Estimation of Q solely using the first guess on K S. A-II : Estimation of Q and K S. August 22, 206 hind.oubanas@irstea.fr 42

43 Experiment (2) : Simultaneous estimation of Q and K S, given exact Z b A-I A-II Q rrmse 2.9% 2.6% K S rrmse 20.4% 3.4% A-I : Estimation of Q solely using the first guess on K S. A-II : Estimation of Q and K S. Experiment (3) : Simultaneous estimation of Q and Z b, given exact K S Estimation of upstream discharge Q under uncertainty in the bed level Z b, assuming nown the friction coefficient K S First guess on Bed level is derived from the perturbed steady flow WSE Friction coefficient and cross-sections shape assumed nown Sequential version (DA sub-window: 75-day period) August 22, 206 hind.oubanas@irstea.fr 43

44 Experiment (2) : Simultaneous estimation of Q and K S, given exact Z b A-I A-II Q rrmse 2.9% 2.6% K S rrmse 20.4% 3.4% A-I : Estimation of Q solely using the first guess on K S. A-II : Estimation of Q and K S. Experiment (3) : Simultaneous estimation of Q and Z b, given exact K S B-I B-II Q rrmse 50% 3.8% Z b rrmse 5.7% 4.9% B-I : Estimation of Q solely using the first guess on Z b. B-II : Estimation of Q and Z b. August 22, 206 hind.oubanas@irstea.fr 44

45 Experiment (4) : Simultaneous estimation of Q, K S and Z b Estimation of upstream discharge Q, under uncertainty in the bed level Z b and the friction coefficient K S Identical twin experiments framewor Assimilation of water surface elevation Z observations Observations error σ = 0 cm Observations time period: day (up to 5 days) Observation spatial sampling: each 0 m The first guess on discharge is taen as the mean annual value First guess on the friction coefficient is taen as a 20% error of the mean value First guess on Bed level is derived from the perturbed steady flow WSE Cross-sections shape assumed nown Sequential version (DA sub-window: 75-day period) August 22, 206 hind.oubanas@irstea.fr 45

46 Experiment (4) : Simultaneous estimation of Q, K S and Z b C-I: Estimation of Q solely using the first guess on K S and Z b C-II: Estimation of Q, K S and Z b C-III: Estimation of Q and Z b, using the first guess on K S rrmse C-I C-II C-III Q 40.5% 7.% 5.% K S 3% 24.4% 3% Z b 5.7% 4.7% 4.5% November 2, 206 PhD Irstea/CLS 46

47 Experiment (4) : Simultaneous estimation of Q, K S and Z b C-I: Estimation of Q solely using the first guess on K S and Z b C-II: Estimation of Q, K S and Z b C-III: Estimation of Q and Z b, using the first guess on K S Equifinality issue!! rrmse C-I C-II C-III Q 40.5% 7.% 5. K S 3% 24.4% 3% Z b 5.7% 4.7% 4.5% November 2, 206 PhD Irstea/CLS 47

48 Control set design: notations

49 Control set design: variational DA

50 Control set design: goal-function covariance

51 Control set design: partial control case

52 Control set design: partial control covariance

53 Control set design: implementation

54 Control set design: algorithm

55 Control set design: test configuration

56 Control set design: NA results, case A

57 Control set design: NA results, case B

58 Conclusion 2 (cadre variationnel). Le concept de choix du vecteur de contrôle a été présenté. Ce concept est utile pour les modèles ayant des incertitudes nombreuses, complexes parmi leurs entrées. En particulier, les erreurs de modèle peuvent être incluses dans ces entrées du modèle. 2. La méthode présentée permet de quantifier la performance obtenue pour toute combinaison d entrées actives (sous-partie du vecteur des entrées), permettant de faire apparaitre les sous-vecteurs suffisants. Le choix entre ces options de sousvecteurs suffisants peut ensuite être fait selon des considérations de solvabilité et de robustesse. 3. La méthode est une généralisation de l approche variationnelle classique de Quantification ou Réduction des Incertitudes. Elle n utilise pas de formalisme matriciel et est donc adaptée aux modèles de grande dimension. 4. La méthode a été appliquée dans le domaine de l hydraulique à surface libre et a prouvé son intérêt. Les résultats obtenus illustrent et justifient toutes les conclusions que nous avions tirées de nos tests d AD avec SIC (et son adjoint). 5. La même approche peut être étendue au cas ou les entrées (vecteur de contrôle) sont divisées entre une part active, passive et de nuisance. Des développements peuvent aussi être proposés pour obtenir une performance globale plutôt que locale.

Thans! Questions? Thans for your attention pierre-olivier.

59 Thans! Questions? Thans for your attention

Outils de Recherche Opérationnelle en Génie MTH Astuce de modélisation en Programmation Linéaire

Outils de Recherche Opérationnelle en Génie MTH 8414 Astuce de modélisation en Programmation Linéaire Résumé Les problèmes ne se présentent pas toujours sous une forme qui soit naturellement linéaire.