An Introduction to Optimal Control of Partial Differential Equations with Real-life Applications

Transcription

1 An Introduction to Optimal Control of Partial Differential Equations with Real-life Applications Hans Josef Pesch Chair of Mathematics in Engineering Sciences University of Bayreuth, Bayreuth, Germany

2 Motivation: Optimal placement of laser beams to avoid hot cracking multi-beam welding [Karkhin, Ploshikin] solidification mushy zone weld pool additional beams weld seam hot crack main laser beam Semi-infinite optimization problem for an elliptic PDE with state constraints compression

3 opening displacement Motivation: Optimal placement of laser beams to avoid hot cracking [Petzet] liquidus and solidus isotherms surrounding the mushy zone hot crack criterium limit zoom weld pool region

4 Motivation: Minimum fuel transcontinental flights at hypersonic speeds Europe - USA in 2 hrs / Europe - Australia in 4.5 hrs quasilinear 2 box constraints heat equation non-linear 1 control-state boundary constraint conditions coupled 1 state constraint with ODE PDE ODE

5 Motivation: Minimum fuel transcontinental flights at hypersonic speeds [Wächter, Chudej, LeBras] velocity [m/s] altitude [10,000 m] flight path angle [deg] [s] temperature [K] temperature [K] temperature [K] limit temperature 1000 K on a boundary arc 1st layer 2nd layer 3rd layer [s]

6 Motivation: Optimal load changes for fuel cell systems Hotmodule [CFC Solutions, IPF Berndt] Molten Carbonate Fuel Cell cell stack

7 Motivation: Optimal load changes for fuel cell systems exhaust 2D cross-flow design recirculation [Sundmacher] [Heidebrecht] air inlet cathode exhaust anode exhaust catalytic burner CO 3 2- mixer anode inlet cathode inlet 28 quasi-linear partial integro-differential-algebraic equations with non-standard non-linear boundary conditions

8 Outline A glimpse on the theory A glimpse on the numerics An application: MCFC Conclusions

9 Outline A glimpse on the theory A glimpse on the numerics An application: MCFC Conclusions

10 A simple elliptic optimal control problems Lions (since 1970s), Casas (1987-), Tröltzsch (1980-) An example: optimal stationary temperature distribution set of admissible controls subject to tracking functional Tikhonov regularization Elliptic optimal control problem with distributed control

11 A simple elliptic optimal control problems An example: optimal stationary temperature distribution set of admissible controls subject to tracking functional Tikhonov regularization Elliptic optimal control problem with boundary control

12 Necessary conditions An example: Optimal stationary temperature distribution subject to Elliptic optimal control problem with distributed control

13 Necessary conditions An Optimization example: Optimal problem stationary in Hilbert space temperature distribution subject with linear to and continuous solution operator Necessary condition: variational inequality bilinear form linear form Elliptic optimal control problem with distributed control

14 Necessary conditions Optimization problem in Hilbert space Necessary condition: variational inequality adjoint operator

15 Necessary conditions Description with the adjoint solution operator adjoint state pointwise evaluation Description with the adjoint state

16 The formal Lagrange technique Defining the Lagrange function and twice formal integration by parts Differentiation in the direction of, resp.

17 Optimality system: semi-linear elliptic, distributed + boundary control

18 Challenges in PDE constrained optimization Functional Analysis Partial Differential Equations Optimization in Banach spaces Numerical Methods of Linear Algebra Optimal Control of PDE Parallel Numerical Methods Numerics of PDE Numerical Methods of Optimization High Performance Scientific Computing

19 Outline A glimpse on the theory A glimpse on the numerics An application Conclusions

20 Methods for PDE constrained optimization The general problem The aims effort of optimization effort of simulation small constant concepts for real-life application

21 First Discretize then Optimize vs. First Optimize then Discretize First discretize then optimize (FDTO) or DIRECT First optimize then discretize (FOTD) or INDIRECT Questions appropriate choice of and ansatz for? appropriate choice of and ansatz for? Solve large capture scale as NLP much structure of ( P ) as possible on Solve discrete coupled level PDE system ( P h ) appropriate ansatz for adjoint variables and multipliers?

22 First Discretize then Optimize vs. First Optimize then Discretize First discretize then optimze (FDTO): replace all quantities of the infinite dimensional optimization problem by finite dimensional substitutes and solve an NLP First optimze then discretize (FOTD): Derive optimality conditions of the infinite dimensional system, discretize the optimality system and find solution of the discretized optimality system In general Ideal: discrete concept for which both approaches commute Discontinuous Galerkin methods

23 Outline A glimpse on the theory A glimpse on the numerics An application optimal control of a molten carbonate fuel cell process control via model reduction techniques Conclusions

24 Configuration and function of MCFC slow fast very fast algebraic state variable exhaust cathode exhaust load changes input 2D cross-flow design recirculation anode exhaust controllable [Heidebrecht] [Sundmacher] catalytic burner air inlet controllable mixer controllable anode inlet cathode inlet boundary conditions by ODAE

25 Optimal load changes. Computation by FDTO cell voltage for a load change using controls [Sternberg] 0.4 sec scaled time using control optimal control simulation 0.8 sec optimal control simulation scaled time

26 Numerical results: simulation of load change (FDTO) [Chudej, Sternberg] anode gas temperature cathode gas temperature [ C] [ C] flow directions reforming reactions are endothermic oxidation reaction is exothermic reduction reaction is endothermic

27 Numerical results: simulation of load change (FDTO) solid temperature [Chudej, Sternberg] [ C] [ C] flow directions in anode and cathode

28 Numerical results: simulation of load change (FDTO) solid temperature [ C] [ C] state constraint would be desirable

29 Numerical results: optimal control of fast load change (FDTO) while temperature gradients stay small Pareto performance index: fast with slow instead of state constraint on on

30 How to apply adjoint-based methods on real-life problems?

31 Configuration of MCFC for 1D counter-flow design 1D counter-flow design CH H 2 O CO H 2 Anode gas channel Reforming reaction Air inlet CH 4 H 2 O Exhaust CH 4 + H 2 O CO + 3H 2 CO + H 2 O CO 2 + H 2 H 2 + CO 2-3 H 2 O + CO 2 + 2e - CO + CO 2-3 2CO 2 + 2e - Anode Elektrolyte CO 3 2- Cathode ½O 2 + CO 2 + 2e - CO 2-3 U e - Mixer O 2 N 2 Catalytic burner Cathode gas channel Recirculation

32 Configuration and function of MCFC 1D counter-flow design Oxidation reaction CH 4 H 2 O Anode gas channel CH 4 + H 2 O CO + 3H 2 CO + H 2 O CO 2 + H 2 H 2 + CO 2-3 H 2 O + CO 2 + 2e - Air inlet O 2 N 2 CO + CO 2-3 2CO 2 + 2e - Anode Elektrolyte CO 3 2- U e - Catalytic burner Cathode Mixer Exhaust ½O 2 + CO 2 + 2e - CO 3 2- Cathode gas channel Recirculation

33 Configuration of MCFC for 1D counter-flow design 1D counter-flow design CH 4 H 2 O Anode gas channel CH 4 + H 2 O CO + 3H 2 CO + H 2 O CO 2 + H 2 H 2 + CO 2-3 H 2 O + CO 2 + 2e - Air inlet O 2 N 2 CO + CO 2-3 2CO 2 + 2e - Anode Elektrolyte CO 3 2- U e - Catalytic burner Cathode Mixer Exhaust ½O 2 + CO 2 + 2e - CO 3 2- Cathode gas channel Reduction reaction Recirculation

34 Configuration of MCFC for 1D counter-flow design 1D counter-flow design CH 4 H 2 O Anode gas channel CH 4 + H 2 O CO + 3H 2 CO + H 2 O CO 2 + H 2 H 2 + CO 2-3 H 2 O + CO 2 + 2e - Air inlet O 2 N 2 CO + CO 2-3 2CO 2 + 2e - Anode Elektrolyte CO 3 2- U e - Catalytic burner Cathode Mixer Exhaust ½O 2 + CO 2 + 2e - CO 3 2- Cathode gas channel Recirculation

35 Configuration of MCFC for 1D counter-flow design 1D counter-flow design Fuel gas CH 4 H 2 O Anode gas channel CH 4 + H 2 O CO + 3H 2 CO + H 2 O CO 2 + H 2 H 2 + CO 2-3 H 2 O + CO 2 + 2e - Air inlet O 2 N 2 CO + CO 2-3 2CO 2 + 2e - Anode Elektrolyte CO 3 2- U e - Catalytic burner Cathode Mixer Exhaust ½O 2 + CO 2 + 2e - CO 3 2- Reactant Cathode gas channel Recirculation

36 Configuration of MCFC for 1D counter-flow design 1D counter-flow design Anode gas channel Air inlet CH 4 H 2 O CH 4 + H 2 O CO + 3H 2 CO + H 2 O CO 2 + H 2 H 2 + CO 2-3 H 2 O + CO 2 + 2e - O 2 N 2 only ions can move through electrolyte CO + CO 2-3 2CO 2 + 2e - Anode Elektrolyte CO 3 2- U e - Catalytic burner German Federal Pollution Exhaust Control Act: Air Cathode ½O 2 + CO 2 + 2e - CO 2-3 Mixer Cathode gas channel Recirculation

37 Configuration of MCFC for 1D counter-flow design 1D counter-flow design 2 Anode gas channel Air inlet 4 CH 4 H 2 O CH 4 + H 2 O CO + 3H 2 CO + H 2 O CO 2 + H 2 H 2 + CO 2-3 H 2 O + CO 2 + 2e - O 2 N 2 CO + CO 2-3 2CO 2 + 2e - Anode e - Catalytic burner controls Elektrolyte CO 3 2- U Cathode Mixer Exhaust ½O 2 + CO 2 + 2e - CO 3 2- Cathode gas channel Recirculation 1

38 The equations: gas channels and solid molar fractions gas temperature molar flow densities solid temperature

39 The equations: burner and mixer The catalytic burner is fed by the anode and cathode outlet The mixer is described by a system of ODAE

40 The equations: potential fields currents current densities cell voltage input data for load changes potentials plus appropriate initial and boundary conditions for all equations

41 Numerical solution via first optimize then discretize (1st part) Numerical methodology for forward solver (fact sheet) States: smooth in space direction, but high gradients in time: semi-discretization in space (N fixed grid points) upwind formulas to preserve the conservation laws adaptive time steps large scale index 1 DAE system of dimension 14N 6 fully implicit multistep variable order method ode15s (MATLAB) with simplified Newton method for the non-linear systems and Jacobian by numerical or automatic differentiation Choice of consistent initial data by computing stationary initial values by a multi-level discretization (from coarse to fine grids) State solver [Rund]

42 Numerical simulation of a load change [Rund] zoom

43 Optimal control problem (fact sheet) Choice of controls: Initial values enter the mixing chamber obey box constraints Choice of objective functional: Minimize L 2 distance to target temperature (reduce overheating) Minimize L 2 value of temperature gradient (reduce thermal stress) Minimize L 2 distance to target cell current (fast load change) Minimize control costs (regularization) Pareto optimal control problem Assumption on existence of optimal solution (because of non-linearity)

44 Necessary conditions (fact sheet) Assumption on existence of multipliers of sufficient regularity formal Lagrange technique (67 multipliers) [Rund] Derivation of directional derivatives partial integration, differentiation, separate contribution of objective, symbolic or automatic differentiation of source terms Variational argument structure of adjoint system (type of PDE/ODE/DAE preserved) with reverse time partial derivatives of states in source terms due to quasilinearity ODEs with spatial integrals in their r.h.s. Coupled staggered system of variational inequalities to determine optimal control laws no projection formulae, but gradient of objective function (for gradient or Newton method)

45 Lagrangian

46 Necessary conditions (summary) control control BC anode PDE PDE OUT AE burner by variational inequalities OUT cathode PDE BC DAE mixer control control OUT anode PDE PDE BC AE burner BC cathode PDE OUT DAE mixer Adjoint state solver

47 Numerical solution via first optimize then discretize (2nd part) Numerical methodology for optimization (fact sheet) Backward sweep method: staggered solution of optimality system efficient for many time steps (different time scales) good initial guesses for non-linear solver drawback: inferior convergence properties Choice of iterative method: Quasi-Newton (use gradient) superlinear convergence no second derivatives [Rund] SQP methods are hardly applicable (2nd order information required)

48 Numerical results first optimize then discretize [Rund] load change after 0.1. sec regularization: 41 lines in space 767 time steps

49 Aim for process control How to apply optimal solutions in practise?

50 Aim for process control measurable: cell voltage, gas temperatures and concentrations at anode and cathode outlet? diserable for process control: information on spatial temperatur and concentration profiles solution ansatz: observer / state estimator Problem: complexity of model?? Remedy: model reduction technique

51 Model reduction by POD: idea Complete model: Ansatz (separation of variables): Method of weighted residuals: orthogonal snapshots Reduced model: low order model: ODAE of index 1

52 Model reduction by POD: computation of snapshots by the complete model test signal orthogonalization by singular value decomposition 1. temperature basis function 2. temperature basis function

53 Model reduction by POD: comparison of reduced vs. complete model [Mangold, Sheng] random variation of cell voltage #eqs vs sec vs. 82 sec 2 < N < 10 perfect coincidence with reference model appropriate for process control

54 Scheme for state estimator for discrete measurements input MCFC? sensors y process state measurement observer correction + - ŷ Simulator observer MCFC model sensor models

55 References Focus on Theory: Tröltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods, and Applications AMS, Graduate Studies in Mathematics, Vol. 112, Focus on Methods: Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints Mathematical Modelling: Theorie and Applications, Vol. 23, Borzi, A., Schulz, V.: Computational Optimization of Systems Governed by Partial Differential Equations SIAM, Philadelphia, Focus on Applications: See my homepage: google: Hans Josef Pesch

56 The Fuel Cell Team Prof. Kai Sundmacher Dr.-Ing.h.c. Joachim Berndt Dr.-Ing. Peter Heidebrecht Prof. Michael Mangold Prof. Kurt Chudej Dr. Kati Sternberg Dr. Armin Rund

57 Conclusions Concerning theory: already well developed Concerning numerics: still improving Concerning applications: has to be intensified one always abuts against limits

58 Thank you for your attention