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Direct (FDTO) Versus Indirect (FOTD) Methods in Real-Life Applications of PDE Optimal Control:

Direct (FDTO) Versus Indirect (FOTD) Methods in Real-Life Applications of PDE Optimal Control: Load Changes for Fuel Cells Hans Josef Pesch jointly with Armin Rund, Kurt Chudej, Johanna Kerler, Kati Sternberg Chair of Mathematics in Engineering Sciences

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Direct (FDTO) Versus Indirect (FOTD) Methods in Real-Life Applications of PDE Optimal Control:

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  1. Direct (FDTO) Versus Indirect (FOTD) Methods in Real-Life Applications of PDE Optimal Control: Load Changes for Fuel Cells Hans Josef Pesch jointly with Armin Rund, Kurt Chudej, Johanna Kerler, Kati Sternberg Chair of Mathematics in Engineering Sciences University of Bayreuth, Bayreuth, Germany hans-josef.pesch@uni-bayreuth.de

  2. 2002-2005 Motivation: Optimal load changes for fuel cell systems Hotmodule [MTU CFC Solutions, IPF Berndt] Molten Carbonate Fuel Cell cell stack

  3. Motivation: Optimal load changes for fuel cell systems [Sundmacher] [Heidebrecht] 2D cross-flow design exhaust air inlet recirculation cathode exhaust anode exhaust catalytic burner CO32- mixer anode solid anode inlet cathode cathode inlet 28 quasi-linear partial integro-differential-algebraic equations with non-standard non-linear boundary conditions

  4. Outline A glimpse on the theory A glimpse on the numerics Direct and indirect solution: MCFC Conclusions

  5. Outline A glimpse on the theory A glimpse on the numerics Direct and indirect solution: MCFC Conclusions

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

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

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

  9. pointwise evaluation adjoint state Description with the adjoint state Necessary conditions Description with the adjoint solution operator

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

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

  12. Outline A glimpse on the theory A glimpse on the numerics Direct and indirect solution: MCFC Conclusions

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

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

  15. 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

  16. Outline A glimpse on the theory A glimpse on the numerics Direct and indirect solution: MCFCModelling aspectsFDTO_2DFDTO_1DFOTD_1D Conclusions

  17. Modelling aspects and the Engineering approach FDTO_2D

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

  19. 0.7 0.6 for a load change cell voltage using controls optimal control simulation 0.4 sec 0 0.1 1.1 11.1 111.1 1111.1 scaled time using control 0.8 sec optimal control simulation 0 0.1 1.1 11.1 111.1 1111.1 scaled time Optimal load changes. Computation by FDTO_2D [Sternberg]

  20. Numerical results: simulation of load change (FDTO_2D) [Chudej, Sternberg] cathode gas temperature anode gas temperature [2.8 ≈ 560 °C] flow directions [3.2 ≈ 680 °C] reforming reactions are endothermic oxidation reaction is exothermic reduction reaction is endothermic

  21. Numerical results: simulation of load change (FDTO_2D) [Chudej, Sternberg] solid temperature [3.2 ≈ 680 °C] [2.8 ≈ 560 °C] flow directions in anode and cathode

  22. fast 0.7 0.6 slow on on Numerical results: optimal control of fast load change (FDTO_2D)while temperature gradients stay small Pareto performance index: instead of state constraint with

  23. Modelling aspects and the Engineering approach FDTO_1D

  24. Reforming reaction CH4 + 2 H2O CO2 + 4 H2 Configuration of MCFC for 1D counter-flow design 1D counter-flow design Anode gas channel Air inlet CH4 H2O CH4 + H2O CO + 3H2 CO + H2O CO2 + H2 O2 N2 H2 +CO32-H2O+CO2+2e- CO+CO32-2CO2+2e- e- Catalytic burner Anode Elektrolyte U CO32- Cathode Mixer Exhaust ½O2 + CO2 + 2e- CO32- Cathode gas channel Recirculation

  25. Oxidation reaction Configuration and function ofMCFC 1D counter-flow design Anode gas channel Air inlet CH4 H2O CH4 + H2O CO + 3H2 CO + H2O CO2 + H2 O2 N2 H2 +CO32-H2O+CO2+2e- CO+CO32-2CO2+2e- e- Catalytic burner Anode Elektrolyte U CO32- Cathode Mixer Exhaust ½O2 + CO2 + 2e- CO32- Cathode gas channel Recirculation

  26. Configuration of MCFC for 1D counter-flow design 1D counter-flow design Anode gas channel Air inlet CH4 H2O CH4 + H2O CO + 3H2 CO + H2O CO2 + H2 O2 N2 H2 +CO32-H2O+CO2+2e- CO+CO32-2CO2+2e- e- Catalytic burner Anode Elektrolyte U CO32- Cathode Mixer Exhaust ½O2 + CO2 + 2e- CO32- Reduction reaction Cathode gas channel Recirculation

  27. Configuration of MCFC for 1D counter-flow design 1D counter-flow design Anode gas channel Air inlet CH4 H2O CH4 + H2O CO + 3H2 CO + H2O CO2 + H2 O2 N2 H2 +CO32-H2O+CO2+2e- CO+CO32-2CO2+2e- e- Catalytic burner Anode Elektrolyte U CO32- Cathode Mixer Exhaust ½O2 + CO2 + 2e- CO32- Cathode gas channel Recirculation

  28. Configuration of MCFC for 1D counter-flow design 1D counter-flow design Fuel gas Anode gas channel Air inlet CH4 H2O CH4 + H2O CO + 3H2 CO + H2O CO2 + H2 O2 N2 H2 +CO32-H2O+CO2+2e- CO+CO32-2CO2+2e- e- Catalytic burner Anode Elektrolyte U CO32- Cathode Mixer Exhaust ½O2 + CO2 + 2e- CO32- Reactant Cathode gas channel Recirculation

  29. only ions can move throughelectrolyte Configuration of MCFC for 1D counter-flow design 1D counter-flow design Anode gas channel Air inlet CH4 H2O CH4 + H2O CO + 3H2 CO + H2O CO2 + H2 O2 N2 H2 +CO32-H2O+CO2+2e- CO+CO32-2CO2+2e- e- Catalytic burner Anode Elektrolyte U CO32- Cathode Mixer German Federal Pollution Control Act: Air Exhaust ½O2 + CO2 + 2e- CO32- Cathode gas channel Recirculation

  30. Configuration of MCFC for 1D counter-flow design 1D counter-flow design 2 Anode gas channel Air inlet CH4 H2O CH4 + H2O CO + 3H2 CO + H2O CO2 + H2 O2 N2 4 H2 +CO32-H2O+CO2+2e- CO+CO32-2CO2+2e- e- Catalytic burner Anode controls Elektrolyte U CO32- Cathode Mixer Exhaust ½O2 + CO2 + 2e- CO32- Cathode gas channel Recirculation 1

  31. Numerical optimization via FDTO_1D (Method 1: semi-discretization) [Rund] Numerical approach (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 fully implicit multistep variable order method ode15s (MATLAB) with simplified Newton method for the non-linear systems and Jacobian by automatic differentiation numerical differentiation of gradient for optimization (Quasi-Newton) • Choice of consistent initial data by computing stationary initial values by a multi-level discretization (from coarse to fine grids) State solver (equivalent to the reduced problem)

  32. Numerical simulation of a load change (state solver) [Rund] zoom

  33. Numerical optimization via FDTO_1D (Method 1: semi-discretization) temperatures anode cathode solid uncontrolled controlled 4 major controls molar fraction H2O cell voltage

  34. The Engineering approach FDTO_1D

  35. Numerical optimization via FDTO_1D (Method 2: full-discretization) Numerical approach (fact sheet) • States: semi-discretization in space (N fixed grid points) upwind formulas to preserve the conservation laws fixed time steps on a logarithmic grid: implicit Euler huge scale index 1 DAE system trajectories are not absolutely continuous functions automatic differentiation of gradient for optimization (AMPL+IPOPT) state constraints • Choice of consistent initial data by computing stationary initial values in the entire space-time cylinder State solver (equivalent to the non-reduced problem)

  36. Numerical optimization via FDTO_1D (Method 2: full-discretization) solid temperature capability of handling state constraints molar flow density molar flow density

  37. How to apply adjoint-based methods on real-life problems? FOTD_1D

  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 input data for load changes cell voltage potentials plus appropriate initial and boundary conditions for all equations

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

  42. Lagrangian parabolic eqs. algebraic and differential- algebraic eqs. hyperbolic eqs. ordinary integr- differential eqs.

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

  44. Numerical solution via FOTD_1D [Rund] 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 • SQP methods are hardly applicable (2nd order information required)

  45. Numerical results via FOTD_1D [Rund] load change after 0.1. sec regularization: 41 lines in space 767 time steps

  46. _ _ _ _ _ _ + + + + + + + . . . . . . . . . . . . ? Conclusion (FDTO-semi / FDTO-full / FOTD) algorithmic efficieny human resources accuracy reliablity miscellaneous application of AD only for implicit solver FDTO-semi application of AD state constraints 1D FDTO-full application of AD state constraints FOTD 2D is a challenge in any case FDTO-semi 2D

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

  48. Thank you for your attention

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