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

International Symposium on Mathematical ProgrammingAugust 20-24, 2012, Berlin, Germany

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

Load Changes for Fuel Cells

Hans Josef Peschjointly with

Armin Rund, Kurt Chudej, Johanna Kerler, Kati Sternberg

Chair of Mathematics in Engineering SciencesUniversity of Bayreuth, Bayreuth, Germany

[email protected]


Motivation: Optimal load changes for fuel cell systems

Molten Carbonate Fuel Cell

cellstack

Hotmodule[MTU CFC Solutions, IPF Berndt]

2002-2005


catalyticburner

mixeranode

anode inlet

cathode inlet

anode exhaust

cathodeexhaust

exhaust air inletrecirculation

cathode

2D cross-flow design

CO32-

solid

Motivation: Optimal load changes for fuel cell systems

28 quasi-linear partial integro-differential-algebraic equations with non-standard non-linear boundary conditions

[Sundmacher][Heidebrecht]


Outline

A glimpse on the theory

A glimpse on the numerics

Direct and indirect solution: MCFC

Conclusions


Outline




Conclusions


subject to

An example: Optimal stationary temperature distribution

Elliptic optimal control problemwith distributed control

Necessary conditions


with linear and continuous solution operatorsubject to

An example: Optimal stationary temperature distribution

Elliptic optimal control problemwith distributed control

Optimization problem in Hilbert space


Necessary condition: variational inequality

bilinear form linear form


Necessary condition: variational inequality

Optimization problem in Hilbert space


adjoint operator


Description with the adjoint solution operator


Description with the adjoint state

adjoint state pointwiseevaluation


The formal Lagrange technique

Defining the Lagrange function and twice formal integration by parts

Differentiation in the direction of , resp.


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


Outline




Conclusions


Methods for PDE constrained optimization

The general problem

The aims

concepts for real-life application

small constanteffort of simulation

effort of optimization


capture as much structure

of ( P )as possibleon discrete

level( Ph )

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 ?

appropriate ansatz for adjoint variables and multipliers?

Solvelarge scale

NLP

Solvecoupled PDE

system


First Discretize then Optimize vs. First Optimize then Discretize

First discretize then optimze (FDTO):replace all quantities of the infinite dimensional optimization problemby 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 discretizedoptimality system

In general

Ideal: discrete concept for which both approaches commuteDiscontinuous Galerkin methods


Outline




Modelling aspects

FDTO_2D

FDTO_1D

FOTD_1D

Conclusions


Modelling aspectsand

the Engineering approachFDTO_2D


catalyticburner

mixeranode

solid

anode inlet

cathode inlet

anode exhaust

cathodeexhaust

exhaust air inletrecirculation

cathode

Configuration and function of MCFC

2D cross-flow design

controllable

controllable

controllable

load changesinput

boundary conditionsby ODAE

slow

statevariable

fastvery fastalgebraic

[Heidebrecht] [Sundmacher]


0 0.1 1.1 11.1 111.1 1111.1

using control

optimal controlsimulation

0.8 sec

0 0.1 1.1 11.1 111.1 1111.1scaled time

using controls

optimal controlsimulation

0.4 sec

scaled time

cell voltage 0.7 0.6for a load change

[Sternberg]

Optimal load changes. Computation by FDTO_2D


anode gas temperature cathode gas temperature

[2.8 ≈ 560 °C]

[3.2 ≈ 680 °C]

Numerical results: simulation of load change (FDTO_2D)

reforming reactions are endothermicoxidation reaction is exothermic

reduction reaction is endothermic

flow directions

[Chudej, Sternberg]


[2.8 ≈ 560 °C]

[3.2 ≈ 680 °C]

solid temperature

Numerical results: simulation of load change (FDTO_2D)

flow directionsin anode

and cathode

[Chudej, Sternberg]


with

Pareto performance index:

Numerical results: optimal control of fast load change (FDTO_2D)while temperature gradients stay small

fast

slow

on

on

0.7 0.6

instead of state constraint


Modelling aspectsand

the Engineering approachFDTO_1D


O2

N2

U

e-

CO32-

H2 + CO32- H2O + CO2 + 2e-

CO + CO32- 2CO2 + 2e-

½O2 + CO2 + 2e- CO32-

CH4 + H2O CO + 3H2

CO + H2O CO2 + H2

CH4

H2O

Cathode

Anode

Elektrolyte

Mixer

Catalyticburner

Anode gas channel

Cathode gas channel

Recirculation

Exhaust

1D counter-flow design

Air inlet

Reforming reactionCH4 + 2 H2O CO2 + 4 H2

Configuration of MCFC for 1D counter-flow design


O2

N2

U

e-

CO32-

H2 + CO32- H2O + CO2 + 2e-

CO + CO32- 2CO2 + 2e-

½O2 + CO2 + 2e- CO32-

CH4 + H2O CO + 3H2

CO + H2O CO2 + H2

Cathode

Elektrolyte

CH4

H2O

Anode

Anode gas channel

Cathode gas channel

Configuration and function of MCFC

Recirculation

Exhaust

Mixer

Catalyticburner

Oxidation reaction


Air inlet


U

e-

CO32-

O2

N2

½O2 + CO2 + 2e- CO32-

CH4 + H2O CO + 3H2

CO + H2O CO2 + H2H2 + CO3

2- H2O + CO2 + 2e-

CO + CO32- 2CO2 + 2e-

CH4

H2O

Cathode

Anode

Elektrolyte

Mixer

Catalyticburner

Anode gas channel

Cathode gas channel

Recirculation

Exhaust

Reduction reaction


Air inlet



U

e-

O2

N2

½O2 + CO2 + 2e- CO32-

CH4 + H2O CO + 3H2


2- H2O + CO2 + 2e-

CO + CO32- 2CO2 + 2e-

CH4

H2O

Cathode

Anode

Elektrolyte

Mixer

Catalyticburner

Anode gas channel

Cathode gas channel

Recirculation

Exhaust


Air inlet

CO32-



U

e-

O2

N2

½O2 + CO2 + 2e- CO32-

CH4 + H2O CO + 3H2


2- H2O + CO2 + 2e-

CO + CO32- 2CO2 + 2e-

CH4

H2O

Cathode

Anode

Elektrolyte

Mixer

Catalyticburner

Anode gas channel

Cathode gas channel

Recirculation

Exhaust

Reactant


Air inlet

Fuel gas

CO32-



CO32-

O2

N2

½O2 + CO2 + 2e- CO32-

CH4 + H2O CO + 3H2


2- H2O + CO2 + 2e-

CO + CO32- 2CO2 + 2e-

U

e-

Recirculation

Exhaust

CH4

H2O

Cathode

Anode

Elektrolyte

Mixer

Catalyticburner

Anode gas channel

Cathode gas channel

only ions can movethrough electrolyte

German Federal Pollution Control Act: Air


Air inlet




CO32-

O2

N2

½O2 + CO2 + 2e- CO32-

CH4 + H2O CO + 3H2


2- H2O + CO2 + 2e-

CO + CO32- 2CO2 + 2e-

U

e-

Recirculation

Exhaust

CH4

H2O

Cathode

Anode

Elektrolyte

Mixer

Catalyticburner

Anode gas channel

Cathode gas channel

Air inlet


controls

4

2

1


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)

Numerical optimization via FDTO_1D (Method 1: semi-discretization)

[Rund]


Numerical simulation of a load change (state solver)

zoom

[Rund]


Numerical optimization via FDTO_1D (Method 1: semi-discretization)

anode cathode solid

temperatures

molar fraction H2O 4 major controls cell voltage

uncontrolled

controlled


The Engineering approachFDTO_1D


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)


Numerical optimization via FDTO_1D (Method 2: full-discretization)

molar flow density

molar flow density

solid temperature

capability of handlingstate constraints


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

FOTD_1D


The equations: gas channels and solid

molar fractions

gas temperature

solid temperature

molar flow densities


The equations: burner and mixer

The catalytic burneris fed by the anode and cathode outlet

The mixer is described by a system of ODAE


The equations: potential fields

currents current densities

cell voltage

potentials

input datafor load changes

plus appropriate initial and boundary conditions for all equations


• 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 sides Coupled staggered system of variational inequalities to determine optimal control laws no projection formulae, but gradient of objective function (for gradient or Newton method)

Necessary conditions (fact sheet)

[Rund]


Lagrangianhy

perb

olic

eqs

.

para

bolic

eqs

.

alge

brai

c an

d

diff

eren

tial-

alge

brai

c eq

s.

ordi

nary

inte

gr-

diff

eren

tial e

qs.


Necessary conditions (summary)

Adjoint state solver

control

control

BC

OUT

OUTcontrol

control

BC

BC

BC

OUT

AE

PDE

PDE

PDE

PDE

PDE

PDE OUT

AE

DAE

DAE

anode

anode

cathode

cathode

burner

burner

mixer

mixerbyvariationalinequalities


Numerical solution via FOTD_1D

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)

[Rund]


Numerical results via FOTD_1D

[Rund]

load change after 0.1. sec

regularization:

41 lines in space

767 time steps


Conclusion (FDTO-semi / FDTO-full / FOTD)

FDTO-semi

FDTO-full

FOTD

hu

man

re

sou

rces

alg

ori

thm

ic

effi

cien

y

reli

abli

ty

accu

racy

1D

application of ADonly for implicit solver

2D FDTO-semi 2D is a challengein any case

application of ADstate constraints

application of AD state constraints

miscellaneous

_

. .

. . +

+

+ +

+

+

+

_ _

_

_

_

?

. .

. . . .

. .


The Fuel Cell Team

Prof. Kai Sundmacher Dr.-Ing.h.c. Joachim Berndt Dr.-Ing. Peter Heidebrecht

Prof. Kurt Chudej Dr. Kati Sternberg

Prof. Michael Mangold†

Dr. Armin Rund Johanna Kerler


Thank you for your attention

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

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