yoed tsur

French-Israeli Workshop on Renewable Energies

Solid oxide fuel cells research Solid oxide fuel cells research

using impedance spectroscopy using impedance spectroscopy Yoed Tsur, Shany Hershkovitz,

and Sioma Baltianski

Department of Chemical Engineering

Technion- Israel institute of Technology, 32000

Haifa, Israel

November 2010, Tel Aviv


Outline• Appetizer• SOFCs• Impedance spectroscopy• Genetic programming

– A very short introduction– Application to our problem

• ISGP



Why are we addicted to fossil fuels?


X one week(~50 work hours)=

(Concept adapted from David Cahen to fit my own family)


How Fuel Cells can help?-High efficiency

-Low ‘local’ pollutants

•4November 2010, Tel Aviv

To the grid


Fuel Cells:Stacks and complexity

•5November 2010, Tel Aviv

Fuel in


A reminder of linear IS

SignalSupply

Z

V

If

( )( ) where 2( )

,

VZ Z iZ fI

and Z Z comply with KK relations


French-Israeli Workshop on Renewable Energies November 2010, Tel Aviv


1 2 3 4

3 1 2 4 2 2 1

22 1

( 1, , , ) ( 2, , , ) if:; (1 / )

and 2 1(1 / )

Z Z R R Z Z R RR R R R R R R

Z Z R R

1

0 1

1( ) a iZb b i



Distribution of time constants

A system with a finite number of time constants can be put into the following form:

01 1

( ) ( ) where 11

n nk

kk kk

gZ Z R gi

Distribution of time constants is the extension: n → ∞. Then we have:

00 0

( )( ) ( ) where ( ) 11g dZ Z R g di



Or in log scaleTaking an arbitrary reference frequency, and

defining:

We get:

0 ( ) ( )

( ) ( )( ) ( )10 10

( , ) ( )

L L L L

L d LZ L Z L R

K L L L

00

log ; log ; ( ) ( )L L L g

0( )( ) ( )

1 10 L L

L dLZ L Z L Ri

And in particular:



Equivalent circuits and DFRT

• DFRT=distribution function of relaxation times. (We also call it “the function”).

• In many cases one can find the DFRT from a given equivalent circuit.– An equivalent circuit like this:has a DFRT of two delta functions.– In most real cases: “distributed elements”

should be used -> DFRT with peaks.• Equivalent circuits are not unique.



Discrepancy-complexity plot

[Baltianski and Tsur, J. Electroceramis 2003]We take the discrepancy between the prediction of the model and the data (e.g., ) on a log scale vs. the model’s complexity (# of adjustable parameters). Each point represents a solution. Look for the “knee” in this plot.

2



Discrepancy-complexity plot

[Baltianski and Tsur, J. Electroceramis 2003]We take the discrepancy between the prediction of the model and the data (e.g., ) on a log scale vs. the model’s complexity (# of adjustable parameters). Each point represents a solution. Look for the “knee” in this plot.

2

Surprisingly amount of info can be inferred from that!

-4

-2

0

0 2 4 6 8 10



Avoid Over-fitting• This is one of the most common

mistakes. – It could be done ad absurdum: data set

of n point can be fitted with no discrepancy by a function with n free parameters.

• Our remedy: we use two data sets and give “merit” according to:

1 21f f f

4

3

2

1

0 x

y

1 2 3 4 5 6

4

3

2

1

0 x

y

1 2 3 4 5 6



Outline• Appetizer• SOFCs• Impedance spectroscopy• Genetic programming

– A very short introduction– Application to our problem

• ISGP



A D CA CA B

How Genetic Programming works

•Mutation •Permutation•Crossover

A B A C

Population Population

11


22


doublesdoubles



Adopting to our case• Pre-knowledge:

– We have decided to look for linear combinations of known peak functions only.

– Simple, no need to check feasibility.• Additional input from the user

– “Expected” and “too high” complexity– What type of peaks to include– Population size and total number of

generations– Normalization, etc.



The main loop:• A generation contains N offsprings.• They “breed” and the population is now 2N

– The population is doubled using a pre-determined reservoir of genes.

– Add a peak/ change a peak/ eliminate a peak• The program finds parameters for each new

offspring, and gives it a figure of merit• The best N-1 offsprings plus a randomly

selected one survive, and become the next generation.

• Plotting discrepancy-complexity, Nyquist &



Let’s see this again



The adaptive pressure

Is achieved by the figure of merit:Compatibility (between 0 and 1) with 2 sets timesPenalty for complexity timesPenalty for not being properly normalized.

0

0 expect

1 2 2

5( )1 exp

0.8 0.2 1 exp (1 )bC n

C

f f

C

f



IS measurements of a system contained a IS measurements of a system contained a MIEC and electrodes in air at the MIEC and electrodes in air at the temperature range of 500-600 °C temperature range of 500-600 °C



Same sample at 550 °C and varying Same sample at 550 °C and varying oxygen partial pressureoxygen partial pressure



Summary• Fuel Cells should be a part of any energy

portfolio• IS – enessential tool to improve them• Discrepancy-complexity plot• The ISGP free program

[electroceramics.technion.ac.il] – Inherently avoid most of the common mistakes that

you can find in literature (over-fitting; what is a “good fit”; “generating” information)

– Can be used both for exploration (new problems) and routinely for systems with a known DFRT shape

– Can also solve Fredholm equations of the 2nd kind


yoed tsur

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