Yoed TSUR

Solid oxide fuel cells research using impedance spectroscopy Yoed Tsur, Shany Hershkovitz, and Sioma Baltianski Department of Chemical Engineering Technion- Israel institute of Technology, 32000 Haifa, Israel French-Israeli Workshop on Renewable Energies

November 2010, Tel Aviv

Outline • Appetizer • SOFCs • Impedance spectroscopy • Genetic programming – A very short introduction – Application to our problem

• ISGP

French-Israeli Workshop on Renewable Energies

November 2010, Tel Aviv

Why are we addicted to fossil fuels?

one week (~50 work hours)= X

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

French-Israeli Workshop on Renewable Energies

November 2010, Tel Aviv

How Fuel Cells can help?

-High efficiency -Low ‘local’ pollutants

To the grid

French-Israeli Workshop on Renewable Energies

November 2010, Tel Aviv

•4

Fuel Cells: Stacks and complexity Fuel in

French-Israeli Workshop on Renewable Energies

November 2010, Tel Aviv

•5

A reminder of linear IS

f Signal Supply

I

Z

V

V (ω ) Z (ω ) = = Z ′ + iZ ′′ where ω = 2π f I (ω ) and Z ′, Z ′′ comply with KK relations French-Israeli Workshop on Renewable Energies

November 2010, Tel Aviv

French-Israeli Workshop on Renewable Energies

November 2010, Tel Aviv

Z ( Z 1, R1 , R2 , ω ) = Z ( Z 2, R3 , R4 , ω ) if: R3 = R1 + R2 ; R4 = R2 (1 + R2 / R1 ) and Z 2 = Z1(1 + R2 / R1 )

2

1 + a1iω Z (ω ) = b0 + b1iω French-Israeli Workshop on Renewable Energies

November 2010, Tel Aviv

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

gk Z (ω ) = Z 0 (ω ) + R ∑ where k =1 1 + iωτ k

n

∑g k =1

k

=1

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

g (τ )dτ Z (ω ) = Z 0 (ω ) + R ∫ where 1 + iωτ 0 French-Israeli Workshop on Renewable Energies

∞

∫ g (τ )dτ = 1 0

November 2010, Tel Aviv

Or in log scale Taking an arbitrary reference frequency, and defining:

ω Lω ≡ log ; Lτ ≡ log ω0τ ; γ ( Lτ ) ≡ τ g (τ ) ω0

We get:

And in particular:

∞

γ ( Lτ )dLτ Z ( Lω ) = Z 0 ( Lω ) + R ∫ ( Lω + Lτ ) −∞ 1 + i10 ∞

γ ( Lτ )d ( Lτ ) − Z ′′( Lω ) + Z 0′′( Lω ) = R ∫ − ( Lτ + Lω ) ( Lτ + Lω ) 10 + 10 −∞ ≡ K ( Lτ , Lω )γ ( Lτ )

French-Israeli Workshop on Renewable Energies

November 2010, Tel Aviv

Equivalent circuits and DFRT • DFRT=distribution function of relaxation times. (We also call it “the γ 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.

French-Israeli Workshop on Renewable Energies

November 2010, Tel Aviv

Discrepancy-complexity plot [Baltianski and Tsur, J. Electroceramis 2003] We take the discrepancy between the prediction of 2 χ 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.

French-Israeli Workshop on Renewable Energies

November 2010, Tel Aviv

Discrepancy-complexity plot [Baltianski and Tsur, J. Electroceramis 2003] 0 We take the discrepancy between the prediction of 2 χ the-2model and the data (e.g., ) on a log scale vs. the model’s complexity (# of adjustable parameters).

Each -4 point represents a solution. Look for the “knee” in this0plot.

2 4 6 8 10

Surprisingly amount of info can be inferred from that!

French-Israeli Workshop on Renewable Energies

November 2010, Tel Aviv

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: f = α f1 + ( 1 − α ) f 2 y

y

4

4

3

3

2

2

1

1

0

1

2

3

4

5

6

x

0

1

2

French-Israeli Workshop on Renewable Energies

3

4

5

6

x

November 2010, Tel Aviv

Outline • Appetizer • SOFCs • Impedance spectroscopy • Genetic programming – A very short introduction – Application to our problem

• ISGP

French-Israeli Workshop on Renewable Energies

November 2010, Tel Aviv

How Genetic Programming works Population 1

A+C

A+ B Mutation Population

A

•

Crossover

A+B

•

A+C

Permutation

•

D +C

doubles

Population 2

French-Israeli Workshop on Renewable Energies

November 2010, Tel Aviv

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.

French-Israeli Workshop on Renewable Energies

November 2010, Tel Aviv

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 & γ French-Israeli Workshop on Renewable Energies

November 2010, Tel Aviv

Let’s see this again

French-Israeli Workshop on Renewable Energies

November 2010, Tel Aviv

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

(

(

0.8 f1 + 0.2 f 2 2 f = × 1 − δ + δ exp −(1 − ∫ γ ) −5(C0 − nb ) 1 + exp C0 − Cexpect French-Israeli Workshop on Renewable Energies

November 2010, Tel Aviv

))

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

French-Israeli Workshop on Renewable Energies

November 2010, Tel Aviv

Same sample at 550 째C and varying oxygen partial pressure

French-Israeli Workshop on Renewable Energies

November 2010, Tel Aviv

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

French-Israeli Workshop on Renewable Energies

November 2010, Tel Aviv