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
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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
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French-Israeli Workshop on Renewable Energies
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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