INVERSION OF ELECTROCHEMICAL IMPEDANCE SPECTROSCOPIC MEASUREMENTS

Transcription

1 BIOFUEL CELL POLARIZATION ESTIMATION INVERSION OF ELECTROCHEMICAL IMPEDANCE SPECTROSCOPIC MEASUREMENTS IMPORTANCE OF MODEL FORMULATION Rosemary Renaut, Mathematics, ASU Jakob Hansen, Jarom Hogue and Grant Sander, Math Undergraduates ASU Sudeep Popat Biodesign Institute ASU MARCH 7, 24: MBI

2 Outline Application: Electrochemical Spectroscopy for Biofuels Mathematical Model Nonlinear Fitting Model Modification Integration by s Linear Fitting Numerical Quadrature Integration by t Right Preconditioner Regularization Simulations Nonnegative Least Squares Simulations Conclusions

3 Physical Experiment: Microbial Electrolysis Cell for anaerobic bacteria (a) Fuel cell (b) Geobacter (c) Microbial electrolysis cell Figure: Geobacter, fuel cells, and waste water: Single-chamber microbial electrolysis cell consisting of a carbon fiber anode with a stainless steel current collector and a stainless steel cathode is used to convert food wastes into hydrogen in a single step.

4 (b) Potential Loss inside ARB cell ARB in fuel cell should be electrically self-sufficient (a) Cyclic Voltammograms (CVs) Figure: (a) CVs of an ARB biofilm during its growth phase ( d). Nernst-Monod relationship compared against the CVs (dotted lines). (b) CV explaining the potential profile for an ARB biofilm. The Nernst-Monod relationship (red dotted line) shows that most potential losses occur inside the ARB s cell.

5 Electrochemical Impedance Spectroscopy Figure: Schematic of the microbial electrochemical cells used for Electrochemical Impedance Spectroscopy (EIS)

6 Nyquist Plot of EIS impedance measurements Figure: Nyquist plot and fitting by equivalent circuit model (ECM) for an ARB biofilm tested at -.35 vs Ag/AgCl.

7 Goals Given EIS measurements can we find. The underlying resistance 2. The equivalent circuit model 3. Number of resistances / processes 4. Chemical environment that minimizes ARB potential loss

8 Mathematical Model Fredholm Integral Equation Small input current and measured output voltage leads to Impedance measurements Z(ω) for angular frequency ω Z(ω) = R + R pol g(t) dt, () + i ωt Kernel h(ω, t) = +i ωt - is not square integrable. R and R pol are unknown parameters. Distribution function of relaxation times (DRT) g(t) g(t)dt = and g(t) > is unknown

9 The DRT g(t): Model Simple Resistance - Capacitance Circuit Cole-Cole or ZARC element. (literature dependent) g (t) = 2πt The impedance is analytically given sin βπ ( ( )) cosh β ln t t + cos βπ Z (ω) = < β < and distribution of time constants t., (2) + (iωt ) β, (3) Constant Phase element Q and resistance R

10 The DRT g(t) : Model Simple network of resistance - capacitance circuits May also be realized by a lognormal distribution for the DRT ) g (t) = ( tσ 2π exp (ln(t) µ)2 2σ 2. (4) No analytic form for the impedance is given. Three lognormals and a weighted linear combination: given yellow can we find black blue and green

11 Comparison of and for one process chosen to align (a) Nyquist Z(ω) (b) Real Z (c) Imag Z 2 Simulated exact data measured at 65 logarithmically spaced points in ω. In each case the solid line indicates the functions and the symbols the functions Goal: Given the data find DRT: g(t) 2

12 Nonlinear Fitting to Z(ω) is known fitting to Z is possible Either use the analytic form for Z or integrate the known DRT What error is incurred by a wrong choice of model? All residuals are small, but for constant for incorrect model and increasing for correct model Mean Fit by Upper 95% bound Lower 95% bound Mean Fit by Upper 95% bound Lower 95% bound 6 8 Mean Fit by Upper 95% bound Lower 95% bound Mean Fit by Upper 95% bound Lower 95% bound (d) Fitting to the (e) Fitting to the Residual norms fitting one process to the impedance spectrum of a DRT consisting of one process with white noise.

13 The impact (f) by 2 (g) by Fitting functions with the consistent mean values obtained over 5 noisy selections.. True β =.72, t =., true σ =.83, t =.. to gives β =.86 and t =.2 to gives σ = and t =.3 Location and height of peak is incorrect Small residual does not imply good fitting

14 Transformation s = ln(t) Z(ω) = h(ω, t)g(t) dt = h(ω, e s )f(s) ds, For the DRTs (2)-(4) we obtain the functions f(s) := tg(t). f (s) = 2π f (s) = σ 2π exp sin(βπ) (cosh(β(s ln(t ))) + cos(βπ)) (s µ)2 ( 2σ 2 (5) ), ln(t ) = µ σ 2, (6) fit fit (h) DRTs: s space 5 5 (i) Fitting by 5 5 (j) Fitting by Location of peak is identified in s space

15 Observations Solve in t space the location of a peak (process) is incorrect. Solve in s space the location of a peak (process) is correct. BUT : Nonlinear fitting requires an assumption of the model You may not know that you have picked the wrong model. The residual is small.

16 Linear Formulation: Integration in t Z(ω) = Z (ω) + iz 2 (ω) ) ( = (R + R pol g(t) ) + ω 2 t 2 dt ir pol ωtg(t) + ω 2 t 2 dt, ) ( ) = (R + R pol h (ω, t)g(t) dt ir pol h 2 (ω, t)g(t) dt. Given N quadrature points and [T min, T max ], Tmax T min g(t)h k (ω, t) dt t For s-space constant step in s smax s min h k (ω, e s )f(s) ds s N n= N n= g(t n )h k (ω, t n ) (7) h k (ω, e sn )f(s n ). (8)

17 Reduced Quadrature Error by s space quadrature h (s) h (s) 2 h (t) h 2 (t) 6 h (s) h 2 (s) h (t) h 2 (t) (k) t =., β = (l) σ = ln(3). Figure: Quadrature error for a single (left) (right) process with N = 65 for quadrature in s and t as a function of ω, plotted on a log-log scale.

18 Discrete Systems and Impact on Conditioning Matrices (A k ) mn = a n h k (ω m, t n ) m M, n N. Measurements (b ) m Z (ω m ) R and (b 2 ) m Z 2 (ω m ) (x) n g(t n ) satisfies the discrete linear systems A k x b k, k =,..., 3, A 3 = [A ; A 2 ] (9) Definining finer resolution also yields square A 4 = [A e, Ae 2 ] Quad Equation A A 2 A 3 A 4 t (7).5e + 3.4e e e + 2 s (8) 2.9e e e e + 8 Comparing condition number of matrices with different quadratures for optimal selection of nodes for t n = /ω n. Transformation acts as a right preconditioner on the system. Use matrix A 3 if sufficient data for resolution of the solution.

19 Systems are Ill-conditioned- the discrete problem is ill-posed [Dataset Rpol R] = [ ] x 5 A x 5 A2 x 4 A3 3 A4 x Figure: An example solution for real data provide no useful information- the solution is sensitive to noise in the data and the floating point arithmetic calculations

20 Regularization The improved matrices are still ill-conditioned and regularization is required. x = arg min{ Ax b 2 + λ 2 Lx 2 } () Tikhonov regularization controls the smoothness of the solution λ is a regularization parameter, many techniques exist to find λ - some require statistical properties of the data measurements L controls the smoothness - standard choices I or D or D 2 - derivative operators. Given λ an explicit solution is given with respect to the basis vectors, and solves the normal equations (A T A + λ 2 L T L)x = A T x

21 Simulations: Generate Data to test approach: solid line, (a) A: g(t) (b) tg(t) (c) Nyquist (d) Z (ω) (e) Z 2(ω) (f) B: g(t) (g) tg(t) (h) Nyquist (i) Z (ω) (j) Z 2(ω) (k) C: g(t) (l) tg(t) (m) Nyquist (n) Z (ω) (o) Z 2(ω)

22 Solutions by LLS Different estimation of parameter λ: Minimum error red, LC blue and NCP green (p) L = I (q) L = L (r) L = L (s) L = I (t) L = L (u) L = L 2 Figure: Mean error and example LLS solutions..% noise, -A data set, matrix A 4.

23 Solutions by LLS: Minimum error red, LC blue and NCP green (a) L = I (b) L = L (c) L = L (d) L = I (e) L = L (f) L = L 2 Figure: Mean error and example LLS solutions..% noise, -A data set, matrix A 4.

24 Observations: Low NOISE. Nonnegativity of g(t) not maintained. 2. Boundary effects are observed 3. Important to impose additional constraints.

25 Nonnegative LS with regularization The DRT satisfies g(t) >. The transformed DRT, f(s) = tg(t) > also. Hence nonnegative least squares: x = arg min{ Ax b 2 + λ 2 Lx 2, s.t. x }, () { ( ) } Ax b 2 = arg min, s.t. x (2) λlx 2 Depends on regularization parameter λ Can be solved using a standard solver - Matlab lsqnonneg Impose bounds on solution x >. Parameter choice - solve for multiple λ and pick solution as for the LS problem.

26 Solutions by NNLS: Minimum error red, LC blue and NCP green (a) L = I (b) L = L (c) L = L (d) L = I (e) L = L (f) L = L 2 Figure: Mean error and example NNLS solutions..% noise, -A data set, matrix A 4.

27 Solutions by NNLS: Minimum error red, LC blue and NCP green (a) L = I (b) L = L (c) L = L (d) L = I (e) L = L (f) L = L 2 Figure: Mean error and example NNLS solutions..% noise, -A data set, matrix A 4.

28 Observations. Nonnegativity of g(t) is maintained. 2. Boundary effects better preserved 3. Additional constraints helpful 4. Parameter choice extends easily to the NNLS case for small scale problem 5. Information on noise level would be helpful - allow other methods UPRE, χ 2 etc

29 Conclusions Small Scale problem - standard parameter choice can be used for regularization parameter estimation in NNLS The model formulation is important for both nonlinear and linear fitting (s space instead of t space) Transformation is equivalent to a right preconditioning The ability to recognize more multiple processes in presence of noise is challenging

30 Acknowledgements Authors Hansen, Hogue and Sander were supported by NSF CSUMS grant DMS 73587: CSUMS: Undergraduate Research Experiences for Computational Math Sciences Majors at ASU. Renaut was supported by NSF MCTP grant DMS 4877: MCTP: Mathematics Mentoring Partnership Between Arizona State University and the Maricopa County Community College District, NSF grant DMS 2655: Novel Numerical Approximation Techniques for Non-Standard Sampling Regimes, and AFOSR grant 2577 Development and Analysis of Non-Classical Numerical Approximation Methods. Popat was supported by ONR grant N42344: Characterizing electron transport resistances from anode-respiring bacteria using electrochemical techniques. Professor César Torres for discussions concerning the EIS modeling.