# Investigation of Electrochemical Processes in Solid Oxide Fuel Cells by Modified Levenberg–Marquardt Algorithm: A New Automatic Update Limit Strategy

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory and Computations

#### 2.1. EEC and Objective Function Used in EIS Study

#### 2.2. Optimization Algorithm Used in This Study

**h**,

**J**,

**w**,

**W, I**,

**y**

^{exp},

**y**

^{com}, and $\lambda $ represent the vector of computed increments in EEC parameters, the Jacobian matrix, the vector containing weights (3), the matrix with columns equal to vector of weights

**w**, the identity matrix, the vector of experimental EIS values, the vector of computed EIS values, and the damping parameter.

**a**is the vector of EEC parameters that define the values of

**y**

^{com}.

**h**is added to the parameter values (${a}^{t}$) from the previous iteration (t):

**a**

^{t}is not updated and λ is increased (see Algorithm 1). The change in the λ value allows LMA to balance between the Gauss–Newton method (when near the solution) and the steepest descent (when far from the solution) [23,24]. For a more comprehensive study related to the $\lambda $ update strategy, the reader is encouraged to inspect the following papers [10,22,23,24].

Algorithm 1. Pseudocode of Levenberg–Marquardt algorithm (adapted from [22,24]). Symbol references: J is the Jacobian matrix, J^{T}(W$\circ $J) is the approximate Hessian matrix; I is the identity matrix, J^{T}(w$\circ $ (y^{exp} − y^{com})) is the gradient vector, w is the vector containing weights (3), and W is the matrix with columns equal to weights (3). |

Levenberg–Marquardt algorithm |

a = a_{0}; λ = λ_{0}; ν = 2 |

repeat |

Solve (J^{T}(W∘J) + λI)h = J^{T}(w $\circ $ (y^{exp} − y^{com})) |

if ρ > 0 |

a = a + h |

λ = λ ∗ max(1/3, 1 − (2ρ − 1)^{3}) |

ν = 2 |

else |

λ = λ ∗ ν |

end if |

until |

#### 2.3. Limit Strategy in EIS Study

#### 2.4. Ordinary Limit Strategy for Levenberg–Marquardt Algorithm

_{int,j}, a

_{ext,j}, lb

_{j}, and ub

_{j}are the jth value of the internal parameter, the jth value of the external parameter, and jth values of lower and upper limits, respectively. The limit values are computed by:

^{+5}is the limit update factor (LUF) that is given a priori. Only in the case of the parameter n, the limit values are set to 0.449 and 0.999.

_{ext,j}is converted (7) into a

_{int,j}(Algorithm 2). However, prior to each EEC model evaluation, a

_{int,j}is converted back into a

_{ext,j}. According to (8), a

_{ext,j}can take on only the value from a limit gap which is characterized by lb

_{j}and ub

_{j}. To rephrase it, the EEC parameters can only have values from a specific and limited region. On the other hand, during the iterations, a

_{int,j}can take on any value from the feasible solution space. Knowing this, the limit gap can also be taken into account as, e.g., a trust region, which is a term commonly used in the trust-region methods [9]. Since the size of this limit gap remains fixed during the whole iteration procedure, we refer to this strategy as the ordinary limit strategy (see Table 1).

Algorithm 2. Pseudocode of Levenberg–Marquardt algorithm, which is coupled by the ordinary limit strategy. Symbol references: J is the Jacobian matrix, J^{T}(W∘J) is the approximate Hessian matrix; I is the identity matrix, J^{T}(w$\circ $ (y^{exp} − y^{com})) is the gradient vector, w is the vector containing weights (3), and W is the matrix with columns equal to weights (3). |

Ordinary limit strategy for the Levenberg-Marquardt algorithm |

a_{ext} := a_{0}; λ := λ_{0}; ν := 2 |

a_{int} := k(a_{ext}) (convert to a_{int} by (7)) |

LUF := 1e5 (limits update factor) |

compute limits ((7),(8),(9)) |

repeat |

Solve (J^{T}(W$\circ $J) + λI)h = J^{T}(w $\circ $ (y^{exp} − y^{com})) (only here use l(a_{int}) (8) instead of a_{int}) |

if ρ > 0 |

a_{int} := a_{int} + h |

λ := λ ∗ max(1/3, 1 – (2ρ – 1)^{3}) |

ν := 2 |

else |

λ := λ * ν |

ν := ν * 2 |

end if |

until |

a_{ext} := l(a_{int}) (convert to a_{ext} by (8) ) |

#### 2.5. Automatic Update (i.e., Adaptive) Limit Strategy for Levenberg–Marquardt Algorithm

Algorithm 3. Pseudocode of the Levenberg–Marquardt algorithm, which is coupled by the automatic update limit strategy (see sub procedure). Symbol references: J is the Jacobian matrix, J^{T}(W$\circ $J) is the approximate Hessian matrix; I is the identity matrix, J^{T}(w $\circ $ (y^{exp} − y^{com})) is the gradient vector, w is the vector containing weights (3), and W is the matrix with columns equal to weights (3). | |

Automatic update limit strategy for the Levenberg-Marquardt algorithm | |

a_{ext} := a_{0}; λ := λ_{0}; ν := 2 | |

g := 0; b :=0 (g and b are number of good and bad iterations) | |

LUF := 1e5 (limits update factor) | |

compute limits ((7),(8),(9)) | |

a_{int}: = k(a_{ext}) (convert to a_{int} by (7))) | sub update_limits |

repeat | if g > 2 |

Solve (J^{T}(W$\circ $J) + λI)h = J^{T}(w $\circ $ (y^{exp} − y^{com})) (use l(a_{int}) (8) instead of a_{int}) | LUF := LUF*0.95; 10 ≤ LUF ≤ 10^{4} |

if ρ > 0 (good iteration) | a_{ext} := l(a_{int}) |

a_{int} := a_{int} + h | compute lb_{i} and ub_{i} by a_{ext,i} |

update_limits (using current l(a_{int}) value) | compute a_{int,i}:= k(a_{ext,i}) by new lb_{i}, ub_{i} |

λ := λ ∗ max(1/3, 1 – (2ρ – 1)^{3}) | end if |

ν := 2 | if b > 2 |

g := g+1; b := 0 | LUF := LUF*2; 10 ≤ LUF ≤ 10^{4} |

else (bad iteration) | a_{ext} := l(a_{int}) |

λ := λ * 2 | compute lb_{i} and ub_{i} by a_{ext,i} |

ν := ν * 2 | compute a_{int,i}:= k(a_{ext,i}) by new lb_{i},ub_{i} |

g := 0; b := b+1 | end if |

end if | end sub |

until | |

a := l(a_{int}) (convert to a_{ext} by (8)) |

^{4}in order to allow both a sufficiently wide gap for convergence and to prevent an increase in the numerical inaccuracy. Please note that values 10 and 10

^{4}used herein were selected during the testing, and their choice is not additionally explained.

## 3. Experimental

#### 3.1. Synthetic Noisy ZARC Data Used in This Study

_{k}, and p are resistances, the angular frequency associated with ith data point, the time constant associated with the kth ZARC process, the parameter associated with the τ distribution of associated kth ZARC process, and the number of ZARCs processes. The data were prepared by using 10 data points per decade. The EEC parameters used for ZARC data generation are given in Table 2.

_{ZARC}) data were polluted by noise:

#### 3.2. Experimental SOFC Data Used in This Study

^{2}. The cells were operated with different synthetic fuels (H

_{2}and CO, CO

_{2}, CH

_{4}, H

_{2}O, and N

_{2}) that have the same composition as gasses from biomass gasification. Such complex composition of the fuel results in numerous electrochemical processes, such as hydrogen oxidation, carbon monoxide oxidation, water gas shift reaction, steam and dry reforming, and many others. In order to investigate the processes within the fuel cell, we collected EIS data in the relevant frequency range from 0.1 Hz to 10 kHz.

#### 3.3. EEC Model Used in This Study

_{R(QR)(QR)(QR)}model was applied that is a serial combination of three (k = 3) parallel QR elements (Figure 1). The model is characterized by three time constants, i.e., by the same number of time-constants that describe ZARC processes in this work (Table 2). Next, the EEC

_{R(QR)(QR)(QR)}model in Figure 1 can be represented by:

_{R(QR)(QR)(QR)}parameter values used to start fits in this study are presented in Table 3. Two types of starting parameter values were used herein. First, a good starting guess (chosen close to the optimal values), which is typical in EIS study, especially as the starting values are vital when commencing LMA fit [8,20] and second, a poor starting guess that was taken by design to be far from the solution to show the robustness gained by the application of the automatic limit update strategy.

#### 3.4. Open-Source Packages Used in This Study

## 4. Results and Discussion

#### 4.1. Impact of Diverse Limit Strategies on LMA Convergence Properties When Fitting ZARC Data by Using Good Starting Parameters

_{R(QR)(QR)(QR)}. The ZARC data in Figure 2 show only one depressed semi-circle; and thus, the number of the involved electrochemical processes has to be determined. Therefore, we applied EEC

_{R(QR)(QR)(QR)}(Figure 1) to test limit strategies; however, remember that the ZARC data are computed by using three ZARC elements that simulate three electrochemical processes.

_{s}, R

_{k}, n

_{k}andτ

_{k}values given in Table 2 and Table 4.

^{−1}) after only 36 iterations. Furthermore, when the ordinary (Figure 2e) and adaptive limit (Figure 2f) strategies were applied, the fits were terminated at the same low S-value (1.310 × 10

^{−6}) but, after 80 and 49 iterations. The lower number of iterations (49 vs. 80) clearly implies that the automatic update (vs. standard) limit strategy has superior convergence properties in the characterization of electrochemical processes by EIS.

#### 4.2. Impact of Ordinary and Automatic Update (i.e., Adaptive) Limit Strategies on LMA Convergence Properties When Fitting ZARC Data by Using Poor Starting Parameters

^{−1}). In the equal fitting conditions, the adaptive limit strategy reached a significantly lower S-value (1.310 × 10

^{−6}) after 65 iterations (Figure 3d). The considerably better convergence properties can be explained by the automatic update of the limit gap (Algorithm 3). Finally, the slow reduction in the S-value in the final stage of the fits can be observed in both Figure 3c,d; and thus, it is not governed by the adaptive strategy.

#### 4.3. Impact of Ordinary and Automatic Update (i.e., Adaptive) Limit Strategies on LMA Convergence Properties When Fitting More Corrupted ZARC Data by Using Poor Starting Parameters

^{−6}). According to Table 5, the computed τ values (2.386 × 10

^{−2}, 7.184 × 10

^{−3}, and 1.096 × 10

^{−3}s) correspond well to the ones (0.01, 0.05, and 0.001) used to prepare the corrupted ZARC data (Table 2). This test confirms that the adaptive limit strategy is especially suitable for describing electrochemical processes with closely distributed τ characteristics.

#### 4.4. Automatic Update of LUF Value during LMA Iteration

#### 4.5. Experimental SOFC Impedance Data

_{R(QR)(QR)}with two (k = 2; see (12)) serial QR circuits. A poor starting EEC parameters guess was used to test the worst possible fitting scenario (see Table 6). In this scenario, when using the ordinary limit strategy, LMA failed to yield a good data match (Figure 6a). On the other hand, the application of the adaptive limit strategy resulted in a good fit (Figure 6b).

^{−3}and 1.078 × 10

^{−1}s). The first depressed semi-circle in the high frequency region is defined by a low n = 0.767 value, which implies that several electrochemical processes (with similar τ ) might occur simultaneously. The second semi-circle in the low frequency region yielded n = 0.999 value, which suggests the presence of only one process.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations and Symbols

SOFC | solid oxide fuel cells |

EIS | electrochemical impedance spectroscopy |

EEC | electrical equivalent circuit |

CNLS | complex nonlinear least-square problem |

LMA | Levenberg–Marquardt algorithm |

LUF | limit update factor |

g | number of good successive iterations |

b | number of bad successive iterations |

J | Jacobian matrix |

C | approximated Hessian matrix (i.e.,$\text{}{J}^{T}\left(W\circ J\right)$) |

h | vector containing computed estimates in EEC parameters |

λ | damping parameter |

R: | resistor |

Q | constant phase element (impedance form: ${Z}_{Q}={\left({Y}_{0}{\left(i\omega \right)}^{n}\right)}^{-1})$ |

QR | parallel QR circuit |

Z_{ZARC} | ZARC data |

S | objective function used for EIS data fitting |

m | number of EIS data points |

f | EEC model |

ω | angular frequency |

y^{exp} | vector containing experimental EIS data |

y^{com} | vector containing computed EIS data |

w | vector containing weights (3) |

W | matrix with columns equal to w |

p | number of ZACR elements |

a | vector containing EEC parameters |

r | number of EEC parameters |

a_{j} | jth EEC parameter |

a_{int,j} | jth internal EEC parameter |

a_{ext,j} | jth external EEC parameter |

lb_{j} | jth lower bound |

ub_{j} | jth upper bound |

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**Scheme 1.**Impact of the automatic update limit strategy on the limit gap during the Levenberg–Marquardt algorithm (LMA) fit.

**Figure 1.**EEC

_{R(QR)(RQ)(RQ)}model used to fit ZARC data. The model is comprised by using three serial QR circuits. Symbol reference: R—resistor (Ω cm

^{2}) and Q—Constant Phase Element (S s

^{n}cm

^{−2}).

**Figure 2.**The impedance spectra of the ZARC data (subplots

**a**–

**c**) and the corresponding S-value data (subplots

**d**–

**f**). EEC

_{R(QR)(QR)(QR)}(Figure 1) was used in the fitting attempts.

**Figure 3.**The impedance spectra of the ZARC data (subplots

**a**,

**b**) and the corresponding S-value data (subplots

**c**,

**d**). EEC

_{R(QR)(QR)(QR)}(Figure 1) was used in the fitting attempts.

**Figure 4.**The impedance spectra of the corrupted ZARC data (subplots

**a**,

**b**) and the corresponding S-value data (subplots

**c**,

**d**). EEC

_{R(QR)(QR)(QR)}(Figure 1) was used in the fitting attempts.

**Figure 6.**The impedance spectra of the experimental solid oxide fuel cells data (subplots

**a**,

**b**). EEC

_{R(QR)(QR)}was used in the fitting attempts. Subplot (

**a**) displays a failed fitting attempt, whilst subplot (

**b**) shows a successful fit. Both fits were commenced by using a poor EEC starting guess (Table 6).

Limit Strategy | Automatic Limits Update | Reported in EIS |
---|---|---|

No limit | No | e.g., [8] |

Ordinary | No | [20] |

Automatic update (i.e., adaptive) | Yes | This work |

**Table 2.**EEC parameter values used to compute the synthetic ZARC data (10) in this work. Values in parentheses present τ values used to prepare ZARC data with a more corrupted local landscape.

EEC Parameters | EEC Parameter Values | |||
---|---|---|---|---|

k | ||||

1 | 2 | 3 | ||

R_{s} (Ω cm^{2}) | 10 | - | - | - |

R_{k} (Ω cm^{2}) | - | 50 | 50 | 50 |

τ_{k} (s) | - | 0.01 (0.01) | 0.001 (0.005) | 0.0001 (0.001) |

n_{k} | - | 0.7 | 0.7 | 0.7 |

**Table 3.**The starting values of the EEC

_{R(QR)(QR)(QR)}parameters used to initiate the fits in this work. The values in parentheses represent poor starting values taken by design to be far from the optimal values.

EEC Parameters | EEC Parameter Values | |||
---|---|---|---|---|

k | ||||

1 | 2 | 3 | ||

R_{s} (Ω cm^{2}) | 10 (1.1) | - | - | - |

Y_{0,k} (S s^{n} cm^{−2}) | - | 0.1 (1.2) | 0.01 (1.3) | 0.001 (1.4) |

n_{k} | - | 0.85 (0.85) | 0.83 (0.83) | 0.87 (0.87) |

R_{k} (Ω cm^{2}) | - | 70 (1.5) | 20 (1.6) | 50 (1.7) |

**Table 4.**Final EEC parameter values obtained by EEC

_{R(QR)(QR)(QR)}and both the ordinary and automatic update limit strategies. The values were obtained by fitting the ZARC data (Figure 2).

EEC Parameters | ||||
---|---|---|---|---|

k | 1 | 2 | 3 | |

R_{s} (Ω cm^{2}) | 9.996 | - | - | - |

Y_{0,k} (S s^{n} cm^{−2}) | - | 6.638 × 10^{−4} | 1.346 × 10^{−4} | 3.126 × 10^{−5} |

n_{k} | - | 0.692 | 0.759 | 0.695 |

R_{k} (Ω cm^{2}) | - | 57.49 | 37.60 | 54.90 |

* τ_{k} (s) | 8.920 × 10^{−3} | 9.420 × 10^{−4} | 1.050 × 10^{−4} |

**Table 5.**Final EEC parameter values obtained by using EEC

_{R(QR)(QR)(QR)}and the automatic update limit strategy. The values are obtained by fitting the ZARC data (Figure 4).

EEC Parameters | ||||
---|---|---|---|---|

k | 1 | 2 | 3 | |

R_{s} (Ω cm^{2}) | 9.99 | - | - | - |

Y_{0,k} (S s^{n} cm^{−2}) | - | 9.995 × 10^{−3} | 3.520 × 10^{−4} | 1.500 × 10^{−4} |

n | - | 0.677 | 0.714 | 0.694 |

R (Ω cm^{2}) | - | 7.959 | 83.441 | 58.549 |

^{a}τ_{k} (s) | 2.386 × 10^{−2} | 7.184 × 10^{−3} | 1.096 × 10^{−3} |

^{a}estimated τ values $\left({\left({R}_{k}{Y}_{0,k}\right)}^{\frac{1}{{n}_{k}}}\right)$.

**Table 6.**Final and poor starting (in parentheses) EEC parameter values obtained by using EEC

_{R(QR)(QR)}and the automatic update limit strategy. EEC

_{R(QR)(QR)}was used only for experimental solid oxide fuel cells data fitting. The values are obtained by fitting the experimental data given in Figure 6.

EEC Parameters | |||
---|---|---|---|

k | |||

1 | 2 | ||

R_{s} (Ω cm^{2}) | 2.29 × 10^{−3}(1.10) | - | - |

Y_{0,k} (S s^{n} cm^{−2}) | - | 1.038 (0.01) | 14.445 (5.00) |

n | - | 0.767 (0.63) | 0.999 (0.73) |

R (Ω cm^{2}) | - | 6.427 × 10^{−3} (0.50) | 7.484 × 10^{−3} (2.50) |

^{a}τ_{κ} (s) | 1.459 × 10^{−3} | 1.078 × 10^{−1} |

^{a}estimated τ values $\left({\left({R}_{k}{Y}_{0,k}\right)}^{\frac{1}{{n}_{k}}}\right)$.

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**MDPI and ACS Style**

Žic, M.; Fajfar, I.; Subotić, V.; Pereverzyev, S.; Kunaver, M.
Investigation of Electrochemical Processes in Solid Oxide Fuel Cells by Modified Levenberg–Marquardt Algorithm: A New Automatic Update Limit Strategy. *Processes* **2021**, *9*, 108.
https://doi.org/10.3390/pr9010108

**AMA Style**

Žic M, Fajfar I, Subotić V, Pereverzyev S, Kunaver M.
Investigation of Electrochemical Processes in Solid Oxide Fuel Cells by Modified Levenberg–Marquardt Algorithm: A New Automatic Update Limit Strategy. *Processes*. 2021; 9(1):108.
https://doi.org/10.3390/pr9010108

**Chicago/Turabian Style**

Žic, Mark, Iztok Fajfar, Vanja Subotić, Sergei Pereverzyev, and Matevž Kunaver.
2021. "Investigation of Electrochemical Processes in Solid Oxide Fuel Cells by Modified Levenberg–Marquardt Algorithm: A New Automatic Update Limit Strategy" *Processes* 9, no. 1: 108.
https://doi.org/10.3390/pr9010108