Parameter Identification of Solid Oxide Fuel Cell Using Elman Neural Network and Dynamic Fitness Distance Balance-Manta Ray Foraging Optimization Algorithm

Abstract

An accurate solid oxide fuel cell model is a prerequisite for optimizing the operation and state estimation of subsequent cell systems. Hence, this work aimed to utilize a vigoroso algorithmic tool, i.e., Elman neural network, for data prediction to enrich cell measurement data and employ the trained network model for noise reduction of voltage–current data. Furthermore, to obtain reliable cell parameters, a novel parameter identification model based on the dynamic fitness distance balance-manta ray foraging optimization (dFDB-MRFO) algorithm is proposed. Two datasets were applied to extract the electrochemical model and simple electrochemical model parameters of the solid oxide fuel cell model. To verify adequately the superiority of this method, which is compared with another seven conventional heuristic algorithms, four performance indicators were selected as evaluation criteria. Comprehensive case studies demonstrated that through data processing, the precision and robustness of identification could be effectively heightened. In general, the model fitting data obtained via parameter identification using dFDB-MRFO have excellent fitting precision contrast with the measured voltage–current data. Notably, the fitting degree obtained by dFDB-MRFO in the simple electrochemical model reached 99.95% and 99.91% under the two datasets, respectively.

1. Introduction

The widespread consumption and continuous rise in prices of fossil fuels accelerate resource depletion and trigger severe energy crises and environmental pollution problems [1,2]. These two challenges are jointly driving humanity to accelerate the search for other clean, renewable, and low-cost new energy sources [3,4]. So far, many renewable energy sources have been developed and utilized to a certain extent [5,6,7], such as solar energy, wind energy, wave energy, and hydrogen energy. Hydrogen energy is currently one of the cleanest, most effective, and most promising renewable energy sources. Particularly, as an outstanding energy storage carrier in the power grid system, can effectively solve the inherent intermittency and volatility problems of renewable energy [8]. The solid oxide fuel cell (SOFC) is an outstanding representative of numerous hydrogen energy applications; it is an electrochemical device that directly converts the chemical energy of fuel into electrical energy [9,10,11], in which H₂ is generally used as fuel, while electrical energy, H₂O, and heat are used as the output of the internal power generation reaction in the cell. In addition, SOFC also has advantages such as high-power conversion efficiency, low pollution, and high waste heat utilization rate [12,13], which give it bright development prospects and application value.

It is worth noting that dependable SOFC models are crucial for achieving optimal control of cells and improving cell performance. Currently, models for SOFC mainly include the electrochemical model (ECM), simple electrochemical model (SECM), steady-state model, and dynamic model. Therefore, ECM and SECM have been widely used due to their simple structure and high accuracy [14]. However, due to the nonlinearity, strong coupling, and multivariable nature of SOFC, it is not easy for analytical methods and traditional optimization algorithms to identify the optimal cell model parameters, making it difficult to achieve accurate cell modeling [15]. Heuristic algorithms can effectively solve these problems, using random search for iterative optimization, which can quickly and accurately solve the problem. An SOFC parameter identification model based on a cooperation search algorithm was proposed by Islam et al. [16], which studied the parameter identification of the steady-state and dynamic models. Wang et al. [17] proposed an identification method based on a modified version of gray wolf optimization (MGWO). By conducting a parameter identification study under various temperature and pressure conditions, the results obtained by MGWO were compared with the other four algorithms, and the results showed that the error of MGWO was minimal in all cases. A novel fractional order dragonfly algorithm model was designed by Guo et al. [18], which uses pressure and temperature as variables to study the impact on the final parameter identification results. Yousri et al. [19] proposed a parameter identification model based on a comprehensive learning dynamic multi-swarm marine predator algorithm, which utilizes dynamic multi-group and comprehensive learning strategies to improve the exploration and development process of the marine predator algorithm. In addition, it is applied to the steady-state model under different temperature conditions. The experimental results show that it has the characteristics of fast convergence speed and high optimization accuracy. Fathy et al. [20] used a political optimizer to study parameter identification of steady-state and dynamic models of SOFC. The sum of mean square errors between experimental voltage and model-calculated voltage data was used as the objective function, and the result showed that the political optimizer had better robustness and accuracy. An SOFC parameter identification model based on artificial ecosystem-based optimization (AEO) and manta ray foraging optimization (MRFO) was proposed by Chen et al. [21], which utilizes a fusion mechanism to significantly improve the algorithm’s global search ability. At the same time, the impact of temperature and pressure changes on parameter identification results was also studied. The results showed that the designed new model can achieve fast, accurate, and stable parameter identification. The methodologies failed to incorporate the ramifications of the quantity and quality of experimental data on the identification outcomes, instead confining their investigation to parameter identification within the confines of a solitary experimental dataset. This study provides a new direction for SOFC parameter identification, fully considering the methods and the impact of experimental data on parameter identification results. Overall, the contributions and innovations of this study can be summarized as follows:

• Two SOFC models were established, which were studied for parameter identification. Two experimental datasets of SOFC were collected and organized
• Elman neural network (ENN) was applied to process two datasets, namely data prediction and data denoising, which were applied to the parameter identification of the two models
• The fitness and distance balance criteria applied to the MRFO algorithm was used to design an SOFC parameter identification model based on the dynamic fitness distance balance-manta ray foraging optimization (dFDB-MRFO) algorithm, and compared with seven other typical heuristic algorithms under four performance indicators
• The performance of dFDB-MRFO was proved via comprehensive comparison with seven heuristic algorithms; it has higher accuracy, better robustness, and faster convergence speed. In addition, it was also proven that through data prediction and noise reduction processing, identification accuracy could be effectively improved as well as the stability of the identification results.

2. SOFC Modelling

In this section, the principle of SOFC and two SOFC models are introduced.

2.1. Mathematical Model

The electrochemical reaction principle inside SOFC is shown in Figure 1, where the anode and cathode chemical reaction mechanisms of the cell are as follows [22]:

Anode

$$2{\mathrm{H}}_2+2{\mathrm{O}}^{2-}\to 2{\mathrm{H}}_2\mathrm{O}+4{\mathrm{e}}^{-}$$

(1)

Cathode

$${\mathrm{O}}_2+4{\mathrm{e}}^{-}\to 2{\mathrm{O}}^{2-}$$

(2)

Here, two types of SOFC models were mainly studied, namely ECM and SECM. SECM is a simplified model of ECM, and the mathematical expressions and parameters to be identified for both models are summarized in

Table 1. Models of SOFC.

Models	Mathematical Equations	Identification Parameters
ECM [23]	$V={N_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}(E}_{\mathrm{o}}-V_{\mathrm{a}\mathrm{c}\mathrm{t}}-V_{\mathrm{o}\mathrm{h}\mathrm{m}}-V_{\mathrm{c}\mathrm{o}\mathrm{n}})$ $={N_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}(E}_{\mathrm{o}}-A{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}}^{-1}\left({\displaystyle \frac{I}{{2I}_{0,\mathrm{a}}}}\right)-A{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}}^{-1}\left({\displaystyle \frac{I}{{2I}_{0,\mathrm{c}}}}\right)$ $+B·\mathrm{l}\mathrm{n}\left(1-{\displaystyle \frac{I}{I_{\mathrm{L}}}}\right)-I·R_{\mathrm{o}\mathrm{h}\mathrm{m}})$	$E_{\mathrm{o}}$, $A$, $R_{\mathrm{o}\mathrm{h}\mathrm{m}}$, $B$, $I_{0,\mathrm{a}}$, $I_{0,\mathrm{c}}$, and $I_{\mathrm{L}}$
SECM [24]	$V={N_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}(E}_{\mathrm{o}}-V_{\mathrm{a}\mathrm{c}t}-V_{\mathrm{o}\mathrm{h}\mathrm{m}}-V_{\mathrm{c}\mathrm{o}\mathrm{n}})$ $={N_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}(E}_{\mathrm{o}}-{A\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}}^{-1}\left({\displaystyle \frac{I}{2I_0}}\right)+B·\mathrm{l}\mathrm{n}(1-{\displaystyle \frac{I}{I_{\mathrm{L}}}})-I·R_{\mathrm{o}\mathrm{h}\mathrm{m}})$	$E_{\mathrm{o}}$, $A$, $R_{\mathrm{o}\mathrm{h}\mathrm{m}}$, $B$, $I_0$, and $I_{\mathrm{L}}$

. Therein, $N_{\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}}$ represents the number of single cell in series in the stack cell; $E_{\mathrm{o}}$ represents the open circuit voltage of the cell, V; $V_{\mathrm{a}\mathrm{c}\mathrm{t}}$ represents the active polarization voltage loss of the cell, V; $V_{\mathrm{o}\mathrm{h}\mathrm{m}}$ represents ohmic voltage loss, V; $V_{\mathrm{c}\mathrm{o}\mathrm{n}}$ represents concentration polarization voltage loss, V; $I_0$ represents the cell exchange current density, A.

In addition, activation voltage loss ${\mathrm{V}}_{\mathrm{a}\mathrm{c}\mathrm{t}}$ can be formulated as below [25,26]:

$${\mathrm{V}}_{\mathrm{act}}=\left\{\begin{array}{c}\mathrm{A}{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}}^{-1}\left({\displaystyle \frac{\mathrm{I}}{{2\mathrm{I}}_{0,\mathrm{a}}}}\right)+\mathrm{A}{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}}^{-1}\left({\displaystyle \frac{\mathrm{I}}{{2\mathrm{I}}_{0,\mathrm{c}}}}\right), for ECM\\ {\mathrm{A}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{h}}^{-1}\left({\displaystyle \frac{\mathrm{I}}{2{\mathrm{I}}_0}}\right), for SECM\end{array}\right\}$$

(3)

where $\mathrm{A}$ represents the slope of Tafel curve, mV/decade; $\mathrm{I}$ represents the load current density, mA/cm²; ${\mathrm{I}}_{0,\mathrm{a}}$ represents the anode exchange current density, mA/cm²; ${\mathrm{I}}_{0,\mathrm{c}}$ represents the cathode exchange current density, mA/cm²; ${\mathrm{I}}_0$ is defined as exchange current density for SECM. Moreover, ohmic voltage loss can be described by

$$V_{\mathrm{ohm}}=I·R_{\mathrm{o}\mathrm{h}\mathrm{m}}$$

(4)

where ${\mathrm{R}}_{\mathrm{o}\mathrm{h}\mathrm{m}}$ represents the equivalent resistance inside the cell, kΩ·cm².

Finally, concentration voltage loss ${\mathrm{V}}_{\mathrm{c}\mathrm{o}\mathrm{n}}$ can be calculated by

$$V_{\mathrm{con}}=-B·\mathrm{l}\mathrm{n}(1-{\displaystyle \frac{I}{I_{\mathrm{L}}}})$$

(5)

where $B$ represents a constant; $I_{\mathrm{L}}$ represents the ultimate current density, mA/cm².

2.2. Objective Function

To achieve accurate extraction of cell parameters, it is necessary to minimize the error between fitting data and actual data. Therefore, this work used root mean square error (RMSE), mean absolute error (MAE), and mean absolute percentage error (MAPE) [27] as fitness functions for SOFC cell parameter identification as follows:

$${\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}}_{\mathrm{m}\mathrm{i}\mathrm{n}}(x)=\sqrt{{\displaystyle \frac{1}{M}}{\sum }_{m=1}^M{\left({V_{\mathrm{f},m}-V}_{\mathrm{a},m}\right)}^2}$$

(6)

$${\mathrm{M}\mathrm{A}\mathrm{E}}_{\mathrm{m}\mathrm{i}\mathrm{n}}\left(x\right)={\displaystyle \frac{1}{M}}{\sum }_{m=1}^M\left|{V_{\mathrm{f},m}-V}_{\mathrm{a},m}\right|$$

(7)

$${\mathrm{M}\mathrm{A}\mathrm{P}\mathrm{E}}_{\mathrm{m}\mathrm{i}\mathrm{n}}\left(x\right)={\displaystyle \frac{100\%}{M}}{\sum }_{m=1}^M\left|{\displaystyle \frac{{V_{\mathrm{f},m}-V}_{\mathrm{a},m}}{V_{\mathrm{a},m}}}\right|$$

(8)

$$\left\{\begin{array}{l}x=\left[E_{\mathrm{o}}, A, R_{\mathrm{o}\mathrm{h}\mathrm{m}}, B, I_{0,\mathrm{a}}, I_{0,\mathrm{c}}, I_{\mathrm{L}}\right],ECM\\ x={[E}_{\mathrm{o}}, A, R_{\mathrm{o}\mathrm{h}\mathrm{m}}, B, I_0, I_{\mathrm{L}}],SECM\end{array}\right.$$

(9)

where $x$ represents the variable to be identified; $K$ is the actual current and voltage data size; $V_{\mathrm{f}, m}$ means the $m$th output voltage in the fitted data; $V_{\mathrm{a}, m}$ indicates the $m$th output voltage in the actual data. Therefore, $V_{\mathrm{f}, m}$ can be calculated based on the mathematical equation of the cell model in Table 1.

3. ENN for Data Process

This section discusses the principle of FNN and the data preprocessing via FNN.

3.1. Principle of ENN

ENN is a special type of artificial neural network that contains an input layer, a hidden layer, and an output layer, as well as a feedback layer, which is its main difference from traditional feedforward neural networks. In ENN, the neurons in the hidden layer receive not only signals from the input layer but also feedback signals from the state of the hidden layer at the previous moment. The structure of ENN [28] is shown in Figure 2. It mainly achieves self-connection through the memory of the feedback layer through the hidden layer, greatly improving the dynamic and time-varying characteristics of the network [29].

The nonlinear state space model of ENN is as follows [30,31,32]:

$$\left\{\begin{array}{l}{y}_k=g(w_3·x_k)\\ x_k=f(w_1·x_{c,k}+w_2·u_{k-1})\\ x_{c,k}=x_{k-1}\end{array}\right.$$

(10)

where $y_k$ represents the output layer vector; $x_k$ represents the hidden layer vector; $u_k$ represents the input layer vector; $x_{c,k}$ represents the feedback layer vector; $w_1$ represents the connection weight value from the feedback layer to the hidden layer; $w_2$ represents the connection weight value from the input layer to the hidden layer; $w_3$ represents the connection weight value from the hidden layer to the output layer; $g(\ast )$ represents the transfer function of the output layer; $f(\ast )$ represents the transfer function of the hidden layer.

3.2. ENN for V-I Data Preprocessing

The ENN algorithm is based on the nearest neighbor (NN) classification method, with its central tenet being the computation of nearest neighbor samples for each instance within the undersampled category, followed by a determination of whether to retain the current instance based on the category of these nearest neighbors. First, for each sample in the dataset, its k nearest neighbors are calculated. Subsequently, based on the category of these nearest neighbor data points, a decision is made regarding the retention of the current data point. Finally, through these steps, the ENN algorithm eliminates instances that are discordant with their neighboring instances, resulting in a refined and purer dataset.

3.2.1. ENN for V-I Data Prediction

Parameter identification heavily relies on the output V-I data of SOFC, while newly manufactured cells often have low parameter identification accuracy and modeling accuracy due to insufficient V-I data provided by the manufacturers, which affects the subsequent study on cells. Moreover, in addition to the lack of data, the direct loss of data and the incompleteness of records will also significantly affect the accuracy of parameter identification. Therefore, this study proposed the use of ENN for predicting and processing current and voltage data to increase the amount of data used and improve the accuracy and reliability of parameter identification.

3.2.2. ENN for V-I Data Denoising

In addition, the measured SOFC current and voltage data will be affected by factors such as operating conditions and environmental conditions during actual operation, resulting in many abnormal values in the measured V-I data, seriously reducing the accuracy of SOFC parameter identification. Therefore, this work studied the parameter identification results under noisy data and utilized ENN to achieve data denoising processing.

4. dFDB-MRFO Based SOFC Parameter Identification

In this section, the parameter identification method is established.

4.1. dFDB-MRFO Algorithm

dFDB-MRFO improves the foraging search method of manta ray based on the MRFO algorithm [33]. The algorithm process is as below.

4.1.1. Population Initialization

In the algorithm, the variable of the problem is the position of the manta ray in space, and the position matrix is the following [34]:

$$P={\left[\begin{array}{ccc}{P}_1^1& \cdots & P_1^d\\ ⋮& \ddots & ⋮\\ P_N^1& \cdots & P_N^d\end{array}\right]}_{N\times d}$$

(11)

where $N$ denotes the number of individual manta rays; $d$ represents the individual dimension of manta rays, i.e., the number of variables; $P_N^d$ indicates the position of the $N$th manta ray individual in the d-dimension.

The fitness of candidate solutions for manta ray individuals is the following:

$$F={\left[\begin{array}{c}{F}_1\\ ⋮\\ F_N\end{array}\right]}_{N\times 1}$$

(12)

where $F_N$ represents the fitness value of the $N$th manta ray individual.

4.1.2. Foraging Search

There are three main foraging behaviors of manta ray individuals, including chain foraging, spiral foraging, and rolling foraging, as shown in Figure 3.

dFDB Strategy

dFDB solves from candidate solutions based on the total score, the fitness value, and the distance of manta ray individuals which together serve as the total score of the candidate solutions [35]. The mathematical model is described below.

Suppose that the $j$th solution is $P_j$, the optimal candidate solution selected based on fitness can be represented as $P_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}$, then the distance between candidate solutions selected based on dFDB strategy is calculated using Euclidean distance, as follows:

$$\left\{\begin{array}{l}{P}_j=[P_1^j,P_2^j,\cdots ,P_N^j]\\ P_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}=[P_{1\left(\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}\right)},P_{2\left(\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}\right)},\cdots ,P_{N\left(\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}\right)}]\\ D_{P_j}=\sqrt{{(P_1^j-P_{1\left(\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}\right)})}^2+{(P_2^j-P_{2\left(\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}\right)})}^2+{{(P}_N^j-P_{N\left(\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}\right)})}^2}\\ j=\mathrm{1,2},\cdots ,d\end{array}\right.$$

(13)

where $j$ represents the number of dimensions; $P_{N\left(\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}\right)}$ represents the position of the $N$th best candidate solution for manta ray based on fitness criteria.

In addition, it is first necessary to normalize the fitness and distance, and then perform weighted summation to obtain the final optimal solution selection criteria, as follows:

$$\left\{\begin{array}{l}{S}_{P_j}=w·\mathrm{norm}\left(F_j\right)+(1-w)·\mathrm{norm}\left(D_{P_j}\right)\\ j=\mathrm{1,2},\cdots ,d\end{array}\right.$$

(14)

where $S_{P_j}$ is the total score using dFDB as the standard; $w$ represents the weight of fitness and distance in the total score; $F_j$ represents the fitness value of the jth solution; $D_{P_j}$ denotes the Euclidean distance calculated from the $j$th solution.

In dFDB strategy, the selection of $w$ is the following:

$$w={\displaystyle \frac{t}{T}}·\left(1-lb\right)+lb$$

(15)

where $t$ is the current number of iterations; $T$ represents the maximum number of iterations; $lb$ represents a lower limit value of $w$, with a range of $0<lb<0.5$.

Chain Foraging

During the chain foraging, the manta ray population arranges into a chain like predatory chain. In this case, the mathematical model for updating the individual position of manta rays is as follows [36]:

$$P_i^d(t+1)=\left\{\begin{array}{l}{P}_i^d\left(t\right)+r·\left(P_{\mathrm{d}\mathrm{F}\mathrm{D}\mathrm{B}}^d(t)-P_i^d\left(t\right)\right)+\alpha ·\left(P_{\mathrm{d}\mathrm{F}\mathrm{D}\mathrm{B}}^d(t)-P_i^d\left(t\right)\right),i=1\\ P_i^d\left(t\right)+r·\left(P_{i-1}^d(t)-P_i^d\left(t\right)\right)+\alpha ·\left(P_{\mathrm{d}\mathrm{F}\mathrm{D}\mathrm{B}}^d(t)-P_i^d\left(t\right)\right),i=2,\cdots ,N\end{array}\right.$$

(16)

$$\alpha =2r·\sqrt{\left|\mathrm{l}\mathrm{o}\mathrm{g}\left(r\right)\right|}$$

(17)

where $P_i^d\left(t\right)$ represents the position of the $i$th individual in the $d$-dimension of the manta ray population during the $t$th iteration; $P_{\mathrm{d}\mathrm{F}\mathrm{D}\mathrm{B}}^d(t)$ means the position of the optimal candidate individual in the $t$th iteration of the improved algorithm for the $d$-dimension of manta ray population; $r$ is a uniform random number on the interval [0, 1]; $\alpha$ represents the weight coefficient.

Cyclone Foraging

Cyclone foraging mainly involves the formation of a foraging chain in the population of manta ray and cyclone towards the target food [37]. The mathematical model is as follows:

$$P_i^d\left(t+1\right)=\left\{\begin{array}{c}{P}_{\mathrm{d}\mathrm{F}\mathrm{D}\mathrm{B}}^d(t)+r·\left(P_{\mathrm{d}\mathrm{F}\mathrm{D}\mathrm{B}}^d(t)-P_i^d\left(t\right)\right)\\ +\beta ·\left(P_{\mathrm{d}\mathrm{F}\mathrm{D}\mathrm{B}}^d(t)-P_i^d\left(t\right)\right),i=1\\ P_{\mathrm{d}\mathrm{F}\mathrm{D}\mathrm{B}}^d\left(t\right)+r·\left(P_{i-1}^d(t)-P_i^d\left(t\right)\right)\\ +\beta ·\left(P_{\mathrm{d}\mathrm{F}\mathrm{D}\mathrm{B}}^d(t)-P_i^d\left(t\right)\right),i=2,\cdots ,N\end{array}\right.$$

(18)

$$\beta =2{\mathrm{e}}^{r_1·{\displaystyle \frac{T-t+1}{T}}}·\mathrm{s}\mathrm{i}\mathrm{n}(2\mathsf{\pi }·r_1)$$

(19)

$$P_{\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}}^d={LB}^d+r·({UB}^d-{LB}^d)$$

(20)

$$P_i^d\left(t+1\right)=\left\{\begin{array}{l}{P}_{\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}}^d+r·\left(P_{\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}}^d-P_i^d\left(t\right)\right)+\beta ·\left(P_{\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}}^d-P_i^d\left(t\right)\right),i=1\\ P_{\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}}^d+r·\left(P_{i-1}^d\left(t\right)-P_i^d\left(t\right)\right)+\beta ·\left(P_{\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}}^d-P_i^d\left(t\right)\right),i=2,\cdots ,N\end{array}\right.$$

(21)

where $\beta$ means the weight coefficient of cyclone foraging; $r_1$ represents a random number on [0, 1]; $P_{\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}}^d$ is the random position of the $d$-dimensional manta ray; $LB$ represents the lower limit value of the variable to be solved; $UB$ represents the upper limit value of the variable to be solved.

Somersault Foraging

Somersault foraging mainly uses food sources as a fulcrum, and manta rays use this fulcrum to roll and find better foraging positions [38]. The mathematical model is as follows:

$$\left\{\begin{array}{l}{P}_i^d\left(t+1\right)=P_i^d\left(t\right)+S·(r_2·P_{\mathrm{d}\mathrm{F}\mathrm{D}\mathrm{B}}^d\left(t\right)-r_3·P_i^d\left(t\right))\\ i=\mathrm{1,2},\cdots ,N\end{array}\right.$$

(22)

where $S$ represents the somersault factor of manta ray; $r_2$ and $r_3$ mean a random number between [0, 1].

4.1.3. Update Iteration

The updated iterative mathematical model is shown in the following equation:

$$\left\{\begin{array}{l}{F}_i\left(t+1\right)>F_i\left(t\right),P_i\left(t\right)=P_i\left(t+1\right)\\ F_i\left(t+1\right)\le F_i\left(t\right),P_i\left(t\right)=P_i\left(t\right)\\ F_i\left(t\right)>F_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}\left(t\right),P_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}\left(t\right)=P_i\left(t\right)\end{array}\right.$$

(23)

where $F_i\left(t\right)$ represents the fitness value of the manta ray individual; $P_i\left(t\right)$ is the position of the manta ray individual.

4.2. dFDB-MRFO Based SOFC Parameter Identification Design

4.2.1. Fitness Function

The goal of using this algorithm for parameter identification is to minimize the difference between the fitted output current and the actual output current. The fitness function of this algorithm uses the three objective functions established in the cell modeling section as measurement indicators, namely RMSE, MAE, MAPE, and the time required for a single run.

4.2.2. Constraints

This study combines the variables to be solved into vectors as the positions of manta rays in the algorithm and then performs iterative optimization. Therefore, each parameter needs to be limited within its upper and lower boundaries as follows:

$$\left\{\begin{array}{l}{x}_{\mathrm{m}\mathrm{i}\mathrm{n}}^h\le x^h\le x_{\mathrm{m}\mathrm{a}\mathrm{x}}^h,h\in H\\ x=\left[E_{\mathrm{o}}, A, R_{\mathrm{o}\mathrm{h}\mathrm{m}}, B, I_{0,\mathrm{a}}, I_{0,\mathrm{c}}, I_{\mathrm{L}}\right],ECM\\ x={[E}_{\mathrm{o}}, A, R_{\mathrm{o}\mathrm{h}\mathrm{m}}, B, I_0, I_{\mathrm{L}}],SECM\end{array}\right.$$

(24)

where $x^h$ denotes the $h$th SOFC parameter to be identified; $x_{\mathrm{m}\mathrm{i}\mathrm{n}}^h$ and $x_{\mathrm{m}\mathrm{a}\mathrm{x}}^h$ are the lower limit and upper limit of the $h$th parameter to be identified, respectively; $H$ represents the number of parameters to be identified.

4.2.3. Execution Process

The complete process of parameter identification can be divided into three parts: data collection, data processing, and parameter extraction, respectively, as shown in Figure 4.

5. Case Studies

The dFDB-MRFO algorithm was applied to two datasets for parameter identification study on two models of SOFC, namely ECM and SECM. The search range of each parameter in both models is shown in

Table 2. The searching range of each unknown parameter for SOFC models.

ECM	Bound	Parameters
		Eo (V)	A (V)	Rohm ($\mathrm{k}\mathsf{\Omega }·{\mathrm{c}\mathrm{m}}^2$)	B (V)	I0,a (mA/cm2)	I0,c (mA/cm2)	IL (mA/cm2)
	Lower	0	0	0	0	0	0	0
	Upper	1.2	1	1	1	300	300	2000
SECM	Bound	Parameters
		Eo (V)	A (V)	Rohm ($\mathrm{k}\mathsf{\Omega }·{\mathrm{c}\mathrm{m}}^2$)	B (V)	I0 (mA/cm2)	IL (mA/cm2)
	Lower	0	0	0	0	0	0
	Upper	1.2	1	1	1	300	2000

. The datasets are respectively 1 kW flat stack cells developed by Huazhong University of Science and Technology (HUST) [39] and the single cell developed by CEREL Company in Poland [40,41], namely HUST and Poland mentioned in the work.

In addition, ENN was applied to two datasets for data processing, namely data prediction and data denoising. The parameter settings of ENN and the situation of each dataset are shown in

Table 3. Model and algorithm parameter setting.

Types		Parameters	Value
ENN		Number of neurons	6
		Training function	Trainscg
		Training times	6000
Eight heuristic algorithms	ABC	Search range expansion factor	1
		Maximum iterations	1000
		Population size	50
	BSA	Perception coefficient	1.5
		Maximum iterations	1000
		Population size	50
		Social acceleration coefficient	1.5
		Flight interval	5
	ALO dFDB-MRFO GWO IWOA MSA WOA	Maximum iterations	1000
		Population size	50
Dataset	HUST	Number of series cells	26
		Pairs of V-I data	18
		Pairs of predicted V-I data	180
	Poland	Number of series cells	1
		Pairs of V-I data	20
		Pairs of predicted V-I data	190

, and the data processing results are shown in Figure 5. The results show that the predicted and denoised data curves are basically consistent with the original data curves, it was fully demonstrated that ENN has high prediction accuracy and fitting accuracy.

Moreover, the dFDB-MRFO algorithm was compared with seven other heuristic algorithms, namely artificial bee colony (ABC) [42], ant lion optimizer (ALO) [43], bird swarm algorithm (BSA) [44], gray wolf optimization (GWO) [45,46], improved whale optimization algorithm (IWOA) [47], moth swarm algorithm (MSA) [48], and whale optimization algorithm (WOA) [49]. ABC imitates the swarm intelligence optimization algorithm of bees’ honey harvesting behavior and finds the optimal solution to the problem through the collaborative work of bee harvesting, observing, and reconnaissance. ALO is an optimization algorithm based on the hunting behavior of ant lions, using their traps and hunting strategies to search for the optimal solution. BSA is an optimization algorithm that mimics the foraging and migration behavior of birds and seeks the optimal solution to the problem through the flight and gathering behavior of birds. GWO is an optimization algorithm based on the predation behavior of grey wolves, which searches for the optimal solution through the social hierarchy and predation strategy of grey wolves. IWOA simulates the hunting and bubble net attack behavior of whales to find the optimal solution, which has better search performance and convergence speed. MSA imitates the behavior of moths using light for navigation and seeks the optimal solution to the problem through their flight and search strategies. WOA is an optimization algorithm based on whale hunting behavior, which searches for the optimal solution by simulating whale hunting strategies and bubble net attack behavior [50]. These seven algorithms all have the advantages of strong global search ability, high parallel computing efficiency, good robustness, and are suitable for solving complex optimization problems. A total of eight algorithms were applied to two models of SOFC, using two datasets for parameter identification study, in four cases. To fairly compare the algorithms, the maximum number of iterations and population of each algorithm was set to the same value. Specifically, the maximum number of iterations is designed to be 1000, the population is designed to be 50, and all eight algorithms underwent 10 independent runs [51]. All case studies were undertaken by SimuNPS, released by Shanghai KeLiang Information Technology Co., Ltd., Shanghai, China through a personal computer with Intel(R) Core (TM) i5 CPU at 3.0 GHz and 64 GB of RAM.

5.1. ECM Parameter Identification

5.1.1. Results of ECM Based on Prediction Data

Table 4. Identified parameter and RMSE results obtained by various algorithms for ECM based on predicted data and original data.

Dataset	Algorithm	Data	Identified Parameters							RMSE
			Eo	A	Rohm	B	I0,a	I0,c	IL
(HUST)	ABC	O	1.0084	0.0476	1.8374 × 10−5	0.3334	217.1714	20.7065	1980.7297	6.0511 × 10−3
		P	0.9886	0.0506	9.3017 × 10−5	0.1906	152.9825	50.9524	1981.6448	4.8217 × 10−3
	ALO	O	1.0121	0.0310	3.2092 × 10−5	0.3981	46.1898	20.7702	2000.0000	6.8041 × 10−3
		P	0.9841	0.1057	6.4114 × 10−5	0.0006	247.6706	122.8542	1754.8986	3.1884 × 10−3
	BSA	O	1.0066	0.0676	7.0196 × 10−5	0.1317	227.2606	42.4770	1994.5771	7.8553 × 10−3
		P	0.9808	0.1342	0.0000	0.0000	261.1027	256.4412	1549.1421	7.9009 × 10−3
	dFDB- MRFO	O	1.0299	0.0301	2.6629 × 10−4	0.0280	130.1199	5.5840	1356.1667	1.7875 × 10−3
		P	1.0280	0.0341	2.9020 × 10−4	0.0036	249.3177	7.4399	1172.6976	1.4748 × 10−3
	GWO	O	1.0254	0.0302	0.0000	0.4172	39.2815	9.9196	1999.9289	3.2237 × 10−3
		P	1.0192	0.0429	0.0000	0.3557	146.2351	14.9770	1967.2670	2.3444 × 10−3
	IWOA	O	0.9873	0.1355	0.0000	0.0000	259.2276	246.1912	864.0600	1.2561 × 10−2
		P	0.9804	0.0490	0.0000	0.3948	284.5482	47.1109	2000.0000	7.9059 × 10−3
	MSA	O	1.0043	0.0329	0.0000	0.3851	41.8693	33.9236	1937.6914	6.2021 × 10−3
		P	0.9854	0.0949	0.0000	0.1068	153.7583	133.1701	1997.2129	7.9444 × 10−3
	WOA	O	0.9642	0.0195	3.9581 × 10−4	0.0304	250.5678	159.2426	1546.5298	9.9934 × 10−3
		P	0.9641	0.0365	0.0000	0.4497	162.8437	159.9484	2000.0000	4.1312 × 10−3
(Poland)	ABC	O	1.0109	0.0153	2.2785 × 10−4	0.1363	253.5995	156.2469	715.2902	6.0578 × 10−3
		P	1.0115	0.0190	1.8986 × 10−4	0.1577	259.2224	185.6259	710.6529	4.6889 × 10−3
	ALO	O	1.0095	0.0000	4.4685 × 10−5	0.3629	300.0000	24.4399	925.8585	4.8371 × 10−3
		P	1.0061	0.0023	1.9289 × 10−4	0.1852	104.3663	101.2819	787.5287	3.7879 × 10−3
	BSA	O	1.0208	0.0012	0.0000	0.3169	299.9991	0.0000	845.6171	6.8994 × 10−3
		P	1.0042	0.0000	2.9484 × 10−5	0.3710	202.0059	116.3580	933.7275	6.8444 × 10−3
	dFDB- MRFO	O	1.0219	0.0209	4.5817 × 10−5	0.1327	48.7848	44.4692	665.4088	1.5902 × 10−3
		P	1.0233	0.0259	2.5100 × 10−5	0.1256	76.6912	41.0389	656.2669	9.5081 × 10−4
	GWO	O	1.0048	0.0033	0.0000	0.3469	266.6240	169.0980	887.0722	8.2074 × 10−3
		P	1.0069	0.0074	0.0000	0.3056	191.8814	49.7938	856.1800	7.1003 × 10−3
	IWOA	O	1.0086	0.0925	0.0000	0.0987	289.9562	259.9562	668.2762	8.5109 × 10−3
		P	1.0196	0.0218	0.0000	0.1678	131.7244	29.4241	703.2403	7.8439 × 10−3
	MSA	O	1.0214	0.0898	0.0000	0.0643	299.9952	132.7677	617.7042	9.7095 × 10−3
		P	1.0257	0.1653	0.0000	0.0294	300.0000	106.7103	593.0896	7.0223 × 10−3
	WOA	O	1.0268	0.0201	0.0000	0.1655	151.1938	14.6126	691.9322	2.1817 × 10−2
		P	1.0089	0.0000	0.0000	0.4771	233.5462	47.9121	1079.1425	7.7361 × 10−3

presents ECM parameter identification and RMSE results obtained from various algorithms in two datasets, where ‘O’ and ‘P’ represent the original data and predicted data, respectively. Apparently, dFDB-MRFO has the highest accuracy in various situations, with a minimum RMSE of 9.5081 × 10⁻⁴. After data prediction, it can effectively improve the accuracy of parameter identification and reduce the performance indicator RMSE. Particularly, WOA has the largest RMSE decrease in the Poland dataset, at 64.5413%. However, it is worth noting that in the HUST dataset, there are anomalies in BSA and MSA, resulting in an increase in RMSE values and a decrease in accuracy after data prediction.

The iterative curves obtained by the eight algorithms on two datasets are shown in Figure 6. dFDB-MRFO has the smallest RMSE, the highest accuracy, and the fastest convergence speed in all cases, while ALO has the slowest convergence speed among the eight algorithms. The box diagram of RMSE obtained by the eight algorithms under 10 independent running conditions is shown in Figure 7. Obviously, the results obtained by dFDB-MRFO have the highest accuracy, stability, and robustness among all the algorithms. In addition, after data prediction, the RMSE of each algorithm showed a certain degree of decrease, and GWO eliminated outliers in RMSE after data prediction. Overall, the values of RMSE obtained by each algorithm decrease as the amount of data increases.

Table 5. Evaluation metrics of eight algorithms for ECM based on predicted data.

Dataset	Algorithm	Data	MAE	MAPE	Tcost (s)
(HUST)	ABC	P	2.2783 × 10−3	0.2418	37.3324
	ALO	P	2.4007 × 10−3	0.2889	23.8300
	BSA	P	1.9064 × 10−3	0.3349	20.6537
	dFDB-MRFO	P	1.2302 × 10−3	0.1460	34.1251
	GWO	P	1.8876 × 10−3	2.0263	18.8555
	IWOA	P	2.2625 × 10−3	0.3925	53.6705
	MSA	P	2.2570 × 10−3	0.2330	19.4584
	WOA	P	4.2883 × 10−3	0.5380	17.8711
(Poland)	ABC	P	2.2092 × 10−3	0.3374	40.3677
	ALO	P	3.4368 × 10−3	0.4518	26.5260
	BSA	P	5.7945 × 10−3	0.2202	19.8924
	dFDB-MRFO	P	6.8940 × 10−4	0.0894	37.3125
	GWO	P	1.2923 × 10−3	0.1528	19.9486
	IWOA	P	4.0978 × 10−3	0.3130	55.8479
	MSA	P	1.1075 × 10−3	0.2375	20.3565
	WOA	P	3.2596 × 10−3	0.3648	19.3635

presents the performance indicators obtained by each algorithm after prediction under two datasets. It shows that dFDB-MRFO yields the smallest MAE and MAPE results, with the highest identification accuracy. However, its disadvantage is that it takes a long time, while WOA takes the shortest time, with a single run time of only 17.8711 s.

The identified parameters of dFDB-MRFO are substituted into the SOFC model to obtain the fitted curve, as shown in Figure 8. In the prediction dataset, the fitting accuracy is very high. Specifically, in the HUST prediction dataset, R² reaches 99.90%, while in the Poland prediction dataset, R² is 99.86%. Figure 9 shows the radar diagrams of the worst and best results of the eight algorithms for RMSE, MAE, MAPE, and time under 10 independent operating conditions. One can easily observe that the first three indicators of dFDB-MRFO are the most stable, while WOA has the shortest time consumption, as shown in Figure 9d. It is evident that the various indicators of IWOA show the worst robustness and the highest fluctuation in algorithm performance.

5.1.2. Results of ECM Based on Denoising Data

Table 6. Identified parameters and RMSE results obtained by various algorithms for ECM based on noised data and denoised data.

Dataset	Algorithm	Data	Identified Parameters							RMSE
			Eo	A	Rohm	B	I0,a	I0,c	IL
(HUST)	ABC	N	1.0408	0.0480	1.7318 × 10−4	0.0473	243.0569	8.8039	1932.7709	2.5196 × 10−2
		DN	1.0173	0.0372	1.0597 × 10−4	0.2728	188.9808	49.8484	2000.0000	6.3903 × 10−3
	ALO	N	1.0531	0.0480	8.2077 × 10−5	0.0311	242.9147	15.0355	2000.0000	2.6823 × 10−2
		DN	1.0221	0.0375	7.4828 × 10−5	0.2839	126.1736	12.3680	2000.0000	6.5182 × 10−3
	BSA	N	0.9407	0.0000	0.0000	0.7348	221.2304	220.4400	2000.0000	2.6894 × 10−2
		DN	1.0062	0.0630	0.0000	0.2332	261.6800	35.2368	1744.5576	1.0173 × 10−2
	dFDB-MRFO	N	1.0556	0.0376	2.2173 × 10−4	0.0032	104.5354	78.4485	1712.0942	2.4767 × 10−2
		DN	1.0290	0.0329	2.3280 × 10−4	0.0918	176.7367	171.1254	1976.2655	1.4120 × 10−3
	GWO	N	1.0577	0.0410	0.0000	0.2977	95.2575	65.3310	1999.6485	2.6910 × 10−2
		DN	1.0145	0.0446	0.0000	0.1703	124.2762	68.1133	1318.8597	7.7041 × 10−3
	IWOA	N	1.0378	0.0545	1.1626 × 10−5	0.0999	57.9673	20.3646	1375.4261	4.6882 × 10−2
		DN	0.9793	0.0659	0.0000	0.3245	154.7226	126.7479	2000.0000	2.9647 × 10−2
	MSA	N	1.0105	0.0919	5.3483 × 10−5	0.0033	248.0689	95.9777	923.7004	2.6578 × 10−2
		DN	1.0107	0.0518	0.0000	0.2929	180.8079	25.2453	1880.1309	4.9883 × 10−3
	WOA	N	1.0204	0.0826	0.0000	0.0758	281.3075	33.4079	1399.6655	4.6882 × 10−2
		DN	0.9469	0.0000	0.0000	0.7489	171.1936	142.3863	2000.0000	2.9645 × 10−2
(Poland)	ABC	N	1.0293	0.0829	1.5714 × 10−4	0.0463	260.5604	206.0742	623.6771	2.2675 × 10−2
		DN	1.0077	0.0104	3.1564 × 10−4	0.0825	296.8653	151.6166	651.2173	5.1109 × 10−3
	ALO	N	1.0194	0.0239	0.0000	0.1540	92.4391	52.9801	683.0828	2.3778 × 10−2
		DN	1.0248	0.1088	3.6119 × 10−5	0.0560	275.6218	191.8501	647.4461	1.0263 × 10−2
	BSA	N	1.0382	0.0120	3.8280 × 10−4	0.0254	299.9480	54.5671	594.9236	2.1734 × 10−2
		DN	1.0142	0.1055	4.6119 × 10−5	0.0665	289.5618	272.4108	623.9358	4.1625 × 10−3
	dFDB- MRFO	N	1.0228	0.0311	3.5181 × 10−4	0.0179	261.3744	88.7520	593.3863	2.1010 × 10−2
		DN	1.0215	0.0196	8.9138 × 10−5	0.1187	49.1925	46.8012	657.3215	1.2957 × 10−3
	GWO	N	1.0124	0.0045	0.0000	0.5224	280.8711	261.7358	1196.1545	2.4765 × 10−2
		DN	1.0171	0.0259	1.1441 × 10−5	0.1481	93.9606	75.6367	683.7487	2.4633 × 10−3
	IWOA	N	1.0198	0.0679	0.0000	0.2447	249.5798	225.9068	975.1523	2.4780 × 10−2
		DN	1.0133	0.0041	0.0000	0.5743	256.7416	229.6859	1251.0259	8.7935 × 10−3
	MSA	N	1.0210	0.0150	0.0000	0.2187	49.1191	32.6179	779.7619	2.4618 × 10−2
		DN	1.0180	0.0784	0.0000	0.0726	148.1870	117.0644	623.1194	3.2388 × 10−3
	WOA	N	1.0190	0.0065	0.0000	0.3209	164.8314	121.9037	900.9027	2.4201 × 10−2
		DN	1.0277	0.1151	0.0000	0.0382	300.0000	149.1398	601.0619	1.1927 × 10−2

presents the parameter identification results of ECM obtained after data denoising processing on two datasets, where ‘N’ and ‘DN’ represent noise data and denoised data, respectively. It shows that the accuracy of each algorithm has been significantly improved after data denoising. Particularly, dFDB-MRFO obtained the smallest RMSE after denoising in the Poland dataset, only 1.2957 × 10⁻³. At the same time, the largest decrease in RMSE is obtained by dFDB-MRFO in the HUST dataset, from 2.4767 × 10⁻² to 1.4120 × 10⁻³, with a decrease of 94.2989%. Second, the same decrease is observed in dFDB-MRFO in the Poland dataset, with a decrease of 93.8329% and the smallest decrease of 36.7625%, for the results obtained by IWOA on the HUST dataset.

Figure 10 shows the iterative curves of the eight algorithms obtained under two datasets. It is evident that after data denoising, the RMSE of each algorithm has significantly decreased, and among the eight algorithms, dFDB-MRFO always obtains the smallest RMSE value and the highest accuracy. In addition, ALO has the slowest convergence speed among the eight algorithms. The boxplot results obtained after 10 independent runs of noise reduction processing are shown in Figure 11. It shows that dFDB-MRFO has the smallest fluctuation in the results under multiple running conditions, while BSA, WOA, and IWOA have larger fluctuations, as shown in Figure 10a. However, it is very obvious that after data denoising, the value of RMSE significantly decreases, indicating that after denoising, the accuracy of parameter identification can be significantly improved.

The fitting V-I curve obtained by dFDB-MRFO is shown in Figure 12, with R² at 99.93% and 99.82%, respectively, indicating high fitting accuracy. Figure 13 shows the radar diagrams of each performance indicator obtained by the eight algorithms. It shows that dFDB-MRFO obtains the best results under noise reduction data conditions, and its worst- and best-case fluctuations are small, making the algorithm relatively stable. However, its disadvantage is that it takes a relatively long time.

5.2. SECM Parameter Identification

5.2.1. Results of SECM Based on Prediction Data

Table 7. Identified parameters and RMSE results obtained by various algorithms for SECM based on predicted data and original data.

Dataset	Algorithm	Data	Identified Parameters						RMSE
			Eo	A	Rohm	B	I0	IL
(HUST)	ABC	O	0.9959	0.0644	1.8789 × 10−4	0.0853	37.9933	1572.5825	8.6550 × 10−3
		P	0.9875	0.1246	3.1201 × 10−5	0.1663	95.7661	1688.1018	5.7883 × 10−3
	ALO	O	0.9824	0.1749	2.2731 × 10−5	0.0701	130.4238	1549.5708	1.2283 × 10−2
		P	1.0214	0.0295	2.9226 × 10−4	0.2421	64.9809	1985.9896	6.2325 × 10−3
	BSA	O	0.9940	0.2286	0.0000	0.0002	146.9997	864.0600	8.0427 × 10−3
		P	0.9892	0.1053	0.0000	0.1857	74.4544	1446.7452	4.9445 × 10−3
	dFDB-MRFO	O	1.0299	0.0359	3.3483 × 10−4	0.0092	27.3804	1963.3014	2.8946 × 10−3
		P	1.0165	0.0428	2.8037 × 10−4	0.0605	14.9765	1914.9696	2.1036 × 10−3
	GWO	O	0.9971	0.1864	0.0000	0.0118	112.5108	874.7816	1.0391 × 10−2
		P	0.9993	0.0953	0.0000	0.2028	56.6599	1483.3104	4.2223 × 10−3
	IWOA	O	0.9872	0.2084	0.0000	0.0094	144.8862	875.5106	1.1768 × 10−2
		P	0.9760	0.1839	0.0000	0.1063	155.5571	1562.4918	7.1351 × 10−3
	MSA	O	0.9907	0.2229	1.3483 × 10−5	0.0038	151.9466	864.6546	1.1419 × 10−2
		P	0.9820	0.1691	0.0000	0.0768	130.5388	1236.5841	6.2959 × 10−3
	WOA	O	0.9759	0.2689	0.0000	0.0945	243.3425	1999.6929	1.4692 × 10−2
		P	0.9851	0.1288	0.0000	0.1839	99.6173	1631.8595	5.6719 × 10−3
(Poland)	ABC	O	1.0087	0.0820	2.0200 × 10−4	0.0939	283.1375	672.0733	6.0210 × 10−3
		P	1.0143	0.0549	2.1321 × 10−4	0.0858	181.8114	642.7615	4.4861 × 10−3
	ALO	O	1.0387	0.2627	1.1876 × 10−4	0.0088	243.3413	592.8920	6.2782 × 10−3
		P	1.0343	0.3179	8.3912 × 10−5	0.0175	300.0000	589.8790	6.6270 × 10−3
	BSA	O	1.0389	0.4277	0.0000	0.0000	300.0000	611.0538	8.3311 × 10−3
		P	1.0282	0.0231	0.0000	0.1968	16.0398	722.5594	1.5154 × 10−2
	dFDB-MRFO	O	1.0219	0.0499	6.6134 × 10−6	0.1350	54.4361	665.8446	1.4992 × 10−3
		P	1.0221	0.0432	9.6881 × 10−5	0.1067	51.9844	643.9199	1.0670 × 10−3
	GWO	O	1.0054	0.0142	0.0000	0.3221	162.6233	866.1572	3.6775 × 10−3
		P	1.0047	0.0034	0.0000	0.3648	75.2659	920.9167	2.4511 × 10−3
	IWOA	O	1.0369	0.4125	0.0000	0.0032	299.5569	592.8920	1.7263 × 10−2
		P	1.0158	0.1303	0.0000	0.2504	299.4474	895.9101	7.8751 × 10−3
	MSA	O	1.0186	0.0128	0.0000	0.2486	10.7416	780.4319	4.9739 × 10−3
		P	1.0098	0.0155	0.0000	0.2661	43.1013	806.2505	4.2821 × 10−3
	WOA	O	1.0138	0.0911	0.0000	0.1212	134.2589	662.8850	3.2853 × 10−3
		P	1.0122	0.0182	3.3970 × 10−5	0.2303	41.8487	773.8104	1.8440 × 10−3

presents the SECM parameter identification results obtained from two datasets. It shows that dFDB-MRFO outperforms other algorithms, with a minimum RMSE of only 1.0670 × 10⁻³. After data prediction, RMSE has been reduced to a certain extent, and the accuracy of parameter identification has been further improved.

Figure 14 compares the iteration curves of SECM under two datasets, and the results show that dFDB-MRFO performs better than other algorithms, and its convergence speed is also faster. Figure 15 shows the boxplot obtained from 10 independent runs, fully demonstrating the better accuracy and robustness of dFDB-MRFO compared to other approaches.

Table 8. Evaluation metrics of eight algorithms for SECM based on predicted data.

Dataset	Algorithm	Data	MAE	MAPE	Tcost (s)
(HUST)	ABC	P	2.1547 × 10−3	0.2634	42.3649
	ALO	P	3.1916 × 10−3	0.3798	24.9545
	BSA	P	2.0090 × 10−3	0.2902	19.8749
	dFDB-MRFO	P	1.4111 × 10−3	0.1825	38.7129
	GWO	P	2.0460 × 10−3	0.2288	19.1030
	IWOA	P	2.1170 × 10−3	0.3501	54.0887
	MSA	P	2.0387 × 10−3	0.2538	19.0055
	WOA	P	2.3336 × 10−3	0.3328	18.9393
(Poland)	ABC	P	3.5170 × 10−3	0.3277	38.7902
	ALO	P	4.4878 × 10−3	0.3709	23.6014
	BSA	P	6.1218 × 10−3	0.7347	18.1037
	dFDB-MRFO	P	6.6031 × 10−4	0.0823	34.0141
	GWO	P	1.9230 × 10−3	0.1560	18.1561
	IWOA	P	3.7055 × 10−3	0.4304	52.9035
	MSA	P	1.4135 × 10−3	0.1892	18.4856
	WOA	P	3.7348 × 10−3	0.1491	17.7537

presents MAE, MAPE, and T_cost obtained by each algorithm for SECM in the two predicted datasets. The results show that the accuracy indicators MAE and MAPE of dFDB-MRFO are both optimal, but the disadvantage is that the running time is relatively long, requiring 38.7129 s in the HUST dataset and 34.0141 s in the Poland dataset. In the same situation, WOA requires the shortest time, only 18.9393 s in HUST data, and only 17.7537 s in Poland data, reducing the time by half.

Figure 16 shows V-I fitting curves obtained by the dFDB-MRFO algorithm under the two types of data, with high fitting accuracy of 99.41% and 99.91%, respectively. Figure 17 shows the optimal and worst performance indicators under 10 independent operating conditions. WOA, BSA, and IWOA show significant fluctuations in RMSE, MAE, and MAPE, indicating the poor robustness of these three algorithms. In addition, IWOA also requires the longest time among the eight methods.

5.2.2. Results of SECM Based on Denoising Data

Table 9. Identified parameters and RMSE results obtained by various algorithms for SECM based on denoised data and noised data.

Dataset	Algorithm	Data	Identified Parameters						RMSE
			Eo	A	Rohm	B	I0	IL
(HUST)	ABC	N	1.0239	0.0902	8.0400 × 10−5	0.1574	33.6532	2000.0000	2.5354 × 10−2
		DN	0.9950	0.1191	2.3572 × 10−5	0.1073	79.3750	1214.9105	9.1871 × 10−3
	ALO	N	1.0285	0.0856	8.6930 × 10−5	0.1039	30.8879	1444.4458	2.6887 × 10−2
		DN	1.0084	0.1618	3.1818 × 10−5	0.0187	89.0017	902.6166	1.1141 × 10−2
	BSA	N	1.0490	0.0691	0.0000	0.3324	13.2586	2000.0000	1.4015 × 10−1
		DN	0.9944	0.2134	0.0000	0.0025	134.9176	864.0600	2.9648 × 10−2
	dFDB-MRFO	N	1.0533	0.0548	2.6043 × 10−4	0.0000	8.6001	1651.7166	2.4958 × 10−2
		DN	1.0290	0.0371	3.3480 × 10−4	0.0032	8.2064	1883.1519	1.5600 × 10−3
	GWO	N	1.0488	0.0690	5.0449 × 10−5	0.2628	13.5532	2000.0000	2.6694 × 10−2
		DN	1.0203	0.0674	5.3987 × 10−5	0.1741	23.9954	1373.0948	4.6941 × 10−3
	IWOA	N	1.0214	0.0932	0.0000	0.2639	35.9610	2000.0000	2.6214 × 10−2
		DN	0.9725	0.0270	3.9130 × 10−4	0.0256	65.2928	2000.0000	1.2417 × 10−2
	MSA	N	1.0137	0.1386	1.0114 × 10−5	0.0432	61.8185	1134.2961	2.7332 × 10−2
		DN	1.0023	0.0937	0.0000	0.2338	53.3175	1628.9630	7.2302 × 10−3
	WOA	N	1.0330	0.0848	0.0000	0.2694	24.9538	1958.2301	2.8336 × 10−2
		DN	0.9831	0.0255	3.5701 × 10−4	0.0760	28.9481	2000.0000	1.4312 × 10−2
(Poland)	ABC	N	1.0183	0.0361	3.8249 × 10−4	0.0332	188.5794	603.8424	2.2084 × 10−2
		DN	1.0152	0.1057	1.5161 × 10−4	0.1030	209.2736	668.6288	5.9881 × 10−3
	ALO	N	1.0299	0.2928	1.2785 × 10−4	0.0113	300.0000	592.8920	2.1858 × 10−2
		DN	1.0099	0.0184	2.0317 × 10−4	0.1557	183.2716	741.5057	6.9660 × 10−3
	BSA	N	1.0444	0.0429	3.5072 × 10−5	0.1004	18.4335	635.7112	2.4820 × 10−2
		DN	1.0198	0.0000	0.0000	0.8499	254.9711	1632.9096	1.1887 × 10−2
	dFDB-MRFO	N	1.0213	0.2069	1.2547 × 10−4	0.0198	238.8162	593.5600	2.1003 × 10−2
		DN	1.0217	0.0526	0.0000	0.1339	57.2269	664.4808	1.1003 × 10−3
	GWO	N	1.0242	0.3572	0.0000	0.0106	300.0000	592.8920	2.4745 × 10−2
		DN	1.0055	0.0015	0.0000	0.3552	35.1242	893.9047	8.0012 × 10−3
	IWOA	N	1.0183	0.1337	0.0000	0.0910	165.2074	648.4477	2.2751 × 10−2
		DN	1.0071	0.0320	0.0000	0.2930	196.4814	848.1346	7.6441 × 10−3
	MSA	N	1.0249	0.1512	1.5676 × 10−5	0.0412	132.3149	599.2817	2.1464 × 10−2
		DN	1.0163	0.0024	4.1476 × 10−5	0.2973	14.1800	842.3375	4.5383 × 10−3
	WOA	N	1.0165	0.0464	0.0000	0.3960	159.9633	1093.5445	2.8834 × 10−2
		DN	1.0087	0.0002	3.3680 × 10−4	0.0902	70.9730	656.2933	4.5140 × 10−3

presents the parameter identification and RMSE results obtained by SECM under data denoising processing. For dFDB-MRFO, the RMSE obtained is the smallest among the eight algorithms. In addition, after data denoising, the RMSE of all eight algorithms decreased to a certain extent. Specifically, dFBD-MRFO decreased from 2.1003 × 10⁻² to 1.1003 × 10⁻³ on the Poland dataset, with a decrease of 94.7612%, while the result obtained by WOA on the HUST dataset showed the smallest decrease, with a decrease of 49.4918%.

Figure 18 depicts the iteration curves obtained by each algorithm before and after data denoising processing. One can easily observe that dFDB-MRFO has good convergence speed and accuracy. Figure 19 shows the boxplot obtained by each algorithm. Among the eight algorithms, dFDB-MRFO has the best robustness and accuracy, while BSA has very poor robustness, and there are many RMSE outliers after 10 runs, as shown in Figure 19b.

Figure 20 shows V-I curve fitting obtained by dFDB-MRFO, with fitting accuracy of 99.95% and 99.91% for both datasets, respectively. Figure 21 shows the optimal and worst results obtained by the eight algorithms for the four performance indicators. It is evident that BSA, IWOA, and WOA exhibit significant volatility, while dFDB-MRFO is more stable.

5.3. Discussions

Overall, after data processing and algorithm improvement, the effectiveness of parameter identification was improved. However, it should not be overlooked that due to the random search characteristics, the identification accuracy of some algorithms unexpectedly decreased after data processing. For instance, in the HUST dataset, the RMSEs obtained by BSA and MSA applied to ECM increased compared to before data prediction, as shown in Table 4. In addition, it should be noted that although the performance of the algorithm was improved by selecting the dFDB strategy, its cost time was not the shortest compared to the other seven algorithms, as shown in Figure 21d.

6. Conclusions

This work adopts a novel hybrid method for SOFC parameter identification study, which contains the following three outcomes/novelties:

• A comprehensive modeling study was conducted on SOFC, resulting in the establishment of two cell models: SECM and ECM. In addition, cell-related experimental data were fully investigated, two V-I datasets of SOFC were collected and organized, and the two datasets were applied to two cell models for parameter identification study
• ENN was used to predict and denoise the datasets, and the effects of cell data number and quality on the final parameter identification results were compared and analyzed. The case results indicate that after data prediction and denoising, more reliable and stable identification results can be extracted. Especially, after data prediction, WOA applied to ECM performs at a 64.5413% increase of accuracy under Poland data, and dFDB-MRFO achieved a 94.7612% decrease for the RMSE of SECM after data denoising under HUST data
• The weighted values of fitness and distance were selected as the optimal individual selection criteria, and the optimization strategy of MRFO was improved. The result obtained by dFDB-MRFO was compared with other heuristic algorithms. A comprehensive comparative analysis was carried out, which proved that dFDB-MRFO has faster convergence speed, higher identification accuracy, and better robustness compared to other algorithms. In particular, the parameters identified by dFDB-MRFO have the highest fitting accuracy of 99.93% for V-I data of ECM. The highest fitting accuracy is 99.95% for V-I data of SECM.

7. Future Work Outlook

Future work can be focused on the following two aspects outlined below.

The work on SOFC should be more in depth, using the established model to research the cell state of charge and the state of health estimation.

Due to the characteristic of random search, heuristic algorithms can significantly reduce the optimization speed but take a longer time when dealing with multiple data situations. Therefore, it is necessary and urgent to develop more parameter identification models based on neural networks and machine learning in the future.