Solid Oxide Fuel Cell Voltage Prediction by a Data-Driven Approach

Abstract

A solid oxide fuel cell (SOFC) is an electrochemical energy conversion device that provides higher thermoelectric efficiency than traditional cogeneration systems. Current research in this field highlights a variety of mathematical models. These models are based on complex physicochemical and electrochemical reactions, enabling accurate simulation and optimal control of fuel cells. However, these models require substantial computational resources, leading to high processing times. White box and gray box models are unable to achieve real-time optimization of control parameters. A potential solution involves using data-driven machine learning (ML) black-box models. This study examines three ML models: artificial neural network (ANN), random forest (RF), and extreme gradient boosting (XGB). The training dataset consisted of experimental results from SOFC laboratory experiments, comprising 32,843 records with 47 control parameters. The study evaluated the effectiveness of input matrix dimensionality reduction using the following feature importance evaluation methods: mean decrease in impurity (MDI), permutation importance (PI), principal component analysis (PCA), and Shapley additive explanations (SHAP). The application of ML models revealed a complex nonlinear relationship between the SOFC output voltage and the control parameters of the system. The default XGB model achieved the optimal balance between accuracy (MSE = 0.9940) and training speed (τ = 0.173 s/it), with performance capabilities that enable real-time enhancement of SOFC thermoelectric characteristics during system operation.

1. Introduction

Solid oxide fuel cells (SOFCs) have attracted significant research attention over the past decades due to their capability to produce electricity and high-grade heat from hydrogen or natural gas [1,2]. This cogeneration system has a higher thermoelectric efficiency than traditional and alternative generation systems [3]. This makes SOFCs a sustainable, green technology aimed at reducing carbon emissions. They also have great potential for use in distributed generation systems [4,5] and new vehicles [5,6].

SOFC performance depends on numerous operational parameters, so long-term testing and substantial financial resources for experimental research are required to ensure stable, safe, and cost-effective operation of these systems [7,8]. The application of mathematical modeling significantly reduces both the time and financial resources required for determining optimal operating parameters [4].

Currently, there are numerous mathematical modeling methods with varying levels of complexity, ranging from zero- to three-dimensional computational fluid dynamics (CFD). A brief overview of these models can be found in the works of Mütter F. et al. [4] and Huo W. et al. [8]. Multidimensional models (2D and 3D) have higher accuracy but require significantly greater computational resources [4,5]. Fuel cells rely on complex multiphysical processes involving heat and mass transfer, electrochemical reactions, and electron conductivity. Therefore, researchers have to make some simplifications in mathematical models [5,9,10], which negatively affects their predictive accuracy [7].

Machine learning (ML) methods can be used to overcome the above limitations. They simplify and accelerate computational efforts while maintaining accuracy in SOFC performance and lifespan prediction using experimental data directly [11]. These methods are easy to implement and often have higher accuracy and wider applicability than mathematical models [5]. Surrogate models trained on SOFC data can perform most calculations within one second, while physical CFD models can take hundreds of hours to perform such calculations [11]. This also leads to a substantial cost reduction when repeating a large number of experiments using ML models [12].

The aim of this study was to identify the most significant features affecting the performance of SOFCs, evaluating the weight of and quantifying their relative contributions to the SOFC output voltage. This would aid in optimizing the system’s thermoelectric efficiency. The work is structured as follows: Section 2 analyzes the existing applications of ML methods to SOFCs, and it evaluates the current state of research in this field, briefly summarizing the most common ML models and evaluation metrics. Section 3 describes the SOFC laboratory setup, experimental conditions, and the obtained monitoring data structure. It also presents a multistage ML model investigation scheme with specified hyperparameter optimization ranges. Section 4 presents the feature value analysis and highlights the most significant factors for predicting SOFC output voltage whilst also comparing and analyzing the accuracy of ML models. The concluding remarks are introduced in Section 5.

2. Current State of the Research Field

Table 1. Review of recent ML-based approaches for SOFC modeling.

Reference	Year	Model (s)	Best Model	Data Size	Proportion Set: Train–Test–Valid	Variables		Best Error Metrics
						Input (Count)	Output
Subramanian Y. et al. [1]	2023	SVM	SVM	16	85:15:0	T, V (2)	J, w	MAPE = 0.0098
Testasecca T. et al. [3]	2024	XGB, RF, LSTM, GB, MLP, PR	RF	>2500	90:10:0	$W,Q_{\mathrm{gas}},W_{\mathrm{gas}},m_{\mathrm{C}{\mathrm{O}}_2\mathrm{emiss}}$$,m_{\mathrm{C}{\mathrm{O}}_2\mathrm{save}}$$,\Sigma m_{\mathrm{C}{\mathrm{O}}_2\mathrm{save}}$, ΔW (7)	ηE	MAE = 0.24 MAPE = 0.04 MSE = 0.14 RMSE = 0.38 R2 = 0.98
Mütter F. et al. [4]	2023	GA-MLP	GA-MLP	534,976	4:1:0	$M_{{\mathrm{H}}_2}$$,M_{{\mathrm{H}}_2\mathrm{O}}$$,M_{\mathrm{C}\mathrm{O}}$$,M_{\mathrm{C}{\mathrm{O}}_2}$$,M_{\mathrm{C}{\mathrm{O}}_4}$$,M_{{\mathrm{N}}_2}$, T, J (8)	V	MSE = 6.384 × 10−7 ± 7.159 × 10−8 RMSE = 0.799 ± 0.268
Huo H. et al. [5]	2025	VARMA, RBF, GRU N-BEATS, LSTM	N-BEATS	3743	90:10:0	–	V	MAE = 0.0225 RMSE = 0.0237 R2 = 0.9889
Rao M. et al. [6]	2022	ARIMA, multi-step LSTM, recursive LSTM	multi-step LSTM	10,323	7000:2323:1000	$V_{\mathrm{rt}},I,{P_{{\mathrm{CH}}_4}}_{\mathrm{in}}$, Pcathod air, Pannod in, Pannod out, Pcathod out (7)	V	RMSE = 0.3444 MAE = 0.1691
Golbabaei M.H. et al. [7]	2022	LR, SVM, GP, DT, RF, GB, KNN, MLP	MLP	403	8:2:0	ASL-, EL-, CFL-thickness, ASL porosity, T, J (6)	V	R2 = 0.998 MSE = 9.6 × 10−5 MAE = 6 × 10−3
Huo W. et al. [8]	2021	DNN, RF-CNN	RF-CNN	>10,000	–	(26)	I-V curve	RMSE = 0.0396 MAE = 0.0355 R2 = 0.9119
Vairo T. et al. [9]	2023	GB	GB	4392	80:20:0	V, I, F, Zr, Zim (5)	2 state	p = 0.99 r = 0.99 F-1 score = 0.99
Natali A. [10]	2023	LR, BRR, PR, DT, GB, RF, Hb-GB, MLP	Hb-GB	10,000	80:20:0	$M_{{\mathrm{H}}_2}$, T, J (3)	V	R2 = 0.972 RMSE = 1.6 × 10−4
Ding R. et al. [12]	2020	DT, XGB, BP-ANN	BP-ANN	>10,000	85:15:0	(26)	w	R2 = 0.9621 RMSE = 58.5
Kim J. et al. [13]	2024	LR, DT, RF, MLP, SVM, XGB	MLP	591	8:2:0	(57)	w	R2 = 0.9251
Li M. et al. [14]	2022	LSTM, ed-LSTM, GRU, ed-GRU	ed-LSTM	10,323	4129:5162: 1032	$V_{\mathrm{rt}},I,{P_{{\mathrm{CH}}_4}}_{\mathrm{in}}$, Pcathod air (4)	V	MSE = 0.014966 MAE = 0.084220 R2 = 0.964618
Chen K. et al. [15]	2024	SVM, BP	BP	2000	8:2:0	$t,V,I,Q_{{\mathrm{H}}_2\mathrm{rt}}$ (4)	$Q_{{\mathrm{H}}_2}$	RMSE = 0.259 MAPE = 0.003334 MAE = 0.017 R2 = 0.9976
Lai M. et al. [16]	2024	MLR, BP	BP	662,327	–	(12)	T, V, W	NRMSE = 0.0066 NME = 0.428 MAE = 1.367
Wu Y. et al. [17]	2023	DAG, DN, BP, SVM, RF, GA-RBF, RBF, GA-BP, LS-SVM	DAG	1099	1000:99:0	Qfuel, Qair, Qsteam, W (4)	ηH, ηE	MAE = 0.0109 RMSE2 = 0.0135
Hou D. et al. [18]	2023	ARX, NLARX	NLARX	3600	–	W, T, Qwater (3)	$Q_{{\mathrm{H}}_2}$	MSE = 0.05266
Sheng C. et al. [19]	2023	KF, LSTM	LSTM	3750	2500:1250:0	J, Vrt, w, I (4)	V, RUL	RMSE = 1.4373 MAE = 0.0015
Song S. et al. [20]	2021	BP, SVM, RF	BP	858	650:208:0	$T,J,Q_{\mathrm{air}},Q_{{\mathrm{H}}_2}$ (4)	V	R2 = 0.999 RMSE = 0.0032 MAE = 0.0769
İskenderoğlu F.C. et al. [21]	2020	SVM, RF	SVM	1272	1122:150:0	T, J, syngas types, etc. (10)	V	MAPE = 0.0092
Keyhanpour M., Ghassemi M. [22]	2025	DNN, RF, LASSO	RF	601	601:30:0	vfuel, vair, T, ASL and CFL porosity (5)	w, T	RMSE = 0.1213 MAE = 0.08853 R2 = 0.9999
Subotić V. et al. [23]	2021	MLP	MLP	2271	80:15:5	T, J, type fuel, etc. (9)	E, Zim, Zr	E/Zr/Zim: MSE = 2.93 × 10−5, 7.12 × 10−7/3.68 × 10−7 MAPE = 0.0034/0.00204/0.369 SMAPE = 0.0034/0.02/0.237
Milewski J., Świrski K. [24]	2009	MLP	MLP	583	1:0:0	J, T, qfuel, qoxidant (4)	V	RE = 1.0%
Sheng C. et al. [25]	2023	ES (ES2/ES3)-R-GM, ANFIS-SC (gp/FCM)	ES3-R-GM + ANFIS-SC	800	500:300:0	Tfuel, Tair, Tstack, Tburn, Pgas, V, I, W, etc. (82)	V	RMSE = 0.1345 R2 = 0.9450
Wu X. et al. [26]	2024	PSO-BP	PSO-BP	7290	70:30:0	$C_{\mathrm{C}}$$,C_{{\mathrm{H}}_2}$$,C_{{\mathrm{O}}_2}$, Tstack, Tanode, Qm, S/B (7)	V, J, ηE, ηCHP	MAPE < 0.06 RMSE < 0.33 R2 > 0.98
Wu X. et al. [27]	2024	ReliefF-mRMR	ReliefF-mRMR	2206	70:30:0	$W,\mathrm{grad}{T}_{\mathrm{cc}},T_{{\mathrm{H}}_2}$ (3)	3 state	CDR1 = 0.98 CDR2 = 0.978 CDR3 = 0.981
Wu X. et al. [28]	2024	MLR, RBF, BP, LSTM, PSO-BP, GA-BP	GA-BP	9104	6300:2804:0	t, Tafterburn, Tstack, I, W (5)	V	MAE = 0.182 MSE = 0.081 R2 = 0.949 RMSE = 0.285
Kheirandish A. et al. [29]	2016	SVM, MLP	SVM	9725	–	J, V, W, ηE (4)	V–I, P–I, ηE–P curve	P–I: MSE = 0.0009 R2 = 0.9952
Chen H., el al. [30]	2021	LS-MLR, KNN, SVM, AdaBoost, RF, Bagging DT, GB	GB	500	80:20:0	λ, T, RH, Panode, J (5)	V	R2 = 0.89609
Raeesi M. et al. [31]	2021	RNN, DNN, LSTM, BLSTM	DNN	6000	5250:750	–	V	MSE = 0.14 R2 = 0.9982
Zheng Y. et al. [32]	2021	PCA-MLP, RF, PCA-SVM	SVM	71,064	70:30:0	Qair,re-burner, Qair,bypass, Qwater, Qfuel,react, Tair,exch, Tafter-burner, Treformer, I, V (9)	3 state	AUC = 0.997 A = 0.9304 F1-score = 0.929
Huo W. et al. [33]	2022	ELM, SVM, GA-SVM	GA-SVM	400	350:50:0	(12)	9 state	A = 0.98
Legala A. et al. [34]	2022	SVM, BP-ANN	BP-ANN	1100	70:30:0	I, T, Pcathod, PO2, PH2, Membrane Hydration (6)	V	MAE = 0.011 R2 = 0.995 RMSE = 0.015
Chauhan V. et al. [35]	2020	LogR, SVM, MLP	MLP	4734	3750:984:0	Extracted channel photo	3 state	A = 0.95
Lin R.H. et al. [36]	2020	DT, RF, KNN, SVM, AdaBoost, ANN	RF-PCA	206,360	75:25:0	(8)	3 state	AUC = 0.99975 F1-score = 0.9989
Lü X. et al. [37]	2022	RF-HPSO	RF-HPSO	200,000	70:30:0	(4)	W	MSE = 47.6444
Han I.S. et al. [38]	2016	SVM, ANN	ANN	1468	923:454:0	PH2, PO2, T, RHc, I (5)	V	R2 = 0.9994 RMSE = 2.4 MAPE = 0.0022
Zhong Z.-D. et al. [39]	2006	SVM	SVM	–	–	T, J, V, etc	I-V curve	MSE = 0.0002 R2 = 0.997%

provides a review of recent ML-based approaches for SOFC modeling. These models can generally be divided into two categories: classification and regression models.

Classification models evaluate SOFC states using operational parameters, detecting conditions such as normal state, air leaks in the air supply manifold, flooding failure in the stack, etc. [33]. Furthermore, these models can be subdivided into binary [9] and multiclass [27,32,33,35,36] classification models.

The performance and accuracy of classification models are most commonly evaluated using the following metrics:

• precision (p), as follows:

$$p={\displaystyle \frac{\mathrm{TP}}{\mathrm{TP}+\mathrm{FP}}}$$

(1)

• recall (r), as follows:

$$r={\displaystyle \frac{\mathrm{TP}}{\mathrm{TP}+\mathrm{FN}}}$$

(2)

• accuracy (A), as follows:

$$\mathrm{A}={\displaystyle \frac{\mathrm{TP}+\mathrm{TN}}{\mathrm{TP}+\mathrm{FN}+\mathrm{FP}+\mathrm{TN}}}$$

(3)

• F1-score, as follows:

$$\mathrm{F}1={\displaystyle \frac{2\cdot p\cdot r}{p+r}}$$

(4)

where TP, FP—true and false positives, respectively; TN, FN—true and false negatives, respectively.

In regression models, input parameters are used to predict continuous output features such as power density, current density, voltage, and others. These models are also subdivided into statistical (LR, LogR, PR, ARIMA, VARMA, etc.), traditional ML (RG, GB, SVM, etc.) and deep learning (DL) (ANN: MLP, LSTM, RNN, etc.) [5]. The performance of DL methods continues to improve as the amount of training data increases [40].

The performance and accuracy of regression models are most commonly evaluated using the following metrics:

• mean absolute error (MAE), as follows:

$$\mathrm{MAE}={\displaystyle \frac{1}{n}}\cdot {\displaystyle \sum _{i=1}^n\left|R_i-P_i\right|}$$

(5)

• mean absolute percentage error (MAPE), as follow:

$$\mathrm{MAPE}={\displaystyle \frac{1}{n}}\cdot {\displaystyle \sum _{i=1}^n\frac{\left|R_i-P_i\right|}{R_i}}$$

(6)

• symmetric mean absolute percentage error (SMAPE), as follow:

$$\mathrm{SMAPE}={\displaystyle \frac{100}{n}}\cdot {\displaystyle \sum _{i=1}^n\frac{\left|R_i-P_i\right|}{\left(\left|R_i\right|+\left|P_i\right|\right)/2}}$$

(7)

• mean squared error (MSE), as follows:

$$\mathrm{MSE}={\displaystyle \frac{1}{n}}\cdot {\displaystyle \sum _{i=1}^n{\left(R_i-P_i\right)}^2}$$

(8)

• root mean square error (RMSE), as follows:

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{n}}\cdot {\displaystyle \sum _{i=1}^n{\left(R_i-P_i\right)}^2}}$$

(9)

• normalized root mean square error (NRMSE), as follows:

$$\mathrm{NRMSE}={\displaystyle \frac{1}{{\overline{R}}_i}}\cdot \sqrt{{\displaystyle \frac{1}{n}}\cdot {\displaystyle \sum _{i=1}^n{\left(R_i-P_i\right)}^2}}$$

(10)

• normalized mean error (NME), as follows:

$$\mathrm{NME}={\displaystyle \frac{1}{n\cdot {\overline{R}}_i}}\cdot {\displaystyle \sum _{i=1}^n\left|R_i-P_i\right|}$$

(11)

• coefficient of determination (R²), as follows:

$${\mathrm{R}}^2=1-{\displaystyle \frac{{\displaystyle \sum _{i=1}^n{\left(R_i-P_i\right)}^2}}{{\displaystyle \sum _{i=1}^n{\left({\overline{R}}_i-P_i\right)}^2}}}$$

(12)

where R_i, P_i—measured and predicted values, respectively; ${\overline{R}}_i$—sample average; n—validation sample size.

Comparative analysis of the regression models efficiency from Table 1 is challenging because the various authors do not have a unified approach to choosing metrics. However, it should be noted that in most cases, the MAPE metric did not exceed 5%, while the R2 score most often tended to 1. Thus, black-box models (ML and DL) are often more accurate and efficient than physical and statistical models (white-box and gray-box models). The authors of the studies came to similar conclusions [5,10,14].

Table 1 indicates that DL models (MLP, LSTM, BP, etc.) generally have higher efficiency according to metrics 5–12. Traditional ML models (RF, GB, SVM, etc.) also show robust and stable results, even on large data samples. For example, in [8,32,36], when analyzing samples consisting of ~10 × 10⁶ records, ML models were found to be more efficient than DL models. This finding partially contradicts the conclusion of Ming and Sun [40] that traditional ML models outperform DL models when training data is limited.

At the same time, statistical models (ARIMA, LR, MLR, etc.) showed worse performance than ML and DL models. This phenomenon occurs because features with a strong relationship (for example, power W and electrical efficiency η_E) are not always strictly correlated with each other. Therefore, their relationships cannot always be reliably described by linear approximations [3]. As exceptions, we would like to highlight Golbabaei M.H. et al. [7] (GP); Hou D., Wu X. et al. [18] (NLARX); and Huo H., Li X. et al. [5] (N-BEATS), where traditional statistical models outperformed ML and DL approaches in predictive accuracy or computational efficiency.

The accuracy and performance of classification and regression models directly depends on the size of the training matrix. The size of the matrix is determined by the volume of records (rows) and the number of input parameters (columns). A large number of input parameters significantly reduces computational speed, while the models themselves often do not have the best metrics (for example, in [8,12,13,25]). Over-reduction of input features may result in the loss of critical relationships in the model itself. Such a model will poorly match the actual operational characteristics of the SOFC. According to Table 1, the median number of input parameters taken into account in the model among the authors was 5, while the average was 11.

Optimization of the model input features is most often performed using the following tools:

• Correlation matrix (Pearson, Kendall tau, Spearman) [3,7,16];
• Covariance ranks matrix [7];
• Shapley additive explanations (SHAP) [13];
• Sequence forward selection (SFS) [6,14] or sequential backward selection (SBS);
• Principal component analysis (PCA) [32,36];
• Mean decrease in impurity (MDI) [7,8,12];
• Permutation importance (PI).

In addition, some researchers also reduce the volume of input data. For example, Rao M., Wang L. et al. [6] and Li X., Wu J. et al. [4] reduced the size of the experimental dataset from 629,873 to 10,323 records in their studies, and Sheng C., Fu J. et al. [25] reduced the sample size from 240,000 to 800 records. Thus, by analyzing Table 1, we can conclude that the optimal input matrix size for data-driven models is approximately (10 × 10⁵) × (5 ÷ 11).

The most frequently predicted continuous target feature is the output voltage (V) generated by the SOFC (Table 1). This is a universal parameter that allows one to evaluate both SOFC performance and operational stability [7]. According to Chen H., Shan W. et al. [30], accurate stack voltage prediction enables optimization of SOFC control and design parameters, which in turn extends the fuel cell service life. As highlighted by Rao M., Wang L. et al. [6], the SOFC output voltage is the most important and valuable state characteristic, outperforming other operational parameters for validating predictive model performance. Jouin M., Bressel M. et al. [41] noted that a voltage drop in cell to a minimum value can lead to automatic shutdown of the entire stack by the safety system. Voltage prediction helps prevent such events. Li M. and Wu J. et al. [14] confirmed these findings, noting that a voltage drop to 70% of the nominal value indicates stack failure. Sheng C., Zheng Y. et al. [19] and Wu X.-L., Li Y. et al. [28] also noted that voltage is an indirect indicator of the SOFC state of health.

Following the example of other authors, this study focuses on stack voltage prediction as the most valuable feature for SOFC performance and reliability assessment. Therefore, only ML regression models have been considered in this work.

The scientific novelty of this work lies in the implementation of extreme gradient boosted random forest (XGBRF) modeling—an approach previously unreported in SOFC-related studies.

3. Materials and Methods

3.1. Laboratory Setup and Data Collection

The data for the study were obtained from a laboratory setup (Figure 1), which consisted of natural gas and nitrogen supply manifolds, a flow distribution system, an SOFC, a battery storage unit, and a steam reforming and preheating system. Gas flow rate, temperature, pressure, and electrical load were controlled by specialist software. The fuel cell unit operated in three power output modes (500 W, 1000 W, and 1500 W), with operational ranges of 600–1000 °C for stack temperature and 4.5–15.5 L/min for gas flow rate.

The stack itself contained 27 cells located in the SOFC hot zone (Figure 2).

A schematic diagram of the laboratory setup is shown in Figure 3.

The initial natural gas flow (A1) was fed from the main line to a pressure reduction unit (A2), then directed to a desulfurizer (A3). Before entering the burner (E2) and heat exchanger (E3), the main flow passed through coarse (A4) and fine (A5) filters, followed by flow splitting according to the operational mode. Nitrogen (B1) was fed to the heat exchanger in parallel with natural gas. The mixture of natural gas and nitrogen (B2) was mixed with steam (C1) superheated by exhaust gases from the burner (E2). The resulting gas mixture was then fed to the heat exchanger, after which it was directed to the reformer (E4), where the mixture reached the operating temperature of the steam reforming reaction. The synthesis gas (C2) produced by reforming was fed to the anode chamber of the fuel cell (E5). The purified air (D1), heated in the heat exchanger (E3), was supplied to the cathode chamber of the fuel cell (E5). The products of the chemical reactions of the cathode and anode chambers (F1), including oxygen and residual methane, were fed to the burner (E2), where the reaction of complete oxidation of methane occurred. The resulting exhaust gases (F2) sequentially transferred heat to the incoming flows in the heat exchanger (E3) and in the carburetor (E1).

To obtain recent on-site monitoring data, 10 experiments with a total duration of 80 h were conducted. Various power generation modes (500, 100, 1500 W) were reached. The steam reformer commissioning and hot standby modes were also reached. During the experiments, the nitrogen supply pressure was varied in the range from 1 to 5 bar with a step of 1 bar to determine the relationship between gas flow rate and SOFC stack heating dynamics. During the process of generating electricity in various modes, the SOFC was disconnected from the battery storage unit, and then an external load with variable resistance was connected (Figure 4).

As a result of the experiments, 32,844 data records were obtained in CSV format, captured via SOFC control software onto a FAT32-formatted external storage device. The complete dataset is publicly available at: https://github.com/caapel/SOFC/tree/master (accessed on 27 March 2025).

3.2. Data Preprocessing

Reading and working with the resulting CSV file were performed using the Pandas and NumPy libraries in the Jupyter Notebook development environment in Python. The monitoring data, converted to a DataFrame object, included 32,844 records with 47 parameters. Subsequent analysis required comprehensive data preprocessing.

At the first preprocessing stage, the DataFrame was reindexed to the value of the “MCGS_TIME” column (date and time of the log entry). The column itself was dropped.

At the second stage, noninformative features (for example, the column with the millisecond value “MCGS_TIMEMS”, etc.) that did not affect the target feature were eliminated.

At the third stage, similar features with identical and/or close values were combined. For example, T3 and T4—representing burner temperatures in the upper and lower sections, respectively—were merged into a common feature T3: “burner temperature”. Features were considered similar when their percentage difference was below 1%. Keeping these features separate could lead to multicollinearity issues. This dimensionality reduction approach eliminated four additional parameters (T4, T24, T26, T28).

At the fourth stage, incorrect “temperature” features (associated with temperature sensors) of which the average value exceeded 2999.9 °C were removed. This filtering process allowed us to reduce additional 12 data columns.

At the fifth stage, the target feature—output voltage (V)—was isolated. As a result, the input feature vector was reduced to 25 elements (

Table 2. Feature notation and interpretations.

Feature	Interpretation	Feature	Interpretation
T3	Burner temperature, °C	T27	Temperature at the right point of the reformer, °C
T5	Temperature at the inlet of the reformer, °C	T30	Cooling water temperature, °C
T7	SOFC exhaust gases temperature, °C	T31	Water tank temperature, °C
T9	Heat exchanger temperature, °C	pump_spd	Peristaltic pump rotation speed, rps
T12	Water temperature for steam reforming, °C	impl_spd1	Cooling fan speed, rps
T16	Temperature at the SOFC left front point, °C	impl_spd2	Main fan speed, rps
T17	Temperature at the SOFC right rear point, °C	Q_CH4	CH4 flow rate, m3/h
T19	Air temperature at the SOFC inlet, °C	Q_CH4_N2	CH4/N2 flow rate, m3/h
T20	Hydrogen temperature at the SOFC inlet, °C	P_NG	Differential natural gas pressure, bar
T21	Air temperature at the SOFC outlet, °C	O2	Oxygen concentration at the burner inlet, %
T22	Hydrogen temperature at the SOFC outlet, °C	W	The stack’s power, Wt
T23	Temperature at the rear point of the reformer, °C	I	The stack’s current, A
T25	Temperature at the left point of the reformer, °C	V	The stack’s voltage, V

).

During the sixth processing stage, 92 records containing “temperature” features (un-removed at the fourth stage) with values of 3000 °C were eliminated.

During the final seventh processing stage, records in which the target feature was equal to zero (representing system startup/shutdown, warm-up, and cooldown periods) were removed. As a result, the number of records in the DataFrame was reduced to 14,427 lines.

The subsequent preprocessing involved input feature normalization to scale the data and prevent features with large values from dominating. This standardization was necessary for proper neural network operation. Some activation functions, such as sigmoid and tanh are most sensitive to input values near zero. If the input values are excessively large or small, activation functions may saturate (produce values close to 0 or 1), resulting in vanishing gradients and slowed training. Normalization adapts input values to an optimal range for these functions [7,34]. Data normalization was performed using the Z-score method (StandardScaler). This normalization process is described by the following equation:

$$z={\displaystyle \frac{x-\overline{x}}{\sigma }}$$

(13)

where x—initial value; z—normalized value; $\overline{x}$—sample average; σ—the standard deviation of the training samples.

The dataset was randomly split into training (for training models) and test (for tuning regressor hyperparameters) samples in a ratio of 80:20 using the train_test_split tool of the Scikit-learn library.

Principal component analysis (PCA) was implemented for some models to further reduce the feature vector and eliminate linear correlations. For implementation details of PCA-based hybrid models, refer to Section 3.3.

3.3. Model Selection and Hyperparameter Tuning

To develop predictive models, ensemble methods of traditional machine learning (extreme gradient boosting and random forest) and a deep learning method (Multilayer perceptron) were used.

Model accuracy and computational performance were evaluated on the test set using the following metrics: R², MSE, MAE, MAPE (%) and τ (s/it).

3.3.1. Extreme Gradient Boosting

Extreme gradient boosting (XGB) is a popular and powerful machine learning algorithms. The XGB architecture is a sequential ensemble of decision trees, where each subsequent model corrects the errors of the previous, weak model [3]. Figure 5 demonstrates an example of such a tree structure.

The XGB model was developed using the XGBRegressor tool of the XGBoost library. The model was optimized by tuning the XGB regressor hyperparameters in the specified scanned ranges:

• max_depth: 2–5;
• learning_rate: (0.01, 0.1, 0.2);
• gamma: (0, 0.1, 1.0, 10.0).

Hyperparameter tuning was performed using the cross-validation method with the help of a hyperparameter grid (GridSearchCV tool, Scikit-learn library). Five data splits were specified in the cross-validation parameters (n_splits = 5). The MAPE metric was selected for evaluating the model’s efficiency.

All other hyperparameters retained their default settings.

Five versions were developed for this model:

• XGB default—default model with a full set of features (25 pcs.);
• XGB + MDI—a model where the control parameter vector is obtained using standard XGBoost’s feature importance (MDI). The number of the strongest initial features varies in the range from 5 to 11;
• XGB + PI—a model where the control parameter vector is obtained using permutation feature importance (PI). The number of the strongest initial features varies in the range from 5 to 11;
• XGB + SHAP—a model where the control parameter vector is obtained using Shapley additive explanation (SHAP) feature importance. The number of the strongest initial features varies in the range from 5 to 11;
• XGB + PCA—a hybrid model with preliminary standardization and PCA data decomposition, where the number of components varies in the range from 5 to 23 with a step of 2.

3.3.2. Random Forest

Random forest (RF) is a parallel ensemble method where multiple weak decision trees of the same type are trained independently and in parallel, and their average output becomes the prediction result. This architecture provides inherent resistance to input noise and mitigates overfitting risks [3].

The RF model was developed using the XGBRFRegressor tool of the XGBoost library. Model optimization involved tuning the max_depth hyperparameter within a specified scanned range of 5–15. The remaining parameters were saved with default settings. The GridSearchCV configuration for the RF model is similar to the XGB model described in Section 3.3.1.

Two versions of this model were developed: the default (XGBRF default), with the full set of 25 features, and the hybrid (XGBRF + PCA), using PCA data decomposition with the number of components varying from 5 to 23 in steps of 2.

3.3.3. Multilayer Perceptron

Artificial neural networks (ANNs) are ML algorithms related to deep learning. The ANN is a mathematical abstraction that models the structure and functioning mechanism of a biological neural network [7].

Figure 6 shows the architecture of the multilayer perceptron (MLP) used for data processing (Section 3.1). This was a feed-forward neural network containing two hidden layers.

The MLP model was developed using the MLPRegressor tool of the Scikit-learn library. The model was optimized by tuning the hyperparameters of the MLP regressor within specified scanned range:

• Feature vectors size: 5–25;
• Hidden layer sizes: (5–45; 5–45);
• Activation functions for hidden layer neurons: logistic, ReLu, tanh.

The hyperparameter grid tuning for the MLP model followed the same approach as for the XGB model in Section 3.3.1.

The static hyperparameters of the MLP model were configured as follows:

• Learning rate: invscaling;
• Learning rate init: 0.055;
• Solver: Adam;
• Maximum iterations: 2000.

Default values were used for all other hyperparameters.

Two model variants were developed: a default version (MLP default) with the full set of 25 features, preprocessed only through standardization, and a hybrid version (MLP + PCA), where the input layer nodes varied from 5 to 23 in increments of 2. Data preprocessing, in addition to standardization, included PCA decomposition.

3.4. Hardware and Software

The following software and libraries were used in this work:

• Windows 10, v. 22H2, build 19045.5608;
• Python, v. 3.10.7;
• Jupyter Notebook, v. 6.4.12;
• SHAP, v. 0.47.0;
• NumPy, v. 1.25.2;
• Pandas, v. 2.2.3;
• Seaborn, v. 0.13.2;
• Graphviz, v. 0.20.1;
• Matplotlib, v. 3.6.2;
• YellowBrick, v. 1.5;
• SKLearn, v. 1.4.2;
• XGBoost, v. 2.1.2.

The calculations were carried out on the following hardware:

• CPU: Intel(R) Core(TM) i5-8300H;
• GPU: NVIDIA GeForce GTX 1050 Ti;
• RAM: DDR4, 16 GB.

4. Results and Discussion

4.1. Feature Selection

Exploratory data analysis was focused on identifying and selecting significant in-put features. It was necessary to reduce training set dimensionality, thereby improving model accuracy and computational efficiency.

At the first stage, a correlation analysis of the data was performed (Figure 7). The absolute values of the correlation coefficients were sorted by column V in descending order for visual interpretation.

The obtained Pearson correlation matrix (Figure 7) revealed strong dependence be-tween the “temperature” features (T19, T20, T21, T22, T23, etc.). According to Lin R.H., Pei Z.H. et al. [36], it is necessary to eliminate such multiple collinearities by reducing the dimensionality of the input feature vector.

“Temperature” features also had a strong correlation with the target feature (voltage, V). Such dominance of the similar features may cause the model to overlook more important but smaller features [7]. The weakest correlations with the target feature were observed for CH₄ flow rate ($Q_{{\mathrm{CH}}_4}$), stack’s current (I), and power (W). However, weak correlations for these features does not imply their insignificance, since the relationships between parameters cannot always be reliably described by linear approximations [3].

The second stage involved feature importance evaluation using the MDI (Figure 8a), PI (Figure 8b), and SHAP (Figure 8c) methods following dataset standardization. The final feature vector was reduced to 11 elements.

The strongest feature sets for the PI and SHAP evaluation methods had matches for all features albeit with different weights for these features. The match with the MDI was partial, with only 8 of 11 features matching. Despite preliminary data standardization, the T20 feature (hydrogen temperature at the SOFC inlet) remained the dominant feature across all evaluation methods.

To mitigate feature dominance and address multicollinearity, PCA data decomposition was applied at the third stage. The features’ importance after using PCA-XGBoost is shown in Figure 9.

4.2. Diagnostics of Models

The results of ML model diagnostics performed on the full set of standardized data using the Learning Curve tool of the Yellowbrick library are presented in Figure 10. The negative mean absolute percentage error was used as the evaluation metric. The lines on figure are the mean score value, and the shaded area around each lines indicate the variance of the model [7].

All models had comparable score gaps between training and validation data. The MLP model demonstrated the highest variance. The XGB model showed overfitting after about 9000 training examples. The XGBRF model achieved the lowest variance. Thus, the XGBRF model showed the greatest stability and predictive accuracy during the training process.

4.3. Results of Model Fitting

Four out of five XGB models used dimensionality reduction of the control parameter vector. The search for the optimal number of features was performed in the following score ranges: (5, 11, 1) for the MDI (Figure 11a), PI (Figure 11b), and SHAP (Figure 11c) feature importance evaluation methods and (5, 23, 2) for PCA (Figure 11d).

The models’ scores decreased monotonically as the number of features increased, as shown in Figure 11a–c. Consequently, reducing the control parameter vector dimensionality did not improve model performance—and therefore, using the MDI, PI, and SHAP evaluation methods to select the strongest features and sample dimensionality reduction was ineffective in this case. For this reason, those three evaluation methods (MDI, PI, and SHAP) were not used for the remaining ML models (RF, MLP).

The hyperparameters for the best XGB models are presented in

Table 3. Hyperparameters of optimal XGB models.

Model	Score	Number of Features/Components	Hyperparameters
			Gamma	Learning Rate	Max Depth
XGB default	MSE	25	0	0.2	5
XGB + MDI	MAPE	10	0	0.2	5
	MSE	11	0.1	0.2	5
XGB + PI	MAPE	11	0.1	0.2	5
	MSE	10	0	0.2	5
XGB + SHAP	MAPE, MSE	11	0.1	0.2	5
XGB + PCI	MAPE	19	0.1	0.2	5
	MSE	17	0.1	0.2	5

.

The results of finding the optimal number of components for PCA decomposition of the RF and MLP models are presented in Figure 12 and Figure 13. The hyperparameters for the best RF and MLP models are given in

Table 4. Hyperparameters of optimal RF models.

Model	Score	Number of Features/Components	Hyperparameters
			Max Depth
XGBRF default	MSE	25	15
XGBRF + PCA	MAPE	13	15
	MSE	17	15

and

Table 5. Hyperparameters of optimal MLP models.

Model	Score	Number of Features/Components	Hyperparameters
			Hidden Layer Size	Activation
MLP default	MAPE	25	(40, 40)	logistic
MLP + PCA	MAPE, MSE	23	(15, 15)	logistic

.

4.4. Discussion

A comparison of metrics of the best ML models is given in

Table 6. Comparison of accuracy and performance of optimized ML models.

Model (Components)	R2	MSE	MAE	MAPE	τ, s/it
Extreme gradient boosting
XGB + default (25)	0.99698	0.9940	0.309	2.63%	0.172760
XGB + PCA (19)	0.99524	1.5670	0.392	3.52%	0.142445
XGB + SHAP (11)	0.99660	1.1180	0.383	3.40%	0.095001
XGB + PI (11)	0.99658	1.1234	0.384	3.40%	0.093776
XGB + MDI (10)	0.99525	1.5604	0.427	3.52%	0.091928
Random forest
XGBRF default (25)	0.99680	1.0546	0.266	1.12%	3.215760
XGBRF + PCA (13)	0.99518	1.5818	0.336	1.55%	2.389213
Multilayer perceptron
MLP default (25)	0.99468	1.7546	0.554	5.89%	2.444528
MLP + PCA (23)	0.99527	1.5553	0.490	4.82%	2.148090

.

The analysis of the metrics in Table 6 revealed that any attempts to reduce the control parameter vector improved model performance but led to a noticeable decrease in accuracy. This confirmed our conclusions from Section 4.2.

Multicollinearity mitigation using PCA decomposition demonstrated improvements in both performance and accuracy only for the MLP model. In this case, the best result was achieved for the extreme value of the score range (n_components = 23, Figure 13). This also confirmed our conclusion that feature dimensionality reduction is inadvisable in this context.

However, PCA decomposition did not provide such a performance boost to models as reduction with the MDI, PI, and SHAP feature importance evaluation methods. PCA was a useless tool in this case. These findings contradicted the results of studies by Golbabaei M.H. et al. [7] and Zheng Y., Li X. et al. [32].

The best results were obtained using the default RF and XGB models, which was consistent with the results of the studies by Testasecca T. et al. [3] and Keyhanpour M. and Ghassemi M. [22] but contradicted the results reported by Natali A. [10], Ding R. et al. [12], and Kim J., Choi M. et al. [13].

The default XGB model showed the best performance and predictive accuracy (by the R² and MSE metrics). The default XGBRF model demonstrated greater stability, accuracy (by the MAE and MAPE metrics), and generalization capability.

The higher performance of the XGB model (0.17 s/it) is explained by the fact that XGB builds an ensemble of trees, where each corrects the errors of previous ones, and each is individually built relatively quickly. However, this speed is negated by greater variance and a tendency to overfit (Figure 10a).

XGBRF is a modification of XGB where each tree’s training uses random data subsampling and random feature subsets. This approach helped reduce variance and improve the model’s generalization capability (Figure 10b), creating a balance between accuracy and overfitting resistance (Table 6). However, it led to significantly reduced performance (3.22 s/it) due to additional computations. Since in our case we had high feature dimensionality (25 features), and each tree was trained on a random feature subset, XGBRF required a larger number of trees to achieve good accuracy, and consequently more time.

The MLP model demonstrated relatively low performance (2.44 s/it) similar to that of the XGBRF model. This can be attributed to several factors. First, the performance of MLP models may be constrained by the backpropagation algorithm, which requires sequential computation of gradients and weight updates across all network layers. While these computations are more complex than building a single decision tree, they remain less intensive than constructing multiple parallel trees in ensemble methods such as XGBRF. The MLP’s computational inefficiency primarily resulted from imperfect hyperparameter tuning (batch size, learning rate, epochs), potentially leading to the model getting stuck in local minima during the training process.

The learning curve (Figure 10c) revealed that the MLP model exhibited overfitting and likely required stronger regularization. Unlike MLP, the default XGB and XGBRF models incorporate built-in L1 and L2 regularization, whereas MLP necessitates manual regularization configuration. Furthermore, the default XGB and XGBRF models possess intrinsic feature importance evaluation mechanisms, enabling automatic selection of relevant features while disregarding noninformative ones. Consequently, feature dimensionality reduction and PCA decomposition failed to enhance accuracy for XGB and XGBRF models, whereas these techniques simultaneously improved both accuracy and performance for the MLP model.

In conclusion, selecting the most suitable approach in our study involves finding a balance between prediction accuracy and computational training time.

Thus, for SOFC output voltage prediction under strict time constraints (including real-time operation scenarios), the authors recommend using the default XGB model with the full feature set as the optimal solution for SOFC performance and reliability evaluation.

5. Conclusions

Based on the results of the study, the following conclusions were drawn:

• SOFC output voltage is the most frequently predicted continuous target feature for assessing fuel cell reliability and performance characteristics;
• Applying dimensionality reduction to sample sets using MDI, PI, and SHAP feature importance evaluation methods in this task improved model performance but significantly reduced their accuracy;
• The positive effect (simultaneous increase in both accuracy and computational performance) from PCA decomposition was obtained only for the MLP model. Therefore, PCA application is not recommended in this case;
• The default XGB model with a full feature set demonstrated the best performance (0.17276 s/it) and accuracy (R² = 0.99698 and MSE = 0.9940);
• The default XGBRF model with a full feature set demonstrated the lowest variance and absolute error (MAE = 0.266 and MAPE = 1.22%), as well as the best generalization capability.

In conclusion, this study successfully achieved its objective: the performance of the developed XGB model enables real-time optimization of the most critical SOFC system control parameters (including inlet air oxygen concentration, peristaltic pump rotation speed, stack’s current, etc.). This capability directly facilitates the improvement of SOFC thermoelectric characteristics during operation.