Application of Multiple Linear Regression and Artificial Neural Networks in Analyses and Predictions of the Thermoelectric Performance of Solid Oxide Fuel Cell Systems

Abstract

Solid oxide fuel cells (SOFCs) are an efficient, reliable and clean source of energy. Predictive modeling and analysis of their performance is becoming increasingly important, especially with the growing emphasis on sustainable development’s requirements. However, mathematical modeling is difficult due to the complexity of its internal structure. In this study, the system’s electricity generating performance and operational characteristics were analyzed using recent on-site monitoring data first. Then, based on Pearson’s correlation coefficient, some of the variables were selected to build two prediction models: an artificial neural network (ANN) model and a multiple linear regression (MLR) model. The models were evaluated on the basis of the normalized mean square error (NRMSE), which was 1.89% for the MLR model and 0.66% for the ANN model, with no overall bias. They were also compared with other existing models, and it was found that the two models used in this study have the advantage of high accuracy and low difficulty. Therefore, the models developed in this study can more accurately and effectively assess the SOFC system’s state and can better support work to improve the thermoelectric performance of SOFC systems.

1. Introduction

The World Energy Outlook 2023, issued by the International Energy Agency, indicates that fuel cells (FCs) will become one of the significant components of energy technology in the future. And a solid oxide fuel cell (SOFC) is a kind of FC, and its system is an efficient energy conversion device [1], which can directly convert the chemical energy in fuel into electric energy through electrochemical reactions rather than a combustion process [2]. SOFCs are important auxiliary equipment of renewable energy technology, which is sustainable [3]. Moreover, the 26th UN Climate Change Conference of the Parties (COP26) noted that global climate change will have far-reaching impacts on present and future generations, indicating that reducing carbon emissions to mitigate global warming is urgent, and that reducing the burning of fossil fuels is one of the ways to do so [4]. SOFCs show great potential to reduce the demand for fossil fuels. The advantage of SOFC technology lies in its high power conversion efficiency and environmentally protective by-product, H₂O. Its advantage shows that accurate and reliable prediction and modeling of SOFCs can guide the improvement of their thermoelectric performance and contribute to the cause of human sustainable development. Experimental studies on the performance of SOFC systems usually require significant time and cost. The data-driven prediction-based approach can significantly reduce the cost of developing new SOFC systems and accelerate the diffusion and application of SOFC technology.

In the field of energy forecasting, three forecasting methods are widely used for SOFC systems: (1) statistical methods, such as linear regression and autoregressive integrated moving average (ARIMA), are commonly used [5,6]; (2) artificial intelligence methods, such as ANN, the Gaussian process (GP), and support vector machines (SVMs) [7,8,9]; and (3) engineering simulation methods, including specialized SOFC simulation software (COMSOL 6.1, MATLAB R2016, Aspen Plus v11) [10,11,12]. Traditionally, the Energy Service Company (ESCO) industry has used statistical methods to establish the modeling and verification baseline of SOFC systems, and standardized guidance documents such as the International Organization for Standardization (ISO) standards have provided a framework for these concepts and procedures.

A number of existing studies have carried out predictive modeling of SOFCs based on a variety of approaches, such as (1) artificial intelligence [13,14,15], (2) least squares support vector machines (LS-SVMs) [16,17,18], (3) improved backpropagation neural networks (BPNNs) [19,20], (4) virtual reality (V-R) technology [21], and (5) the Takagi–Sugeno (T-S) fuzzy model [22]. According to the research literature, the backpropagation neural network (BPNN) is one of the most accurate methods for predicting the hourly performance of fuel cell systems [23,24,25]. Although there are many methods, each forecasting method has its own advantages and applicability.

In addition, many literature reviews have proved that ANN can be successfully applied to analyses of thermoelectric performance and predictive modeling of an SOFC system. The SOFC system is a highly complex and nonlinear system, so it is necessary to find predictive methods that are compatible with such systems. ANN can simulate the behavior of an object and predict the parameters of the SOFC system in a process of operation based on available experimental data when a large number of physical quantities are unknown, thus achieving high-precision modeling of the SOFC system [26,27,28]. However, ANN suffers from the problem of overfitting, so there is also a need to find a method that is free from the overfitting problem. However, existing studies based on the MLR methodology for modeling polymer electrolyte membrane fuel cells (PEMFC) have shown good modeling results and no overfitting problems [29,30]. Therefore, we can draw on PEMFC’s predictive methodology and choose MLR and ANN models. Meanwhile, these two methods are widely used in the study of energy and electricity, and have proved to be very accurate and reliable [31,32,33,34]. Therefore, this study adopted two methods, MLR and ANN, for data-driven modeling of SOFC systems.

The focus of this study was to quantitatively analyze the performance influencing variables of the SOFC system (kilowatt-scale, flat-paneled stacks, methane as a fuel, with a voltage range of 0–27 V) so as to obtain prediction of the performance based on minutes, which could provide guidance for improving the thermoelectric performance of the system. Thus, this article consists of the following parts. Initially, the text provides an overview of the SOFC system and dataset. Then the techniques used for data preprocessing, predictive modeling, the performance assessment, and an evaluation of the predictive accuracy within the context of the SOFC system are presented. Section 3 introduces and analyzes the research results. Finally, the full text is summarized and suggestions for future work are put forward. Meanwhile, in Section 4 and Section 5, this study compares and analyses the accuracy of MLR and ANN models to find a simple prediction method (ANN) that is compatible with SOFC systems with cogeneration. It was compared with other existing methods, and we found that the method used in this study had higher prediction accuracy and, at the same time, was simpler.

2. Materials and Methods

2.1. The SOFC System

The SOFC system has been put into use in the fuel cell center of Huazhong University of Science and Technology [34,35], providing efficient and environmentally friendly electricity and heat energy. The system covers an area of 20 square meters, and the core components of the stack include a high-performance ceramic electrolyte and advanced anode/cathode materials, which can run stably at temperatures as high as 850 °C. Figure 1 shows the architecture of the SOFC system, and the SOFC system consists of a stack module, which includes an electric stack, a reformer, a heat exchanger, and so on. The stack module is composed of 27 layers of cells, each of which is made of an anode layer, a cathode layer, and an electrolyte layer. In the reformer, the following two chemical reactions (Equations (1) and (2)) take place and result in the production of hydrogen, in which Pt is selected as the catalytic metal to optimize the fuel conversion efficiency.

Table 1. The SOFC system’s parameter settings.

Parameter	Value or Type
Number of cells	27
Coverage (m3)	20
CH4 flow rate (NL/min)	0–10.0
water-carbon ratio	1–2
Operating temperature of the stack (°C)	650–850
Air flow (NL/min)	0–400
Stack’s nominal power (kW)	1
Catalyst	Pt
Catalytic processor	Allothermic
Electrolyte	Precious metals
Type of cell	Flat-paneled
Fuel	CH4
variable voltage range (V)	0–27

gives the parameters and configuration of the SOFC system studied in this experiment.

$$C{H}_4+H_{2}O\to CO+3H_2$$

(1)

$$CO+H_{2}O\to C{O}_2+H_2$$

(2)

When the SOFC system is in operation, water and a portion of the hydrogen in the cold box are introduced into the reforming chamber after preheating in the evaporator, while another portion of the hydrogen enters the reforming combustion chamber and undergoes a combustion reaction with the air, thus providing continuous high temperature conditions for the subsequent reforming reaction. In the hot box, the air–exhaust heat exchanger exchanges the high temperature exhaust gas from the exhaust combustion chamber with the cathode’s cold air introduced into the system, so that the temperature of the cathode’s air rises to about 600 °C after the first heat exchange. The fuel–air heat exchanger exchanges the reformed high-temperature fuel gas with the cathode’s air after the first heat exchange to reduce the temperature difference between the anode’s gas and the cathode’s air.

In order to maximize the energy output and reduce the overall cost of the system, the SOFC system has adopted an innovative thermal management strategy, including combined heat and power (CHP) technology, to recover the waste heat generated by the SOFC stack and tail gas combustion chamber for heating or other industrial processes. In addition, the system design emphasizes modularity and expansibility, and is easy to maintain and upgrade. The SOFC system not only improves energy efficiency but also reduces the dependence on fossil fuels, which is consistent with the goal of low-carbon energy policies. By using pure hydrogen or natural gas as fuel, the system reduces harmful emissions and contributes to sustainable industrial development.

Figure 2 shows the configuration of the SOFC system used in this study. An SOFC system with 3 kW of power runs in parallel with the hot water system, and the stack is connected with the corresponding 1 kW tail gas combustion chamber. The heat exchanger can be categorized into two parts, which are used for preheating cold air and fuel–air temperature equalization before their entrance into the stack. Prior to conducting the experiment, the measuring devices and monitoring system underwent a thorough inspection and recalibration to verify the accuracy and reliability of the measurements.

2.2. Dataset

The original dataset used in this study consisted of 82 variables that collected 662,327 observations from the real-time monitoring system at 1-s intervals, based on the data of five recent experiments [35,36]. In this process, the temperature, flow rate, and power consumption are monitored at 1 s intervals utilizing a PT 100 thermometer with a precision of 0.2%, an electronic load with an accuracy of ±0.2%, and a flow meter with an accuracy of ±0.5%.

There are too many variables in the original dataset, shown in

Table 2. Original variables and their interpretation.

Name	Interpretation	Name	Interpretation
I	The stack’s current	Premix air pressure	Value of air pressure in the straight-through exhaust gas combustion chamber
Taferburner	The temperature of the exhaust gas combustion chamber	TC17	Temperature of the exhaust gas combustion chamber’s front end
Burner CH4 PV	The methane flow rate of the reforming combustion chamber	TC19	Middle temperature of the exhaust gas combustion chamber
Stack air PV	Air velocity of the reactor cathode	AnIn Pressure	Reactor cathode’s inlet pressure
Feed water pump SV	Set flow rate of the water	TC9	Exhaust gas temperature at the inlet of the exhaust gas heat exchanger
Premix air PV	Air flow rate directly through the exhaust gas combustion chamber	TC4	Temperature of the reformer outlet’s fuel
Burner air PV	Air flow rate of the reforming combustion chamber	Heater PV	The electric heater’s pressure value and feedback value
Feed CH4 PV	Feedback value for the flow rate of methane supply	TC1	Temperature of the reformer’s inlet
TC22	Temperature of the reformer’s exhaust gas	TC24	Temperature of the stack’s anode inlet
TC3	Temperature of the reformer’s combustion chamber	TC14	Temperature of the cold gas premixing chamber of the exhaust gas combustion chamber
TC2	Temperature of the reformer’s reforming chamber	TC6	Temperature of the exhaust gas outlet of the tail gas heat exchanger
TC10	Temperature of the combustion heat exchanger’s inlet	TC0	Reformer’s ignition temperature
Stack air pressure	The stack’s air pressure	TC5	Temperature of the exhaust gas combustion chamber’s outlet
TC21	Temperature of the exhaust gas combustion chamber’s end	TC23	Final temperature of exhaust gases
Forming gas pressure	Pressure of hydrogen and nitrogen (95% nitrogen, 5% hydrogen)	TC13	Temperature of the combustion heat exchanger’s air outlet
CH4 pressure	Pressure of methane	TC11	Temperature of the fuel outlet of the combustion heat exchanger
TC15	Exit temperature of the cold gas premixing chamber of the exhaust gas combustion chamber	Tstack	The stack’s temperature
TC20	Temperature in the middle and back of the exhaust gas combustion chamber	U	The stack’s voltage
CaOut pressure	Pressure of the stack’s cathode outlet	P	Power
TC12	Temperature of the combustion heat exchanger’s air inlet

. Therefore, the data’s dimensionality was reduced using the correlation variable method and PCA (2.3 (3)), to obtain 12 variables (T_afterburner, feed CH₄ PV, burner CH₄ PV, stack air PV, feed water pump SV, premix air PV, burner air PV, I, TC22, TC3, TC2, and TC10) as the input variables and three variables (stack temperature, voltage, and temperature) as the predicted output variables; all are shown in

Table 3. Input and output variables.

Name	Units and Measuring Instruments	Typology
I	A, electronic load	Input variable
TAFERBURNER	°C, PT 100 thermometer	Input variable
BURNER CH4 PV	NL/min, flow meter	Input variable
STACK AIR PV	NL/min, flow meter	Input variable
FEED WATER PUMP SV	NL/min, flow meter	Input variable
PREMIX AIR PV	NL/min, flow meter	Input variable
BURNER AIR PV	NL/min, flow meter	Input variable
FEED CH4 PV	NL/min, flow meter	Input variable
TC22	°C, PT 100 thermometer	Input variable
TC3	°C, PT 100 thermometer	Input variable
TC2	°C, PT 100 thermometer	Input variable
TC10	°C, PT 100 thermometer	Input variable
TSTACK	°C, PT 100 thermometer	Output variable
U	V, electronic load	Output variable
P	J, P = UI	Output variable

. Meanwhile, Table 3 gives the unit of each variable, the method of obtaining it, and the type of variable it is in this study.

The power consumption of parasitic components was not included in this study. In this study, the power consumption data (kW), temperature data (°C), and airflow velocity (NL/min) of the SOFC system and its corresponding combustion chamber and hot water system were measured. At the same time, the water demand data at 1 h intervals were obtained from the water tank’s flow sensor and combined into the dataset.

2.3. Data Processing

(1)

General

Figure 3 illustrates a flow chart of the research, outlining the process of predicting the performance of the SOFC system. The appropriate input variables were identified through principal component analysis (PCA), followed by an analysis of the system’s performance and its associated variables. In terms of predictive modeling, an initial multiple linear regression (MLR I) model was established to quantitatively assess the factors influencing the system’s performance. If the assumptions of the MLR model were not met, a second MLR model (MLR II) was developed by incorporating additional variables to enhance the predictions’ accuracy. Subsequently, the optimal structure of a neural network model was determined using the same variables as the MLR model to capture any nonlinear relationships between the response and explanatory variables. Finally, the model demonstrating strong predictive capabilities was utilized as the benchmark for predicting the thermoelectric performance of the SOFC system.

(2)

Preliminary data processing

For the cogeneration of the SOFC system, we can use Equation (3) to calculate the total thermal energy output ($Q$) and electric energy output ($P$) at an interval of 1 h, combining the temperature difference between the fuel inlet and outlet ($\Delta T_{fuel}$, $k$), the increase in the temperature of the water ($\Delta T_{water}$, $k$), the unit conversion factor ($A$, $\mathrm{m}\mathrm{o}\mathrm{l}/\mathrm{L}$), and the rate of fuel flowing ($N_{fuel}$, $\mathrm{NL}/\min$) through the SOFC. The subscript “$k$”represents different cogeneration units. The molar mass ($M_{fuel}$) and specific heat capacity ($C_p$$,75.3 \mathrm{J}/\mathrm{m}\mathrm{o}\mathrm{l}·\mathrm{K}$) of water were assumed to be constant under the operating conditions, and “$t_0$” and “$t_n$” are the start and end data of the corresponding period.

$$\left\{\begin{array}{l}Q=A\times C_p\times \Delta T_k{\displaystyle \sum _{i=t_0}^{t_n}{\stackrel{\cdot }{{\displaystyle N}}}_{k,i},k=fuel, or water}\\ P={\displaystyle \sum _{i=t_0}^{t_n}{U}_i{I}_i}\end{array}\right.$$

(3)

Analysis of the performance and predictive modeling of the SOFC system were carried out by averaging the scale datasets of each time and every hour. Only when the cogeneration mode of the SOFC system was in the start-up state, we could carry out time-weighted averages for intensive variables such as fuel, the air flow rate, and temperature. For thermoelectric output variables, whether the system was running or not, these should be continuously accumulated to calculate the overall thermoelectric output. The time series of changes, the system’s operation mode, the frequency distribution, and basic statistical data of the related variables were analyzed for predictive modeling of the performance and interpretation of the results.

Given the different dimensions of the original data, we first used the mapminmax function to normalize the data, as shown in Equation (4), where $x_o$ represents the raw data, $\min$ represents the minimum value in the raw data, $\max$ represents the maximum value in the raw data, and $Y_{\max }$ and $Y_{\min }$ represent the normalized maximum and minimum values, respectively.

$$Y=(x_o-\min )\times {\displaystyle \frac{Y_{\max }-Y_{\min }}{\max -\min }}+Y_{\min }$$

(4)

(3)

PCA method

The method of principal component analysis (PCA) offers a robust mathematical structure for pinpointing the principal trends within datasets with multiple variables. By detecting inter-variable correlations within the defined state space, PCA facilitates the discovery of a new coordinate framework that aligns with the greatest variations in the dataset. This alignment enables the dismissal of subsidiary dimensions while preserving the dataset’s fundamental structure. The intricacies of the PCA’s process of reducing the dimensionality have been previously detailed in [37]. In this discussion, the essence of PCA is succinctly revisited, and a comprehensive analysis of how preprocessing and post-processing of the data influence PCA’s effectiveness will be undertaken.

The PCA methodology involves mapping n instances of $Q$ attributes onto an $(n\times Q)$ dimensional matrix $X$, with each row capturing a distinct instance of the $Q$ attributes $X$. In the context of the combustion-related examples addressed within this study, the $Q$ attributes in the matrix $X$ correspond to temperature measurements and species’ mass fraction data. The PCA technique transforms $X$ through basic rotation, which is computed via decomposition of the eigenvalues of the $(Q\times Q)$ dimensional covariance matrix.

$$M={(n-1)}^{-1}{X}^{T}X=AK{A}^T$$

(5)

wherein $A$ represents the eigenvectors and $K$ signifies the eigenvalues derived from $M$. The reoriented basis, constituted by the eigenvectors $A$, can be condensed to encapsulate primarily the most dynamic directions (the specific columns of $A$ that correspond to the most substantial eigenvalues in $M$). This results in the formation of a rectangular matrix $A_q$, onto which the initial dataset is mapped to generate the principal components (PCs), denoted $Y_q$.

Equation (5) can be reversed to yield a reconstructed approximation of the initial sample with the dimensions $(n\times Q)$.

Equation (6) facilitates a linear method of reconstruction. The fundamental linear nature of PCA poses a significant limitation when applied to highly nonlinear phenomena such as combustion. Nonetheless, this constraint can be mitigated to some extent through the use of localized PCA techniques. In contrast, nonlinear methods of reconstruction (Equation (5)) could yield more precise correlations between $Y_q$ and $X_q$.

$$Y_q=X{A}_q$$

(6)

$$X_q=Y_q{A}_q^T$$

(7)

2.4. Calculation of the System’s Performance

Through data preprocessing, it was found that the remaining key variables could be used to calculate the performance of the SOFC system. The efficiency (${\eta }_{sys}$) of the SOFC system was calculated by using the fuel consumption ($N_i$) of the SOFC, the power ($P$) generated, and the increasing trend of the cooling water’s temperature ($T$) generated by the data at 1-h intervals, as shown in Equation (8), where the subscript “$i$” is the observed data variable. The formula for calculating the efficiency of the SOFC system is as follows:

$${\eta }_{sys}={\displaystyle \frac{{\displaystyle {\sum }_{t_0}^{t_n}{P}_i}}{{\displaystyle {\sum }_{t_0}^{t_n}{N}_i\cdot LH{V}_i}}}\times 100\%$$

(8)

Among these, if $i$ is $H_2$, $LH{V}_{H_2}$ is the low calorific value of hydrogen, and the value is 241.83 kJ mol⁻¹. Only the efficiency (${\eta }_{sys}$) of the SOFC system can be calculated by Equation (8), and fuel consumption is used as the denominator. Given the performance parameters corresponding to the cooling water tank, Equation (8) can be applied to obtain the heating efficiency of the SOFC system. If the electric power required by the external load is given, the power generating efficiency of the SOFC system can be calculated by Equation (8).

2.5. Predictive Models of Performance

(1)

MLR

MLR is a model where the dependent variable ($y_i$) is determined by a linear combination of multiple independent variables or explanatory variables ($x_{ik}$), as depicted in Equation (9).

$$y_i={\beta }_0+{\beta }_1{x}_{i1}+{\beta }_2{x}_{i2}+\cdot \cdot \cdot +{\beta }_k{x}_{ik}+{\epsilon }_i$$

(9)

The parameters in Equation (9) are computed through the OLS technique, and it identifies optimal solution of the residual sum of squares ($RSS$) by Equation (10); $y_i$ represents the predicted response variable, and $E$ denotes the model’s residual.

$$RSS={\displaystyle \sum _{i=1}^n{{\displaystyle \widehat{\epsilon }}}_i^2}={\displaystyle \sum _{i=1}^n{(y_i-{\widehat{y}}_i)}^2}$$

(10)

In order for the multiple regression (MR) model to achieve the best linear unbiased estimation (BLUE), certain assumptions must be met. These include: (1) the residuals should follow a normal distribution with an expected value of zero; (2) nonautocorrelation, i.e., the residuals should not be correlated with each other; (3) homoscedasticity, i.e., each residual should have the same finite variance; and (4) among the explanatory variables, there should be no exact multicollinearity. If the assumptions of nonautocorrelation and homoscedasticity are violated, the estimation’s coefficients will remain consistent and unbiased. However, the estimated variance of the OLS estimator will be biased, leading to unreliable results in T tests, F tests, and the determination coefficient (R). The MLR model should adhere to the “saving principle” and strive for a good fit, that is, the R standard should be greatly adjusted and it should be consistent with relevant theories.

(2)

ANN

ANN is a computational method used in numerical analysis to estimate the relationship between input variables and output variables. The neural network of MLP comprises neurons, synapses, and layers. The input and output layers comprise every independent variable, which function as the neurons, and every respective corresponding dependent variable, and additional concealed layers containing different quantities of neurons can be inserted between them. A deep neural network (DNN), which has multiple hidden layers, is an artificial neural network. In both the input and hidden layers, a bias term is incorporated, and the synapses are exclusively linked to the subsequent layer. For $j$ neurons and a hidden layer, the relationship between the explanatory variable ($x_k$) and the predicted response variable ($y$) can be expressed using Equation (11) involving weight parameters (${\omega }_{kj}$ and ${\omega }_j$), constant terms (${\omega }_0$ and ${\omega }_{01}$), and an activation function ($f$) that is differentiable, and it is monotonically increasing. In this research, the hyperbolic tangent function was utilized as the activation function.

$$\widehat{y}={\omega }_0+{\displaystyle \sum _{j=1}^J({\omega }_j\times f({\omega }_{01}+{\displaystyle \sum _{k=1}^K{\omega }_{kj}{x}_k}))}$$

(11)

Regression analysis determines the weight parameters of ANN model through finding the optimal solution, minimizing the $RSS$, as defined in Equation (7). We used the BP algorithm as outlined in Equation (12) to update the parameters until the objective function, represented by the RSS, reached a specified convergence threshold. Here, the variable “$\eta$” denotes the learning rate, and “$i$” signifies the current iteration step.

$${\omega }_{kj}^{(i+1)}={\omega }_{kj}^{(i)}-\eta \times ({\displaystyle \frac{\partial RS{S}^{(i)}}{\partial {\omega }_{kj}^{(i)}}})$$

(12)

In this study, we improved the limitations of the backpropagation (BP) algorithm through the application of the gradient descent method. These limitations include sensitivity to the initial weights and abnormal data points, the possibility of staying at the local minimum, overfitting the training data, and slow calculation efficiency. In order to overcome these problems, the elastic propagation (Rprop) algorithm including a backtracking mechanism and the DBN pretraining technology was adopted to construct the neural network model. The Rprop algorithm is considered to be one of the fastest and most stable methods for updating weights. It is different from the traditional BP algorithm because it adjusts the learning rate for each parameter in each iteration, which is based on whether a partial derivative sign of a certain weight has changed in two successive iterations. In addition, the backtracking mechanism is a powerful technology which helps to avoid crossing the local minimum by reducing the weight’s updated amount when the symbol changes and retreating to the previous iteration when necessary. On the other hand, the data bank network (DBN) sets the initial values for the weights through an unsupervised pretraining process, which is regarded as a strategy to prevent the optimization process from falling into the local minimum. This kind of pretraining has been found to promote regularization and to initialize the parameters of the deep neural network (DNN) to a region close to the ideal local minimum, thus helping to improve the ability for generalization in global supervised learning. In order to verify the external predictive ability of the model and exclude the possibility of overfitting, this study used the cross-validation (CV) technique. When determining the optimal structure of the neural network and estimating the weight parameters, several different initial weights were tested.

2.6. Evaluation of Predictive Accuracy

The evaluation of the predictive accuracy of ANN and MLR models was conducted through the use of the normalized mean error ($NME$) and the normalized root mean square error coefficient ($NRMSE$). $NME$ and $NRMSE$ were derived through dividing the mean absolute error ($MAE$) and root mean square error ($RMSE$) by the response variables’ ($X$) average value, thereby rendering both indices independent of variable scales. $NME$ serves as an indicator of the overall deviation of predicted values, while $NRMSE$ assesses the mean standard error of predicted values, as delineated in Equations (13) and (14), correspondingly. In these formulas, $x_j$ denotes the measured value of the j-th variable, $y_j$ represents the predicted value of the j-th variable, and $m$ signifies the total number of observations. A nonzero $NME$ value indicates that the total error will not be offset. Equation (15) utilizes MAE to compare the predictions’ errors with those observed in other studies.

$$NME={\displaystyle \frac{{\displaystyle \sum _{j=1}^m\left|x_j-y_j\right|}}{m\overline{x}}}\times 100\%$$

(13)

$$NRMSE={\displaystyle \frac{1}{\overline{x}}}\sqrt{{\displaystyle \frac{{\displaystyle \sum _{j=1}^m{(x_j-y_j)}^2}}{m}}}\times 100\%$$

(14)

$$MAE={\displaystyle \frac{{\displaystyle \sum _{j=1}^m\left|x_j-y_j\right|}}{m}}$$

(15)

3. Results

This experiment was completed on the MATLAB software platform. The computer for analysis was the Windows 11 system, the central processor was an Intel CORE i5, the memory capacity was 16 G, and the hard disk capacity was 512 G.

3.1. Statistical Results of Thermoelectric Performance Variables

When evaluating the cogeneration performance of the SOFC system, the combined heat and power efficiency (CHP) of the SOFC system was 1.85 according to Equation (6). In a set evaluation period, the power consumption of the system per hour was 1 kWh, which only considered the power consumption of the SOFC system when generating electricity and heating at the same time, and did not include the power consumption of any auxiliary equipment such as the circulating pump. In contrast, under the standard test conditions of a fuel input temperature (FIT) of 650 °C and an exhaust temperature (ET) of 60 °C, the rated electrical efficiency (EE) of the SOFC system was 50%, while the thermal recovery efficiency (TRE) was 30%.

Table 4. Performance and operation of the combined heat and power supply of the SOFC system each time.

Time	Median Power Generation (W)	Quantity of Heating Water (L)	Peak Water Temperature (°C)	Running Time (h)
1	720	3000	65	298
2	755	2900	70	382
3	910	3800	68	503
4	810	3100	65	438
5	730	2900	65	357

summarizes the performance parameters (TPP) of the SOFC system and the average value of heating water consumed for five experiments.

Figure 4 shows the peak water temperature, quantity of heating water, median power, running time, and so on of the five experiments. Among them, the average power generated was 785 W. The average amount of heating water was 3140 L, the average peak temperature of water was 66.6 $\mathrm{℃}$, and the average running time was 395.6 h. There was a certain relationship between both the peak water temperature and water consumption, and the season in which the experiment was carried out. For example, the water consumption in winter is relatively small and the peak water temperature is low. In summer, the water consumption is greater and the peak water temperature is higher. This can explain the relationship between water consumption and peak temperature, but in order to simplify the research, this study did not consider the seasonal influence on the SOFC system for the time being.

3.2. Analysis of Related Variables Affecting the System’s Performance

According to the operation data of the SOFC system in Figure 4 above, combined with the detailed flow rate of air supply at multiple positions, temperature, air pressure and load parameters of the system, correlation analysis of the variables was carried out, as shown in Figure 5. The process of the correlation analysis of the variables was as follows.

(1)

According to the operating data of the SOFC system for five experiments, shown in Figure 4, we extracted the parameters of the fuel/air flow rate and fuel/air pressure of multiple components during each operation, as well as the load power, the stack’s temperature, the combustion chamber’s temperature, and the change in the cooling water’s temperature.

(2)

We drew a chart of the fluctuation trend of each parameter according to the time series (the trend chart has been omitted in this article) and preprocessed the data. The standard deviation method was used to eliminate outliers in during preprocessing, and the nearest neighbor interpolation method was used for missing values.

(3)

We carried out a correlation analysis of the data, sorted out the variables with large correlations and the variables with small correlations, and prepared for reducing the data’s dimensionality.

(4)

Data normalization was realized by minimum–maximum normalization to obtain data that could be used for reducing the data’s dimensionality.

(5)

Principal component analysis (PCA) was used to reduce the dimensions; the specific method is shown in Equations (5)–(7).

Through the steps above, a diagram of the variables’ correlation, as shown in Figure 5, was obtained. See the Table 2 for specific explanations of each variable.

The fuel utilization rate, the stack’s current, and the temperature of the heating water of the SOFC system are necessary conditions for understanding the system’s operating characteristics and establishing a predictive model of the thermoelectric performance. Figure 6 shows the voltage, current, and power curves of five experiments caused by factors such as the fuel’s flow rate and fluctuations in the water temperature. The rapid cooling of water expressed the process of changing the water, and the volume of water changed was 100 L each time. Although the volatility was high at this time, the process of exchanging the water was ignored in this study.

Through an analysis of Figure 6, it was found that there were differences in the duration, and the range of variation in the voltage and power during each operation of the SOFC system, which were mainly due to factors such as the inconsistent state of the used stack and the external environment.

According to the “on” or “off” state of the switch, the value of each BOP and component of the stack was set to 1 or 0, and the values of eight system subcomponents were added to obtain the parameters of the system’s components that operated together every minute, that is, an integer between 0 and 8. This parameter was named “NK” in this study, and NK represents the average number of the system’s subcomponents running at full load every hour. When making a predictive model of the performance, we used the average value of NK as one of the process’s variables.

3.3. Quantitative Analysis of the Variables of Interest

Table 5. Correlation matrix of the key variables of the thermoelectric performance of the SOFC system.

PCC	I	Tafterburner	Burner CH4 PV	Stack Air PV	Feed Water Pump SV	Premix Air PV	Burner Air PV	Feed CH4 PV	TC22	TC3	TC2	TC10	TC25	TC18	Premix Air Pressure	TC17	TC19	AnIn Pressure	TC9
Tstack	0.974	0.969	0.901	0.883	0.842	0.81	0.808	0.806	0.78	0.768	0.765	0.749	0.708	0.651	0.636	0.631	0.588	0.537	0.536
U	0.919	0.841	0.87	0.843	0.892	0.805	0.845	0.909	0.587	0.598	0.581	0.519	0.491	0.5	0.635	0.691	0.579	0.427	0.564
P	0.999	0.972	0.903	0.921	0.908	0.824	0.825	0.875	0.701	0.683	0.677	0.658	0.703	0.614	0.58	0.696	0.585	0.653	0.571
PCC	TC4	Heater PV	TC1	TC24	TC14	TC6	TC0	Stack Air Pressure	TC21	Forming Gas Pressure	CH4 Pressure	TC15	TC20	CaOut Pressure	TC12	TC5	TC23	TC13	TC11
Tstack	0.533	0.528	0.514	0.496	0.495	0.414	0.389	0.377	0.364	0.301	0.225	0.224	0.221	0.182	0.169	0.147	0.141	0.118	0.116
U	0.303	0.555	0.292	0.531	0.555	0.268	0.334	0.314	0.461	0.156	0.349	0.388	0.279	0.272	0.342	0.296	0.091	0.188	0.03
P	0.431	0.562	0.401	0.367	0.578	0.419	0.377	0.37	0.416	0.223	0.243	0.193	0.237	0.295	0.264	0.261	0.062	0.072	0.147

is the tabular form of Figure 5. Table 5 shows the Pearson correlation coefficients (PCCs) which mainly affected the performance and variables of the SOFC system. The closer a PCC’s value is to 1, the stronger the correlation of the two variables.

For the sake of quantitatively analyzing the influencing factors of the thermal and electrical performance of SOFC systems, a MLR model that demonstrated the statistical significance was established, namely Equation (16), after evaluating different combinations of functional forms and explanatory variables, including logarithmic, polynomial, and reciprocal functions. The effectiveness and predictive accuracy of the model were evaluated on the basis of the standards described in Section 2.5 and Section 2.6. To eliminate the uncertainty caused by the system’s startup and shutdown, outliers were excluded from the original 1978 datasets, accounting for 3.7%, including 12 Boolean data columns. Based on value of PCC, we chose the top 12 most correlated observed variables to model.

$$P{F}_{hourly,i}={\displaystyle \sum _{j=1}^2{\displaystyle \sum _{k=1}^2{\alpha }_{jk}{D}_j{D}_k+{\displaystyle \sum _{j=1}^2{\beta }_j{D}_j{Q}_i+{\gamma }_{1}ES{T}_i+{\gamma }_{2}EL{T}_i+{\gamma }_3{T}_{outside,i}+{\displaystyle \sum _{k=1}^2{\delta }_k{D}_{k}N{o}_{\cdot }H{p}_{on,i}+{\epsilon }_i}}}}$$

(16)

On the basis of engineering theory, we introduced four core variables that can affect the thermoelectric performance of a SOFC system into the model: external load (EL), the fuel’s flow rate (FLR), water consumption (UUV), and water temperature (WT). When the external load ran within the available power range of the SOFC, the thermoelectric performance of the SOFC system was improved under the conditions of high FLR and high UUV and WT. The linear combination of the four core variables and three outputs of the system could explain the power generation performance of the SOFC system, and all variables, including constant terms, demonstrated statistical significance, as shown in

Table 6. The MLR model’s coefficients with statistical significance.

Times	Value	b1	b2	b3	b4	b5	b6	b7	b8	b9
Time 1	Stack temperature	259.3013	−25.2129	−9.7954	0.0393	−1.5448	2.0382	−3.6983	2.5273	1.0347
	Voltage	17.5669	0.1855	1.9084	−0.0103	−0.1083	0.0656	−0.1474	−0.2017	0.0116
	Power	136.0201	−25.7975	−105.1033	1.2978	−3.3728	1.4678	−1.3815	17.0507	0.5091
Time 2	Stack temperature	525.5366	−29.2521	14.3873	0.2598	−0.1497	0.7433	0.9772	1.7931	0.0621
	Voltage	22.6174	−1.596	1.3378	0.0164	−0.0582	−0.0146	0.2514	−0.0276	−0.0008
	Power	197.8473	−3.0413	30.2764	0.03	−3.1234	−0.3156	1.0768	13.6352	−0.057
Time 3	Stack temperature	1798.0234	13.1742	−78.1279	−1.3623	0.0566	−1.1655	−18.1553	0.9617	−0.2705
	Voltage	33.7386	−0.0398	−0.2589	−0.0128	0.001	−0.0306	−0.511	−0.0357	−0.0021
	Power	978.5543	1.5459	−15.9767	−0.8382	0.1815	−1.0462	−33.1387	12.5398	−0.0507
Time 4	Stack temperature	277.1368	−5.7635	14.5393	−1.3032	0.0498	3.3466	−1.205	1.473	0.624
	Voltage	32.7131	1.2716	−2.0299	−0.0676	−0.0384	0.0829	0.4116	0.095	−0.022
	Power	596.7318	47.0304	−66.1091	−2.7356	−1.5946	3.7635	14.7878	19.7956	−0.8187
Time 5	Stack temperature	269.2489	−21.5247	20.8656	−0.0221	0.3768	1.0195	5.187	0.7194	0.4849
	Voltage	2.5948	−0.3633	−0.3121	−0.0158	−0.044	0.009	0.0676	−0.07	0.0407
	Power	−411.928	10.4022	−12.8524	−0.1902	−1.153	0.4861	−10.7301	13.44	1.1031

y = b1 + b2 × 1 + b3 × 2 +…+ b9 × 8 (×1…×8 in order for the first eight outputs, and b1…b9 are constant coefficients).

; b1–b9 are constant coefficients.

Identifying the variables that exerted a substantial influence on performance through correlation analysis posed challenges due to its limitations in capturing the effects of other variables. Correlation serves as a measure of the linear association between two variables, but it does not account for the influence of other variables that are held constant. According to the results above, we could see which variables had a significant impact on performance, so there was no need to add variables to build the MLR II model.

3.4. Results of the Predictive Model of the SOFC System’s Performance

Ongoing research initiatives are being conducted utilizing “Matlab”, which is an open-source programming language for graphical and statistical computations.

3.4.1. MLR

To construct a MLR model for forecasting the thermoelectric efficiency of solid oxide fuel cell (SOFC) systems, the following variables were also implicated, according to Equation (14): (1) the methane inlet’s flow rate, the combustion chamber’s methane flow rate, the stack’s air flow rate, the combustion chamber’s air flow rate, the air flow rate, the flow rate of deionized water, the current, and the combustion chamber’s temperature, etc., and (2) consumption of cooling water and change in temperature. These variables were selected according to the results of the principal component analysis and whether the NRMSE of the test set was improved through cross-validation. According to the analytical model of the factors influencing performance, the minute-level predictive model of the performance in power generation was the final model. We defined MLRX (X = 1, 2, 3, 4, 5) to represent the model of five training rounds.

Table 7. Minute-level predictive accuracy of the MLR model of performance obtained by cross-validation.

Model: MLR	Statistics	Stack Temperature	Voltage	Power
1	MAE	2.28	0.19	2.93
	NME	0.33	0.95	0.63
	NRMSE	0.44	1.45	0.86
2	MAE	2.03	0.18	5.64
	NME	0.33	1.22	1.18
	NRMSE	0.42	1.59	1.49
3	MAE	4.3	0.13	6.46
	NME	0.65	0.91	0.98
	NRMSE	0.98	1.28	1.44
4	MAE	2.04	0.56	20.99
	NME	0.31	3.51	3.64
	NRMSE	0.52	5.20	5.46
5	MAE	2.62	0.37	11.21
	NME	0.41	2.45	2.61
	NRMSE	0.60	3.23	3.38

shows the five MLR models’ results after cross-validation, and the results in the table are based on the whole process, with both training and testing phases being included. The best prediction for each output variable in the table has been bolded. Figure 7 shows the comparison between the predicted value of MLR model and the measured value, in which red represents the measured data, blue represents the training data, and purple represents the test data. It can be seen that the predicted value is basically consistent with the measured value, indicating that the predictive model is reliable. By averaging, applying the final MLR model, the prediction’s accuracy in terms of NRMSE improved to 1.89%.

In order to visualize the predicted effects more visually using images, Figure 8 shows the residual analysis of MLR for each round. As can be seen from Figure 8, the errors of MLR1 in predicting the temperature, voltage, and power of the stack were all within 5%, and a good predictive effect was achieved. Although the error of MLR2 for predicting the temperature, voltage, and power of the stack fluctuated greatly, the overall predictive error was kept within 10%, which could basically realize the prediction and tracking of the target value. There were some errors in the predictions of temperature, voltage, and power of MLR3, but the overall error was not high. The predictive error of MLR4 regarding the temperature, voltage, and power of the stack was large, and the overall error exceeded 20%, so the predictive effect was not ideal, which will be explained in Section 4. The predictive error of MLR5 for the temperature, voltage, and power of the stack fluctuated greatly, and the predictive effect was average.

3.4.2. ANN

In this section, the same input variables as those in the MLR model in Section 3.4.1 were selected, and the predicted values of five corresponding output variables were obtained through fitting and training the neural network. Simultaneously, we established TimeX (where X ranged from 1 to 5) to align with MLRX (where X ranged from 1 to 5). By changing the number of hidden layers and neurons by trial and error, the optimal structure of the ANN model (3 hidden layers and 10 neurons) was determined, so that the best predictive value could be obtained without overfitting.

Table 8. Minute-level predictive accuracy of performance for the ANN model obtained by cross-validation.

Model: ANN	Statistics	Stack Temperature	Voltage	Power
1	MAE	0.70	0.07	1.45
	NME	0.10	0.34	0.31
	NRMSE	0.14	0.48	0.44
2	MAE	0.98	0.07	2.82
	NME	0.16	0.51	0.59
	NRMSE	0.22	0.73	0.88
3	MAE	0.91	0.03	2.10
	NME	0.14	0.22	0.32
	NRMSE	0.24	0.34	0.52
4	MAE	0.94	0.16	5.98
	NME	0.14	0.97	1.04
	NRMSE	0.22	1.50	1.67
5	MAE	1.14	0.10	3.06
	NME	0.18	0.69	0.71
	NRMSE	0.27	1.14	1.18

shows the minute-level predictive accuracy of performance by cross-validation of the neural network model, and the best predictions for each output variable are bolded in the table. Figure 9 shows the comparison between the predicted values of the ANN model and the measured values, where red, blue and purple represent the measured data, the training data, and the test data, respectively. The experimental results showed that the predicted values of ANN were basically consistent with the measured values, and the MAE and RMSE of the three output variables were within the acceptable range, which showed that it is reliable for predicting the thermoelectric output of the SOFC system. And by averaging, the predictive accuracy in terms of NRMSE improved to 0.66%.

3.4.3. Comparison of the Predictive Accuracy

In a comparison of the predictive accuracy of these two prediction methods for the same output variable, it was clearly seen that the predictive accuracy of the ANN model was better than that of the MLR model for the five groups of results. Especially in the fourth group, it was obvious that the MAE, NME, and NRMSE of the MLR model were about 0.56, 3.51, and 5.20, respectively, but for the ANN model, they reduced to 0.16, 0.97, and 1.50, respectively. In terms of predicted power, the MAE, NME, and NRMSE of the MLR model were about 20.99, 3.64, and 5.46, respectively, but they reduced to 5.98, 1.04, and 1.67 in the ANN model. Generally speaking, in terms of the stack’s temperature, the MAE value, NME value, and NEMSE value of the ANN model were kept to 0.7–1.2, 0.1, and 0.3, respectively, with a small error. The MAE value, NME value, and NRMSE value of the MLR model remained at 0.3–0.7 and 0.3–1.0. From the above, it can be seen that the predictive accuracy of the ANN model was significantly higher than that of the MLR model. This suggests a more complicated association between the independent and dependent variables that exceeds the explanatory capacity of the current MLR framework. While the predictive accuracy of the MLR was is inferior to that of the ANN model, the predictive accuracy of the MLR model for the SOFC’s thermoelectric performance was still high.

4. Discussion

By analyzing the model coefficients in Table 6, it can be found that the parameters in the SOFC system, such as methane inlet’s flow rate, the flow rate of methane in the combustion chamber, the stack’s air flow rate, the combustion chamber’s air flow rate, the air flow rate, the flow rate of deionized water, the current, and the combustion chamber’s temperature, had significant effects on the stack’s temperature, voltage, and power output. In different models, the methane inlet’s flow rate, the flow rate of methane in the combustion chamber, the stack’s air flow rate, and the combustion chamber’s air flow rate had negative effects on the combustion chamber’s temperature. Reducing these flow rates is beneficial for reducing the stack’s temperature and realize the control of the temperature of the SOFC’s stack. The air flow rate, the flow rate of deionized water, and the combustion chamber’s temperature had positive effects on the stack’s temperature, and controlling these parameters is beneficial for controlling the stack’s temperature. However, the methane inlet’s flow rate, the combustion chamber’s methane flow rate, and the stack’s air flow rate had positive effects on voltage in most models, and increasing these flow rates appropriately would be beneficial for improving the voltage output. The air flow rate, the flow rate of deionized water, and the current had little influence on the voltage and had different effects. The direction and magnitude of the influence of air velocity, the current, and the combustion chamber’s temperature were different in different models, which indicates that their relationship with the power output is complex and may be influenced by other parameters and working conditions of the system. The key parameters above have an important influence on the temperature, voltage, and power of the stack, and the specific situation may need to be further studied in combination with specific working conditions. These analytical results provide theoretical guidance for optimization of the SOFC system’s operation and may help to improve the performance and efficiency of the stack by fine-tuning various parameters.

Further discussing the correlations between the variables,

Table 9. Values of selected variables at different flow rates of methane.

CH4 Flow Rates (NL/min)	Combustion Chamber’s Temperature (°C)	Stack’s Temperature (°C)	P (J)
6.1	688	550	18.974
6.3	683	549	68.284
6.6	691	555	100.643
6.9	685	567	134.159
7	684	564	133.735
7.3	765	679	806.654
7.4	690	564	159.621
7.7	751	702	788.277
7.8	767	639	765.666
8	756	580	265.082
8.2	801	640	654.654
8.9	800	604	478.241
9	798	607	493.068

lists the combustion chamber’s temperature, the stack’s temperature, and power at different flow rates of methane in this study. It can be observed that as the methane’s flow rate increases, the changes in the other variables are not linear, especially the change in power. This is because a change in quantity will inevitably cause changes in other variables, and a variable is influenced by many factors. The variables do not have a simple linear relationship with each other. This is the dynamic nonlinearity within the SOFC system. Therefore, machine learning methods are needed. This can help us accurately predict the changes in the output of complex systems.

In addition, from the results of the two different forecasting methods, MLR and ANN, as seen in Table 7 and Table 8, the accuracy of the predicted voltage was higher than the accuracy of predicted power. This is because the power is the product of the current and the voltage, and the current is larger, so the error of the output power will become larger. Moreover, through the results in Section 3.4, it was found that regardless of the method, the fourth group of data predicted significantly worse results than the other groups. This is because the machine malfunctioned when carrying out the experiments for that group, making the actual values obtained somewhat wrong. This is why the predictions were not as good as they could have been. Nonetheless, the predictions were still roughly in line with the trend of the measured data. This shows that the prediction models built by the two methods above with the wrong data still have a certain degree of accuracy, indicating their robustness.

The prediction models built by both methods used in this study had high average predictive accuracy, with a mean NRMSE of 1.89% for the MLR model and 0.66% for the ANN model. In comparison with the other prediction methods available, the GA-BP algorithm model had the lowest predictive accuracy of 1% [7], and the ANN model in this study was significantly better than this. If the fourth set of data with errors is removed, then the predictive accuracy in terms of NRMSE for the ANN model was reduced to less than 0.5%. Compared with to the LSTM method used in [6], the ANN method used in this study was simpler and more efficient, with similar predictive accuracy. In addition, compared with the BPNN method [38], the advantages of the two methods used in this study were more obvious [38]. Overall, the two models used in this study, the ANN model and the MLR model, not only had high predictive accuracy but were also simpler and more efficient than other prediction models that have been used.

Meanwhile, in this study, temperature, flow rate, and power consumption were monitored at 1 s intervals utilizing a PT 100 thermometer with a precision of 0.2%, an electronic load with an accuracy of ±0.2%, and a flow meter with an accuracy of ±0.5%. These measuring instruments had a certain amount of error anyway, which made this study have a certain amount of uncertainty. In order to analyze this, Equation (17) was used to calculate the total uncertainty of this study [39], where $E_1$, $E_2$, and $E_3$ represent the error of each measuring instrument, respectively. and $\overline{E}$ represents the total uncertainty of this study. After calculation, it was found that the uncertainty of this study was 0.57%.

$$\overline{E}=\sqrt{{(E_1)}^2+{(E_2)}^2+{(E_3)}^2}$$

(17)

5. Conclusions

By integrating field monitoring data and using the techniques exploratory data analysis, the thermoelectric performance of a single SOFC system (comprising a combustion chamber and a cooling water supply subsystem) was computed, and the system’s operating patterns were determined on the basis of five instances of the system’s operating data. The MLR model, demonstrating statistical significance, reliably elucidated the quantitative impacts of the influencing variables on the system’s performance, providing logical grounds for analyzing variations in the system’s performance, such as identifying the key factors affecting the characteristics of the system’s output through an analysis of correlated variables. The MLR (1.89%) and ANN (0.66%) models developed in this study exhibited satisfactory precision in predicting the minute-level thermoelectric performance of the system, serving as references for real-time monitoring of the SOFC system’s performance. And it can provide guidance for improving the thermoelectric performance of SOFC systems, so as to contribute to the cause of sustainable development. This study underscores the advantage of MLR in quantitatively analyzing performance-influencing factors, while also demonstrating the superior predictive capability of ANN in forecasting the system’s performance. Although this study focused on SOFC systems, the same methodology and procedures can be extended to develop predictive models for other types of energy facilities and corresponding time periods.