Application of Artificial Intelligence Methods in the Analysis of the Cyclic Durability of Superconducting Fault Current Limiters Used in Smart Power Systems

Abstract

This article presents a preliminary study on the potential application of artificial intelligence methods for assessing the durability of HTS tapes in superconducting fault current limiters (SFCLs). Despite their importance for the selectivity and reliability of power networks, these devices remain at the prototype testing stage, and the phenomena occurring in HTS tapes during their operation—particularly the degradation of tapes due to cyclic transitions into the resistive state—are difficult to model owing to their highly non-linear and dynamic nature. A concept of an engineering decision support system (EDSS) has been proposed, which, based on macroscopically measurable parameters (dissipated energy and the number of transitions), aims to enable the prediction of tape parameter degradation. Within the scope of the study, five approaches were tested and compared: Gaussian process regression (GPR) with various kernel functions, k-nearest neighbours (k-NN) regression, the random forest (RF) algorithm, piecewise cubic hermite interpolating polynomial (PCHIP) interpolation, and polynomial approximation. All models were trained on a limited set of experimental data. Despite the quantitative limitations and simplicity of the adopted methods, the results indicate that even simple GPR models can support the detection of HTS tape degradation in scenarios where direct measurement of the critical current is not feasible. This work constitutes a first step towards the construction of a complete EDSS and outlines directions for further research, including the need to expand the dataset, improve validation, analyse uncertainty, and incorporate physical constraints into the models.

1. Introduction

The analysis of measurement data obtained from experiments on electrical materials constitutes an important component of contemporary research in the fields of materials engineering and electrical engineering [1]. Such data often reflect complex physical phenomena that are difficult to describe unambiguously using classical analytical methods. This necessitates an approach capable of identifying patterns and relationships hidden within the data, even when the number of measurement points is limited. In response to these challenges, artificial intelligence methods—particularly machine learning algorithms—are increasingly being employed, as they enable the modelling of complex interdependencies between variables and the prediction of the behaviour of the systems under investigation [2]. These tools allow not only for the efficient processing of data, but also for the development of systems that support the interpretation of experimental results and the prediction of selected material parameters [3]. One of the areas where artificial intelligence methods show particular promise is in modern smart power systems. In the face of the ongoing energy transition, the growing share of renewable energy sources, and the need to ensure the reliability and security of grid operation, the integration of advanced data analysis tools is becoming a key element in the development of power infrastructure [3].

The dynamic development of technology across various industrial sectors, as well as the rising standard of living in societies, is leading to a systematic increase in the demand for electrical energy [4]. Simultaneously, growing environmental awareness is driving a shift away from conventional energy sources based on fossil fuels in favour of renewable energy sources (RES) [5]. This type of energy transition necessitates the continuous expansion and modernisation of the power infrastructure. The integration of new generating units into the existing power system is transforming its structure from a hierarchical to a distributed one, where energy sources are connected at deeper points within the grid. In particular, the connection of RES units to medium-voltage grids results in an increase in short-circuit power at the point of integration, which consequently leads to a rise in short-circuit current levels. Elevated prospective short-circuit currents can pose a threat to the existing technical infrastructure due to the adverse dynamic and thermal effects arising from their flow [6]. Therefore, in modern medium-voltage smart power grids, methods for limiting short-circuit currents are applied, based either on increasing the short-circuit impedance or on directly reducing the magnitude of the short-circuit current [7]. One of the promising approaches to short-circuit current limitation in contemporary power systems involves solutions based on the concept of superconducting fault current limiters (SFCLs). SFCLs can serve as advanced components of smart power grids, playing a key role in enhancing grid safety and reliability by rapidly and effectively limiting short-circuit currents [7]. SFCLs enable the safe integration of RES by mitigating the adverse effects of faults, which is essential in the context of the energy transition and the expansion of RES [8]. Their fast response time (in the range of 2–3 ms) and virtually maintenance-free operation make SFCLs an excellent alternative to conventional methods of current limitation. However, their technology readiness level remains relatively low. The literature indicates that superconducting tapes used in the construction of SFCLs may be subject to mechanical damage (such as deformation [9,10], delamination of the stabiliser layer [11,12], or failures at soldering points [10,13]) as well as degradation of the critical current [14,15,16,17], all of which affect the device’s durability [18].

Despite the wide implementation of SFCLs tests in medium-voltage grids (MV) [19,20], and even high-voltage grids (HV) [21], these devices still remain in the prototype phase and require further research to improve their reliability and increase their technology readiness level. Research aimed at estimating the service life and assessing the functional performance of these devices is particularly important to prevent unexpected failures. Equally crucial are studies focused on the superconducting tapes used in SFCL constructions. Of particular significance—though often overlooked—is the issue of the device’s cyclic durability (the number of transitions between the superconducting and normal state without degradation of the superconducting tape parameters), as well as the resilience of the tapes to rapid changes in temperature and current during SFCL operation [8].

The superconducting tapes used in SFCLs are manufactured as multilayer, second-generation, high-temperature superconducting (HTS 2G) tapes [22]. These are employed in SFCLs due to their ability to transition abruptly from the superconducting state to the resistive state, where they are capable of limiting short-circuit currents upon the exceedance of a critical parameter—the critical current, $I_C$. One of the most important parameters of the HTS tapes is their critical current, since it is this value that triggers the operation of the SFCL—that is, the transition from the superconducting to the resistive state. Studies have shown that this parameter may degrade as a result of exposure to short-circuit currents. The degradation mechanism of $I_C$ is complex and depends on several factors: the type and structure of the HTS tape, the energy dissipated during the surge current, the duration of SFCL operation during the fault event, and the number of times the tape is driven out of the superconducting state by a prospective short-circuit current [18].

The main contribution of this article lies in the application of machine learning methods to the analysis of the cyclic durability of SFCLs. The presented approach is based on the practical use of artificial intelligence algorithms, such as random forest (RF), k-nearest neighbours (k-NN), and Gaussian process regression (GPR), for modelling and predicting changes in operational parameters. The article focuses on the development of an engineering decision support system (EDSS), which functions as a predictive system integrating conventional machine learning techniques.

The main aim of the EDSS is to support engineers in utilising measurement data to address technical challenges. The system will perform predictions of the expected change in the critical current of the SFCL. If the critical current $I_C$, decreases due to the operational use of the SFCL, the limiter will begin to activate at lower values of the prospective short-circuit current. This will necessitate adjustments in the settings of the power system protection (PSP) [23]. Furthermore, within the adopted prediction algorithms, it is possible—over a limited number of operations—to forecast changes in the critical current $I_C$, decreases due to the operational use of the SFCL, where the limiter will begin to activate at lower values of the prospective short-circuit current. This will necessitate adjustments in the settings of the PSP. More advanced predictive algorithms based on measurement data must be employed.

The approach presented in this work addresses the need to develop advanced analytical tools capable of efficiently processing data obtained from experiments and measurements of real-world physical systems. The proposed system will enable the functional assessment and prediction of the cyclic durability of SFCLs.

However, this study does not present a complete engineering decision support system. Instead, experimental research was conducted on the application of artificial intelligence methods as predictors of critical current ($I_C$) degradation in HTS tapes, which could potentially be integrated into a future EDSS. The aim was to verify whether it is possible to obtain reliable estimates of operational parameters—particularly, the value of $I_C$—based on operationally available data, such as the number of transition cycles and the dissipated energy. The analysis, therefore, constitutes an initial stage—an attempt to develop a predictive model that could serve as a component of a future real-time SFCL condition monitoring system. The use of such a model is justified by the lack of technical feasibility of directly measuring the $I_C$ value under real operating conditions, as well as by the growing need for dynamic coordination between SFCL operation and PSP schemes. The proposed approach is not intended for immediate implementation but, rather, forms the basis for further research and development work on a complete decision support system.

2. Characteristics of the Tested HTS Tapes

The study employed two high-temperature second-generation (2G HTS) superconducting tapes of the SF12100-CF type, manufactured by SuperPower. Both tapes were fabricated using thin-film technology and are suitable for application in SFCL systems. They feature a 100 $\mathrm{\mu }\mathrm{m}$ thick Hastelloy substrate. The substrate is separated from the superconducting layer by a diffusion barrier, consisting of buffer layers, the structure of which is presented in

Table 1. Buffer layer structure in the HTS 2G tapes.

Buffer Layers	Layer Thickness
alumina	∼80 $\mathrm{nm}$
yttria YSZ	∼7 $\mathrm{nm}$
IBAD MgO	∼10 $\mathrm{nm}$
Homo-epi MgO	∼20 $\mathrm{nm}$
LMO $\left({\mathrm{LaMnO}}_3\right)$	∼30 $\mathrm{nm}$

. Hastelloy accounts for over 94% of the constituent materials of the HTS tapes and is a non-magnetic alloy with high resistivity, composed of Ni 57%, Mo 15%, Cr 15.5%, Fe 5.5%, W 4%, and Co 2.5% [24]. The differences between tapes TP1 and TP2 concern the thickness of the silver layer and the critical current value declared by the manufacturer. The selection of the samples was intentional—it enabled the investigation of the impact of structural differences between nominally similar tapes on their resistance to cyclic overloading and the effectiveness of degradation parameter prediction. The parameters of the HTS tapes examined in this study are presented in

Table 2. The HTS 2G tape parameters without copper stabilizer.

HTS Tape	TP1	TP2
silver layer thickness	42 µm	22 µm
YBCO layer thickness	12 µm	12 µm
width	12 mm	12 mm
thickness	0.105 mm	0.105 mm
substrate thickness (Hastelloy)	0.1 mm	0.1 mm
minimum critical current $I_{C\left(77K\right)}$	312 A	281 A

.

3. Measurement Setup and Research Methodology

In order to obtain data for the development of the recommendation system, it was necessary to evaluate the degradation of the critical current value in the selected HTS tapes. The tests were carried out under laboratory conditions on 10-centimetre samples of the HTS tapes. The influence of surge currents on the HTS tape samples was considered, corresponding to prospective short-circuit currents in terms of the energy dissipated during the surge event and the number of transitions from the superconducting state (the number of repetitions).

The measurement setup (Figure 1) was equipped with current transformers capable of setting an output current of up to 1 kA (which corresponds to short circuit currents) with an open circuit voltage at the transformer output of 6 V. In addition, the measuring system includes a digital oscilloscope, current probe with 1 kA measumerent range, voltage probe with voltage gain 1:1, and cryostat enabling operation with liquid nitrogen. The current transformers are controlled by a programmable voltage source (IT7626) with a power rating of 3 kVA. The measurement setup in a general diagram is shown in Figure 2. The HTS tape samples were mounted in measurement holders (Figure 3) and cooled in a liquid nitrogen bath. A detailed description of the test scope and methodology is provided in [18].

To explain the research methodology, first, we describe the signals in the test setup. The first signal is the prospective short-circuit current ($I_p$). This is the current that would flow in a given circuit with an HTS tape or SFCL arrester if it were not limited by the tape’s transition from the superconducting state. In other words, it would be the current that would flow in the circuit when the superconducting resistance were zero—operation in the superconducting state. The $I_p$ impulse is called a surge current impulse. If the surge current reaches the HTS tape’s critical current $I_C$, it will cause the tape to transition from the superconducting state. The critical current of the HTS tape is a technical parameter of the tape defining the current below which the tape is in the superconducting state. Exceeding the current value $I_C$ causes the tape to transition to a resistive state (the HTS tape’s resistance is different from zero).

The duration of the surge current impulse was set at 0.2 s. The amount of energy dissipated in the sample during the surge was calculated based on the measurement of instantaneous current and voltage values:

$$E={\displaystyle \frac{\Delta t}{n}}\sum _{k=1}^n{u}_k·i_k$$

(1)

where E—the amount of energy dissipated in the sample, $\Delta t$—the duration of the impulse, $u_k$—the instantaneous value of the voltage across the tape, $i_k$—the instantaneous value of the current, and n—the number of samples.

Because the critical current of the tape varies along successive sections of the tape, the critical current of the tested tape fragment must be precisely determined. Direct current (DC) and alternating current (AC) methods are used to determine currents $I_C$ in HTS tapes [25]. Since the HTS tape sample will be used in an AC power application, the AC method for determining current $I_C$ was choosen. This method utilizes a sinusoidal excitation current with a frequency of 50 Hz and an adjustable amplitude of the test current I. By varying the current I value, the current $I_C$ of the tested samples must be determined as the minimum amplitude of current I that initiated the sample’s transition from the superconducting state. Figure 4 presents idealized time histories of currents and voltages in the system in the superconducting and resistive states. The transition of the tape to the resistive state is illustrated by an increase in the voltage across the sample and/or a distortion of the voltage waveform, along with a decrease in the amplitude of the current I.

Figure 5 presents the real mesurement waveforms of current and voltage recorded on the oscilloscope in experimental setup from Figure 1, for samples in the superconducting state (Figure 5a) and the resistive state (Figure 5b). In Figure 5b, where the HTS tape sample is in the resistive state, a decrease in measured current and distortion of the voltage waveform (peak deformation of the sine wave) can be observed. In Figure 5a, where the tape is in the superconducting state, the current remains constant and the voltage retains a sinusoidal shape.

The study was carried out for two HTS tapes, designated TP1 and TP2, whose rated parameters are listed in Table 1. Eight samples were tested for each HTS tape. After determining the initial critical current value $I_{C0}$, the samples were subjected to surge current impulses $I_S$. Each sample was exposed to a surge current selected so as to achieve different energy levels dissipated in the sample during the impulse duration. Subsequent measurements of the $I_C$ value, performed in accordance with the method shown in Figure 4 and Figure 5, were taken after 3, 10, and 25 transitions from the superconducting state. After each surge, a pause followed during which the HTS tape sample was cooled and returned to the superconducting state. The obtained results clearly indicate a degradation in the $I_C$ of the HTS tapes under repeated exposure to surge currents $I_S$. The results showing the percentage change in critical current for both HTS tapes are presented in Figure 6 and Figure 7, respectively.

A decrease in the critical current $I_C$ was observed for both examined HTS tapes, progressively increasing with the number of transitions from the superconducting state triggered by impulse current surges. For tape TP1, which featured a silver layer thickness of 4 $\mathsf{\mu }\mathrm{m}$, the decrease in $I_C$ value was below 3% if the energy released on the sample during the impact did not exceed 158.47 J, regardless of the number of sample leads out of the superconducting state. In other cases, at higher energy levels, the rate of $I_C$, reduction was strongly dependent on the number of such transitions and reached as much as 45.28% at the highest tested energy for 25 transitions. For the TP2 tape with a 2 $\mathsf{\mu }\mathrm{m}$ thick silver layer, changes in the $I_C$ value below 3% were observed for energy levels up to 90.73 J and up to 25 transitions from the superconducting state. At higher energy levels, a similarly small drop in $I_C$ occurred only after three transitions from the superconducting state. For 10 transitions, the reduction remained below 5%, and for 25, it reached approximately 8% throughout the tested energy range.

Because the value of energy generated in the sample as a result of the action of the surge current impulse is associated with higher temperature values reached by the samples, it can be concluded that a process of thermal ageing occurred in the tested tapes. The intensity of this process in the tested tapes is determined by the thickness of the silver layer, because the resistance of the HTS tape after leaving the superconducting state is a function of this thickness. Tape TP2 has nearly twice the value of resistance, which causes it to heat up to higher temperature values during the surge (Figure 8). That is why a greater percentage decrease in the value of $I_C$ is observed for tape TP2 (in comparison with tape TP1), even at lower values of generated energy. The process of cooling the samples in liquid nitrogen is also of significance. The intensity of heat dissipation from the HTS tape into the liquid nitrogen depends on the temperature difference between the sample surface and the liquid nitrogen (Leidenfrost effect) [26,27], which is shown in Figure 9. Therefore, different operating conditions during a fault should be expected for both HTS tapes, and consequently, a different course of changes in the value of $I_C$ should be observed, as shown in Figure 6 and Figure 7.

4. Methodology of Experimental Data Processing and Analysis

In order to analyse the degradation of the $I_C$ in the HTS tapes as a function of the generated energy in the tapes and the number of repetitions of surge current impulses, a predictive system was developed and implemented in the MATLAB environment. The system incorporates three approaches: interpolation, surface approximation, and machine learning models (random forest, k-NN, GPR). The input data include experimental values of the relative decrease in critical current, recorded for four levels of the number of repetitions (0, 3, 10, and 25) and eight values of the impulse energy. The degradation of the superconducting properties of the HTS tapes was described using the indicator $\Delta I_C$, denoting the relative decrease in the $I_C$ value with respect to its initial value $I_{C0}$. It was defined as

$$\Delta I_C={\displaystyle \frac{I_C}{I_{C0}}}$$

(2)

or, if the result is expressed as a percentage,

$$\Delta I_C={\displaystyle \frac{I_C}{I_{C0}}}100\%$$

(3)

4.1. Interpolation Method

In the first stage, two-dimensional interpolation was performed. The data were initially interpolated along the axis representing the number of repetitions using the interp1 function with the PCHIP method (piecewise cubic Hermite interpolating polynomial), which preserves monotonicity and the continuity of the first derivative of the data [28]. The use of the function enabled the generation of intermediate values for the range of 0–25 repetitions. Next, for each of the new points, interpolation with respect to the energy values was carried out, resulting in a refined data grid with 100 points along the energy axis. This made it possible to estimate the values of the function at locations not covered by the experimental measurements, while preserving the shape of the original data. The results are presented in the form of a three-dimensional plot.

4.2. Surface Approximation Method

In the second stage, surface approximation was performed based on a second-degree polynomial with two variables (energy and number of repetitions), using the fit function and the poly22. This type of approach is widely used in materials engineering to develop simple predictive models based on a limited number of measurement points [29]. The approximating model was subsequently applied to the full data grid (energy vs. number of repetitions), allowing for the estimation of the decrease in $I_C/I_{C0}$ at points not covered by the experimental measurements.

4.3. Application of Machine Learning

The third stage involved the implementation of three machine learning algorithms.

• Random Forest (RF): the random forest model is based on a committee of decision trees aggregated to reduce variance and improve prediction accuracy. In the analysis, 50 decision trees were used (TreeBagger in regression mode). This algorithm is resistant to overfitting and performs well with nonlinear data and noise [30].
• k-Nearest Neighbours (k-NN): the k-NN algorithm (in regression mode) assigns the output value as the average of the values of the k nearest training points. In the analysis, $k=3$. This algorithm is characterised by its simplicity and intuitiveness, but it is sensitive to the distribution of data in the feature space [31].
• Gaussian Process Regression (GPR): Gaussian process regression is a probabilistic model that provides not only the predicted value but also its associated uncertainty. In this study, a linear basis function and a squared exponential kernel were used (squaredexponential) [32]. The use of alternative kernel functions (e.g., rational quadratic or Matern32), which may better handle nonlinear trends and a certain degree of extrapolation, could support the extension of the predictive system beyond the experimental range of energy values and the number of transitions of HTS tapes from the superconducting state. GPR models are widely used in the analysis of experimental data and the modelling of physical processes due to their high accuracy and ability to capture nonlinearity.

The prediction results were transformed into a matrix format consistent with the interpolation grid and mapped onto three-dimensional surface plots.

5. Artificial Intelligence-Based Machine Learning in the Analysis of Cyclic Durability of Superconducting Fault Current Limiters

A predictive system architecture was designed, consisting of several key components: the selection of input data, the choice of analysis methods, and forecast generation. The system operates on experimental data describing the percentage decrease in the value of the critical current $I_C$ of the HTS tapes as a function of the energy generated during the surge and the number of transitions of the HTS tape from the superconducting state. The experimental data were subjected to multidimensional interpolation in order to obtain a dense grid of values, assigning to each combination of input parameters (energy and number of repetitions) the corresponding percentage decrease in the $I_C$. Interpolation methods, surface approximation techniques (including polynomial surface fitting), and machine learning algorithms for regression—such as RF, k-NN, and GPR—were employed.

Based on the specified parameter values, the system predicts the percentage decrease in the $I_C$, enabling the forecasting of material behaviour beyond the range of the original measurement data (extrapolation). Appropriate validation of the results was carried out in the regression models, along with visualisation of the fit to the experimental data. To ensure the reliability of the predictions, the input data were subjected to an initial quality analysis; however, no automatic data cleaning was performed at the current stage. The applied regression methods allow for the accurate modelling of nonlinear relationships between the input parameters and the decrease in the value of the $I_C$. A simplified diagram is presented in Figure 10 to illustrate the main mechanisms of the system.

5.1. k-Nearest Neighbours (k-NN) Algorithm

The k-NN algorithm is a machine learning method used for both classification and regression, operating on the principle of analogy: In order to predict the outcome for a new, unknown data point, it searches for the k most similar cases among the existing training data and formulates a response based on them. The operation of the algorithm is based on mathematical relationships: a distance metric and a prediction function for regression.

A distance metric is a mathematical function that defines the similarity between observations in the feature space. In the analysis of experimental data concerning the degradation of superconducting materials, the Euclidean metric can be applied. For two points $p=(p_1,p_2)$ and $q=(q_1,q_2)$ in a two-dimensional feature space (generated energy and number of transitions from the superconducting state), the Euclidean distance is defined as

$$d(p,q)=\sqrt{{(p_1-q_1)}^2+{(p_2-q_2)}^2}$$

(4)

where $p_1,q_1$ represent the values of generated energy, and $p_2,q_2$ correspond to the number of transitions for the respective points. This metric measures the geometric distance between experimental conditions in the parameter space. For normalised data (when both features are on a comparable scale), minimising this distance corresponds to identifying conditions with the most similar influence on the degradation of the critical current.

The prediction function for k-NN regression with $k=3$ nearest neighbours (as adopted in the calculations) can be defined as follows for each new point:

$${\mathbf{x}}_{new}=(E_{new},R_{new})$$

(5)

where $E_{new}$ is the generated energy and $R_{new}$ is the number of transitions from the superconducting state. The prediction function is expressed by the formula

$${\widehat{y}}_{new}={\displaystyle \frac{1}{k}}\sum _{i=1}^k{y}_i$$

(6)

where $y_i$ denotes the percentage decrease in the value of the critical current corresponding to each of the k nearest points ${\mathbf{x}}_i$ in the training dataset. The value of the parameter $k=3$ was selected experimentally. It yielded the lowest prediction error in validation tests for the available dataset. In the case of a small number of samples, low k values allow the model to retain local sensitivity while simultaneously reducing the risk of excessive smoothing of the results.

In the proposed predictive system, the k-NN algorithm was, therefore, used for regression, i.e., for predicting continuous values across the full range of energy generated in the HTS tapes and the number of transitions from the superconducting state (from 0 to 25). Its purpose was to predict the percentage decrease in the value of the critical current based on these two features. The system averages the critical current degradation values (%) from the three most similar experimental conditions—those with the smallest Euclidean distance in the energy–transition space. The choice of the k-NN algorithm is justified by the nature of the experimental data and the specifics of their analysis. The main advantage of the k-NN in this context is that it does not require any assumptions regarding the distribution of the data or the mathematical form of the relationship between variables—this is essential, as experimental data often exhibit complex, nonlinear relationships that are difficult to describe using a simple analytical model. However, the method is sensitive to the choice of the parameter k. A value of k that is too small (e.g., 1) leads to overfitting, as the model becomes overly responsive to noise, while a value that is too large (e.g., 10) results in excessive smoothing of the output. The choice of $k=3$ was, therefore, motivated by the need to maintain sensitivity to local variations in the measurement data. With this setting, the algorithm was better able to capture differences in material degradation for specific combinations of generated energy and the number of transitions, which is particularly important in the case of highly variable data. At the same time, this choice increased the model’s sensitivity to potential measurement errors or noise in the data, which represents a typical trade-off when using this method.

5.2. Random Forest Algorithm

Another algorithm used was RF, which served as a machine learning method for modelling nonlinear relationships between the experimental parameters and the degradation of the superconducting properties of the HTS tapes, expressed as the percentage decrease in the critical current $I_C$. The number of decision trees in the forest was set to 50. Preliminary analysis showed that increasing this number (e.g., to 100) did not lead to a significant improvement in performance metrics (i.e., RMSE, MAE), while considerably increasing the model training time. Therefore, a compromise value was adopted to ensure result stability and a reasonable computation time. The RF is an ensemble learning method in which a collection of T independent decision trees is constructed, each trained on a randomly selected subset of the training data with replacement (known as bootstrap sampling). In the system presented here, $T=50$ regression trees were used, trained on a dataset containing $n=36$ measurement points.

During the construction of each tree, at each split node, a random subset of m input features was considered when selecting the splitting rule. In the analysed case, where the number of input features is two (generated energy E and number of transitions from the superconducting state R), $m=\sqrt{2}\approx 1$. For each of the considered features, the algorithm determines a threshold value that splits the data in a way that minimises the prediction error. In regression problems, the quality of a split is evaluated by the reduction in the mean squared error (MSE). For a given split into two subsets—the left subset (denoted as L) and the right subset (denoted as P)—the MSE is calculated according to the following formula:

$$\mathrm{MSE}={\displaystyle \frac{1}{n_L+n_p}}(\sum _{i=1}^{n_L}{(y_i-{\overline{y}}_L)}^2+\sum _{j=1}^{n_p}{(y_j-{\overline{y}}_p)}^2)$$

(7)

where $n_L$ and $n_P$ are the numbers of samples in the left and right subsets, respectively, $y_i$ is the target value for the i-th sample in the left subset (i.e., the predicted percentage decrease in the critical current), $y_j$ is the target value for the j-th sample in the right subset, ${\overline{y}}_L$ is the mean target value in the left subset, and ${\overline{y}}_p$ is the mean target value in the right subset. ${\overline{y}}_L$ i ${\overline{y}}_p$ are calculated using the following formulas:

$${\overline{y}}_L={\displaystyle \frac{1}{n_L}}\sum _{i=1}^{n_L}{y}_i$$

(8)

$${\overline{y}}_p={\displaystyle \frac{1}{n_P}}\sum _{j=1}^{n_L}{y}_j$$

(9)

The MSE value represents the total prediction error for a given data split. During tree training, the algorithm selects the split for which this value is minimised, which corresponds to the greatest reduction in prediction uncertainty. The trees were constructed recursively until one of the stopping criteria was met, such as the number of samples in a node being smaller than a minimum threshold (default: 5), no further splits resulting in a significant error reduction, or the samples in the node being homogeneous.

In the analysed case, no additional pruning was applied, which allowed each tree to grow to its maximum depth according to its structure and the content of the data. Although individual trees may be prone to overfitting, aggregating their outputs by averaging effectively reduces the overall model variance. The final prediction of the ensemble of decision trees (random forest) for a given feature vector $\mathbf{x}=[E,R]$, expressed as the mean of the predictions obtained from all trees in the forest:

$$\widehat{z}\left(\mathbf{x}\right)={\displaystyle \frac{1}{T}}\sum _{t=1}^T{h}_t\left(\mathbf{x}\right)$$

(10)

where $\widehat{z}\left(\mathbf{x}\right)$ is the final predicted value (the percentage decrease in the value of the $I_C$) for the given input point $\mathbf{x},T$ is the total number of decision trees in the forest (50 in the analysed case), $h_t\left(\mathbf{x}\right)$ is the prediction obtained from the t-th decision tree for the point $\mathbf{x}$, and $\mathbf{x}=[E,R]$ is the input feature vector. Aggregating predictions by arithmetic averaging reduces the impact of individual erroneous models and minimises variance, which leads to a more robust and generalisable final prediction.

5.3. Gaussian Process Regression Algorithm

The GPR algorithm was applied as one of the machine learning methods for modelling the nonlinear relationship between the experimental parameters and the decrease in the $I_C$ value of the HTS tapes. As in the previous cases, the purpose of using the algorithm was to predict the percentage decrease in the critical current as a function of the generated energy E and the number of transitions from the superconducting state R, within the range $R\in [0;25]$.

The GPR is a supervised learning method used for nonlinear regression. It assumes that the output data (the percentage decrease in the critical current) are samples drawn from a multivariate normal distribution—the so-called Gaussian process. In this approach, a single explicit prediction function is not defined; instead, a probability distribution is assumed over a set of possible functions that describe the data. Each of these functions is evaluated by a covariance function (kernel), which models the similarity between input points.

In the analysed case, the vector $\mathbf{y}$ contains known target values (labels) corresponding to the training points ${\mathbf{x}}_i=[E_i,R_i]$, (feature vector), that is, the observed percentage values of the decrease in the $I_C$. The objective of training was to find the relationship between the input features ${\mathbf{x}}_i$ and the output values $y_i$, and then use this relationship to predict the value ${\widehat{y}}_{new}$ (the percentage decrease in the critical current value) for a new point ${\mathbf{x}}_{new}=\left[E_{new,}{R}_{new}\right]$.

Training the GPR model involved using the training data set

$$D={\{{\mathbf{x}}_i,y_i\}}_{i=1}^n=\{([E_1,R_1],y_1),([E_2,R_2],y_2),\dots ,([E_n,R_n],y_n)\}$$

(11)

where $n=36$ is the number of all input data points obtained from measurements (the number of training points).

The covariance function (kernel), which describes the similarity between input points $\mathbf{x}=[E,R]$ and ${\mathbf{x}}^{\prime }=[E^{\prime },R^{\prime }]$, assumes that the more similar these points are, the more similar the corresponding output values y will be. In the analysed case, the following covariance function was applied:

$$k(\mathbf{x},{\mathbf{x}}^{\prime })={\sigma }_f^2·exp\left(-{\displaystyle \frac{1}{2}}{(\mathbf{x}-{\mathbf{x}}^{\prime })}^T{\Lambda }^{-1}(\mathbf{x}-{\mathbf{x}}^{\prime })\right)$$

(12)

where $\mathbf{x}=[E,R]$ and ${\mathbf{x}}^{\prime }=[E^{\prime },R^{\prime }]$ are two arbitrary input points, ${\sigma }_f^2$ is the process variance (scaling the range of function variation), and $\Lambda =diag(l_1^2,l_2^2)$ is a diagonal matrix. The parameters $l_1$ i $l_2$ determine how quickly the similarity decreases as the points become more distant in a given direction (in E or R). The covariance function, thus, serves as a “distance measure” in the feature space. The smaller the distance between the input points, the higher the value of the covariance function, indicating greater similarity and a stronger correlation between the corresponding y values. It is assumed that the data originate from a zero-mean Gaussian process, and the correlation between observations is modelled using the covariance (kernel) function $k(\mathbf{x},{\mathbf{x}}^{\prime })$.

Based on the training set, the covariance matrix $\mathbf{K}\in {\mathbb{R}}^{nxn}$ was constructed, where $K_{ij}=k({\mathbf{x}}_i,{\mathbf{x}}_j)$. For a new input point ${\mathbf{x}}_{new}=[E_{new},R_{new}]$, the similarity vector ${\mathbf{k}}_{new}\in {\mathbb{R}}^n$, was computed, where $k_{new,i}=k({\mathbf{x}}_i,{\mathbf{x}}_{new})$ along with the self-covariance value $k({\mathbf{x}}_{new},{\mathbf{x}}_{new})$. The predicted percentage decrease in the critical current for the new point ${\mathbf{x}}_{new}$ is calculated as the expected value of the conditional distribution:

$${\widehat{y}}_{new}={\mathbf{k}}_{new}^T{(\mathbf{K}+{\sigma }_n^2\mathbf{I})}^{-1}\mathbf{y}$$

(13)

where $\mathbf{y}={[y_1,y_2,...,y_n]}^T$ is the vector of observed percentage decreases in the critical current, ${\sigma }_n^2$ is the noise variance representing measurement uncertainty, and $\mathbf{I}$ is the identity matrix.

The GPR model was created in the MATLAB environment using the fitrgp function with a default linear basis function (‘Basis’, ‘linear’) and kernel functions of the ‘squaredexponential’(EqExp) i ‘rationalquadratic’ (RatQuad). The choice of kernel functions used in the analysis was based on their widespread application in modelling non-linear relationships and their favourable smoothing properties. Attempts to apply other kernels (e.g., Matérn, exponential) did not improve the prediction quality and, in some cases, led to computational instability due to the limited size of the dataset. Based on the trained model, the prediction of the percentage decrease in the value of the critical current ${\widehat{y}}_{new}$ was performed for the entire mesh grid of points ${\mathbf{x}}_{new}=[E_{new},R_{new}]$. The main advantage of the GPR method lies in its ability to provide not only the predicted value but also its uncertainty, which makes this method particularly attractive in experimental applications, where the assessment of result reliability is essential.

5.4. Cross-Validation of the Applied Machine Learning Models

In order to objectively assess the generalisation capability of the predictive models applied in this study, a k-fold cross-validation procedure was performed. This method allows the estimation of model performance not only on the training data, but also on independent data, which constitutes an essential aspect of evaluating the effectiveness of machine learning algorithms.

5.4.1. Cross-Validation Methodology

Cross-validation involves repeatedly partitioning the available data into training and test sets. In each iteration, one of the k segments (folds) serves as the test set, while the remaining $k1$ segments are used for training the model. This process is repeated k times, ensuring that each observation appears exactly once in the test set. The resulting error metrics (RMSE, $R^2$, MAE) are then averaged and reported together with the standard deviation. In this study, 5-fold cross-validation ($k=5$) was applied, in accordance with commonly accepted practices in the literature [33,34]. No separate test set was designated in the study. Due to the limited size of the available experimental dataset (36 samples), applying a traditional split into training and test data would have significantly reduced the number of samples available for model training, adversely affecting their quality. To prevent this, a 5-fold cross-validation procedure was employed. In each iteration, the data were divided into five subsets (folds), with four used for training the model and the fifth for testing. This process was repeated five times, with each subset serving once as the test set. As a result, each sample was used both for training and for testing, but never simultaneously. This approach allows the evaluation of model generalisability while maximising the use of available data and reducing the risk of overfitting.

5.4.2. Validation Procedure

For each tape (TP1 and TP2), cross-validation was carried out separately for each of the three selected machine learning models:

• RF—random forests with 50 regression trees;
• k-NN—k-nearest neighbours regression with $k=3$;
• GPR—Gaussian process regression using squared exponential and rational quadratic kernels.

The models were trained on all available experimental data points (energy dissipated in the HTS tape and the number of transitions from the superconducting state), and subsequently validated according to the procedure described above. The implementation was carried out using the cvpartition and crossval functions available in the MATLAB R2025a environment.

5.4.3. Discussion

The analysis of results indicates that all applied algorithms achieved a satisfactory prediction performance. However, Gaussian process regression with the rational quadratic kernel demonstrated the highest stability and the lowest prediction error. It should be emphasised that the use of cross-validation allowed the avoidance of model overfitting and enabled a reliable assessment of its generalisation ability with respect to new, previously unseen measurement data.

It should be noted that the folds used in the 5-fold cross-validation procedure were generated randomly, without explicit balancing of the input variables. As a result, the distribution of dissipated energy and the number of transitions may not have been evenly represented across all subsets. No separate analysis of fold-wise performance variability was carried out in this study. The reported metrics reflect the average performance across all folds and should be interpreted accordingly.

6. Analysis of the Results

In order to develop the foundations of an EDSS, predictive results obtained using various numerical methods were analysed. Based on these results, the usefulness of each method was assessed. To visualise the impact of the energy generated in the HTS tape samples during the action of surge current impulses and the number of transitions from the superconducting state on the percentage decrease in the value of the $I_C$, MATLAB code was used to generate 3D plots (

Table 3. A compilation of plots presenting the relationships between the percentage decrease in the critical current value in the HTS tapes and the generated energy and number of surge current impulse repetitions, obtained using various numerical methods, interpolation, surface approximation, and machine learning algorithms, for the tested HTS tapes.

Tape TP1	Tape TP2

). The X-axis represents the value of generated energy, the Y-axis the number of repetitions, and the Z-axis the percentage decrease in the value of the critical current. For all cases, interpolated values for a denser point grid were obtained within the range from 0 to 25 repetitions and for energy values within the scope of the experimental data. The curves show the predicted decrease in critical current as a function of energy for a fixed number of repetitions (Table 3):

• The first row indicates the result columns for tape TP1 and tape TP2, respectively;
• The second row presents the results of interpolation using the PCHIP method;
• The third row presents the surface obtained by fitting a second-order polynomial model with respect to both input variables (poly22 model)—this type of approximation enables data smoothing and allows extrapolation beyond the measurement range;
• The subsequent rows present the corresponding surfaces for the RF, k-NN, and GPR (SqExp) algorithms—the GPR with a squared exponential kernel—as well as the GPR (RatQuad), i.e., the GPR with a rational quadratic kernel, respectively.

6.1. Performance Metrics for the Analysed Methods

Based on the results of 5-fold cross-validation, quality metrics were calculated for each method: the RMSE (root mean square error), MAE (mean absolute error), Max Error, and $R^2$ (coefficient of determination) [35,36]. A comparison of these metrics enables the evaluation of model accuracy and the selection of the most precise method for predicting the decrease in critical current within the analysed range of energy and number of transitions (

Table 4. Performance metrics for the analysed methods—TP1.

Method	RMSE	${\mathit{R}}^{\mathbf{2}}$	MAE	Max Error
Interpolation	0.0108	1.0000	0.0037	0.0576
Polynomial Approx.	5.0662	0.5772	2.9966	24.4002
Random Forest	$1.095\pm 0.128$	$0.832\pm 0.045$	$0.718\pm 0.093$	$2.861\pm 0.487$
k-NN	$0.997\pm 0.112$	$0.860\pm 0.038$	$0.653\pm 0.088$	$2.543\pm 0.423$
GPR (SqExp)	$0.138\pm 0.044$	$0.990\pm 0.005$	$0.098\pm 0.027$	$0.363\pm 0.112$
GPR (RatQuad)	$0.126\pm 0.039$	$0.992\pm 0.004$	$0.092\pm 0.025$	$0.316\pm 0.098$

and

Table 5. Performance metrics for the analysed methods—TP2.

Method	RMSE	${\mathit{R}}^{\mathbf{2}}$	MAE	Max Error
Interpolation	0.0026	1.0000	0.0012	0.0102
Polynomial Approx.	0.8299	0.8939	0.6269	2.6216
Random Forest	$0.866\pm 0.101$	$0.902\pm 0.034$	$0.534\pm 0.082$	$2.157\pm 0.391$
k-NN	$0.738\pm 0.099$	$0.916\pm 0.029$	$0.506\pm 0.074$	$1.903\pm 0.357$
GPR (SqExp)	$0.441\pm 0.063$	$0.969\pm 0.007$	$0.312\pm 0.034$	$1.308\pm 0.201$
GPR (RatQuad)	$0.428\pm 0.061$	$0.972\pm 0.006$	$0.308\pm 0.031$	$1.278\pm 0.189$

Note: The results for the “Interpolation” method in Table 4 and Table 5 are not based on cross-validation and were calculated using direct fitting to the full dataset. They are presented for comparative purposes only, as an upper bound of fitting performance.

). The RMSE indicates the average magnitude of deviation between predicted and actual values and is particularly sensitive to large errors. The $R^2$ is a commonly used measure of how well a regression model fits empirical data [37]. The MAE represents the average absolute difference between the actual and predicted values. The Max Error denotes the largest single difference between actual and predicted values. In some cases, the Max Error exceeded 2%. These occurrences are typically related to edge cases in the dataset and may reflect local irregularities in the degradation process. Although these errors remain relatively small in absolute terms, they should be considered in practical applications, especially when using the system as a basis for critical decisions.

For both tapes TP1 and TP2, evaluated across the full range of impulse energy and number of transitions (0–25), the applied machine learning algorithms demonstrated high predictive effectiveness. The best performance (lowest RMSE and MAE and highest ${\mathrm{R}}^2$) was achieved using Gaussian process regression (GPR) with both squared exponential and rational quadratic kernels. For TP1, the GPR model with the rational quadratic kernel yielded the lowest maximum prediction error. The results for TP2 were comparable, with a slightly lower maximum error and very high model stability.

It should be emphasised that the proposed predictive models operate reliably only within the experimental data range. Outside this domain, some outputs may become physically inconsistent (e.g., negative values or degradation exceeding 100%). The current system does not include internal physical constraints, and addressing this limitation remains part of future development efforts.

6.2. Comparison of the Predictive and Extrapolative Capabilities of the Investigated Models

The main premise of the EDSS is to base its operation on a predictive system capable of forecasting the degree of degradation of the HTS tapes in SFCLs with high probability. Ensuring the proper preparation of input data, which subsequently serve as training data for the investigated models, is intended to enable the correct functioning of the system. For the purposes of the predictive system, the HTS tape samples were tested in the range of generated energies from 0 to 207 J and the number of transitions from the superconducting state (0–25). A key question was whether the prepared training dataset was sufficient for the extrapolative purposes of the system with respect to values of energy and the number of transitions beyond the investigated range.

Table 6. Summary of exemplary results of the percentage decrease in the critical current value for energy values and the number of transitions from the superconducting state due to a surge current impulse, both within the investigated range (prediction) and beyond it (extrapolation).

Method	Percentage Decrease in the ${\mathit{I}}_{\mathit{C}}$ Value (%) (Prediction)		Percentage Decrease in the ${\mathit{I}}_{\mathit{C}}$ Value (%) (Extrapolation)
	E = 150 J, R = 12		E = 250 J,  R = 30
	Tape TP1	Tape TP2	Tape TP1	Tape TP2
Interpolation	99.03%	95.11%	−130.02%	156.06%
Polynomial Approximation	97.83%	95.31%	61.39%	82.58%
Random Forest	98.63%	95.89%	88.73%	95.26%
k-NN	98.95%	96.32%	81.34%	95.69%
GPR (SqExp)	99.21% ± 1.80	94.94% ± 0.67	82.55% ± 7.89	92.68% ± 1.89
GPR (RatQuad)	99.11% ± 1.80	94.94% ± 0.67	79.90% ± 8.59	92.68% ± 1.89

presents a summary of exemplary results of the percentage decrease in the critical current value for energy values and the number of transitions from the superconducting state caused by a surge current impulse, both within the investigated range (prediction) and beyond it (extrapolation). As can be clearly observed, the extrapolation results require further analysis.

6.2.1. Prediction Results

As a first step, the results for the percentage decrease in the value of the critical current were analysed for values of energy and number of transitions from the superconducting state caused by a surge current, within the range covered by the experimental study. An analysis of the plots presented in Table 3 for both tapes reveals that, for certain regions defined by the XYZ planes, the predicted values of the percentage decrease in the critical current $I_C/I_{C0}$ exceed 100%. In Figure 11 and Figure 12, the reference plane $z(x,y)=100$, was plotted to identify erroneous predictions of the percentage decrease in the critical current for each method. An erroneous result is understood as a point located above the plane $z(x,y)=100$. Such errors were not observed in the interpolation method or in the predictions obtained using the RF and k-NN algorithms.

Table 7. Percentage summary of erroneous predicted values of the percentage decrease in the critical current for each method for tapes TP1 and TP2.

Method	Percentage of Points Above the Reference Plane z(x,y) = 100
	Tape TP1	Tape TP2
Interpolation	0%	0%
Polynomial Approximation	57.423%	10.846%
Random Forest	0%	0%
k-NN	0%	0%
GPR (SqExp)	25.346%	13.962%
GPR (RatQuad)	23.731%	13.962%

presents a summary of the percentage of erroneous predicted values (i.e., those exceeding 100%) for each method and for both HTS tapes (TP1 and TP2), expressed as the ratio of points lying above the $z(x,y)=100$ plane to those lying below it.

In Table 6 and Table 7, several extreme errors can be observed, particularly in extrapolation cases. These outliers are typically caused by the inability of certain models (especially GPR) to maintain physical consistency outside the training domain. Some predictions show an increase in critical current, which is non-physical and stems from overfitting or numerical instability. These phenomena underline the importance of introducing physical boundaries or hybrid models in the next steps of development.

The results showed that all models—to a greater or lesser extent—generate values exceeding 100%, which is non-physical in the context of describing the decrease in $I_C$. The value of the critical current as a result of tape operation cannot “improve” and, thus, cannot exceed its initial value; therefore, the $I_C/I_{C0}$ ratio cannot be greater than 100%. The presence of points above this limit indicates that the model is not aligned with the physical constraints of the data.

As demonstrated in this case, it is important to evaluate models not only based on RMSE or ${\mathrm{R}}^2$, but also in terms of their consistency with domain knowledge (e.g., physical constraints). Interpolation methods such as PCHIP and polynomial approximation (e.g., poly22) may result in the excessive “bulging” of curves, particularly near the boundaries of the mesh grid. It is advisable to monitor such effects and to apply more conservative interpolation approaches (e.g., makima or constrained spline interpolation). Models such as GPR and RF do not inherently account for physical constraints. Therefore, it is worth considering their adaptation or the use of probabilistic modelling with constraints (e.g., by conditioning the GPR). For this type of data, the use of a hybrid model that combines experimental data with physical knowledge appears to be a reasonable approach.

6.2.2. Interpolation Results

The extrapolation results (Table 6) indicate issues leading to predictions with non-physical values. Such cases occur for interpolation and polynomial approximation. Slightly greater stability is exhibited by the RF and k-NN algorithms. GPR demonstrates smaller deviations compared to other methods; however, it can still predict values below 0%, which is physically impossible. In order to improve the reliability of predictions across a broader range of energy and number of repetitions, it is necessary to experimentally extend the dataset within these regions.

Table 8. Comparison of predictive models with respect to accuracy, extrapolation behaviour, and physical realism.

Model	Accuracy	Extrapolation	Realism	Comments
RF	High	Moderately stable	Good (within range)	Physically consistent predictions; no unrealistic outputs observed.
k-NN	Medium	Strongly limited	High	Effective within experimental range; not suitable for extrapolation.
GPR	Variable	Unstable	Often violated	Produces non-physical results outside training data; excluded from final interpretation.

summarises the comparative strengths and weaknesses of the three tested models with respect to accuracy, extrapolation capabilities, and physical consistency.

6.3. Input Variable Sensitivity Analysis

To determine the relative influence of each input variable on the predicted degradation of the critical current value, a feature importance analysis was performed using the random forest model. In this approach, the importance of a given input variable is quantified by the total reduction in the mean squared error (MSE) that results from using that variable to split decision nodes in all trees of the ensemble. This method enables the assessment of which parameters—dissipated energy or the number of transitions from the superconducting state—have a greater impact on the degradation process of HTS tapes.

The raw feature importance values were normalised so that their sum equals 1, enabling a direct comparison between variables of different scales. The resulting values, therefore, represent the relative importance of each input variable. It is important to note that no standardisation or scaling was applied to the input variables themselves.

Feature importance was determined separately for tapes TP1 and TP2. For TP1, dissipated energy played a dominant role, with a relative importance estimated at 0.73, while the value for the number of transitions was 0.27 (Figure 13). In the case of TP2, the influence of both variables was more balanced: The importance of energy was 0.61, and that of the number of transitions was 0.39 (Figure 14).

These results indicate that, for tape TP1, the key factor influencing the degradation of critical current was the energy generated during transitions to the resistive state. For TP2, the number of transitions from the superconducting state had greater significance, which may suggest structural or material differences between the examined tapes. This analysis provides valuable insights for the design of SFCL protection strategies aimed at preventing excessive degradation.

7. Discussion

The development of a recommendation system aimed at predicting changes in the critical current $I_C$ value of the HTS tapes used in superconducting fault current limiters (SFCLs) is justified both practically and strategically, particularly in the context of ensuring the reliability and predictability of intelligent power systems. During each activation of an SFCL, the superconducting tapes are exposed to short-circuit currents that exceed the rated $I_C$ value of the limiter, to rapid temperature rises, and to dynamic mechanical stress variations. Such factors may lead to partial degradation of the superconducting material, resulting in a gradual decrease in the critical current value, deterioration of the rated parameters of the SFCL, and, consequently, a shortening of the device’s service life. This process may progress unnoticed under operational conditions, creating the need for the implementation of predictive mechanisms for assessing the condition of the material based on operational data.

The $I_C$ value is a key parameter that determines the point at which the HTS tape transitions from the superconducting state to the resistive state, thereby triggering the activation of the SFCL. A decrease in this parameter shifts the activation threshold towards lower short-circuit current values, which may result in earlier activation of the device. In cases where the SFCL operates in coordination with power system protection, such a shift may alter the operating zone of instantaneous overcurrent protection devices (as analysed in [23]. Therefore, monitoring and predicting changes in $I_C$ is essential to maintain the predictability of limiter operation and the coordination of protection within the power system.

In this context, a predictive system was developed based on experimental data, enabling the modelling of the decrease in the value of the $I_C$ as a function of the generated energy and the number of transitions from the superconducting state triggered by a surge current impulse. The choice of appropriate analytical methods is of key importance—both in terms of prediction accuracy and the interpretability of the results.

The highest prediction quality was achieved by GPR algorithms, particularly those using a rational quadratic kernel. This method is characterised by high stability, low errors (i.e., RMSE, MAE), and robustness to noise in the data. GPR also enables the estimation of prediction uncertainty, which is important in engineering decision-making applications. Interpolation (using the PCHIP method) also performed very well in local analyses—offering high accuracy at low computational cost, provided that the input data originate from a regular measurement grid.

In contrast to the above methods, polynomial approximation and RF demonstrated lower effectiveness, particularly in the case of the TP1 dataset, where signs of model overfitting or underfitting were observed. Although RF handles data with heterogeneous structure well, it proved less accurate in this case due to the limited size of the training dataset. The k-NN algorithm, on the other hand, produced moderate results—it is easy to implement and interpret, but does not offer high accuracy across the entire range of analysed data.

A fundamental limitation of all the applied methods lies in their range of operation—based on training data within the intervals $E\in [0,207]$ and $R\in [0,25]$. Beyond this range, prediction quality may deteriorate significantly, leading to erroneous and non-physical conclusions. Therefore, the unconditional application of these models for extrapolation is not recommended without additional experimental validation or an extension of the dataset. Extrapolative operation requires the introduction of physical constraints (e.g., preservation of the monotonic decrease in the $I_C$, value, physical limits of the HTS material), as well as consideration of hybrid models that combine expert knowledge with measurement data.

As a result, the predictive system can serve as a practical tool to support engineering decisions concerning the continued operation of the SFCL: determining the appropriate moment for recalibrating the activation threshold, planning maintenance inspections, or withdrawing the device from service. Such an approach supports strategies for managing technical infrastructure, based on anticipating potential failures and preventing them (condition-based maintenance). This is implemented through monitoring the condition of the device and data analysis (predictive maintenance). Such actions, which form the foundation of Smart Grid systems, thus fit into the long-term vision of intelligent energy development.

Based on the obtained research results, the following recommendations have been indicated.

1.

For engineering and predictive applications requiring high accuracy, particularly in the case of extrapolation or working with incomplete data, it is recommended to use GPR, especially with a rational quadratic kernel. This method provides very low errors and stable predictions even with different datasets.

2.

Interpolation (PCHIP) is also highly effective, particularly when the analysed data originate from a regular measurement grid. It can be recommended for local analyses and for the quick estimation of values between known points.

3.

Polynomial approximation and the random forest model are not recommended for this specific application due to their high susceptibility to overfitting or underfitting and poor data fitting (e.g., results for TP1).

4.

The k-NN algorithm can be considered a compromise option—it offers moderate accuracy and easy interpretability, but does not match GPR in terms of prediction quality. Due to the instability and physical inconsistency of its extrapolation results, the GPR model was excluded from the final interpretation and visual analysis.

5.

None of the applied models provides guaranteed result quality outside the range of the training data. Predictions beyond the energy range $E\in [0,207]$ and $R\in [0,25]$ should be interpreted based on expert knowledge, supported by additional data or their verification.

6.

For a more accurate representation of the behaviour of the HTS tapes under operational conditions, the development of a hybrid model should be considered.

8. Conclusions

The development of modern intelligent power grids assumes the integration of a large number of RES with variable energy generation. This variability necessitates increased energy flexibility of the system, as well as of the protective components that ensure its correct operation under both normal and fault conditions. One such fault condition is a short circuit in the system, during which it is essential to limit the short-circuit current value or to disconnect it rapidly. One of the devices used to limit current during a short circuit is the superconducting fault current limiter. Its construction technology and potential applications are still at the prototype or laboratory research stage. Its application potential is limited due to damage to the superconducting tapes from which the limiter is built. Therefore, an important issue is the prediction of the degradation process of the device’s parameters and the estimation of its potential service time under degradation processes associated with the reduction in the critical current value of the superconducting tape. This article proposes a system that enables the prediction of the percentage decrease in the critical current value of the HTS tape for different numbers of transitions under various levels of energy generated in the tapes. Such prediction is an important element in engineering decision support systems concerning the operation of devices in critical infrastructure systems, such as the power system.

The approach presented in the article for estimating changes in the parameters of superconducting tapes employed various methods, including artificial intelligence techniques. Interpolation methods and GPR provide the highest accuracy in modelling the decrease in the critical current in the HTS tapes for input data limited to the range of energy and number of transitions examined in the experiment. Therefore, they are recommended approaches for the analysis of this type of experimental data. In order to obtain precise predictions of the critical current value, for example, in scientific research or engineering applications, GPR (RatQuad or SqExp) or interpolation should be used—provided that appropriately prepared input data are used to build the EDSS and that the system does not extrapolate beyond the experimentally investigated range of energy and number of transitions of the HTS tape from the superconducting state. Extrapolation based on trained statistical or ML models without additional experimental validation may lead to incorrect and non-physical conclusions. It is recommended to limit the interpretation range of the models to the intervals covered by the training data and to apply mechanisms that verify the physical correctness of the forecasts. The development of an EDSS based on a hybrid model using various algorithms (including algorithms that reflect physical processes) should be considered.

The decision support system proposed in the article enables the prediction of the percentage decrease in the critical current value, under the assumption that the subsequent values of energy generated during the surge in the tested sample remain constant. Naturally, this is an assumption adopted to simplify the considered system. During SFCL operations, the energies will vary and, thus, the decrease in the $I_C$ value may also differ. It should be noted that during a surge, micro-damages may occur within the internal structure of the HTS tapes, which can affect parameter changes. This issue has been described in detail in [38]. The occurrence of micro-damages may accelerate the degradation process of the $I_C$ value.

It should be emphasised that the current system is not intended as a ready-to-use industrial tool. The models were trained and validated solely on laboratory-scale experimental data, and their predictive power is limited to the range of conditions tested. Extrapolation beyond this range may lead to non-physical predictions. As such, the system should be considered as a prototype to support the future development of data-driven diagnostic tools in superconducting applications.

A key direction for further development of the EDSS involves the incorporation of internal physical constraints into the predictive models. This step is crucial for ensuring physically consistent behaviour, especially when models operate near or beyond the limits of the experimental domain. One possible solution is to apply constrained Gaussian process regression (bounded GPR), which explicitly limits the predicted output range based on physical feasibility. Another promising approach involves the integration of hybrid models, combining machine learning with rule-based or physically informed components. Such enhancements are expected to increase the robustness of extrapolation and improve the system’s reliability in practical diagnostic and control applications.

In order to improve the accuracy of the model, a larger number of measurement points for different energy values and a larger number of ribbon transitions from the superconducting state can also be included as part of future studies.