Advanced experimental-based data-driven model for the electromechanical behavior of twisted YBCO tapes considering thermomagnetic constraints

Data-driven models can predict, estimate, and monitor any highly nonlinear and multi-variable behaviour of high-temperature superconducting (HTS) materials, and superconducting devices to analyse their characteristics with a very high accuracy in an almost real-time procedure, which is a significant figure of merit as compared with traditional numerical approaches. The electromechanical behaviour of twisted HTS tapes under different strains, magnetic fields, and temperatures is a complicated problem to be solved using conventional approaches, including finite element-based methods, otherwise, experimental testing is needed to characterise it. This paper aims to offer a data-driven model based on artificial intelligence techniques to predict the electromechanical behaviour of HTS tapes operating under various thermomagnetic conditions. By using the proposed model, normalised critical current value and stress of twisted tapes can be predicted under different temperatures and magnetic flux densities. For this purpose, experimental data were used as inputs to design an adaptive neuro-fuzzy inference system (ANFIS). To achieve the best performance of the prediction system, multiple clustering methods were used, such as the grid partitioning method, fuzzy c-means clustering method, and sub-clustering method. Sensitivity analyses were conducted to find the best architecture of ANFIS to predict and model electromechanical behaviour of twisted tapes with high accuracy.


Introduction
The discovery of high-temperature superconducting (HTS) materials was the origination of many investigations to take advantage of superconductors in engineering fields including large-scale power applications [1,2]. These materials in the form of coated conductors are employed to fabricate superconducting cables [3,4], superconducting transformers [5,6], superconducting fault current limiters [7,8], superconducting busbars, and many other high-power electric devices in power grids. In addition, another promising and fast-growing application of HTS materials is in cryoelectrifications of modern transportation systems such as electric aircraft [9] and marine applications [10]. In many power applications, tapes are usually twisted around a former to minimise AC loss. Twisting imposes mechanical tension to the HTS tapes and causes a critical current reduction [11]. This mechanical tension could damage the brittle superconducting layer, and as a result, generate weak points along the length of HTS tapes in such a way that even during normal operation, this could lead to malfunction or to complete failure of HTS device.
The characterisation of the critical currents and the applied mechanical stresses of twisted superconducting tapes is an electromagnetic coupled with thermomechanical problem. To solve this, there are analytical methods with low accuracy and difficulties for dealing with complicated geometries. To address these issues, finite element method (FEM) was proposed. Although FEM-based approaches have higher accuracy, their computation time and cost are extremely high. The high computation burden makes FEM-based approaches inappropriate for applications that require a fast computation response (FCR) or real-time response (RTR). One may propose experimental analyses for solving the aforementioned problem [12][13][14]. These methods are more accurate and faster than previous methods, i.e., analytical and FEM-based methods. However, sometimes the time spent preparing and developing an experimental set-up is considerable. Moreover, in FCR and RTR systems, especially those implemented in cryoelectrified aircraft or other transportation units, there is no time/room for tests and superconducting tapes have to operate just right, considering safety concerns. So, artificial intelligence (AI) techniques can be developed and used as a solution to the aforementioned problems with acceptable accuracy and very low computation time. AI-based methods can be implemented in applications with a requirement for FCR or RTR systems to estimate the behaviour of any kind of HTS tape [15]. For instance, and in the coming future, by developing an accurate surrogate model using AI techniques, the electromagneto-thermo-mechanical properties of HTS tapes/devices would be estimated in a few milliseconds while the HTS device is operating in any cryo-electrified transportation system.
In this paper, a data-driven model based on the adaptive neuro-fuzzy inference system (ANFIS) is proposed to estimate the electromechanical characteristic of Second generation (2G) HTS tapes as an intelligent package with an easily implementable structure, very high accuracy, and ultra-high estimation speed. The proposed method can characterise the electromechanical behaviour of HTS tapes with respect to thermomagnetic considerations. So, the inputs of the critical current estimation are width, thickness, magnetic flux density, temperature, and strain value (ϵ) and the output is the minimum critical current of tape. For the estimation of the stress, inputs are reduced to four. This is because of the independency of the stress from the magnetic flux density. In this paper, electromechanical behaviour of multiple Yttrium Barium Copper Oxide (YBCO) tapes (produced by different manufacturers) is estimated by ANFIS in MATLAB software package using experimental results as input data.

Electromechanical behaviour of HTS tapes subjected to twisting
The AC loss in the superconducting tapes could be affected and intensified by many factors, namely temperature increase [16], harmonic distortions [17], level of carrying current, frequency change, level of external magnetic field, among others. The poor dissipation of AC loss leads to heat accumulation, and consequently temperature increase of HTS tapes, as well as efficiency reduction of the cooling unit and the whole device [18]. As a way to overcome this problem, tapes are wounded on the former to reduce the AC loss and diminish the temperature rise. It should be noted that the concept of twisting is usually used in HTS cables, and magnets. Figure 1(a) presents a twisted tape and two important twisting parameters, pitch angle and pitch length. Twisting pitch length (TPL) is defined as the length within which an HTS tape gets back to its initial relative position and the twisting pitch angle (TPA) is defined by equation (1) [19]: where ℓ is TPL and R is the radius of the former on which the HTS tape is twisted. Twisting causes a specific type of deformation in HTS tape, known as strain. As a matter of fact, strain is the displacement between the particles in the body of the HTS tape relative to its length [12]. When twisting applies a pure torsional load to the HTS tape, the value of the strain is not constant on the surface of HTS tapes. In fact, the strain is distributed nonuniformly at different locations of tape. So, strain is a function of the location of the calculation point, TPA, TPL, temperature, and the structure of the HTS tape [12]. The reduction of critical current and the increase of mechanical stress are the consequences of twisting of the tapes around a former and the resulted strain. Thus, AC loss would be reduced but this advantage is achieved at the cost of generating some weak points on HTS tapes, if twisting is not applied under proper tension and angle. These weak points have a lower critical current than the expected value, locally. So, more heat would be generated in these points, which causes a significant temperature increase, making these points a good candidate for establishing hotspots. In fact, if TPA surpasses a specific value, HTS tapes experience a significant reduction in critical current and may even cause thermomagnetic collapse of the tapes. Another impact of twisting is the applied mechanical stress on the tape which is a function of the strain. Possible locations for critical current degradation along the length of superconducting tape are shown in figure 1(b). A simple schematic of a twisted HTS tape in a superconducting cable is also shown in figure 1(c).

The proposed ANFIS methodology
ANFIS is a highly accurate estimation and prediction AIbased method that operates as a first-order Takagi-Sugeno-Kang (TSK) fuzzy system. In fact, a fuzzy interference system is combined with adaptive neurons to create a structure for prediction and estimation known as ANFIS. Firstly, ANFIS obtains a group of data as train inputs and outputs. In this stage, ANFIS uses learning rules such as back propagation, gradient descent, and the least square method to map the inputs to the outputs. After gaining a proper structure for the problem, test inputs are inserted to prove the capability of the model in predicting the test outputs. The objective is to reduce the error function to the lowest possible value [20][21][22] i.e. minimize it. This procedure is shown in figure 2 for a general structure of ANFIS with one input and output that tends to reduce the error. For this purpose, the mean square error (MSE) index is defined by equation (2): where, N is the number of training data, e i is the difference between each real value and its estimated value, t i is the targets and y i is the output of the TSK fuzzy system. The relation between input (x i ) and output (y i ) of the fuzzy system is defined by equation (3): where, m l i , σ l i are parameters of Gaussian membership functions that are regulated by the gradient descent method to minimize the MSE [23,24] where, k is number of iterations and α is tune coefficient. There are three different methods to generate a preliminary fuzzy system which are discussed in the following subsections.

Grid partition method (GPM)
The GPM operates based on the creation of fuzzy rules from numerical pairs. The minimum and the maximum of each input vector are calculated according to equation (6) and their differences are classified with respect to membership functions [25]: At the final step with respect to the input vector and membership functions, the inputs domain is partitioned. Then, the fuzzy rules of the TSK fuzzy system are written based on input data as equation (7) [25]: where, R j is the jth fuzzy rule and A j N is the jth linguistic variable for Nth input. A schematic of the GPM clustering method is shown in figure 3. As shown in this figure, by dividing the input vector into many fuzzy parts, GPM generates partitions.

Fuzzy C-means clustering method (FCM)
In FCM, every data point belongs to multiple clusters with different membership orders. The FCM performs based on minimizing the objective function of equation (8) [26,27]: where, N C is the number of clusters, c j is the centre of jth cluster and µ m ij is the membership degree of x i to the membership function of jth cluster, and m is fuzzy separation matrix.
All cluster centres are updated with equation (9): The value of µ ij is also obtained from equation (10): where m determines the fuzzy degree of the borders between clusters. Updating cluster centres with equation (9) and determining the data membership degree in each cluster with equation (10) is repeated until centres of clusters do not have a tangible change. Figure 4 represents the FCM method as another option in clustering.

Subtractive clustering method (SCM)
In the SCM, it is assumed that all data could be the centre of the cluster and the score of each point as a centre is calculated. The influence of clustering in the problem space is determined by radius (r) and for each point, the mean distance to all points in the radius of that point is considered as a factor for scoring. In the end, the best centre of the cluster is selected based on the score points. The score of each point is calculated by equation (11) [28]: Based on the considered radius of influence, the data that are dominated by the cluster centre are removed. Then for the remaining data, another cluster centre is selected, and this continues until the end of all data clustering. The data membership degree in each cluster is determined by equation (12): where c j is the cluster centre defined by equation (13): This clustering method is depicted in figure 5.

General remarks on GPM, SCM, and FCM
In GPM, fuzzy rules are designed for all cases which may never be used during the estimation and increase the computation cost. While in clustering techniques, the number of fuzzy rules significantly decreases. Instead of writing fuzzy rules for each of the data that causes complexity, a fuzzy rule is considered for each cluster that simplifies the calculations. In SCM, unlike FCM, the number of clusters cannot be directly specified. But by changing the radius of influence, a control the number of clusters can be achieved. However, determining a small radius of influence usually produces more clusters and more fuzzy rules.

Results and discussions
The ANFIS-similar to any other data-driven methodrequires some input data. In this paper, results of experimental tests were used as input data to ANFIS. In [29][30][31][32][33][34][35][36][37], the experimental data are attained with respect to the electromechanical behaviour of different YBCO tapes under various thermomagnetic conditions based on the procedure shown in figure 6. The first step is to apply a mechanical load to the coated conductor.
Step-motors are normally used to apply mechanical loads. These loads cause mechanical stresses on tape which results in the critical current reduction and imposing stress to the tape. Stress can be measured by an extensometer. The value of critical current can be measured by measuring the current and voltage, and temperature [38]. Table 1 tabulates the specifications of the selected HTS tapes. To create a model based on ANFIS, data were gathered under various temperatures, fields, and strains. The aforementioned gathering data phase is depicted as the experimental data acquisition phase in figure 7. About 50% of these data (total number of 125) was chosen randomly (to eliminate any chance of being biased to any specific data) to use in the training phase as inputs and outputs. At this level, the model is trained to characterise the electromechanical behaviour of YBCO tapes. After reaching the minimum error function, the model proceeds to the next phase. In this phase, the established model obtains some input data as test inputs. The test inputs are those which have remained from the random selection process, i.e., the other 50% of the experimental data (125 data points). This means that neither of the test inputs is the same as the training inputs. In other words, the ANFIS never saw the test data during the training phase. This is shown to be the test phase in figure 7. Finally, the system estimates the test outputs with respect to the laws, rules, and clusters gained from the training phase of the modelling system.
It should be noted that, for the sake of comparison among different methods, root mean square error (RMSE) and correlation coefficient (R 2 ) are analysed as the most famous and common error criteria, which are expressed in equations (14) and (15). RMSE is usually used to show the difference between an estimated value and real data while the correlation coefficient is a statistical quantity that shows the strength of the correlation between the real values and predicted ones: where N is the number of data, A k is the value of real experimental data, F k is the value of the forecasted data,Ā is the mean of experimental data, andF is the mean of forecasted data. Figure 8 presents the estimated values of the normalized critical current with respect to variations of strain, magnetic flux density, and temperature by ANFIS for different tapes. It is worth noting that the value of the magnetic flux density is 19 T at 4.2 K and the self-field (no external magnetic flux density) at 77 K. Figure 8 is produced based on three mentioned clustering methods, namely FCM, GPM, and SCM. To plot figure 8, the  fastest clustering methods among FCM, SCM, and GPM are chosen. A comprehensive sensitivity analysis on ANFIS parameters were done and results were listed in table 2. Table 2 summarises the values of RMSE, computation time, and correlation coefficient (R 2 ) for different clustering methods. It is worth noting that the computation time depends on the configuration of the computer that is used for computing; in our case, the specification of the computing system is as follows: RAM: 16 GB-DDR3, CPU: Corei7-3612QM-2.1 GH. It is worth noting that all computations were done using same computer for all different methods; therefore, the relative difference between them from the computation time viewpoint is still valid. Speed of estimation or computation time is a crucial factor for applications that require a fast prediction of the characteristics of HTS tapes. On the other hand, accuracy is another vital factor to appropriately characterise the electromechanical behaviour of HTS tapes. Among all clustering methods, FCM3 with three membership functions has the highest estimation speed. For the sake of more assessments, the FCM with six (FCM6) and nine (FCM9) membership functions were tested, as well. By doing this, it is found that the computation time of FCM6 and FCM9 compared with FCM3 is 76.6% and 150.5% increased, respectively. Also, for more investigation, FCM3 was compared with respect to GPM and SCM. In comparison to the fastest GPM and SCM, FCM3 has a 1062% and 22.1% higher estimation speed, respectively. Thus, for FCR or RTR systems, GPM is completely out of list due to low speed or high computation time.

Smart estimation of critical currents (Ic)
Considering the accuracy of the estimations, sub-clustering with a radius of 0.5 (SCM0.5) has the lowest RMSE and highest R 2 . The SCM was also investigated with different clustering radii of 0.1 (SCM0.1) and 1 (SCM1). The results show that the accuracy of the SCM was reduced by 20% using SCM0.1 and by 93.5% using SCM1 compared to SCM0.5. Therefore, SCM0.5 is recommended for achieving highest possible accuracy with SCM.
Another factor that can be effective in choosing the appropriate ANFIS method is to consider accuracy and speed simultaneously. However, in some applications some levels of trade-off between estimation accuracy and speed maybe needed. Analysing the results of the best accuracy with the highest speed shows that although FCM3 has a 360% higher speed than SCM0.5, the RMSE value of SCM0.5 is 40.51% lower and R 2 is 25.91% higher than FCM3. Therefore, if one considers the speed and accuracy simultaneously, more discussions are needed on different considerations. In accordance with this approach, figure 9 illustrates the different aspects of choosing a clustering method and it shows which of these methods can fulfil the speed and accuracy considerations.
As shown in figure 9, the methods which lay in the green zone are selected as the fastest methods while the methods in the purple zone have the highest accuracy. If one just considers the estimation speed, the fastest results are extracted from FCM3, SCM1, FCM6, and FCM9, respectively, while the methods with the highest accuracy are SCM0.5, GPM1, SCM0.1, and FCM9, respectively. The overlap of these two zones is FCM9 with RMSE of 0.0579, and R 2 of 0.811. FCM9 has 150% higher computation time than FCM3 and 23% higher RMSE than SCM0.5. So, FCM9 is as the best method of clustering when I c is  The bold values show the best performance of a scenario of a clustering method, when compare it to other scenarios of the same clustering method.   the experimental which should technically give us the y = x line. This figure is an index for analysing the proper performance of three methods which were discussed before against the experimental results. The first method is SCM0.5 with the highest accuracy among all other methods which shows a very similar characteristic to the pattern of experimental results. After that FCM3 is depicted by yellow colour with the highest speed of estimation. As it can be observed, figure 10(a) has the lowest similarity to the first figure, i.e. experimental results pattern. At last, there is the chosen method that meets both speed and accuracy. Figures 10(e)-(h) represents the histogram plot of data in different clustering methods. The subplot with more similar behaviour to figure 10(e) has higher accuracy (i.e. SCM0.5). Figure 11 shows the mean and standard deviation of predicted data for three different ANFIS methods. The mean of the estimated values of the three clustering methods are very close to each other while SCM0.5 has the lowest standard deviation due to its high accuracy; after that there are FCM9 and FCM3, respectively.

Smart estimation of stress
Temperature and strain are two important parameters that can change the stress on HTS tapes. Figure 12 depicts the values of estimated stresses and compares them with the experimental values. These values are estimated for different types of tapes with different clustering methods. It is worth noting that data were reported in two temperatures, i.e. 4.2 K and 77 K. For stress estimation, FCM3, GPM1, and SCM0.1 are the most  The bold values show the best performance of a scenario of a clustering method, when compare it to other scenarios of the same clustering method. accurate clustering methods which are used to plot the data in figure 12.
The computation time, RMSE, and R 2 indices are tabulated in table 3 for the stress estimation process. According to table 3, the fastest method is SCM1 with a speed between 1x-11667x faster than other methods while the most accurate method based on R 2 is FCM3 with an R 2 between 5.5% and 27.6% higher than other approaches. If speed is considered for selecting a method, GPMs are completely out of list. However, if both speed and accuracy are considered together, according to figure 13, two categories of solutions can be formed. SCM1, FCM3, FCM6, and FCM9 are the fastest methods, respectively and FCM3, SCM0.1, GPM1, and FCM6 have the highest accuracy, respectively. Accordingly with respect to table 3 and figure 13, it can be stated that FCM3 has the highest trade-off of speed and accuracy, simultaneously. Therefore, FCM3 is the best clustering method for the estimation of the stress characteristic in YBCO tapes. Figure 14 compares the methods which meet the speed and the accuracy constraints. Figure 14  standard deviation and therefore, higher accuracy in comparison to SCM1.

Verification of model
In general, this section is provided to show the effectiveness and capability of the proposed model in estimating the values of critical current versus the magnetic flux density out of the training bound. By the capability of the model to accurately estimate the electromechanical characteristic, a real-time implementation of such model in near future would be possible, given the fact that high performance computational resources will be more accessible soon. To do this, the SuperPower-tape b is selected and magnetic flux density of 0, 0.1, 0.3, 0.5, 1, 2, 3 T are applied, which was not among any of the previous data sets. Figure 16 presents the estimated values for different clustering methods. This is also simulated by FCM9, GPM1, and SCM0. 5    The bold values show the best performance of a scenario of a clustering method, when compare it to other scenarios of the same clustering method.
FCM9 is 16% higher than FCM3, as the fastest method, and computation time of the FCM9 is 45% lower than SCM0.5, as the most accurate method. Thus, it must be claimed that the FCM9 is the best choice for the critical current estimation. The structure of the FCM9 is shown in figure 17(a) in which for every input, nine membership functions are considered. They are tape thickness, tape width, strain, magnetic flux density, and temperature. Figure 17(b) presents the range of variations for each input whereas the membership functions gain a value between 0 and 1, known as membership degree. The summation of membership degrees for each input must be equal to 1. Different membership functions are related to each other by fuzzy rules to identify the value of inputs.

Conclusion
The estimation of the critical current and the stress of the HTS tapes is a problem dealing with electromechanical considerations and thermomagnetic conditions. To solve such problem, finite element-based methods could be applied, however, they need a long time to compute the electromechanical characteristic of the HTS tapes. On the other hand, any action to reduce their computation time may compromise the accuracy. There are also other methods like equivalent circuit models or stochastic predictions. These methods are faster than FEMs with a lower accuracy. Therefore, neither of these methods can be implemented in real-time which need a fast and highly accurate result. Thus, data-driven models based on AI techniques are becoming of interest which is due to their very high computational speed and high accuracy. This paper has proposed a model based on adaptive Neuro-fuzzy inference systems to estimate the electromechanical characteristic of twisted YBCO taps while the temperature, magnetic flux density, and strain are imposed as thermomagnetic conditions. The proposed method is a combination of fuzzy systems and neural networks. This causes higher computational speed and accuracy in comparison to other aforementioned methods. The impact of multiple membership functions and clustering methods were tested on the accuracy and the speed of estimation. The input data bank was established based on experimental tests of published papers reported in literature.
The most important findings of this paper are summarised as below: • For the estimation of critical current, fuzzy clustering method with nine membership functions fulfils the accuracy and speed constraints while the most accurate method is sub-clustering method with clustering radius of 0.5 with R 2 and RMSE values of 0.047 and 0.92 and the fastest method is fuzzy clustering method with three membership functions and a computation time of 0.628 s. • For the estimation of stress, fuzzy clustering method with three membership functions is the best approach considering both speed and accuracy constraints while the fastest method is sub-clustering method with clustering radius of 1 and estimation time 0.689 s. • By applying the magnetic flux density out of the training range (i.e. data between 0-3 T) to the model, critical current was estimated accurately with an R 2 value of 0.72-0.957. It proves the effectiveness of the proposed ANFIS model for estimating data which it never saw before. This technically simulate the real-time condition.
It is worth noting that, the proposed model is still offline, however, in the future, it could be adjusted into a real-time method for estimating in an online manner. The only requirement for this is a high-performance computational system to bring the estimation time around couple of milliseconds.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).