Sensitivity analysis and machine learning modelling for the output characteristics of rotary HTS flux pumps

Various formulations have been implemented into commercial software, e.g. COMSOL Multiphysics, for HTS modelling, including coupled H–A formulation [30], H-formulation with shell current [25–27], segregated H-formulation [31], minimum electromagnetic entropy production [32, 33], coupled T–A formulation [34, 35], integral equation [36], volume integral equation-based equivalent circuit [37]. Adequate details about comparison for each formulation can be found in recent benchmark research [28]. The unique feature of coupled T–A formulation is that it simulates the superconductor (SC) as a 1D object in this 2D problem, and it was chosen to simplify the model complexity, as illustrated in figure 1.

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The coupled T–A formulation was firstly proposed by Zhang et al [34], and validated to calculate the electromagnetic characteristics of HTS tape stacks and coils. For the superconductor domain, the current vector potential is chosen as the solved variable, and the governing equations are:

$$\begin{align} J = \nabla \times T\end{align} \tag{ 1 }$$

$$\begin{align} \nabla \times \left( {\rho \nabla \times T} \right) = - \frac{{{\text{d}}B}}{{{\text{d}}t}}\end{align} \tag{ 2 }$$

where J, T, ρ and B represent the current density, current vector potential, resistivity, and magnetic flux density, respectively. According to the thin strip approximation, only the superconductor layer of HTS coated conductors is taken into account and represented by a line without thickness, which assumes that the distribution of current density in the horizontal (${J_x}$) and vertical (${J_y}$) direction are zero, simplifying the calculation to a 1D problem. Therefore, (1) and (2) can be simplified as follows:

$$\begin{align} {J_z} = \frac{{\partial \left( {T \cdot {n_y}} \right)}}{{\partial x}}\end{align} \tag{ 3 }$$

$$\begin{align} - \frac{{\rho \cdot \partial \left( {T \cdot {n_y}} \right)}}{{{\partial _x}^2}} = - \frac{{\partial {B_y}}}{{\partial t}} .\end{align} \tag{ 4 }$$

For the whole space, the magnetic vector potential A is introduced to link the distribution of sheet current density in the superconductors, and hence comes to the A-formulation part:

$$\begin{align} \nabla \times \left( {\frac{1}{\mu }\nabla \times A} \right) = J\end{align} \tag{ 5 }$$

$$\begin{align} B = \nabla \times A .\end{align} \tag{ 6 }$$

The current density in the non-superconducting region is not of interest, since that would not affect the open circuit voltage induced on the SC tape in this model. Instead of introducing the magnetic vector potential A in the non-superconducting parts, the magnetic scalar potential V_m can be utilized:

$$\begin{align} \nabla \times \left( { - \nabla {V_{\text{m}}}} \right) = J .\end{align} \tag{ 7 }$$

By replacing A with V_m, the solution time can be dramatically reduced, because the computational cost of scalar is much less compared to vector.

In addition, the E–J power law was employed to represent the electrical characteristics of superconductors:

$$\begin{align}{E_{{\text{z}}}} = \unicode{x03C1} J_{{\text{z}}} = {E_{{\text{c}}}} \left(\frac{J_{{\text{z}}}}{J_{{\text{c}}}(B)}\right)\left(\frac{|J_{{\text{z}}}|}{J_{{\text{c}}}(B)}\right)^{(n-1)} \end{align} \tag{ 8 }$$

where E_c and n were chosen as 1 × 10⁻⁴ V m⁻¹ and 20, respectively. The critical current density J_c(B) is a magnetic field dependent variable, which is commonly described by an anisotropic Kim-like model [38]. In order to compare our results to the work presented in [25], the magnetic field dependence of the critical current density is represented by a practical interpolation function directly obtained from the experimental measurements in [25]. The instantaneous equivalent voltage V_eq and open circuit voltage V_oc are defined as follows:

$$\begin{align}{V_{{\text{eq}}}}\left( t \right) = - L \cdot \frac{1}{{{W_{\text{s}}}}}\int\limits_0^{{W_{\text{s}}}} {{E_Z}} \left( {x,t} \right){\text{d}}l\end{align} \tag{ 9 }$$

$$\begin{align}{V_{{\text{oc}}}}\left( t \right) = \frac{1}{T}\int\limits_T^{2T} {{V_{{\text{eq}}}}} \left( t \right){\text{d}}t.\end{align} \tag{ 10 }$$

Note that L is the active length of the dynamo, and the 2nd cycle was chosen to perform the calculations, avoiding any initial transient effect that may be present in the 1st cycle.

Regression is a popular method for data processing, which is found to be very effective in exploring the underlying relations between a set of predictor variables and corresponding response variables. The aim of regression analysis is to predict outputs by feeding relevant inputs, which can be achieved either by parametric regression or non-parametric regression. Parametric regression is the process to describe the correlations among all inputs and outputs by mathematical expressions, and a typical model for a polynomial regression can be expressed as:

$$\begin{align} \widehat {{y^i}} = \,\alpha + {\beta _1}x_1^i + {\beta _2}x_2^i + \cdots + {\beta _n}x_n^i + {\epsilon ^i}\end{align} \tag{ 11 }$$

where $\alpha$ is the intercept to describe a constant factor that can be inherently included in the expression, ${\beta _n}$ is the coefficient for nth independent predictor variable ${x_n}$, $\widehat {{y^i}}$ and ${\epsilon ^i}$ are the predictive response variable and relative error for ith observation, respectively. In most cases, parametric regression is preferred, because it gives intuitive hints about what roles each of the inputs play in determining the output and provides full details about the numerical relations. Nevertheless, parametric regression is feasible only if the investigated data correlations can be described by explicit formulations. When the underlying relations among a set of input and output data involves highly complex mathematical transformations, it is challenging and inefficient to obtain analytical formulations. Fortunately, a modern artificial intelligence-based approach enables non-parametric regression, which is the process to summarize the key features about the dependence of outputs on the inputs in a certain manner that can be understood by computers, and hence the prediction of outputs by feeding relevant inputs is achieved without deriving explicit expressions.

The five popular artificial intelligence-based methods have been employed via Python Scikit-learn 5.1 environment [39]: multi-layer perception neuron network (MLP), support vector machine (SVM), Gaussian process regression (GPR), decision tree (DT) and k-nearest neighbours (KNNs). A typical MLP neuron network comprises three layers at least: the input layer, hidden layer and output layer. The number of neurons in the input layer and output layer are predefined by the features of given problem, namely the number of predictor and independent variables. It is the process that takes place in the hidden layers links the output layer to the input layer, where each neuron takes values from the previous layer and conducts a weighted linear summation by a transformation with specific activation functions. An identity function is adopted as the activation function in the output layer to receive results from the last hidden layer, and MLP uses the squared error loss function given as:

$$\begin{align} {\text{Loss}}\left( {\hat y,{\text{ }}y,{\text{ }}W} \right) = \frac{1}{2} \| \hat{y} - y\|_2^2 + \frac{\alpha }{2}\left\| W \right\|_2^2 \end{align} \tag{ 12 }$$

where $\hat y$ signifies the predicted values, $W$ is the weight coefficients matrix and $\alpha$ is a non-negative hyperparameter that puts penalty on complex models and hence suppresses overfitting.

In the field of non-parametric regression, GPR is considered as a highly effective approach because it can provide a framework to approximate sophisticated nonlinear functions with probabilistic estimates of the unseen uncertainty. The most important step in GPR is to train the specified data and find the optimized best hyperparameters θ comprised by parameters in the kernel function and a manually editable noise level $\sigma$, by the means of maximizing the log-marginal-likelihood for a GP as expressed in [40]:

$$\begin{align} \log p\left( {y{\text{|}}X,\theta } \right) & = - \frac{1}{2}{y^{\text{T}}}{{\left( {K\left( {X,X} \right) + {\sigma ^2}{I_N}} \right)}^{ - 1}}y \nonumber\\ &\quad - \frac{1}{2}\log \left| {K\left( {X,X} \right) + {\sigma ^2}{I_N}} \right| - \frac{N}{2}\log 2\pi \end{align} \tag{ 13 }$$

where ${I_N}$ denotes the $N \times N$ identity matrix, and $N$ is the number of inputs.

SVM allows the declaration of an allowable error rate $\varepsilon$ that tolerates all predicted values that are deviated from real values less than $\varepsilon$. In order to fairly take every data sample into consideration, one can introduce the slack variable $\xi$ and ${\xi ^{\text{*}}}$, which effectively releases the error rate so that all data samples can be potentially included in the safety margin. Meanwhile, any data point that lies beyond the margin bounds are undesired, so a coefficient $C$ is necessary to impose penalties on such data points to minimize their numbers. The object of this regression process is now to ensure the flatness of the hyperplane, or alternatively speaking that the coefficient vector w^T must be constrained to restrict its impacts on the regression results and hence generalize the problem, which can be formulated as [41]:

$$\begin{align} {\text{MIN}}{\text{ }}\frac{1}{2}\left\| w \right\|_2^2 + C\mathop \sum \limits_{i = 1}^l \left( {\xi + {\xi ^*}} \right)\end{align} \tag{ 14 }$$

$$\begin{align} {\text{st }}\left\{ {\begin{array}{*{20}{c}} {y - \langle w,x\rangle - b \leqslant \varepsilon + \xi } \\ {\langle w,x\rangle + b - y \leqslant \varepsilon + {\xi ^*}} \\ {\xi ,{\xi ^*} \geqslant 0} \end{array}} \right. .\end{align} \tag{ 15 }$$

The main idea of DT algorithm is to divide all data samples into smaller subsets by certain threshold criteria subsequently until decisions can be made on each data sample with enough accuracy. The finalized structure of model assembles a bottom-up tree consists of a root, certain amount of branch nodes and leaves. The critical step in DT regression is to select proper threshold criteria, which must ensure every split of the data samples pushes the model closer to the optimal. Assume there are ${N_m}$ samples at node $m$ represented by ${Q_m}$, a possible split $\theta \left( {j,{t_m}} \right)$, where $j$ is the feature dimension and ${t_m}$ is the threshold value, can partition ${Q_m}$ into the left subset $Q_m^{{\text{left}}}$ and right subset $Q_m^{{\text{right}}}$:

$$\begin{align} Q_m^{{\text{left}}}\left( \theta \right) = \left\{ {\left( {x,y} \right)|{x_j} \leqslant {t_m}} \right\}\end{align} \tag{ 16 }$$

$$\begin{align} Q_m^{{\text{right}}}\left( \theta \right) = Q_m\backslash Q_m^{{\text{left}}}\left( \theta \right) .\end{align} \tag{ 17 }$$

Then the quality of this split can be evaluated by employing the impurity function $P()$, which depends on the task to be solved:

$$\begin{align} M\left( {{Q_m},\theta } \right) = \frac{{N_m^{{\text{left}}}}}{{{N_m}}}P\left( {Q_m^{{\text{left}}}\left( \theta \right)} \right) + \frac{{N_m^{{\text{right}}}}}{{{N_m}}}P\left( {Q_m^{{\text{right}}}\left( \theta \right)} \right) .\end{align} \tag{ 18 }$$

The algorithms mentioned above all implement supervised machine learning process that relies on a specific objective function in those models, i.e. the loss function in MLP and impurity function in DT, which essentially converts the regression to an optimization problem of such a function. On the contrary, KNN model does not count on an objective function to converge the regression problem to an optimization process. Instead, KNN strictly relates the dependent variables to their corresponding inputs through an intuitive convention: any data samples that have the same values of dependent variables will also share the same values of independent variables.

In this paper, firstly it will be shown that parametric regression is not suitable for relating the predicted variable V_oc to the four parameters investigated here, namely $f$, $g$, W_s and, and then secondly conduct non-parametric regression via each of the approaches mentioned above. The adapted Nash–Sutcliffe model efficiency coefficient (NSE) is employed to evaluate the goodness-of-fit, and other top three most popular metrics [42], namely root mean square error (RMSE), mean absolute error (MAE) and mean absolute percentage error (MAPE) are included as complementary statistics to assess the regression results from different aspects. These numerical criteria are centred on prediction error, whose values can be interpreted mathematically and translated to the effectiveness of a certain regression. For example, the best possible score for NSE is 1, which implies that the regression model reaches the highest correctness level so that any unseen observations can be predicted with full confidence. In addition, graphical criteria proposed in [43] have been implemented, such as prediction distribution versus observations and the prediction rate curves, to illustrate the broader and more complex aspects of the relationship between the models and data, from which the most effective one can be selected.

$$\begin{align}{\text{NSE}} = 1 - \frac{{\sum\nolimits_{i = 1}^n {{{\left( {{y_i} - \widehat {{y_i}}} \right)}^2}} }}{{\sum\nolimits_{i = 1}^n {{{\left( {{y_i} - \bar y} \right)}^2}} }}\end{align} \tag{ 19 }$$

$$\begin{align}{\text{RMSE}} = \sqrt {\frac{1}{n}\sum\limits_{i = 1}^n {{{\left( {{y_i} - \widehat {{y_i}}} \right)}^2}} } \end{align} \tag{ 20 }$$

$$\begin{align}{\text{MAE}} = \frac{1}{n}\sum\limits_{i = 1}^n {\left| {{y_i} - \widehat {{y_i}}} \right|} \end{align} \tag{ 21 }$$

$$\begin{align}{\text{MAPE}} = \frac{{100}}{n}\sum\limits_{i = 1}^n {\left| {\frac{{{y_i} - \widehat {{y_i}}}}{{{y_i}}}} \right|} .\end{align} \tag{ 22 }$$

In this paper, a numerical model was built to be solved in the commercially available FEM software COMSOL 5.5. Different state variables are chosen to solve Maxwell equations in different regions: the current vector potential A was solved only in the superconductor domain, and the magnetic scalar potential V_m was solved in the whole space. However, due to the internal software settings that forbids existence of current within any closed loop in the V_m region, A was solved in a small area with close proximity to the superconducting domain. In order to verify the applicability, this model was adapted to represent the experimental set up described in [25]: a 12 mm wide ReBCO tape (Superpower SF12050F) is separated from a NdFeB magnet by an air gap of 3.7 mm. The magnet has a remanent flux density of 1.25 T and is embedded in a cylindrical disc with radius 35 mm. The HTS dynamo is operated at low frequency 4.25 Hz. The simulation results have been compared with data measured in experiments as shown in figure 2, in which excellent correlation can be observed. It should be pointed out that the distinct four peaks and noticeable left-to-right asymmetry observed in experiments have been reproduced qualitatively, and the quantitative agreement between the experimental results and simulation results from the model that takes the critical current density dependence on magnetic field density into account is very good. The up-biased open circuit voltage obtained from the model with constant critical current density can be readily understood by considering the dynamic resistance effect [44–46]. As stated in [25, 44], the open circuit voltage of HTS dynamo originates from the actions of circulating current within the tape, which can be equivalently reflected by the voltage division on the backward and forward eddy current regions, and it is the electric field generated on the forward eddy current region that essentially forms the DC output voltage. The backward region, aligned beneath the PM position, would demonstrate a significantly higher dynamic resistance if the field dependence is accounted, and hence more proportions of the conserved EMF (by Faraday's law) are distributed on this region. As a combinational result, higher DC output voltage is generated eventually.

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Four groups of simulations were arranged to primarily illustrate the relations between the controllable design parameters and the open circuit voltage of the rotary HTS flux pump, where in each group only one of the four aforementioned parameters is altered while the others remain constant. Figure 3(a) shows the frequency response within the input range from 10 to 100 Hz, aligning with the frequency range used in experimental results in [19]. It is worthwhile mentioning that a perfect linear relation was observed in [19] as the frequency increased from zero until a turning point was captured at approximately 100 Hz, which can attribute to the current interactions between different layers of the HTS tape [46–49] under high frequency. The single layer numerical model is not suitable for manifesting the multi-layer current interactions related phenomenon, hence the frequency is limited to 100 Hz to avoid this issue. Figures 3(b) and (c) illustrate how the output voltage varies with air gap and width of superconductor tape respectively, namely the output voltage decreases towards zero along the increase of air gap and increase from zero along the increase superconductor tape width, and both of them show acceptable agreement with previous work in [18, 20], which further validates the model. The impact of the remanent flux density of the permanent magnets upon the open circuit voltage is included, and initial results implies it has a clear positive correlation with the open circuit voltage. In summary, within the ranges investigated in our work, it was found the open circuit voltage tends to only monotonically decrease with the air gap, whilst increasing with the other parameters. In particular, the response variable of open circuit voltage demonstrates obvious linearity with frequency, and non-linearity was observed for all the rest parameters. As for the variations of SC tape width, it appears that there exists an optimal point, after which its positive correlation against the open circuit voltage becomes negative, which will be discussed in more detail later.

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Inspired by the observation in figure 3 that V_oc increases linearly with frequency, it is reasonable to believe that the function that describes V_oc about can be easily formulated by a straight line passing through the origin (based on the fact that no output is expected if the device is not operating at all) under the condition that all other parameters remain the same. Hence, it leads to a critical question: does this linear relation hold through the variation of other parameters? In order to investigate this problem, each of the other three parameters were varied under different frequencies, so that the frequency-normalized output profile V_oc/f can be plotted for each of them. It can be seen from figure 4 that the response level of V_oc against each of the parameters does vary under different frequencies as demonstrated by the separated solid lines, whilst the frequency-normalized output profiles collapse together as indicated by the dash lines in each subplot. Taking figure 3(a) into consideration, the curve behaviour mentioned above can happen only if the frequency acts as an 'amplifier' in the determination of V_oc, which implies that the linear relation between V_oc and $f$ holds despite any changes from other parameters. That is to say, the four-variable problem concerned in this paper can now be reduced to a three-variable problem by a simple frequency normalization as shown in following:

$$\begin{align} {V_{{\text{oc}}}}\left( {f,g,{W_{\text{s}}},{B_{\text{r}}}} \right) & = f\frac{{{V_{{\text{oc}}}}\left( {f,g,{W_{\text{s}}},{B_{\text{r}}}} \right)}}{f} \nonumber\\ & = f \cdot {V_{{\text{oc}}}}\left( {g,{W_{\text{s}}},{B_{\text{r}}}} \right).\end{align} \tag{ 23 }$$

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The consequence of adjusting $g$ or B_r will be reflected by the changing of peak flux density on the surface of the tape, as well as the flux passing through it. More precisely, increasing $g$ or decreasing B_r will lead to a smaller magnetic field seen by the tape surface, and less flux can be captured by the close loop within the tape, e.g. if the magnet is moved away from the tape or replaced by another one with smaller remanent flux density, the tape is then effectively exposed to a smaller magnetic field, and vice versa. According to Faraday's law, the electromotive force around a closed path is equal to the negative of the time rate of change of the magnetic flux enclosed by the path. Under the condition of fixed frequency, the magnetic flux transversing the tape per unit time is determined by the remanent flux density as well as air gap. Hence, the open circuit voltage V_oc, which is time integration of the electromotive force, should drop according to an increase in $g$ or decrease in, due to the reduction of flux along with its changing rate. This is clearly revealed by the results in figures 3(b)–(d), where it was shown that V_oc drops from 800 μV with a 2 mm air gap and approaching zero after the gap distance exceeds 8 mm, while V_oc climbs from approximately 0–1000 μV for values of B_r varying between 0.15 and 1.5 T. It is worthwhile mentioning that both extreme cases, where the gap distance goes to infinity and the remanent flux density reaches zero (the magnet loses magnetism) are well demonstrated in the results by confidently implying a zero output in those conditions.

Even though the individual effect of $g$ and B_r are clearly revealed and can be well explained as stated above, it is challenging to test whether or not the same effect can be influenced by changing other parameters. Unlike frequency, its effect on other parameters can be verified by a simple normalizing transformation about response versus each of the other parameters. The non-linear relationship between V_oc and $g$ and B_r make it difficult to achieve the same validation easily, and hence an analytical formulation that assembles (23) for $g$ and B_r is not obtainable. As a result, parametric regressions on $g$ and B_r are not preferred here, leading to the consideration of non-parametric regressions as an alternative.

As indicated in figure 3(b), V_oc increases gradually as the tape width increases from 6 to 30 mm, after which it starts to decrease if W_s is further increased. This matches the experimental results presented in [20], where the V–I curve intercept on the x-axis firstly became more positive and then less positive for the tape width increasing from 6 mm to 46 mm. Yet the V–I curve from our model plotted in figure 5 differs from the one presented in [20], in which a straight line is interpolated for each value of tape width. In fact, in their experiments, the net transport current of the tape is constrained by the critical current of the superconducting coil connected to the tape, so for the section of curve with the range of current which exceeds the critical current of the connected coils, it can only be interpolated by 1st few measurements where the transport current is below coil critical current. In figure 5, the limits of coil critical current have been removed so that the full profile of V–I curve for each tape width can be obtained. The intercept on the y-axis of each curve represents the theoretical maximum current that can be pumped if the connected coil has infinite critical current. It can be seen from figure 5 that the V–I curves tend to show non-linearity for large tape width, which is in alignment with the results in [26].

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The characteristics of this V–I curve can be further explored by analysing I_sc and V_oc separately. It is not surprising that the short circuit current I_sc increases with the tape width, because wider tape tends to have higher critical current. Two f-ratios are defined to measure the proportion of the capacity of the tape occupied by the transporting current, $\alpha$ and $\beta$ given as:

$$\begin{align}\alpha = \frac{{{I_{{\text{sc}}}}}}{{{I_{{\text{c,self}}}}}}\end{align} \tag{ 24 }$$

$$\begin{align}\beta = \frac{{{I_{{\text{sc}}}}}}{{{I_{{\text{c,min}}}}}}\end{align} \tag{ 25 }$$

where I_c,self denotes the critical current when the tape is not exposed to any external magnetic field, and I_c,min refers to the minimum value of immediate critical current as the applied magnetic field varies. Figure 6(b) illustrated that I_sc can reach up to about 55% of the self-field critical current and 75% of the minimum critical current for a 36 mm wide tape, and then stay at this value. Moreover, if we consider this dynamo as an electric circuit comprised by a voltage source with an internal resistor, then its resistance can be derived from the slope of the V–I curve. Please note that due to the existence of non-linearity for some curves, a data point based differential slope has been used to calculate the resistance for each curve. Figure 6(c) plots the resistance R_sc and R_oc for each tape width when the tape is shorted (e.g. there is no voltage drop across the tape V_oc = 0) and when the tape is isolated (e.g. the net transport current I = 0), whose values indicate the resistive properties of the tape under high and low current level respectively. It can be seen from figure 6 that R_sc and R_oc both drop as the tape width increases, meaning that the wider the tape the more superconductivity can be utilized.

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The most noticeable feature of V_oc versus tape width curve is that it increases first and then decreases as the tape gets wider, which makes W_s the only parameter that has a non-monotonical relationship with V_oc out of the four parameters investigated in this paper. In order to understand why this occurs, we need to go back to the origin of V_oc. The uneven and nonlinear resistivity distribution of the tape can produce an asymmetric electric field induced by the eddy current under a changing magnetic field as shown in figure 2(a), which can result in a nonzero time integration and hence an averaged dc voltage output. That is to say, the instantaneous electric field ${E_z}$ distribution, or equivalently the averaged instantaneous voltage V_eq, has direct impact on the determination of V_oc. Based on this idea, it is reasonable to deduce that the value of V_oc can be roughly reflected by the magnitude of the 1st peak of V_eq curve. As it tells in equation (9), for each instant V_eq is calculated by averaging the integration of the electric field along the tape, and it is the net remanent integration of the negative electric filed that determines the final value of V_eq. Thus, we will need to know how the electric field is distributed along the tape to investigate the formation of V_eq. The distribution profiles have been captured for selected tape width in figure 7 when the 1st peak appears. It can be seen from figures 7(a)–(c) that the integration over the range for negative electric field (the region marked as ${N_1}$ in figure 7(d)) increases significantly as the tape becomes wider, which contributes to a larger value of V_eq. As illustrated from figures 7(d)–(f), once the tape reaches a certain width, the integration for electric field in region ${N_1}$ remains approximately constant, and further width increment will only raise the integration over the range for positive electric field (the regions marked as ${N_2}$ in figure 7(d)). Consequently, a fixed integration over the negative region will be cancelled more and more by an increasing integration over the positive region, leading to a small V_eq as a combinational effect. This smaller V_eq peak value implies that less V_oc is obtainable through the time integration of V_eq over the whole cycle, which explains why V_oc can be promoted by the tape width but suppressed if the tape becomes too wide. Besides, the definition of 'too wide' here is restricted by the sole variation of tape width, but the effects described above can be also influenced by the geometry of the device, e.g. the airgap between the magnet and tape, the dimension of the magnet and even the radius of the rotary disc. More comprehensive analysis is required to fully investigate the 'width effect' and complete the definition of 'too wide' by cooperating all relevant parameters, which is beyond the scope of this paper and will be addressed in the future.

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Although the HTS dynamo can now be simulated by various numerical models via FEM programs, including the T–A formulation based model proposed in this paper, which enables us to obtain considerable knowledge of the physical mechanism that takes place during the operation, but in terms of industrial design, the ultimate goal is to make use of the functionality of certain devices. As for this HTS dynamo, it is expected to be utilized as an energisation source that can pump current into load coils and hence form a HTS magnet. From this point of view, it is the averaged output dc voltage V_oc that matters. There is no doubt that V_oc can be measured from experiments, but these are often time-consuming and expensive and not accessible to all occasions. Valid FEM simulations are considered to be a promising substitution for setting up real experiments to get comprehensive results in less time and saved cost. However, these FEM models can sometimes require relatively long time for hours, and the applicability is also limited by user knowledge and accessibility to a sophisticated FEM program, which is not guaranteed. Artificial intelligence methods are powerful tools to accurately model certain data behaviour, which have been recently involved in solving applied superconductivity problems [50]. For instance, artificial neural network (ANN) has been utilized in [51–54] to estimate the calculation of AC losses in specific SC system and aid the design of SC facilities. As an alternative, we have explored the feasibility of constructing a statistical model, which can give accurate V_oc values according to the given controllable parameters within seconds, and hence provides primary guidance for the design of such devices in practical applications.

Thousand sets of different parameters, in which each of them lies in the rational range for a practical design process, have been randomly chosen and simulated by the numerical model described above to derive V_oc for each set of parameters. As suggested by equation (23), all output voltage variations that are solely affected by frequency can be easily quantified by simple normalization calculation, so we have precluded the frequency parameter $f$ in following investigations, e.g. all parameter settings share the same value of frequency (50 Hz), in order to simplify the modelling work. According to the conventional 8–2 rule, data samples are evenly divided into five folders for cross validation, in which four folders would be utilized as training group and keep the remaining one as a testing group for each validation.

3.5.1. Neuron network model.

3.5.2. Gaussian process model.

3.5.3. SVM model.

3.5.4. DT model.

3.5.5. KNNs model.

3.5.6. Model selection.

Five machine learning based regression models have been primarily tested under various settings, and each of them have also been evaluated by numerical metrics to judge the goodness of such a model in predicting the HTS output voltage based on the four design parameters. Yet, numerical criteria just show the general aspect of a model, which are not enough in practice. As stated earlier, in order to illustrate the broader and more complicated aspects of the relationships between models and data, graphical criteria are indispensable. Let us define the samples with relative prediction errors below a given level as 'effective-predicted' samples, and we can evaluate the predicting effectiveness of different models by the proportion of the effective-predicted samples with respect to the relative prediction error, which is referred as 'prediction rate'. In order to achieve this, we define ${\delta _i} = \left| {\frac{{{y_{i - }} - \widehat {{y_i}}}}{{{y_i}}}} \right|$ and then plot the additive rate of n samples, e.g. from 0 to 1 by step of 1/n with respect to the sorted ${\delta _i}$ from zero to maximum relative error. The prediction rate curve of a model is obtained by joining those points. The ratio of effective-predicted samples can be accessed from vertical axis as relative error being horizontal axis. The introduction of this graph allows us to examine more details about the performance of each model. The area between the prediction rate curve and the vertical axis, which aggregates the relative errors, provides a linkage to the numerical cross-validation criteria formerly discussed. In general, a model can be considered good if this area is smaller, and it is visually easy to compare models with this plot. According to former analysis, the best candidate from each model group are chosen to plot their prediction rate curve.

As shown in figure 8, the prediction rate curves for MLP and DT and KNN models lie completely below those for GPR and SVM, which means that for any given error level, less samples can be effectively predicted by MLP, DT and KNN models than the others. Besides, MLP, DT and KNN curves cover a much wider range of errors than the other two curves, indicating that they have worse error containing ability. For instance, the maximum relative error generated by MLP, DT and KNN models are around 154%, 123% and 80%, respectively, and this strengthens the conclusion that MLP, DT and KNN models are not effective in providing stable and trustable predictions (note that the actual V_oc values for those samples being predicted range from 0 to 1971 μV). As for the GPR and SVM models, though they both demonstrated good capability of maintaining the relative errors well below 12%, the GPR model is believed to restrict the error level under a lower level as it arrives the full proportions slightly faster than the SVM model. Meanwhile, the curve of SVM almost consistently grows beneath that of GPR, which can be interpreted as aforementioned: more samples can be accurately predicted by GPR model than SVM model given a certain error threshold level. It is true that the capability of producing minor error predictions is not significantly different for GPR and SVM models, yet their solution time are relatively distinct. The SVM model requires more than 2 min to be solved, while the GPR model only takes few seconds, which is ten times less than the former one. Therefore, it is reasonable to select the GPR model as the optimal candidate to model the HTS dynamo output voltage, not only for its excellent accuracy but also due to the promise of saving considerable solution cost.

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To verify this hypothesis, we tried to apply this GPR model to reproduce figure 3 by feeding three sets of parameters (except for the frequency $f$) as inputs and extract the values of V_oc as outputs. The results are presented in figure 9(a), where the blue dots V_oc values come from real simulations, e.g. those plotted in figure 3, and the red dots include V_oc values predicted by our GPR model. It is clear that the output voltage dependence against all three parameters are well described by our predictions, more precisely, V_oc drops over the increase of air gap (from 1st to 10th run), V_oc firstly increases then decreases over the increase of SC tape width (from 11th to 20th run), and V_oc climbs over the increase of the remanent flux density of the permanent magnet (from 21st to 30th run). As further illustrated in figure 9(b) that the average error rate for all predictions is 2.98%, which demonstrates the accuracy of our model. It should be noted that there are several predictions that have noticeable errors, such as the 21st, 22nd runs, which are corresponding to extreme conditions, e.g. B_r equals to 0.15 T and 0.3 T. Under such extreme conditions, weak output voltages are expected and hence those small values are highly sensitive to any mismatch. However, in practice B_r is commonly around 1 T, so that the incompetence of our GPR model for predicting extreme conditions will not undermine its performance. It can then be concluded that our GPR model is capable of predicting accurate output characteristics for HTS rotary flux pumps in general, and hence can be utilized as a reliable design tool for such devices.

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It has so far been revealed that the operating frequency $f$, air gap $g$, SC tape width W_s and remanent flux density of the permanent magnet B_r all have influences on the HTS dynamo output voltage V_oc, after which a GPR based statistical model was proposed to accurately predict V_oc by feeding each of parameters as inputs. However, to what extent does each parameter impact the output voltage still remains unclear. In order to fulfil the aim of this paper, which is to provide compressive guidance for HTS rotary flux pump design, the sensitivity of each parameter was further explored through an analysis of variance (ANOVA). As before, the operating frequency $f$ is excluded from the analysis list to reduce the workload. As a as result, the number of second order interaction terms is reduced from $C_4^2$ to $C_3^2$, namely g × W_s, g × B_r and W_s × B_r. We have set the values of each parameter to three levels, because the relation between V_oc and W_s is not monotonic so that at least three values are required to represent the variation trend, which comprises of both increasing and decreasing parts. The orthogonal tables are believed to effectively reduce the cost of experimental schedules and mathematical calculations, by the means of only considering the representative values of each parameter in a proper way. The ${L_{27}}\left( {313} \right)$ Taguchi array has been adapted to include the three parameters and their corresponding interaction terms. The ANOVA results are represented in table 11, which interprets each of variable via F-test and the statistical hypothesis (under a 95% confidence level) test. It is shown that all three parameters are significant as their F-values are far greater than the criterion ${F_{0.05}}\left( {2,8} \right) = 4.46$. The remanent flux density seems to be the prior dominant factor for determining the output voltage, and the importance of air gap is slightly lower while the importance of SC tape width is dramatically less with only 40% of the F-value for the remanent flux density. As for the interaction terms, it is believed that the interactions between $g$ and B_r are very significant as its F-value suppresses the criterion ${F_{0.05}}\left( {4,8} \right) = 3.84$. Meanwhile, the other two interaction terms are relatively significant, because their F-values are located between the criterion interval ${F_{0.05}}\left( {4,8} \right) = 3.84$ and ${F_{0.1}}\left( {4,8} \right) = 2.81$. In a summary, the importance rank for each parameter is: ${B_{\text{r}}} > g > {W_{\text{s}}}$, and all 2nd order interaction effects between them are found to be influential as well. This can be interpreted as that the individual effect of each parameter should be considered first during the design and optimization of HTS rotary flux pumps, whilst the designer should be aware of the existence of interaction effects since they possess unignorable influences upon the output characteristics.

Table 11. ANOVA results.

Source	Df	F-value	p-value
$g$	2	23.668	0.000437
Ws	2	10.136	0.006
Br	2	25.342	0.000345
g × Ws	4	3.261	0.073
Ws × Br	4	3.835	0.050
Br × g	4	8.039	0.007