Contents
Introduction....................................................................................................... 2
Modelling open-circuit voltage behaviour................................................................................. 2
A new benchmark problem........................................................................................ 3
Efficient 3D models............................................................................................. 5
Investigating key HTS dynamo parameters................................................................................ 6
V-I characterisation.............................................................................................. 6
Gap dependence of the open-circuit voltage............................................................................ 6
Influence of the HTS wire width.................................................................................... 7
Frequency dependence: Considering the full HTS wire architecture, coupled with a thermal model................................... 8
Modelling dynamic coil charging behaviour................................................................................ 9
AC loss and energy considerations....................................................................................... 10
Conclusion and view towards the future.................................................................................. 12
Exploring the key parameter space further & using artificial intelligence to accelerate HTS dynamo design optimisation................... 12
More elaborated models towards HTS dynamo design for large-scale applications................................................ 13
Coupling numerical models and simplified circuit models................................................................. 14
Declaration of Competing Interest..................................................................................... 15
References...................................................................................................... 15
0 Introduction
High-temperature superconducting (HTS) flux pumps exploit the nonlinear resistivity of an HTS coated-conductor wire to generate a DC voltage when subjected to a varying magnetic field. This effect enables injection of large DC currents into a superconducting coil connected to the flux pump, without the need for thermally-inefficient current leads. Because of this important advantage, there is significant interest in using such technology to energise superconducting coils in superconducting rotating machines [1], [2] - including wind turbine generators [3], [4] - and NMR [5] and MRI [6] magnets. In the HTS dynamo-type flux pump (hereafter, HTS dynamo), the source of this varying magnetic field is a permanent magnet(s) (PM) that transits past a stationary HTS wire(s) [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18]. Because of its simple construction (similar to a conventional PM alternator) and ease of operation compared to other types of flux pumps, it has drawn significant attention.
Although there has been extensive experimental work carried out over the past decade, there has been some confusion and debate regarding the physical origin of the HTS dynamo’s DC output voltage. Despite the geometric simplicity of the device, quantitatively-accurate, predictive calculations have been difficult to achieve because the solution is actually rather complex, comprising time-varying, spatially-inhomogeneous currents and magnetic fields throughout the superconducting volume [19]. Numerical modelling has played a key role in elucidating the underlying physics of such devices. Several different numerical models have now been developed as useful and cost-effective tools to not only explain and further examine experimental results, but also optimise and improve HTS dynamo designs.
Here, all of the developments in this important area over recent years are summarised, noting that this review focuses specifically on modelling HTS dynamos. An interested reader may refer to the review papers [20], [21], [22], [23] for a broader overview of other types of HTS flux pumps. Firstly, in Section “Modelling open-circuit voltage behaviour”, numerical modelling techniques that can accurately reproduce the observed open-circuit voltage behaviour of the HTS dynamo are described, including the definition of a new benchmark problem for the HTS modelling community which is used to critically compare the performance of state-of-the-art methods for modelling superconductors. Efficient 3D modelling techniques that have been proposed to take into account important 3D considerations not possible with 2D simplifications are also described. In Section “Investigating key HTS dynamo parameters”, the application of numerical models to investigate key HTS dynamo parameters is described, including V-I characterisation, the gap dependence of the open-circuit voltage, the influence of the width of the HTS wire and the frequency dependence - considering the full HTS wire architecture, coupled with a thermal model. Properly modelling the dynamic behaviour of the HTS dynamo while charging a coil is explored in Section “Modelling dynamic coil charging behaviour”, which is of great interest from the perspective of practical applications. As with many other applications of HTS materials where time-varying currents and/or magnetic fields are involved, understanding the nature and magnitude of the losses in the HTS dynamo - including the AC loss - during its operation is important, and Section “AC loss and energy considerations” details how numerical models can be used for this purpose. Concluding the paper, in Section “Conclusion and view towards the future”, a view towards the future is provided, including the outstanding challenges and the developments required to address these.
2 Modelling open-circuit voltage behaviour
Until recently, the underlying physical origin of the HTS dynamo’s peculiar electromagnetic behaviour was not well understood. The central point of confusion is that a DC open-circuit voltage can be generated from a varying magnetic field - despite the fact it is topologically identical to the AC alternator described by Faraday [24] - in apparent contravention of Faraday’s law. This has led to some exotic explanations for its behaviour such as quantum flux coupling [25] and flux ratcheting [26].
A great step forward in understanding the behaviour of HTS flux pumps in general was made by Geng et al. [27] who described in detail - based on analytical considerations and focused on the travelling wave flux pump - that the varying, nonlinear resistivity of type II superconductors due to flux flow is the origin of the DC voltage. The authors also describe how the variation in resistivity, and hence flux pumping, is influenced by the current density, magnetic flux density and the field rate of change. In [11], Bumby et al. suggested that the DC voltage can be explained by a model first proposed by Giaever [28], [29] that considers the impact of time-varying circulating eddy currents within the superconductor, which are partially rectified as the PM transits past the HTS wire.
However, a comprehensive modelling framework that could provide quantitative, predictive results that matched experimental data was still in demand. Mataira et al. [30] showed, via a 2D numerical model, that the open-circuit voltage of the HTS dynamo can be explained well - importantly, with results that quantitatively agree with experimental data - using classical electromagnetic theory. The dynamo topology modelled in this work, shown in Fig. 1, is based on the experimental setup presented in [15] and consists of a single rectangular Nd-Fe-B PM rotating past a single 12 mm-wide HTS coated-conductor wire manufactured by SuperPower.
Fig. 1. Schematic of the HTS dynamo modelled in Mataira et al. [30], consisting of a single rectangular Nd-Fe-B permanent magnet rotating past a single 12 mm-wide HTS coated-conductor wire. |
The 2D finite-element model in [30] implements the well-known H-formulation [31], [32] for the dynamo’s HTS stator wire, where the independent variables are the components of the magnetic field strength H = [Hx, Hy, 0] and the governing equations are derived from Ampere’s and Faraday’s laws. The crucial difference, in comparison to a material that obeys Ohm’s law, is the superconductor’s nonlinear resistivity, ρ(J), which is described by the E-J power law [33], [34], [35]:
$\mathbf{E}=\frac{E_{0}}{J_{c}}\left|\frac{J}{J_{c}}\right|^{n-1} \mathbf{J}$
where the electric field E = [0, 0, Ez], the current density J = [0, 0, Jz], E0 = 1 µV/cm is the characteristic electric field, Jc is the critical current density of the superconductor and n defines the steepness of the transition between the superconducting state and the normal state. The PM rotor is represented, using a conventional technique, by a shell current that is rotated around the rotor boundary. In addition, under open-circuit conditions, no net transport current flows, so an appropriate current constraint is implemented.
Fig. 2 (a) shows the open-circuit voltage waveforms presented in [30], assuming measured in-field Jc(B, θ) data for the HTS wire or a constant Jc, compared with experimental data. The results are presented for the 2nd transit of the PM past the HTS wire, ignoring any initial transient effects that may be present in the 1st cycle. The voltage ΔV(t), which realises the DC output, is derived from the electric field averaged over the cross-section of the wire, Eav(t), multiplied by the active length of the dynamo, l (i.e., the active length (depth) of the PM). In other words,
$\Delta V(t)=-l \cdot E_{a v}(t)=-\frac{l}{S} \iint_{S} E_{z}(x, y, t) d S$
Fig. 2. (a) Open-circuit ΔV(t) voltage waveforms, and (b) their cumulative time-averages, for the HTS dynamo modelled in [30] using the H-formulation + shell current model. |
This ΔV(t) corresponds to V77 K(t) - V300 K(t), where the wire is in its superconducting (asymmetric behaviour) and normal (symmetric behaviour) states, respectively [11]. The actual total measured voltage, V77 K(t), includes ∂tA contributions from both the PM and the current density in the superconductor (where A is the magnetic vector potential). However, it is pointed out in [30] that because A is periodic over each cycle of rotation, the time-averaged (DC) value of ∂tA must always be zero. Later, in Section “Modelling dynamic coil charging behaviour”, the total measured voltage will be described in more detail and why the ∂tA contributions need to be taken into account when considering the dynamic behaviour of the HTS dynamo when used to charge a coil. Fig. 2 (b) then shows the cumulative time-average for each waveform in Fig. 2 (a), which converges to VDC, the DC output voltage, as t → ∞.
It is important to note here that, although the assumption of realistic (i.e., measured) Jc(B, θ) data for the particular HTS wire is needed to obtain a numerical result comparable to experiment, the constant Jc assumption still produces a non-zero DC output. Thus, the nonlinear resistivity described by (1) is all that is required to generate the DC output. The authors emphasise this point in [30], showing that VDC = 0 for n = 1 (linear Ohmic behaviour), and for n > 1, VDC ≠ 0, increasing as n increases (i.e., for greater nonlinearity in resistivity). Hence, the effect is a ‘classical’ one and can be explained using classical electromagnetic theory. It arises naturally in the HTS dynamo from a local rectification effect caused by overcritical eddy currents [30], [36], which is an effect that has actually been observed in HTS materials as far back as Vysotsky et al. [37].
2.1 A new benchmark problem
This work of Mataira et al. [30] generated significant interest amongst the HTS modelling community to apply different frameworks to solve the problem and led to the definition of a new benchmark problem [19]. Benchmark problems are important tools for the evaluation of numerical modelling techniques with respect to accuracy and computational efficiency. Based on a specific simplified geometry with well-defined inputs (i.e., assumptions), it allows a model to be validated against an expected set of outputs (i.e., the solution). The model’s performance can then be critically compared with other state-of-the-art methods for modelling superconductors.
The geometry of the benchmark problem for the HTS dynamo is shown in Fig. 3, assuming for simplicity the 2D case. A PM with a remanent flux density Br, of width a and height b, rotates anticlockwise past the stationary HTS wire at the top, and the face of the PM is located at a radius, Rrotor. The initial position of the PM is such that the centre of its outer face is at (0, -Rrotor), i.e., θM(t = 0) = -π/2. The HTS wire has a width e and thickness f and is positioned such that its inner face is located at (0, Rrotor + airgap). Jc is assumed to be constant - since it was shown in [30] (see Section “Modelling open-circuit voltage behaviour”) that this assumption does not impact the essential dynamics to deliver a DC voltage, i.e., the nonlinear resistivity via the E-J power law - and corresponds to Ic [self-field, 77 K] = 283 A.
Fig. 3. Geometry of the HTS dynamo benchmark problem [19]. A permanent magnet rotates anticlockwise past a (stationary) HTS wire. |
This benchmark problem was implemented in [19] using several different methods, including the H-formulation + shell current (H + SC) method [30] described in the previous Section “Modelling open-circuit voltage behaviour”, the segregated H-formulation (SEG-H) [38], coupled H-A [39] and T-A [40], [41] formulations, the Minimum Electromagnetic Entropy Production (MEMEP) method [42], and integral equation (IE) [43] and volume integral equation-based equivalent circuit (VIE) [44] methods. Each of these approaches - as shown in Fig. 4 - show excellent qualitative and quantitative agreement for the open-circuit equivalent instantaneous voltage, where Veq(t) = ΔV(t) as described by (2). The results are presented for the 2nd transit of the PM past the HTS wire, ignoring any initial transient effects that may be present in the 1st cycle.
Fig. 4. Open-circuit equivalent instantaneous voltage waveforms calculated by each of the models for the HTS dynamo benchmark problem [19]. |
A corrigendum was published in [45] to correct the result from the MEMEP method in [19] as the voltage definition differed slightly to (2) and included the ∂tA contribution from the current density in the superconductor. Although both definitions result in the same DC voltage, the instantaneous voltage waveform differs slightly (see Section “Modelling dynamic coil charging behaviour” for more details).
The average cumulative time-averaged equivalent voltage between all the models was -9.41 µV with a standard deviation of 0.34 µV. There was also excellent agreement between all models with respect to the current density and electric field distributions within the HTS wire at key positions during the PM transit, as shown in Fig. 5.
Fig. 5. Distributions within the HTS wire calculated by each of the models for the HTS dynamo benchmark problem [19], for the normalised current density, J/Jc0 (top), and electric field, E (bottom), for three key PM positions. |
A critical analysis and comparison of each of the models is also presented in [19], based on the following key metrics: number of mesh elements in the HTS wire; total number of mesh elements in the model; number of degrees of freedom (DOFs); tolerance settings; and approximate time taken per cycle. The clear winner in terms of computational speed was the MEMEP method, with the entire 10 cycles of the benchmark problem taking around two minutes to solve. This is due to the efficiency of the variational method used as well as the reduced number of DOFs because only the HTS wire needs to be meshed. The next best performing methods - SEG-H and VIE - also emphasised a reduced number of mesh elements, and several models took advantage of an artificial expansion technique to increase the HTS wire thickness from 1 μm to 100 μm, improving the computational speed without compromising accuracy.
There is also a distinct advantage of self-programmed techniques - like the MEMEP and VIE techniques, and those proposed by Prigozhin and Sokolovsky later in this section and for the 3D problem in Section “Efficient 3D models” - in terms of computational efficiency. The user also has complete control over the implementation of the model. While models implemented in commercial software packages, e.g., COMSOL Multiphysics™, often have a much gentler learning curve and example models can be easily shared, some of the programming cannot be accessed easily, if at all, and there are implementation overheads that generally result in less efficient and comparatively slower models. The choice of formulation is also important: some formulations are inherently more stable than others, e.g., H-A in comparison to T-A, as described in detail in [46].
The benchmark problem was subsequently successfully implemented by Prigozhin and Sokolovsky [47] using expansions in Chebyshev polynomials for approximation in space and the method of lines for integration in time. A similar approach was proposed in [48] for HTS coated-conductor wires, stacks and coils, based on two factors: fast convergence of the interpolation by Chebyshev polynomial expansions and the ease of using such expansions for solving integral equations with a singular Cauchy-type kernel. This method is shown to be faster than all of the methods in [19], taking as little as several seconds per cycle to solve for N = 200 mesh points (the HTS wire is treated as a 1D line in this framework).
The model is then extended to incorporate a field-dependent Jc(B). The impact of this on the open-circuit voltage is demonstrated in Fig. 6 (a), where h0 is a fitting parameter such that the smaller h0 is, the stronger the dependence of Jc on magnetic field (for h0 → ∞, Jc ≈ Jc0). The impact of a finite transport current flowing in the HTS wire on the HTS dynamo’s output voltage is also investigated, as shown in Fig. 6 (b). The importance of knowing this behaviour is discussed in more detail in Section “V-I characterisation”, but it allows one to determine the HTS dynamo’s V-I characteristics as a DC voltage source.
Fig. 6. Calculated results from [47] using expansions in Chebyshev polynomials and the methods of lines: (a) influence of a field-dependent Jc(B) on the open-circuit voltage, (b) influence of a finite transport current flowing in the HTS wire on the HTS dynamo’s output voltage. |
2.2 Efficient 3D models
Efficient 3D modelling techniques have also been proposed to take into account important 3D considerations not possible with the 2D simplification described in previous sections. For example, in Section “Modelling open-circuit voltage behaviour”, the active length (depth) of the PM, l, was used to manipulate the results from 2D models because the experimental setup [15] that these were based on - and others such as [7], [8], [9], [14], [15], [16], [17] - utilised rectangular PMs. However, several experimental works in the literature [10], [11], [12], [13], [18] utilise cylindrical PMs, for which such a simplified assumption is not so easy to make and for which 3D models are needed. Furthermore, 2D models cannot consider the complete return path for the current flow, i.e., the component of the current that flows across the tape width, perpendicular to the current flow along the length that was described in detail earlier (see Fig. 5). Thus, a 3D model can provide us with this complete picture, as well as allow us to deal with geometries that do not lend themselves well to a 2D simplification.
The first such model, proposed by Ghabeli, Pardo and Kapolka in [49], is based on the MEMEP 3D method [50], [51], [52], [53]. This has same advantages in 3D as described for the 2D case: it is a variational method that only requires the HTS wire to be meshed, resulting in a reduced number of DOFs. The model in [49] is based on the experimental setup reported in [11]. To obtain a qualitative sense of 3D effects and avoid the complication of Jc(B), initially a constant Jc for the HTS wire is assumed. Fig. 7 shows the calculated results for the Jc-normalised current density distributions for an airgap of 3.3 mm for several key magnet positions, including step ‘0’ when the PM is very far from the tape at the beginning of the 2nd cycle. It is clear from Fig. 7 (b) how the current induced in the wire flows in 3D, including the impact of the finite size of the PM with respect to the HTS wire along its length and the J component that flows across the tape width.
Fig. 7. Calculated results from [49], using the MEMEP 3D method, for the current density distribution for an airgap of 3.3 mm and constant Jc: (a) Key magnet positions, (b) Jc-normalised current modulus maps and current lines with respect to the key magnet positions, (c) Jc-normalised current profiles with respect to the key magnet positions along the mid-plane of the HTS wire. |
The impact of the wire length is then explored to determine the minimum efficient length of wire that could be employed, as well as the appropriate distance between voltage taps required to fully capture the voltage signals experimentally. It is found that for this setup, which has a 10 mm-diameter PM, that a minimum wire length of 20 mm is needed to generate the full, expected voltage, but this length increases as the airgap increases because the length of wire over which the current is induced increases. Subsequently, the distance between the voltage taps should be at least 2.5 times the PM ‘length’ along the wire for airgaps up to 6 mm, increasing for an increasing airgap.
Finally, the model is validated quantitatively against the experimental results in [11] by assuming realistic Jc(B, θ) data for the particular HTS wire used (SuperPower SF12050CF). Fig. 8 shows a comparison between the MEMEP 3D model in [49] and the MEMEP 2D model of the DC open-circuit voltage for airgaps between 2.4 and 10.4 mm reported in [54]. The 3D model presents very good agreement, even better than the 2D model. One particular reason for this is that the magnetic field from the PM decays with distance, r, by 1/r2 (infinitely long dipole), whereas in reality (in 3D) it decays by 1/r3 (point dipole) [55]. Fig. 8 also highlights the sensitivity of the DC output on the airgap distance, which is described in more detail in Section “Investigating key HTS dynamo parameters”, where a summary of a number of numerical studies investigating key dynamo parameters is provided.
In [56], Prigozhin and Sokolovsky provide two alternative methods of modelling the HTS dynamo in 3D, based on the mixed finite element method in [57] and the modified fast Fourier transform (FFT) method in [58]. The model geometry and assumed parameters are based on the (2D) benchmark problem (see Section “A new benchmark problem”), but includes a finite length for the rectangular PM (12.7 mm, the ‘active length’ multiplier in the benchmark) and HTS wire (48 mm). In addition, the thickness of the HTS wire is neglected such that it is treated as a 2D object (of finite width and length).
Fig. 9 shows the calculated results from [56], using the mixed finite element method, for the electric field component in the direction parallel to the length (x-axis) of the HTS wire and the sheet current density flowing in the wire, for three rotor positions. Similar to Fig. 7 for the MEMEP 3D method, one can clearly observe the current flow including the component of the current flowing across the width of the wire. It is noted that the corresponding electric field in this direction (y-axis) remained less than 15 mV/m for the entire simulation, comparatively much smaller than the x-direction component (parallel to the length of the wire). The study is extended to investigate the influence of the PM length on the accuracy of the solution, and it is shown that if the PM is not sufficiently long enough, then there can be a significant error in the calculated voltage from 2D models using the ‘active length’ multiplier, which is avoided in the 3D case.
Fig. 9. Calculated results from [56], using the mixed finite element method in [57], for the electric field component in the direction parallel to the length (x-axis) of the HTS wire (left; insets show the corresponding PM positions - yellow, with respect to the HTS wire - blue) and the sheet current density flowing in the wire (right), for three rotor positions (top to bottom). |
Another 3D modelling framework was proposed by Sokolovsky and Prigozhin in [59]: a pseudospectral method based on bivariate expansions in Chebyshev polynomials and Hermite functions. The Hermite-Chebyshev method is applied to the same 3D extension of the HTS dynamo benchmark problem [56] and is shown to be fully competitive with, and perhaps even more efficient than, the mixed finite element method used in [57] in terms of speed and accuracy.
3 Investigating key HTS dynamo parameters
3.1 V-I characterisation
As mentioned in Section “A new benchmark problem”, understanding the impact of a finite transport current flowing in the HTS wire on the HTS dynamo’s output voltage allows the dynamo’s electrical (V-I) characteristics as a DC voltage source to be determined. For many configurations [36], [60], these V-I characteristics are linear and the HTS dynamo can thus be modelled electrically as a (current-controlled) DC voltage source, Voc, in series with an effective (internal) resistance, Rint, as shown in Fig. 10. Also included in Fig. 10 are the current source, IT, and the current leads, termination blocks and joints combined into a single resistance, Rj. Such a setup is used experimentally to determine the HTS dynamo’s V-I characteristics. In a numerical model, a transport current can be imposed through an appropriate current constraint such that
$I=\int_{S} \boldsymbol{J} \cdot d \boldsymbol{S}=I_{T}$
where IT = 0 corresponds Voc, the open-circuit condition described in Section “Modelling open-circuit voltage behaviour”, and the current can be incrementally increased until VDC = 0, which corresponds to the short-circuit current, Isc. Isc is hence an important limiting, operational parameter for the HTS dynamo, where its output voltage becomes negligible.
Fig. 10. Electrical circuit diagram showing the dynamo as a DC voltage source, Voc, and an effective (internal) resistance, Rint. Also included are the current source, IT, and the current leads, termination blocks and joints combined into a single resistance, Rj [36]. |
Fig. 11 shows an example of the DC electrical characterisation of the HTS dynamo - presented in [36] using the modelling framework of [30] but also including two copper stabiliser layers on either side of the HTS wire - comparing (a) experimental and (b) modelling results, for rotational frequencies of 2.97 - 24.83 Hz. There is excellent agreement between the experimental and modelling results, with only a 5% deviation for the five highest frequencies. The parameters Voc and Isc can be extracted from these curves for IT = 0 and VDC = 0, respectively, and Rint can be determined from the slope of each curve, i.e., -dVDC/dIT. It is also observed in Fig. 11 that there is a frequency dependence of the HTS dynamo characteristics and this is discussed further in Section “Frequency dependence: considering the full HTS wire architecture, coupled with a thermal model”.
3.2 Gap dependence of the open-circuit voltage
As seen in Fig. 8 in Section “Efficient 3D models”, there is an obvious dependence of the HTS dynamo open-circuit voltage on the gap between the rotating PM and the HTS wire; thus, this gap is a key design parameter that can significantly affect HTS dynamo performance. The magnitude of the magnetic field impinging on the HTS wire plays an important role in the generated voltage, as detailed in [54]; by increasing the gap, the magnetic flux density seen by the HTS wire is lower and there is a corresponding reduction in output voltage.
It also is shown in [54] that the perpendicular component of the magnetic field, which affects Jc(B) the most in HTS coated-conductor wires, decreases sharply with increasing airgap. Not only the gap but the Jc(B) characteristics of the wire are also a key design parameter, as mentioned briefly in Section “A new benchmark problem” with respect to [47], although this has not been a focus of any numerical studies to date. It is also mentioned in [30] that a significant increase in VDC can be achieved by using a wire that exhibits a strong decrease in Jc under a perpendicular magnetic field, and that attempts by commercial manufacturers to reduce this effect (through the addition of artificial pinning centres and so on) for improved performance in many other applications are likely to be detrimental to HTS dynamo performance.
The gap dependence of the HTS dynamo can also be alleviated through the judicious use of iron in a ferromagnetic (FM) circuit linking the PM and HTS wire. It has been demonstrated experimentally that the use of iron can increase the gap over which the HTS dynamo can operate efficiently, even across a cryostat wall with the PM(s) and rotor located outside the cryostat [1], [60], [61], [62], [63]. Recently, [64] explored the use of a magnetic coupler to achieve such ‘through-wall’ operation of the dynamo, and investigated the charging performance with and without an FM slice present on either side of the HTS wire. The authors demonstrate, using a numerical model based on the H-formulation, that when the FM slice is located on the opposite side of the wire to the PM, the magnetic field seen by the wire can be increased, resulting in improved performance - as long as the FM slice does not saturate - for the same gap. This may also have important implications for the use of an HTS wire with a magnetic substrate.
3.3 Influence of the HTS wire width
It was described earlier in Section “Efficient 3D models” that the length of the HTS wire with respect to the PM length can affect the HTS dynamo’s output voltage, as well as the accuracy of the numerical solution when comparing 2D and 3D models. Another related, key dynamo parameter is the stator (HTS wire) width with respect to the PM width because - as shown in Fig. 7, Fig. 9 - the overcritical currents must recirculate within in the HTS wire.
Mataira et al. investigate the influence of this parameter in [65], using again the modelling framework of [30], to determine how wide the HTS wire must be to support both the induced current from the PM’s transit and any transport current flowing, without saturating the wire with overcritical currents. Previous studies - both experimental and numerical - had only considered HTS wires of a similar width to the PM. Fig. 12 shows the modelled V-I curves in [65], for varying HTS wire widths from 6-60 mm and assuming a 6 mm-wide PM. For narrower HTS wire widths (≤18 mm in this case), the V-I curves remain approximately linear. However, these become highly nonlinear as the HTS wire becomes wider. The authors also show that Isc increases approximately linearly with HTS wire width, but Voc approaches a limiting value, Vlim, with increasing width. This suggests that HTS dynamos employing narrower HTS wires have a restricted ability to efficiently rectify the available emf, limiting Voc [65]. It is shown that given enough space (i.e., selecting a wire of sufficiently large width), the induced overcritical currents can co-exist with any transport current, without driving the entire wire into the flux flow regime.
3.4 Frequency dependence: Considering the full HTS wire architecture, coupled with a thermal model
A frequency dependence of Voc has been reported in a number of experimental works in the literature [14], [61], [62], [63], where it was observed that the rate at which Voc increases reduces with increasing frequency, in contrast to the expected linear behaviour. The common explanation for this behaviour is that the heat generated in the HTS wire is the cause; however, Ainslie et al. [66] use numerical modelling to offer an alternative explanation based on the interaction between and current flow in different layers of the HTS wire as the frequency of the HTS dynamo increases.
To support their hypothesis, the segregated H-formulation method [38] as applied to the benchmark problem (see Section “A new benchmark problem”) is used, including the full HTS wire architecture (with the substrate and stabiliser layers) and coupling with a thermal model - with a linear Jc(T) assumption - to examine the effect of (AC loss-related) heat generated in the wire. Fig. 13 shows a comparison of the modelled results for Voc for three different models: the HTS layer only (“HTS only”), the full wire architecture (“Full wire”) and the latter coupled with the thermal model (“Full wire, heat”). There is little difference observed with and without heating, but there is a significant decrease in Voc with increasing frequency for the “Full wire” models. This suggests that the behaviour is due to a mechanism related to the wire architecture, which is known to impact the electromagnetic behaviour of HTS wires, especially at high frequencies [67], [68], [69], [70].
Fig. 13. Comparison of the modelled results in [66] for Voc for three different models investigating the frequency dependence of the HTS dynamo. |
Fig. 14 shows the calculated results from [66] for the average electric field, Ez, and normalised current density, Jz/Jc0, as the PM approaches the HTS wire (t = 1.47 cycles; θM = 0.44π = 79.2° in Fig. 3), comparing the “HTS only” model for rotational frequencies of 10 and 500 Hz and each of the wire layers for the “Full wire” model for 500 Hz. There are a number of key differences between the two models that combine to reduce the output voltage of the dynamo. In the “Full wire” model, a substantial current now flows in the top and bottom copper stabiliser layers (J ≈ 0.2Jc0 at the edge closest to the PM). Jz/Jc0 in the HTS layer in the region close to the PM is slightly reduced, but also flows over a reduced width of wire, suggesting the HTS layer becomes increasingly shielded by the stabiliser layers for higher frequencies. Jz/Jc0 is higher for the return current and flows over a wider region, suggesting an increase in return current. Ez is then substantially reduced in the region close to the PM and increases for the return path, thus reducing the voltage derived from (2).
Fig. 14. Calculated results from [66] for the average electric field, Ez (top), and normalised current density, Jz/Jc0 (bottom), as the PM approaches the HTS wire, comparing (a) the “HTS only” model for rotational frequencies of 10 and 500 Hz and (b) each of the wire layers for the “Full wire” model for 500 Hz. |
4 Modelling dynamic coil charging behaviour
Of great interest from the perspective of practical applications is the proper modelling of the dynamic behaviour of the HTS dynamo while charging a coil [71]. As described in Section “V-I characterisation” and shown in Fig. 10, the HTS dynamo can be modelled electrically as a (current-controlled) DC voltage source, Voc, in series with an effective (internal) resistance, Rint. To analyse the coil charging behaviour, the dynamo can be connected to an ideal coil with inductance, L, via a circuit resistance, Rc (resistance of soldered joints), as shown in Fig. 15. The current flowing in the circuit in Fig. 15 can be calculated as
$i(t)=I_{\text {sai }}\left[1-e^{-t / \tau}\right]$
where Isat = Voc / (Rc + Rint) is the saturation current (i.e., the maximum current the dynamo can deliver) and τ = L / (Rc + Rint) is the time constant of the circuit, which determines its charging rate. Note that this assumes an ideal coil without considering its critical current (i.e., i(t) << Ic,coil [10]). To consider saturation of the pumped current when limited by Ic,coil, one can include Vcoil(I), the current-voltage relationship for the coil, in the governing equation for the RL circuit, as described in [10].
Fig. 15. Equivalent electrical circuit model of an HTS dynamo connected to an ideal coil of inductance L via a circuit resistance, Rc, representing soldered joints [71]. |
While this electrical circuit model can accurately predict the overall charging behaviour, it does not take into account the dynamic behaviour within each cycle of PM rotation - in particular, the current ripples that can generate AC loss in the superconducting parts of the circuit. In [71], the authors use two numerical models - the segregated H-formulation and MEMEP method - to investigate this dynamic coil charging behaviour, and compare their results to the analytical results from (4). The benchmark problem (see Section “A new benchmark problem”) is used as a basis for the problem configuration, and the models are firstly used to obtain the V-I curves for three airgaps (1, 2 and 3.7 mm) and three frequencies (4.25, 25 and 50 Hz), which provide the necessary parameters Voc and Rint for (4) for each case. Next, the numerical (dynamo) models are coupled to a lumped parameter element RcL. To do so requires redefining the voltage of the dynamo to include not only ΔV(t) - as defined by (2) in Section “Modelling open-circuit voltage behaviour” - but also the magnetic vector potential contributions from the PM, AM, and the superconducting current induced in the HTS wire, AJ. Thus, the total output voltage of the HTS dynamo is given by
$V(t)=-l \cdot\left[E_{a v}(t)+\partial_{t} A_{M, a v}+\partial_{t} A_{J, a v}\right]$
where the subscript ‘av’ refers to the average value over the cross-section of the HTS wire.
Fig. 16 shows a comparison of the three components of the output voltage, as calculated in [71], for an airgap of 3.7 mm and a rotational frequency of 25 Hz. Fig. 17 then shows the dynamic charging current curve, over the first five cycles, of the modelled coil. The results of the two numerical models are compared with analytical results, again corresponding to an airgap of 3.7 mm and a rotational frequency of 25 Hz. There is excellent agreement between all three methods and the results clearly show the current ripples within each cycle of PM rotation that cannot be captured analytically.
Fig. 16. Comparison of the three components of the output voltage - l·Eav and the two contributions to the magnetic vector potential A |
Fig. 17. Dynamic charging current curve, over the first five cycles, of the modelled coil in [71]. The results of the two numerical methods (segregated H-formulation and MEMEP) are compared with analytical results, for an airgap of 3.7 mm and a rotational frequency of 25 Hz. |
Since the full charging process of a superconducting coil may require 1000s of cycles of rotation - in contrast to the other studies presented in previous sections, for which only a few cycles can be sufficient - it is crucial that any technique used to model such dynamic charging behaviour is as fast and efficient as possible. This, and related challenges, are discussed further in Section “Coupling numerical models & simplified circuit models”.
5 AC loss and energy considerations
AC loss is generated in a superconductor when it experiences time-varying currents and/or magnetic fields due to the movement of magnetic flux vortices [72]. The cryogenic system must then extract the resulting heat load, which impacts the energy efficiency of the system. Thus, understanding the nature and magnitude of the AC loss in the HTS dynamo during its operation is important. To this end, numerical models play a useful role as a cost-effective and efficient tool for analysis and prediction of such loss.
In [56], Prigozhin and Sokolovsky calculate the local instantaneous power loss density within their 3D mixed finite element method model of the benchmark problem via p(t, r) = E·J and the results are shown in Fig. 18. The plots correspond to the same three rotor positions shown previously in Fig. 9 for E and J. The total power loss can then be calculated a
The calculated loss power curve is then shown in Fig. 19, which for the 2nd cycle is estimated as W = 4.3 mJ; the time-averaged loss power is therefore, fW = 18.3 mW, for the assumed rotational frequency of 4.25 Hz.
Fig. 19. Calculated loss power curve from [56], using (6), during the 2nd cycle. |
Ghabeli et al. [71] calculate the AC loss during coil charging via the same method but using the 2D segregated H-formulation and MEMEP (see Section “Modelling dynamic coil charging behaviour”). The average loss power in the first five cycles (where negligible transport current flows in the wire) is calculated, for a rotational frequency of 25 Hz, to be 135.4 mW and 135.7 mW for the two models. The average loss power for the 5001st cycle (close to pumping saturation of the dynamo) is 135.7 mW and 135.9 mW for the two models. This suggests that for a given frequency the calculated loss in the dynamo is almost constant during the whole charging period of the coil and interrogation of the models shows that this can be understood from the current density and electric field distributions, which do not change significantly between I = 0 and Isat. This agrees with the experimental results presented in [17] for a multi-stator squirrel cage HTS dynamo.
In [73], Morandi et al. further explore the energy behaviour, efficiency and operational limits of the dynamo using the VIE method [44]. Firstly, based on the benchmark problem and a rotational frequency of 25 Hz, the V-I characteristics of the dynamo are calculated. These results show excellent agreement with the curve in [71], but the curve is extended beyond the usual I = [0, Isc] and VDC = [Voc, 0] limits (see Fig. 11, for example), as shown in Fig. 20. Operation within these limits is described as ‘generator mode,’ where a positive average power is delivered by the dynamo over one period. In the adjacent quadrants of the V-I curve, where these limits are exceeded, the dynamo moves to a ‘dissipative mode,’ where it absorbs power rather than generating, as demonstrated experimentally in [13].
Fig. 20. Extended V-I characteristics calculated in [73] using the volume integral equation-based method, based on the benchmark problem and a rotational frequency of 25 Hz. |
Although the equivalent electrical circuit of the HTS dynamo, presented earlier in Sections “V-I characterisation” and “Modelling dynamic coil charging behaviour”, is able to closely reproduce its behaviour in terms of the DC output voltage and coil charging, it is pointed out in [73] that this is unable to account for the dissipation that occurs under open-circuit conditions. A revised equivalent circuit is proposed whereby a resistor, Rintrinsic, is added in parallel to the voltage source, as shown in Fig. 21 (note that Reffective here is equivalent to the effective (internal) resistance, Rint, described previously - see Fig. 10, Fig. 15). The value of Rintrinsic can be calculated simply as
$R_{\text {intrinsic }}=\frac{V_{\text {rms } 0}^{2}}{P_{\text {joula } 00}}$
where Vrms0 is the rms value of the open-circuit voltage V(t, I = 0) and Pjoule0 is the average joule dissipation over one cycle under open-circuit (i.e., no load) conditions. The addition of Rintrinsic to the equivalent circuit does not effect the behaviour at the terminals of the dynamo and thus the charging profiles for an RL load (see Section “Modelling dynamic coil charging behaviour”) can be calculated as usual. However, a more complete picture of the dissipative components is now realised, which allows the energy balance and efficiency to be considered more accurately.
Fig. 21. Revised equivalent electrical circuit of the HTS dynamo proposed in [73]. A resistor, Rintrinsic, is added in parallel to the voltage source, which takes into account the dissipation that occurs in the dynamo under open-circuit conditions without impacting the behaviour at its terminals. |
6 Conclusion and view towards the future
A variety of numerical models have been developed over the past few years to accurately model the HTS dynamo, validated against experimental results and/or the results of the benchmark problem. Such models have provided a clear understanding of the HTS dynamo's underlying physical behaviour, which has been a source of considerable confusion and debate, and are cost- and time-effective tools to explore the key parameter space to design and optimise the HTS dynamo.
Modelling frameworks that emphasise reducing the number of DOFs have shown the best computational efficiency. The thin-sheet approximation and artificial thickness expansion techniques have also been used effectively for the HTS wire to improve computational efficiency. Self-programmed techniques - like MEMEP and the methods proposed by Prigozhin and Solokovsky in [47], [56], [58] - have demonstrated the highest efficiency in solving the problem, even for the more complicated 3D geometry, including more efficient implementation of the underlying physics. However, the implementation of these methods generally have steeper learning curves because they are self-programmed. Models implemented in commercial software like COMSOL have much shallower learning curves and can be shared more easily amongst the research community (where COMSOL is a dominant software package). However, such models have implementation overheads that generally result in less efficient and comparatively slower models, and the modeller has less control over their implementation. The choice of formulation is also important, with some more stable than others, e.g., H-A (more stable) versus T-A (less stable).
The models proposed so far have been used to carry out a wide variety of analyses related to the HTS dynamo, including modelling the open-circuit voltage behaviour in 2D and 3D (Section “Modelling open-circuit voltage behaviour”), investigating key dynamo parameters (Section “Investigating key HTS dynamo parameters”), modelling dynamic coil charging behaviour (Section “Modelling dynamic coil charging behaviour”) and calculating losses (Section “AC loss and energy considerations”). While great progress has been made, there are a number of outstanding challenges and developments required to address these, which are outlined as follows.
6.1 Exploring the key parameter space further & using artificial intelligence to accelerate HTS dynamo design optimisation
As described in Section “Investigating key HTS dynamo parameters”, there is a large parameter space for the design and optimisation of the HTS dynamo: the rotor radius and airgap; the HTS wire dimensions (most notably its width), its architecture and its superconducting properties (i.e., Jc(B)); the PM dimensions and its remanent flux density; the rotational frequency; and also, for more complicated designs (see Section “More elaborated models towards HTS dynamo design for large-scale applications”), the number of PMs and HTS wires, as well as their relative spacing.
While some of these parameters have already been investigated to some degree, both experimentally and numerically, there is still much room to explore. For example, the Jc(B) characteristics of the HTS wire have an important influence on dynamo performance, and as mentioned in Section “Gap dependence of the open-circuit voltage”, a strong decrease in Jc under a perpendicular magnetic field is desirable. By better understanding Jc(B) with respect to dynamo performance, which is simple to analyse in numerical models by merely changing the underlying assumption for Jc, the most optimal wire may be used, which may contradict conventional wire requirements and need specific engineering by wire manufacturers.
In order to accelerate HTS dynamo design, researchers have also begun to look at the application of artificial intelligence (AI) techniques to explore the parameter space in a fast and efficient manner, which is an emerging research area of interest in the superconductivity modelling community in general - see, for example, the recent Focus on Artificial Intelligence and Big Data for Superconductivity [74]. In [75], Wen et al. use the simulated results from a 2D finite-element model of the dynamo, with different four input parameters - frequency, airgap, Br of the PM and the HTS wire width e (see Fig. 3) - as the input data for regression analysis using several machine learning (ML) techniques: multi-layer perception neural network (MLP), support vector machine (SVM), Gaussian process regression (GPR), decision tree (DT) and k-nearest neighbours (KNNs). The authors demonstrate that GPR performs best in terms of both speed and accuracy, and this is then used to perform a sensitivity analysis of the key parameters through an analysis of variance (ANOVA). The importance rank for each parameter is deduced, such that Br > airgap > e. The 2nd order interactions of these are also influential, particularly Br × airgap.
In [76], a deep-learning neural network (DNN) with back-propagation algorithms is proposed. Considering six design parameters (the four above analysed in [75], plus the PM width and rotor radius), and training the DNN with a data set generated by a benchmarked H-A formulation-based numerical model, the output voltage of the dynamo could be predicted instantly with an overall accuracy of 98% with respect to the simulated values with all design parameters explicitly specified. The authors then demonstrate how the tool may be used effectively to provide recommended design parameters according to a given specification and set of constraints.
6.2 More elaborated models towards HTS dynamo design for large-scale applications
Many of the numerical models developed to date have focussed on the simplest form of the dynamo, namely a single PM rotating past a single HTS wire. However, the output voltage (on the order of µV in the case of the benchmark problem) and short-circuit current (much less than the Ic of a single wire [30], [65]) are small. Experimentally, more advanced dynamo designs have been developed - with multiple PMs and HTS stator wires - such that large-scale superconducting applications like rotating machines can fully exploit a dynamo’s ability to inject a DC current at the practical level required for such applications. Indeed, it was demonstrated recently in [2], [77], for example, that the HTS dynamo can be used as a contactless field exciter for an HTS rotating machine.
The extension of the more simplistic HTS dynamo modelling frameworks summarised in this review will be useful for this purpose and will allow large-scale (kA-class) dynamos to be designed and analysed in a cost-effective manner. Some of the practical dynamo designs will require complicated 3D models, for which the models described in Section “Efficient 3D models” will provide a useful starting point. Furthermore, the models may need to incorporate iron or other ferromagnetic materials to improve performance (see Section “Gap dependence of the open-circuit voltage”) as well as be coupled with thermal models (see Section “Frequency dependence: considering the full HTS wire architecture, coupled with a thermal model”).
Russo and Morandi [78] have recently made an attempt at this challenge in their numerical study on the energisation of the field coils of a full-size wind turbine with different types of flux pumps, including the HTS dynamo and both ‘warm’ (rectifier at room temperature) and ‘cold’ (rectifier is integrated into the cryostat) rectifier-type flux pumps. The analyses are based on injecting and maintaining the rated current in the HTS field coils of the 3.6 MW EcoSwing superconducting wind turbine generator [79], [80], as shown schematically in Fig. 22. The HTS dynamo design for this purpose, shown in Fig. 23, consists of 20 rectangular PMs and three 12 mm-wide Theva Pro-Line 2G HTS wires. The HTS wires are fixed with respect to the HTS field winding and move with the generator’s rotor, and the PMs are mounted on a separate assembly, controlled independently and free to rotate at its own rotational velocity. The HTS dynamo is characterised using the modelling framework of [73], but assuming realistic Jc(B, θ) characteristics for the HTS wires, to obtain the relevant parameters for the equivalent circuit shown in Fig. 21. The analyses of [78] show that the cooling power required by the HTS dynamo solution is 26% of that of the state-of-the-art solution with current leads and slip rings, with the additional benefit that slip rings are not required. There is also room for improvement in terms of the magnetic circuit (no iron or other ferromagnetic material is used) to widen the airgap and/or improve performance. The rectifier-type flux pumps, however, due to additional losses in the diodes, result in increased losses in comparison to the state-of-the-art.
Fig. 22. Contactless power supply system for the rotor HTS field winding of the EcoSwing superconducting wind turbine generator, based on the HTS dynamo-type flux pump [78]. |
Fig. 23. 3D schematic of the HTS dynamo-type flux pump analysed in [78], consisting of 20 rectangular PMs and three 12 mm-wide Theva Pro-Line 2G HTS wires. |
6.3 Coupling numerical models and simplified circuit models
One limitation of the models in Section “Modelling dynamic coil charging behaviour” (and the analyses of [78] described in Section “More elaborated models towards HTS dynamo design for large-scale applications”) is that the superconducting coil is assumed as a lumped parameter element consisting of an ideal inductor with some joint resistance. In addition to the AC loss of the dynamo during its operation and its dynamic charging behaviour, the same is of interest from the perspective of the superconducting coil(s) being charged. This leads to the need to develop coupled numerical models - e.g., coupling a dynamo finite element method (FEM) model with a superconducting coil FEM model - to analyse the dynamics of the complete coupled system with more realistic models and assumptions for all constituent parts.
As an intermediate step, one could also consider coupling an electrical circuit representation of the dynamo, in terms of V(t, IT), to the superconducting coil FEM model. Zhou et al. [81] reported a first attempt at the coupling of dynamo and superconducting coil FEM models via their lumped parameters, following a similar method to that proposed recently in [82] in the context of a saturated iron core superconducting fault current limiter. More generally, the development of such techniques will permit accurate modelling of the characteristics and behaviour of interconnected superconducting devices and components. Depending on the nature of the aspect(s) being investigated, a right balance must be struck between the complexity of numerical models - which can provide detailed information on the physics of the system but come at the cost of computational speed and effort - and more simplified but fast models, like the equivalent electrical circuit representation. The full charging process of a superconducting coil, for example, may require 1000s of cycles of rotation.
In [27], Geng et al. proposed a general circuit analogy for travelling wave flux pumps and provided a detailed mathematical analysis to explain the DC voltage. Campbell [83] then proposed an analytic circuit model based on Bean’s critical state model [84] and application of Faraday’s law. An overview of Campbell’s model is shown in Fig. 24. While the model does not obtain the same quantitative agreement as the numerical models described earlier, it does yield a similar insight into the role of the superconductor’s nonlinear resistivity and its rectification effect. The extension of such work warrants further consideration - there is a possibility that powerful and elegant circuit models of the HTS dynamo could be developed in this way in the future.
Fig. 24. An overview of the analytic circuit model of the HTS dynamo proposed by Campbell in [83]. (a) Flux moves into a loop. (b) The flux moves from the loop to the load circuit. (c) The flux is turned off. (a)-(c) A flux Φ moves moves into a square loop of side a and inductance L1 which puts current into a load magnet of inductance L2. |
7 Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
