Review of the AC Loss Computation for HTS using the H-formulation
RReview of the AC loss computation for HTSusing H formulation Boyang Shen , Francesco Grilli , and Tim Coombs Electrical Engineering Division, Department of Engineering, University ofCambridge, Cambridge CB3 0FA, United Kingdom Institute for Technical Physics, Karlsruhe Institute of Technology, 76131Karlsruhe, Germany
Abstract
This article presents a review of the finite-element method (FEM)model based on the H formulation of Maxwell’s equations used tocalculate AC losses in high-temperature superconductor (HTS) tapes,cables and windings for different applications. This model, which usesthe components of the magnetic field as state variables, has been gain-ing a great popularity and has been in use in tens of research groupsaround the world. This contribution first reviews the equations onwhich the model is based and their implementation in FEMfinite-element programs for different cases, such as 2D longitudinal andaxis-symmetric geometries, and 3D geometries. Modeling strategiesto tackle large number of HTS tapes, such as multi-scale and homog-enization methods, are also introduced. Then, the second part of thearticle reviews the applications for which the H formulation has beenused to calculate AC losses, ranging from individual tapes to complexcables and large magnet windings. Afterwards, a section is dedicatedto the discussion of the H formulation in terms of computational ef-ficiency and easiness of implementation. Its pros and cons are listed.Finally, the last section draws the main conclusions. a r X i v : . [ c ond - m a t . s up r- c on ] F e b Introduction
High-temperature superconductors (HTS) are recognized as the solution forfuture superconducting applications, because of their high current and powerdensity, good in-field behavior, and mechanical strength [1, 2]. In recentyears, the performance of HTS has been enhancing, while their price has beengradually decreasing, and several types of conductors have been proposed fordifferent applications [3, 4]. All these advantages make HTS a technicallyviable solution for future superconducting power applications, such as high-field magnets [5], motors and generators [6, 7], transformers [8, 9], powertransmission cables [10], fault current limiters [11].AC losses, however, represent an important limiting factor for the com-mercialization of HTS, especially when solutions based on conventional ma-terials are available. One has to remember that superconductors operate atvery low temperatures, and that a power dissipation that would not con-stitute a concern at room temperature can represent a very serious burdenwhen refrigeration costs are taken into account. For example, the Carnotspecific power for a temperature of 77 K (boiling temperature of liquid nitro-gen at atmospheric pressure) is about 2.9, but realistic estimates must takeinto account the efficiency of the cooling systems, which is in the range of10-20 % [12]. Therefore, scientists and engineers need to estimate the amountof AC losses and take the corresponding actions. Different types of modelscan be used for this purpose.In the past decades, several analytical models have been developed forcalculating AC losses in superconductors by means of simple mathematicalformulas. Notable examples are the expressions for the transport losses of su-perconducting tapes with elliptical or infinitely thin cross section [13] and themagnetization losses of superconducting tapes with infinitely thin cross sec-tion [14, 15, 16]. In the case of infinitely thin tapes, expressions for AC losseshave been derived also for infinite stacks or arrays with transport current orsubjected to an external magnetic field [17, 18, 19, 20], and for individualtapes with substrate of infinite magnetic permeability. A comprehensive re-view of these and other expressions is given in [21]. Typical limitations ofthese analytical models are the restriction to simple geometries and the use ofthe critical state model as constitutive law of the superconductors. Numeri-cal models, such as the FEM model based on the H formulation of Maxwell’sequations that is the topic of this review, allow overcoming these limits andinvestigating the electrodynamic response (and in particular the AC losses)2 − − − − − − − Current (A) A C l o ss ( J m − ) Measurement H formulation2 x Norris strip2 x infinite stack Figure 1: Transport AC losses of an HTS double pancake coil (2 ×
18 turns).Experimental results are compared to those calculated with the H formula-tion and with analytical models for a single tape [13] and an infinite stack oftapes [18, 19]. The shaded area emphasizes the several orders of magnitudeof difference for the losses calculated with the two analytical models. Datataken from [22].of realistic applications. As an example, figure 1 shows the transport AClosses of an HTS double pancake coil. Analytical models can calculate thelosses of a single turn [13] or of an infinite number of turns [18, 19]: theseestimations can only provide lower and upper limits for the AC losses andare not accurate for a real coil made of a finite number of turns. Numericalmodels such as the H formulation can simulate the realistic geometry, thusproviding an estimation of the AC losses much closer to the measured data.The idea of simulating the electromagnetic behavior of superconductors intime-dependent problems by using finite elements with the magnetic field asstate variable was first proposed in 2003 by Kajikawa et al. [23] and Pecher etal. [24]. In both cases, the formulation was implemented in home-made codes.A few years later, Hong et al. [25] and Brambilla et al. [26] independentlyimplemented the H formulation in the commercial FEM software COMSOLMultiphysics [27]. This model became quickly very popular and it is nowthe de facto standard in the applied superconductivity community. A searchon the Web of Science based on the citations of [25, 26] revealed that, atthe time of writing, the H formulation has been used by at least 45 research3roups worldwide.This review article is organized as follows. First, in section 2, the mathe-matical implementation of the H formulation is discussed, first for its basicform, then for its extension and adaption for solving problems of increasingcomplexity. Then, in section 3, the application of the H formulation to casesof practical interest is reviewed. Further, section 4 is dedicated to discussingthe reasons of the popularity of this formulation, its implementation in differ-ent programming environment, its advantages and drawbacks. Finally, themain conclusions of this review article are summarized in section 5.This review is specifically dedicated to the use of the H formulation forinvestigating the AC losses of HTS in a variety of geometric arrangementsand working conditions at constant temperature. The coupling with thermalmodels is therefore not considered here. H formulation The model solves Faraday’s equation in a finite-element environment, usingthe magnetic field components as state variables ∇ × E = − ∂ B ∂ t (1)where B = µ H , with µ = µ r µ . The lower critical field below which type-IIsuperconductors are in the Meissner state is usually very low (mT range), sofor most practical cases of power applications one can assume µ r =1 for thesuperconductor material.Since E = ρ J and J = ∇× H , one can rewrite (1) in terms of the magneticfield as ∇ × ( ρ ∇ × H ) = − ∂ ( µ H ) ∂ t , (2)where the magnetic field needs also to obey Gauss’s law ∇ · ( µ H ) = . (3)The simultaneous solution of equations (2) and (3) presents a problem,as it involves an over-constrained system in the general case. This issue was The first part of this section is mostly taken from [77]. ∇ · [ ∇ × ( ρ ∇ × H )] = ∇ · (cid:18) − ( µ H ) ∂t (cid:19) . (4)The left-hand side of equation (4) is identically zero and, after exchangingthe order of time and spatial derivatives, it is easy to see that ∇· B = ∇· ( µ H )is constant in time. Consequently, if ∇ · B = at a given time t , then ∇ · B = will hold at any other instant. So, if initial conditions are chosensuch that ∇ · ( µ H ) | t = t = (5)then ∇ · ( µ H ) = will hold at all times. It is important to remark thatthe divergence-free characteristic of the magnetic flux density B is enforcedsolely by analytical construction, independently of the type of elements usedin the FEM implementation. However, this way to impose the divergence-freecharacteristic on B can be sensitive to errors if the time integration solver isnot sufficiently robust, and makes some type of elements preferable to others.In particular, for the implementation of this model in the commercial softwarepackage COMSOL Multiphysics [27], first-order edge elements are preferable.A study of the effect of different types (Lagrange or edge) and order (first orsecond) of elements is presented in [29].A Dirichlet boundary condition allows modeling instances where magneticfield, transport current or a combination of both are considered. In the case ofa transport current flowing in several conductors, a set of integral constraintsallows fixing the net current in each conductor. Therefore a net current I k ( t )can be imposed in the k th conductor by enforcing I k ( t ) = (cid:90) k J · d A k , (6)where A k denotes any open surface that completely intersects the k th con-ductor alone. Figure 2 shows an example of how current constraints can beused to control the current distribution in a three-phase Cross-Conductor(CroCo) cable. 5igure 2: Normalized current density distribution of a three-phase Cross-Conductor (CroCo) cable [30] at a given instant during the AC cycle. Three120 ◦ -shifted sinusoidal currents are imposed in each CroCo cable. Since suchcables are made of non-transposed tapes, the current is let free to distributeamong the 32 tapes composing each cable. If the tapes were fully transposed(such as in a Roebel cable), further current constraints would be neededin order to have all tapes carry identical currents. The arrows indicate thedirection of the generated magnetic field, their magnitude being proportionalto the field’s magnitude.The superconductor is modeled as a material with nonlinear electricalresistivity, usually in the form of a power-law ρ ( J ) = E c J c (cid:18) JJ c ( B ) (cid:19) n ( B ) − , (7)where J is the magnitude of the current density, E c is usually set equal to1 × − V m − , J c is the superconductor’s critical current density, and n isthe power-index describing the flux creep. Such a nonlinear resistivity mirrorsthe nonlinear voltage-current relationship observed in the characterization ofHTS tapes [31], where the critical current I c is defined as the current at whicha threshold voltage is reached.In general, the critical current density J c depends on the magnetic fieldamplitude and orientation [32, 33], and this dependence can assume fairlycomplicated forms [34]. The power index n too depends on the magneticfield amplitude and orientation, although in most simulations its field de-pendence is disregarded, as it is expected to have a less important influence6n the results than that of J c . The dependence of J c on the position in-side the tape, for example as a result of the manufacturing process of HTStapes [35], can also be easily included. The space around the conductorsis usually modeled as a material characterized by very high resistivity (e.g.1 Ω m), so that no current flows there. In most cases, the tapes’ or device’sbehavior is determined by the superconducting parts only, and the othermaterials composing the superconducting tapes are ignored. However, theycan be easily included and assigned the proper values of electric resistivityand magnetic permeability. In some cases, magnetic materials need to bemodeled. This constitutes a challenge because their magnetic permeabilityis usually not constant and, in particular, it depends on the magnetic field.Since the magnetic field varies with time, this means that in equation (2) theterm µ cannot be simply taken out of the time derivative. As a consequence,the expression of Faraday’s equation to be implemented in the model needsto be properly changed, as described in [36]. Details about the practical im-plementation in COMSOL Multiphysics, including hints to help convergence,are given in [37].The model can also be modified to take into account the presence of resis-tive components in series with the superconductors [38]. This is for exampleuseful for simulating (in 2D) the current distribution in cables composed ofseveral tapes, each characterized by a different termination resistance. In thiscase, an additional term R i I i must be added to the electric field contributionof each superconductor, where R i and I i are the termination resistance andcurrent of the i th superconducting tape. The final model therefore consistsof equation (2), initial conditions fulfilling equation (3) and a set of appro-priate boundary conditions and/or constraints. One should note that thetreatment above is invariant as it does not rely on the choice of any specificcoordinate system or reduced dimensionality; therefore it holds for rectan-gular, cylindrical, spherical or any other coordinate system in one, two, orthree dimensions.The model can be used to compute AC losses of HTS tapes and devicesin a variety of configurations and operating conditions. Typically, situationswith an AC transport current, an AC magnetic field, or a combination of thetwo are simulated. A sinusoidal current and/or field excitation is imposed(with the appropriate set of current constraints and boundary conditions) andthe AC losses are calculated by integrating the instantaneous local dissipation J · E over the superconducting domain and averaging it over a cycle. For suchcalculation, the first quarter of the sinusoidal cycle must be avoided, because7t is not representative of the AC regime due to the occurring transient. Thecyclic AC losses can therefore be computed on the second half of the firstcycle as Q = 2 /f (cid:90) / (2 f ) (cid:90) Ω J · E d Ω d t (8)where f is the frequency of the AC source and Ω the superconducting domain.This expression can be adapted to calculate the losses for arbitrary time-dependent excitations. A large number of applications can be modeled as a set of long straight con-ductors or as windings with cylindrical symmetry, for which a 2D descriptionconsidering the transversal cross section of the superconductors is sufficient.In this case, the state variables are the two components of the magneticfield in the considered cross section; the current can only flow perpendicularto them. This 2D approximation automatically implies that the materials’properties are assumed to be constant along the conductors’ length. There-fore, for example, the variation of J c along a tape’s length – which is observedexperimentally [39] – cannot be taken directly into account. In the case ofinfinitely long conductors (like straight tapes or cables), it is sufficient toconsider their transversal cross section ( xy plane). In this case, the magneticfield has only two components H = [ H x , H y , J = [0 , , J z ], which, since J = ∇ × H , can be expressed in 2D as J z = ∂H y ∂x − ∂H x ∂y . (9)In the case of conductors exhibiting cylindrical symmetry (like solenoids,pancake coils) the same concept applies. The only difference is that theequations need to be written in cylindrical coordinates.In cylindrical coordinates ( r, θ, z ), the three components of equation (1)are 1 r ∂E z ∂θ − ∂E θ ∂z = − ∂B r ∂t (10) ∂E r ∂z − ∂E z ∂r = − ∂B θ ∂t (11)8 r (cid:20) ∂∂r ( rE θ ) − ∂E r ∂θ (cid:21) = − ∂B z ∂t . (12)For axisymmetric problems, the equations above can be simplified. In par-ticular, the current flows along the θ direction only and the magnetic fieldhas the r and z components, therefore B θ = 0; there is only one component( E θ ) of the electric field (which is parallel to the current density J θ ) and E r = E z = 0; finally, all the derivatives with respect to θ can be put equalto zero.Using B = µ H , one is left with the following two governing equations − ∂E θ ∂z = ∂ ( µH r ) ∂t (13) E θ r + ∂E θ ∂r = − ∂ ( µH r ) ∂t . (14)The current density is given by J θ = − ∂H z ∂r + ∂H r ∂z . (15) Certain applications cannot be accurately modeled in 2D and require theuse of a full 3D model. Implementing equation 1 in 3D is quite straightfor-ward [40]. One difference with respect to the 2D model is that the currentdensity and the electric field do not necessarily need to be parallel, which al-lows exploring the so-called force-free configuration [41]. 3D modeling can beused to explore the end effects in finite geometries, such as superconductingbulks [42] and stacks of HTS tapes [43].When cables are modeled in 3D, measures need to be taken in order tokeep the size of the problem manageable. This typically involves avoiding thesimulation of the whole geometry and focusing on a representative periodic‘cell’ of the cable. In that case, periodicity conditions need to be applied atthe two ends of the simulated cell.The 3D model also allows introducing spatial variation of the physicalproperties along the conductor’s length. In particular, the effect of localizeddefects can be studied [44]. 9 .4 Homogenized and multi-scale models
In certain applications, like large coils, the number of turns to simulate canbe very large, in the range of hundreds or even thousands. Simulating all ofthem as individual objects can rapidly become a daunting task. One solutionto the problem is to ‘homogenize’ the cross section of the coil as one bulkconductor [45]. The superconductor’s critical current density J c is ‘diluted’ totake into account the distance between the layers of superconducting materialin the actual stack. In the case of HTS coated conductors, this distanceincludes all the composing layers (superconducting layers, substrate, bufferlayers, stabilizer) as well as the separation between the tapes (e.g. electricalinsulation). In the case of a coil, each tape in the simulated cross sectionrepresents a turn of the coil, and all the simulated tapes must carry the samecurrent. This means that in the homogenized bulk the current cannot belet free to distribute in the whole cross section, but must be constrained tomirror the real situation, where each tape (turn) carries the same current.In the numerical model, one must impose that the integral of the currentdensity in the direction of the tape’s width is always the same: this can bedone by using an appropriate current constraint, as explained in [45].The advantage of the homogenization becomes evident for the simulationof large systems or when the distance between the turns becomes very small(so that it would require a very fine mesh if all the turns had to be simulatedindividually). For example, in [46] it was shown that, for the case of five100-turn pancake coils, the homogenized model is 50 times faster than themodel simulating all the tapes, with no appreciable difference (less than 1 %)on the AC losses.The homogenized problem is much simpler to solve because one can usea relatively coarse mesh (resulting in a much lower number of degrees offreedom) and less current constraints (whose number influences the speed ofthe solving process). The main drawback is that this method cannot takeinto account magnetization currents caused by a magnetic field parallel tothe flat face of the tapes.The homogenization method was extended to 3D in order to simulate aracetrack coil made of HTS coated conductors in [47]. The idea is the sameas in the 2D version of the method. However, the imposition of the constraintfor having the same current flowing in each turn is more challenging. Thecurrent density has to follow the curvature of the tapes at the coils’ ends.In [47], three strategies were proposed: 2D integral constraints, anisotropic10esistivity, manual discretization of the anisotropic bulk. The last one wasused because of better computational stability.Another approach for modeling large numbers of interacting tapes is themulti-scale method [48]: the idea is to simulate one conductor at a time us-ing a boundary condition for the field it is subjected to as the result of theenvironment (for example, the field created by the other conductors). Thismagnetic field can be calculated in a relatively simple way, with magneto-static models. If one is for example interested in calculating the AC losses ofthe whole device, the position of the tape of interest can then be moved anda ‘map’ of the AC loss distribution in the device created. The advantage isthat the simulations involving one tape are very fast and they can be trulyparallelized. In addition, it is not necessary to consider all the positions ofthe tape in the device, but only some key ones. In [46], it was shown that 25equally distributed positions are used to produce a sufficiently accurate ACloss map of five 100-turn pancake coils. In the case considered in [46], it wasfound that the current distribution used for the magnetostatic calculationhas an important influence on the final AC loss value and hence on the accu-racy of the results: for example, starting with a uniform current distributionin the superconductors is too rough an assumption and produces errors inthe losses as large as 20 %. This section reviews the works that use the different forms of the H formu-lation discussed in section 2 for calculating AC losses in various applicationsand scenarios. A challenge of modeling HTS coated conductor is the very large width-to-thickness ratio of the superconducting layer, which typically results in a verydense mesh and a very large number of degrees of freedom. In [49] it wasfound that artificially expanding the thickness (and correspondingly reducing J c ) results in much faster simulation without compromising the accuracy ofthe results. In [50], the authors proposed the use of a structured mesh forthe superconducting layer, with elongated rectangular elements. This alsoallowed significantly reducing the size of the problem and is now the standard11 − − − − − − current (A) A C l o ss ( J m − ) measuredHTSsubstratetotalthin stripHTSsubstrate Figure 3: Transport losses in an HTS tape with magnetic substrate calculatedby H formulation, showing the good agreement with measurement, and theseparation of the losses into different components which are not directlyavailable from measurements. Note that the magnetic substrate modifies thelosses of the HTS material, which are different from those of a tape withoutmagnetic substrate (indicated by the ‘thin strip’ curve). Replotted from [36].method for simulating coated conductors. A discretization of 50-100 elementsalong the tape’s width is generally sufficient. Adding a discretization of afew elements along the superconducting layer’s thickness does not usuallyresult in an excessive number of degrees of freedom. Since in most casesa 1D description of the superconducting material is sufficient, one elementalong the thickness is often used.Shen et al. [51] used the H formulation to calculate the eddy current ACloss in commercial non-stabilized and copper-stabilized HTS tapes, and theycompared the results with experiments. They found that the eddy currentlosses in the copper have the expected dependence on frequency (power lossproportional to the square of the frequency) and give a substantial contribu-tion from frequencies above 1 kHz.Nguyen et al. [36] calculated and measured the transport AC losses ofHTS tapes with magnetic substrate, showing that at low current the ferro-magnetic losses in the substrate give the more important contribution to thetotal losses (figure 3). As the current is increased, the ferromagnetic losses12aturate and the losses of the superconducting layer are dominant.In [52, 53] the H formulation was used to calculate the AC losses of bifilarcoils made of HTS tapes with and without magnetic substrate to be used forfault current limiter applications. The simulated configuration, consistingof a coil where the current of adjacent turns flows in opposite directions, isa typical example where the use of current constraints (see equation (6)) isparticularly handy.In [54], Zhang et al. used an axisymmetric H formulation model to cal-culate the AC losses in HTS pancake coils made of tape with magnetic sub-strate and compared the results with experimental measurements, findinggood agreement. They also found that the presence of a magnetic substratedoes not influence the critical current of the coil; however, it affects the lossprofile of each turn inside the coil under DC conditions, when the currentapproaches the critical current. This is due to the changes of the magneticfield profile caused by the magnetic substrate.In [55, 56], Ainslie et al. used the 2D H formulation to investigate thepossibility of reducing the transport AC loss of HTS coils by means of fluxdiverters made with both weakly and strongly magnetic materials. The re-sults show that significant loss reductions can be achieved and that the idealdiverter material should have a high saturation field and a low remnant field.On the same topic, Liu et al [57] used numerical simulations to find theeffects of the geometrical parameters of the flux diverters on the transportAC losses of an HTS coil.In [58], de Bruyn et al. presented a simplified method for calculatingAC losses in stacks of superconducting tapes. The method aggregates thesuperconducting, substrate, and copper layers of several windings of the su-perconducting tape to form bulk elements. The resulting model is able toevaluate the AC losses in the superconducting layer faster than the full model.The accuracy of the proposed method depends on the current level at whichthe tapes operate.Liang et al. [59] studied the AC losses in two types of low-inductancesolenoidal HTS coils using 2D axisymmetric H formulation and experiments,and proved that the braid type coil could be a suitable choice for supercon-ducting fault current limiters as it has lower AC loss than the regular typecoils.Shen et al.[60] used the 2D axisymmetric H formulation to investigate thepower dissipation circular HTS coils under the action of different oscillatingfields and currents, and found that more AC losses were generated with the13aster variation in waveform gradient within a certain time interval.Zhang et al. [40] developed a 3D model of the H formulation and used itfor calculating the transport AC losses of a cable composed of one layer of he-lically wound HTS tapes. They found that the transport losses increase withthe twist angle, which they ascribed to the generation of an extra shieldingcurrent (caused by the twisting) flowing in the direction of transport current.A full 3D FEM model of three twisted superconducting wires operatingwith AC transport current was presented in [44]: the results show the effectof the twisting on the current density distribution and agree well with thoseobtained with a 2D model that uses a change of coordinates to take thetwist into account. In the same article, the potential of 3D simulationsis shown for different examples: i) the coupling currents between two thinsuperconducting filaments as a function of the resistivity of the materialbetween them; ii) the effect of a localized defect on the current flow in atwo-filament structure; iii) the effect of a varying cross section on the currentdistribution of a round wire.Lyly et al. [61] used the 3D H formulation to model the losses of twistedNbTi wires with various wire geometries under different magnetic fields, andstudied the operation conditions and coupling effects of filament bundles.Lahtinen et al. [62] compared three FEM methods, A − V − J formulation, T − ϕ formulation and H formulation, for the calculation of hysteresis loss ofa round superconducting wire, and the results showed that the H formulationhas the advantage of reasonable computation speed and boundary conditionsettings.Lyly et al. [63] introduced a time harmonic method to calculate the eddycurrent losses in twisted superconducting wires. The results were comparedto those from the 3D H formulation models, and good agreement was foundin certain conditions.Stenvall et al. [64] performed a similar comparison between the 3D H formulation FEM model and a home-brewed code using Gmsh with C ++ , forthe loss analysis of HTS twisted wire with varying applied magnetic field.They found that the H formulation may underestimate the AC losses in the3D HTS twisted wire model because of a current leakage to the air domain, iftoo low values of the air resistivity are used. An example is shown in figure 4.In [47], the 3D homogenization technique was used to model stacks andcoils of HTS tapes. For validation, the 3D homogenization method was suc-cessfully tested against a 2D model considering all the individual conductorsand enforcing all the individual currents. Both methods provided a remark-14igure 4: 3D simulations of two twisted superconducting filaments carryingAC transport current. The current streamlines demonstrate how part of thecurrent can flow between the air to the superconductors. This can be avoidedby increasing the resistivity of the air (which however tends to result in slowercalculations) or by using cohomology functions in the so-called H − ϕ − Ψ formulation developed in [64].able good agreement over a large range of applied currents and over morethan two orders of magnitude for the calculated AC losses. Then, the 3Dmodel was used to simulate the more complex case of a racetrack coil, and itwas found that the AC loss results agree with those of a planar 2D model onlyfor low current fractions of I c . For medium and high current, the losses cal-culated with the planar 2D model diverge and the use of the 3D homogenizedmodel becomes necessary.Eddy current models such as the H formulation can in principle be usedto estimate the losses caused by AC ripples on top of DC excitations in HTS.However, this kind of modeling presents some challenges, which have beendescribed and discussed with examples in [65].Firstly, a finite n exponent, even though as high as 40 (or, in general, asmooth E − J constitutive law for the superconductor) causes the currentpenetration profile to ’relax’ with time (going toward a homogeneous distri-bution), when there are no longer any changes in the net current or external15eld. This leads to very slowly descending loss curves and the evaluation ofthe cyclic losses caused by the ripples depends on where along those curvesthe evaluation is performed. This was confirmed by simulations of a coatedconductor carrying DC current with AC ripples in [66], where the details ofthe current density distribution along the tape’s width at different instantswere studied and put in relation to the instantaneous power dissipation.Secondly, in the case of a multi-filamentary wire modeled in 2D, therequirement of filaments being uncoupled with respect to the external fieldis contradictory with the requirement of parallel connected filaments in theeddy current models, although this obstacle could be surpassed by usingcurrent constraints.In [67], the authors investigated the effects of AC ripple fields on an HTSracetrack coil subjected to DC background field or DC transport current, twoconditions that can be met in the field windings of electrical machines. Theresults show that both the DC background field and transport current wouldsignificantly increase the values of AC losses, especially when the AC ripplefield is small. Similar results were found in [68], where different ratios of DCand AC currents were simulated and measured. The authors also found thata small value of DC offset in AC current would not affect the AC losses.In [69], the authors presented a comprehensive study of the AC ripplelosses of an HTS coated conductor for a variety of combinations of AC andDC background fields and transport current. They used two numerical mod-els – the H formulation with power-law and the Minimum Magnetic EnergyVariation (MMEV) method with the critical state – and performed exper-imental measurements. For pure AC cases and DC-AC cases with the ACfield significant enough compared to the DC field, the agreement betweenthe models (and with measurements) is good. For some of the studied cases,CSM and ECM yield different predictions for the behavior of DC biased su-perconductors, which are ascribed to the different E − J relations used inthe models – see also [65] discussed above. The experiments seem to suggestthat, for a given current density J , the electric field E is lower than thatpredicted with the power-law E − J relation, and therefore indicate an E − J relation closer to the critical state model for low E. However, the authorsacknowledge that further research on the topic – e.g. by means of higher res-olution field map measurements – is needed, because neither the power-lawnor the critical state model are able to predict the observed behavior.16 .2 High-current cables The H formulation has been used to calculate the AC losses in different typesof cables, composed of HTS tapes. Roebel cables are made of intertwined meander-shaped strands obtained fromHTS coated conductors [70]. For most AC loss calculation purposes, it issufficient to consider their transversal cross section, so that the cables canbe simulated as two stacks of tapes [71].In [72], the authors used this 2D approximation to calculate the transportand magnetization losses. The results of the H formulation were successfullycompared with those obtained with the MMEV model. Two different scenar-ios of strand coupling were considered, which provided two limits for the AClosses of the actual cable. In [73], the same authors extended their models totake into account the angular dependence of the critical current density onthe magnetic field. Situations where the anisotropy of such dependence canplay a role were identified.The AC losses of a Roebel cable under the simultaneous action of ACtransport current and AC magnetic field were calculated in [74]. Two ca-bles made of tapes from different manufacturers, characterized by differentangular dependence of J c , were analyzed and the AC loss results comparedwith experimental data obtained with a calorimetric method measuring theevaporation of liquid nitrogen.In [75], the authors studied the frequency dependence of the transportlosses of an 8-strand Roebel cable by simulations and experiments. Thefrequency was varied between 50 Hz and 10 kHz. They found that for lowand medium current amplitudes the AC losses per cycle decrease as f − /n with n = 26, whereas for high current amplitudes they are proportional to1 /f .In [76, 34], the authors studied the transport losses of different pancakecoils assembled from the same 5 m-long Roebel cable. The coils differ in termsof number of turns and turn-to-turn separation. The experiments revealedthat at low and medium current amplitudes the losses are dominated by thedissipation occurring in the copper current lead used to inject the currentin the cable. Once this contribution was taken into account by means of adedicated 3D simulation and added to the results obtained for the coils with17he H formulation, it was possible to match the experimental data.In [77], Zermeno et al. developed a full 3D H formulation FEM modelof a Roebel cable with 14 strands. The 3D model simulates a periodic cellrepresenting the various positions that a given strand occupies along thelength of the cable. The calculated AC losses were similar to those obtainedwith 2D simulations. The 3D model revealed the presence of high dissipationnear the corners of the strands. Although this localized dissipation doesnot contribute significantly to the cyclic losses of the whole cable, it canrepresent a potential stability issue for this type of cable. To this date, thisis the only full 3D model of a Roebel cable, where also the thickness of thesuperconductor layer is simulated. R (cid:13) cables CORC R (cid:13) cables are assembled by winding multiple layers of HTS coatedconductor on a narrow former [78].In [79], Majoros et al. investigated the magnetization AC losses and heatgeneration in the HTS CORC R (cid:13) cables by experiments and 2D H formulationFEM models. In the simulations they observed the screening effects thatmake the AC losses decrease with higher frequencies.Sheng et al. [80] developed a 3D H formulation model for computing themagnetization AC losses in a CORC R (cid:13) cable, which can account for fullycoupled or fully uncoupled strands. The latter situation provided a bet-ter match with experimental results. The simulations also suggested that astronger shielding effect can be achieved by increasing the coverage ratio ofthe HTS tapes and that the end effects cannot be omitted in the study ofmagnetization loss of short CORC R (cid:13) cable. In particular, a correction coef-ficient should be used when experimental results of short CORC R (cid:13) samplesare used to predict the magnetization loss of very long cables.Terzioglu et al. [81] studied experimentally the AC losses of a CORC R (cid:13) cable with copper former caused by transport AC current, external AC fieldand their combination. They found that the magnetization AC losses increasedue to losses in the metallic former, but that, at low field amplitudes, themagnetization AC loss of the complete cable is lower than the loss in the bareformer. They ascribed this result to the shielding of the magnetic field bythe superconductor, and they used numerical simulations based on the 3D H formulation to support this explanation. The authors also suggest thatthe losses can be reduced by using a material with low electrical conductivity18nd high thermal conductivity for the former. In [82], Ainslie et al. studied the transport AC loss of stacks of HTS coatedconductors, with and without magnetic substrate. When investigating theeffect of a magnetic substrate, it was found that the transport AC loss issignificantly increased, especially in the central region of the stack, due toan increased localized magnetic flux density. The ferromagnetic loss of thesubstrate itself is found to be negligible in most cases, except for small mag-nitudes of current where the substrate is not yet saturated.In [83, 84], the authors showed by simulations that in the case of a twistedHTS stacked-tape cable subjected to a transversal AC field, the magnetiza-tion losses are determined by the field component perpendicular to the tape.As a consequence, they can be calculated on a straight HTS stacked-tapecable (which can be simulated in 2D) and scaled by a factor 2 /π . Termi-nation resistances can be accounted for by simulating two non-connecteddomains [38], one for the resistances and one for the HTS tapes. The currentis imposed on one side of the resistances (in parallel). On the other side, thecurrent exiting a particular resistance is made flow into the correspondingtape by means of appropriate constraints. The model for the terminationresistances requires only a minimal geometrical layout with a small mesh inthe discretization and a consequently low number of degrees of freedom. Inthe specific case of a single HTS stacked-tape, the 3D model is not necessarybecause the contribution of contact resistances can be directly inserted in a2D model [38]. However, this approach can be of interest for more complexsituations for which the 2D approximation is not valid.In [85], Kan et al. carried out an AC loss study of quasi-isotropic strandscable manufactured by HTS. The idea is to build a stacked-tape cable withtapes oriented in two different ways, in order to reduce the effects of thestrongly anisotropic angular dependence of J c typical of HTS coated conduc-tors. They calculated the magnetization AC loss at 4 . .2.4 3D modeling of cables with twisted structures In [86], Makong et al. developed a simplified version of the 3D H formu-lation FEM model (implemented in GetDP) to simulate the performanceand calculate the AC loss of twisted multi-filamentary HTS wires subjectedto transverse magnetic fields. The adopted geometric transformation allowsstudying the wires in the Frenet frame moving along the helical trajectoryresulting from twisting filaments and saves considerable computation time.The accuracy of the method showed some dependence on the twist pitch andon the magnitude of the external field.In [87], Escamez et al. used the 3D H formulation implemented in DarylMaxwell to analyze the AC losses of the MgB wires with 6, 18 and 36filaments. The authors demonstrated – by means of comparison with 2Dmodels – the necessity of a 3D model in order to compute correctly the cou-pling losses, which contribute significantly to the losses, especially for highapplied field values. They also considered the impact of the nonlinear per-meability of the nickel matrix, which generates substantial additional losses.The authors underlined the necessity of speeding up the solving process forsimulations of such complexity, especially for the 36-filament case. Xia et al. [88] used the homogenized H formulation to investigate the electro-magnetic behavior of the HTS prototype coils of the National High MagneticField Laboratory 32 T all-superconducting magnet. Beside AC loss calcu-lation, that approach can be used to study the central magnetic field drift,magnetic field deviation, and to optimize the design of high field coils.Qu´eval et al. [46] used the H formulation to calculate the losses of a largecoil composed of ten 200-turn pancakes (reduced to five 100-turn pancakesthanks to symmetries). They used both the homogenized and multi-scaleapproaches and compared the results with the reference case obtained bysimulating all the tapes. The homogenized and multi-scale approaches allowsaving a great amount of time, with no significant loss of accuracy. In theconsidered case, the homogenization resulted to be the faster (50-60 timesfaster than the reference case), although the full potential of the multi-scaleapproach obtainable with full parallelization was not tested. An example isshown in figure 5. 20igure 5: Simulation of a matrix of 10 ×
200 tapes, representative of the turnsin the cross section of a high-field magnet. Due to the symmetry of the prob-lem, only 5 ×
100 tapes are simulated. The top figures represent the currentdensity and magnetic flux density distributions obtained by simulating alltapes with the time-dependent H formulation of Maxwell’s equations [26].The bottom figures represent the same quantities obtained with a homoge-neous bulk approximation: the results are very similar, but the computationis 50-60 times faster. Reprinted with permission from [46].21 .3.2 Electrical machines Ainslie et al. [89] used an improved H formulation FEM model with struc-tured mesh to calculate the transport AC loss in the HTS coils for largeelectric machines, including the loss contribution of the ferromagnetic tapesubstrate.Zhang et al. [90] calculated the AC losses in HTS racetrack coils to beused as armature windings in electrical machines. The 2D H formulationmodel, which considers the tapes’ anisotropic angular dependence of J c on themagnetic field, was successfully validated against experimental measurementsof AC losses performed with both the electric and calorimetric methods. Theauthors calculated the transport loss of the HTS armature winding in terms ofelectrical loading of the machine, and pointed out that a distributed windingwould be a feasible way to reduce transport loss.Zhang et al. [91] performed a calorimetric measurement of the total ACloss of a 2G HTS racetrack coil subjected to both an applied current and anexternal magnetic field. They found that when the external magnetic fieldis perpendicular to the tape surface, the total AC loss show a modulation inregard to the phase shift between the applied current and external magneticfield. The authors then used the H formulation to support the experimentaldata and to study in detail the influence of the phase shift between thecurrent and the field on the total AC loss.Qu´eval et al. [48] used the H formulation FEM model to estimate theAC losses of a superconducting wind turbine generator (employing Bi-2223tapes) connected to the grid. For this study, the authors performed single-tape simulations on a fraction of the tapes of the coils’ cross section. Theycalculated the magnetic field impinging on the tape by means of an unidirec-tional coupling between the machine model and the HTS tape model. Withthis approximation, they found that the steady-state AC loss of a 10 MW-class wind turbine generator could be under 60 W.Li et al. [92] did a study of HTS armature windings in a 15 kW-class fullyHTS synchronous generator. They used the 2D H formulation on a simplifiedgeometry and, by applying the minimum and maximum homogenous leakagemagnetic fields to the boundary of the simulated domain, they estimated thelower and upper limits for the AC losses.In [93], de Bruyn et al. used the homogenized H formulation to predictAC losses resulting from non-sinusoidal transport currents as are present inhighly dynamic motors with AC armature coils, finding good agreement with22xperimental data. Although modeling the current limiting behavior of superconducting faultcurrent limiters requires thermal models, electromagnetic calculations at con-stant temperature can be used to evaluate the AC losses of these devicesduring normal operation.Hong et al. [94] investigated the AC loss in a 10 kV resistive-type super-conducting fault current limiter, using both experiments and H formulationcalculations. The numerical model was used to calculate the AC loss duringnormal operation, for different values of the transport current. Good agree-ment was found between measurements and simulations for currents above50 % of the critical current.Jia et al. [95] performed a numerical analysis of a saturated core typesuperconducting fault current limiter and partly with loss analysis using H formulation. They found that in the DC biasing coil the outermost HTSlayers have much higher losses than the central layers, and therefore thequench is more likely to occur in the outermost layers. In addition, the totalloss can be estimated by interpolating the loss in several key layers in theoutermost, middle, and innermost layers.Shen et al. [30] used the H formulation to investigate the details of thepower dissipation of a three-phase 35 kV/90 MVA saturated iron core super-conducting fault current limiter. The estimated losses are up to the kW leveland can be even higher, depending on the amplitude of the used DC biascurrent, which should therefore be taken into high consideration during thedesign of these devices. Song et al. [96] used the H formulation with homogenization to calculate theAC losses of a 1 MVA HTS transformer with approximately one thousandturns in the HV winding and with solenoid LV windings, each phase with20 turns of 15/5 Roebel cable. They also modeled a standalone solenoidcoil with the same geometry as the LV winding. The authors successfullyreproduced the main features of AC losses as well as the current and themagnetic flux distributions that were previously calculated with the MMEVmethod in [97]. The disagreement, at rated current, between the AC loss23stimation with the H formulation and the experimental results as well asthe predictions of the MMEV method is less than 20% without optimizingthe mesh.In [98], Wang et al. did an AC loss study of a hybrid HTS magnetof approximately 7000 turns for superconducting magnetic energy storage(SMES) by using the H formulation with homogenization method, whichwas first validated by comparing its results with those of the full model forboth YBCO coils and BSCCO coils of smaller size. The AC losses of theSMES were studied and analyzed during various power exchange conditions.The results show that AC loss is concentrated at the top and bottom ends ofthe magnet and that the higher the steady-state current, dynamic current,and current ramp rate, the higher the AC loss power. The authors also foundthat, for the SMES under study, a steady-state current of less than 100 Acould be an appropriate choice for reducing AC loss.In [99], Morandi et al. used 2D simulations of a Roebel cable to estimatethe dissipation occurring during the charging and discharging of a 1 MW/5 sSMES with solenoidal and toroidal geometry. The simulation showed that thelosses are dominated by the applied field. As a figure of merit for comparingthe loss performance of the two topologies, the authors combined the loss perunit length corresponding to each of the levels of perpendicular field with thelength of conductor exposed to that field, thus obtaining a quantitative “lossindicator”, which resulted to be higher for the toroidal geometry. As demonstrated in the previous sections, the H formulation FEM model hasshown its powerful capability to estimate the AC losses for a wide range ofHTS topologies and in a large variety of operating scenarios. The overwhelm-ing majority of results published in the literature uses the implementationin the FEM software package COMSOL Multiphysics. In truth, other im-plementations of the H formulation FEM model have been proposed. Theseinclude commercial software packages, like FlexPDE [100] and Matlab [62],open-source environments like GetDP [101], and home-made FEM codes likeDaryl Maxwell [87]. However, they represent a minority. It would be na¨ıvenot to recognize that one of the reasons (if not the main one) of the popular-ity of the use of COMSOL Multiphysics is the easiness of implementation ofthe model. In recent years, things have been made even easier thanks to the24uilt-in ‘MFH module’, where equation (1) is already implemented, and onedoes not need to write it in the ‘General PDE module’ for partial differentialequations anymore.The easiness of implementation is particularly attractive because it allowsnew users to get up to speed in a relatively short time. In research groups, italso allows a rapid and efficient transfer of knowledge when personnel comeand leave.Another reason of the popularity is that it is quite easy to exchange mod-els between users, and that the files for numerous topologies and applicationscenarios are publicly available [102]. These include not only the basic imple-mentation of the model, but quite advanced models for cables and windings.In terms of accuracy when compared against experimental data, the H formulation has a performance that is consistent with that of similar numer-ical models. In general, the calculated AC losses are in good agreement withthe experiments, and the accuracy can be quantified as varying between 10 %and 50 %. This can be considered satisfactory, for several reasons: • It is not always possible to have a direct precise characterization of theproperties of the superconductor tape under analysis, for example theangular dependence of J c on the magnetic field; • The tapes’ properties are almost always assumed to be uniform alongthe tape’s length, but variations in the order of 5-10 % are very com-mon; • In the case of tape assemblies like cables and coils, the tapes presentsome degree of misalignment with respect to their theoretical position; • The AC loss measurements are prone to significant disturbances ofdifferent type, for example: spurious electromagnetic signals, difficultyin isolating a (very often small) voltage component in phase with areference signal, extremely low levels of evaporated cryogenic liquid,etc.The H formulation has also important drawbacks. In terms of use, the im-plementation in COMSOL Multiphysics, has some of the typical problemsrelated to using a commercial software. The code is not accessible, and forcertain aspects the code remains a sort of ‘black box’ to the user. In addition,the license price could be not affordable for everyone. On the other hand,25he implementation in open-source codes is not trivial, and in general it isvery time consuming, especially for first-time users. In terms of computa-tional efficiency, the H formulation FEM model is not the best choice. Thishas primarily to do with the fact that the air domains need to be simulatedas well, which may waste a lot of degrees of freedom. For example, in [72]the computation times of the H formulation were reported to be one orderof magnitude longer than those obtained with a homemade code based ona variational method. On the other hand, in [103] the H formulation wasreported to be faster than another FEM method based on T − Ω formula-tion implemented in FLUX. Another pitfall is that the H formulation FEMmodel implemented in COMSOL Multiphysics does not take advantage ofcomputing parallelization. So, having access to large computer clusters doesnot necessarily help. For the simulation of complex systems with coatedconductors, the recently developed T − A formulation seems to be a betteralternative [104], as long as the superconductor layer can be approximatedas an infinitely thin object. This article reviewed the calculation of the AC losses for various HTS topolo-gies based on the H formulation FEM model. A massive number of studiesdemonstrate that the H formulation is one of the most widespread modelsused to calculate AC losses in HTS and has become the de facto standardnumerical tool for that purpose. 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