numerical modeling of hts applications · abstract—high-temperature superconductor (hts)...

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Numerical Modeling of HTS ApplicationsFrancesco Grilli

Abstract—High-temperature superconductor (HTS) applica-tions have reached a mature status, with several prototypesand devices in pre-commercial stage installed around the world.Following this development, numerical models able to simu-late their electromagnetic and thermal behavior have becomeindispensable tools to improve their design and predict theirperformance. In this contribution, I show how numerical modelshave evolved from simple models able to calculate the currentdistribution in one individual tape to powerful instruments able tosimulate complex geometries, material properties and operatingconditions. Then, I focus on the state-of-the-art of numericalmodeling of specific applications, such as cables, magnets, andelectrical machines. Finally, I indicate the challenges that arestill open and the possibility of establishing a collective effort tospeed-up the advance of this important research topic.

Index Terms—Numerical modeling, HTS applications, numer-ical methods, AC losses.

I. INTRODUCTION

NUMERICAL models have become very popular toolsto investigate the electromagnetic and thermal behavior

of superconducting devices, also thanks to the availability ofrelatively cheap computing power. In particular, the modelsthat have seen a booming expansions in recent years arethose dedicated to study the electromagnetic behavior ofhigh-temperature superconductors (HTS). This is because ofdifferent reasons: on one side, HTS technology is becomingmature and, with many applications in a pre-commercial stage,there is a strong demand for tools able to optimize theirdesign and reliably predict their performance. On the otherside, the electromagnetic behavior of HTS is not simpleto simulate, especially in applications characterized by thepresence of varying magnetic fields: one needs to be able totrack the movement of the magnetic flux front. To a certainextent, this can be done analytically, but those models areusually limited to simple geometries and simplified materials’properties. In addition, things are made more difficult if oneneeds to take into account the dependence of the criticalcurrent density Jc on the amplitude and orientation of themagnetic field, on the temperature, and on the position –this latter being a consequence of the difficulty of producingextremely homogeneous HTS materials.

Most electromagnetic numerical models of HTS have beendeveloped with the purpose of calculating AC losses in avariety of working conditions (sinusoidal cycles of transportcurrent, external magnetic field, or combination of the above).As a consequence of the non-linear relation between theelectric field and the current density, steady state modelsare not usually applicable and time-dependent simulations,

F. Grilli is with the Karlsruhe Institute of Technology, Karlsruhe Germany.Corresponding author’s e-mail: [email protected].

Manuscript received September 8, 2015.

which calculate the current and magnetic field distribution(and hence the instantaneous dissipation) at each time step ofthe sinusoidal cycle have been developed. Those models canalso be used to calculate the dissipation in HTS in conditionsdifferent from pure sinusoidal excitations, for example thelinear ramp of current charging a magnet or ripples comingfrom rectification of AC currents. In short, all the situationscharacterized by time-varying magnetic fields.

This contribution does not want to be a review of availablemodels for HTS. For this purpose, the reader can refer to tworecently published topical papers on analytical [1] and numer-ical [2] models, and also to one focusing on magnetization ofHTS bulks [3]. The purpose of this work is rather

• to show how numerical models of HTS have evolvedin recent years and what degree of complexity they canhandle;

• to show how real applications can be simulated.The paper is therefore structured in two main sections

that follow these objectives. A concluding section serves asa summary of this work and discusses possible direction ofdevelopment.

II. EVOLUTION OF NUMERICAL MODELS FROM SINGLETAPES TO COMPLEX DEVICES

The first numerical model that was used to systematicallyinvestigate current and field distribution inside HTS tape andto calculate AC losses in a variety of working condition wasproposed by Amemiya in 1998 [4]. The model, based onthe finite-element method and on a 2-D T-φ formulationof Maxwell’s equation, was written in a proprietary code.Two years later, a group at the Polytechnique Grenoble testeddifferent formulations of Maxwell’s equations with supercon-ductors [5], and they successively developed (in conjunctionwith Cedrat and EPFL) a module for superconductors in thecommercial finite-element program Flux2D (currently calledsimply Flux [6]), based on the A-V formulation [7]. Inboth cases the superconductor was modeled with a power-lawresistivity, with the possibility of including the dependence ofits parameters on the magnetic field and on the position

Their differences notwithstanding, these two models repre-sent a common way of studying the electromagnetic responseof HTS: use finite elements to solve a differential form of time-dependent Maxwell’s equations with a non-linear resistivity(or conductivity) to describe the superconductor’s electricalbehavior. A summary of the different formulations commonlyused to follow this approach is given in Table I.

Other methods have been developed to simulate HTS [2], forexample integral and variational methods. Among the former,there are for example the Brandt’s method [8] and the integralequation method for thin tapes solved with finite-elements by

IEEE/CSC & ESAS SUPERCONDUCTIVITY NEWS FORUM (global edition), October 2015. Invited keynote presentation 3A-LS-O1.2 given at EUCAS 2015; Lyon, France, September 6 – 10, 2015.

A manuscript was submitted to IEEE Trans. Appl. Supercond. for possible publication.

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TABLE IDIFFERENT FORMULATIONS COMMONLY USED TO SOLVE MAXWELL’S

EQUATIONS WITH NUMERICAL MODELS.

Formulation Equations Definitions

∇2A = µσ(∂A∂t

+∇V)

B = ∇×A

A− V ∇ ·(σ ∂A∂t

+ σ∇V)= 0 E = − ∂A

∂t−∇V

σ = σ(E)

∇× ρ∇×T = −µ ∂(T−∇φ)∂t

J = ∇×T

T− φ ∇2φ = 0 H = T−∇φ

ρ = ρ(J)

E-field ∇×∇×E = −µ ∂(σE)∂t

∂B∂t

= −∇×E

σ = σ(E)

H-field ∇× ρ∇×H = −µ ∂H∂t

J = ∇×H

ρ = ρ(J)

Brambilla et al. [9]. Among the latter, one can mention theJ formulation proposed by Prigozhin [10] and the MinimumMagnetic Energy Variation (MMEV) method by Pardo [11].The MMEV method underwent a series of constants improve-ments, including the recently added possibility of using asmooth current-voltage relation (instead of the critical statemodel) and is now called Minimum Electro-Magnetic EntropyProduction (MEMEP) method [12].

These and other numerical models of superconductors havegreatly improved over the past few years and most of them arenow able to simulate complex problems. This section providesa series of “highlights” of the degree of complexity that it ispossible to handle at present.

A. Interacting Superconductors

When moving from individual HTS tapes to devices, thefirst obvious complication is that devices are usually madeof several tapes (or turns in the case of coils), so thatmultiple superconducting domains need to be modeled. Inmost applications, the current is not free to distribute amongthe different simulated taper or turns, but it is somehowconstrained. Typical examples are multi-layer cables (due toadjusted twist pitch), cables made of transposed tapes (becausethe tapes are electromagnetically identical), and superconduct-ing coils (because the turns are made from the same tape).In other situations (multi-phase cables), currents are equal inamplitude but shifted in phase. Generally speaking, in mostcases the model needs to be able to control the current flowingin different parts of the device. This goal can be attainedin different ways, depending on the utilized models: typicalmethods are current constraints [13], coupling between thefinite-element structure and an external circuit [14], settingdifferent electrical potentials in different conductors [15], [16].An exemplary result of interaction between HTS tapes is givenin Fig. 1, which shows the current density distribution in fourstacked pancake coils, each composed of 24 turns, calculatedwith an axis-symmetric model. The figure, which refers tothe instant of the peak of the transport current, reveals the

Fig. 1. Current density distribution in a stack of four 24-turn pancakecoils made of HTS coated conductor. The figure refers to the peak of thetransport current and shows the presence of current density with oppositesign from the transport current at the top and bottom pancakes for the four-pancake stack. For better visualization, the color-graded lines representing thesuperconducting layer are shown with a larger thickness than the one used inthe simulations. Reprinted with permission from [20].

existence of magnetization currents – visible from the fact thatin some parts of the top and bottom coils the current densityhas opposite sign with respect to that of the transport current inthe two central coils. With this kind of approach, it is possibleto have access to the instantaneous current density and fielddistribution inside each tape. In the case represented in thefigure, the number of simulated tapes is 96. Larger numbersof tapes have already been simulated: for example 130 in [17]and 406 in [18]. A recent work based on the MEMEP modelreported the simulation of up to 10,000 tapes [19].

When tapes are compactly and regularly arranged (as in thecase shown in Fig. 1), an alternative approach to simulatingall the tapes is to make an anisotropic continuous mediumapproximation, “diluting” the individual tapes into one bulkconductor, which is a much easier object to simulate. Onehas however to remember the configuration of the originalgeometry: for example, in a coil the same current flows in allthe turns, and this fact must be mirrored in the simplified bulkconductor. This approach was originally proposed for a stackof thin tapes by Clem [21], and successively refined by others.Some models are based on the critical state model and use apredefined distribution of current in the stacks under analysisby means of either a straight line [21] or a parabola [22]to manually separate regions with positive, negative or zero



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current density. A model that does not make any a prioriassumptions about the current distribution in the homogenizedstack is shown in [23]; however, being also based on thecritical state, that work does not consider overcritical currents.A later model that does not make any a priori assumptionsabout the current distribution and that allows for overcriticallocal currents was presented in [24].

Another approach to model large numbers of interactingtapes is the so-called multi-scale [25]: the idea is to simulateone conductor at the time using a boundary condition forthe field it is subjected to as the result of the environment(for example, the field created by the other conductors). Thismagnetic field can be calculated in a relatively simple way,e.g. with magnetostatic calculations assuming uniform currentdistributions in the conductors. If one is for example interestedin calculating the AC loss of the whole device, the position ofthe tape of interest can then be moved and a “map” of the ACloss distribution in the device created. The advantage is thatthe simulations involving one tape are very fast and, when onewants to produce the map, they can be truly parallelized. Inaddition, it is not necessary to consider all the positions of thetape in the device, but only some key ones. This method isillustrated in [26] for a matrix of 5×100 tapes (representing 5pancake coils of 100 turns each), where 25 equally distributedpositions are used to produce a sufficiently accurate AC lossmap.

B. 3-D Modeling

Another degree of complexity of HTS devices is representedby the fact that sometimes 2-D approximations are not suffi-cient. Even at the level of simple HTS wires, certain effectssuch as the coupling between filaments are inherently 3-D.In fact, a 3-D model was developed to investigate this effectalready back in 2003 [27]. That work revealed the heavy com-putation burden of extending the numerical models to the thirddimensions, with reported computation times between 4 and 48hours for the investigated cases. Several other full 3-D modelshave been developed since then [28], [14], [29], [30], [31],but seldom utilized to simulate realistic devices, apart fromHTS bulks [32], [33], [34] or other simple geometries [35].There are however two exceptions. The simulation of a cellunit representing a Roebel cable composed of 14 strands waspresented in [36]. The problem had a number of degrees offreedom and computation times in the order of several hundredthousands and a few days, respectively. Another remarkableexample of full 3-D modeling is a triangular coil assembledfrom HTS tape and consisting of 37 turns [37]. The modelincludes the angular dependence of the critical current densityand, using the natural symmetry of the considered triangularcoil, simulates one sixth of the actual geometry. Particularattention is paid to the mesh: as detailed in [37] mesh scalingand mapped meshing are both good methods to improve theconvergence speed and reduce the simulation time required,which in this case is 10-15 hours for 1.5 AC cycle and 120,000degrees of freedom [38].

The two cases mentioned above clearly show that full 3-D time-dependent simulations of HTS cables and coils are

computationally very demanding. For the simulations of suchobjects in 3-D, it is therefore convenient to introduce somekind of simplifying assumptions, if possible. Typically, one cantry to simulate 3-D effects in a 2-D computational domain. Forexample, using an anisotropic conductivity to simulate powercables composed of layers with different twist pitches [39];introducing an infinitely-thin-conductor approximation to sim-ulate Roebel cables [40]; introducing a change of coordinatesto simulate twisted conductors [41].

C. Magnetic Materials

Certain applications are characterized by the presence ofmagnetic materials, either because these materials are neces-sary constituents of the HTS wires or because they can beused to divert magnetic flux away from the superconductor(hence increasing its critical current and reducing the AClosses) [42], [43], [44] or to improve the magnetic fluxtrapping properties of bulks [3]. In addition, materials withferromagnetic properties can be used for realizing magneticcloaks as well as for focusing or transporting magnetic field.The main issue of modeling those materials is their non-linear and hysteretic magnetic permeability, which requiresa specific formulation of the magnetic constitutive equation.Implementing magnetic hysteresis cycles in time-dependentelectromagnetic problems is not an easy task [45]. If no simul-taneous electromagnetic and thermal coupling is required, theAC losses in the ferromagnetic materials can be calculated aposteriori by considering only the maximum field experiencedby the ferromagnetic material and by using an empiricalrelation linking it to the losses, as done in [46].

III. CURRENT STATUS OF MODELING HTS APPLICATIONS

This section describes the state-of-the-art of numerical mod-eling for some specific HTS applications.

A. Cables

HTS cables for power transmission are composed by oneor more layers of HTS tapes and are usually simulated in2-D, i.e. considering only their transversal cross-section andneglecting the axial magnetic field. Due to the relatively lowpitch angle (the angle between the axis of the tape and theaxis of the cable), the latter is usually significantly lower thanthe tangential magnetic field, so that the introduced error isnot too large. The problem thus reduces to simulate a certainnumber of HTS tapes interacting electromagnetically. In thecase of multi-layer cables, the current is usually not let freeto distribute, because it would flow mostly in the outer layer,rapidly saturating it and causing very large dissipation there. Inreal cables, the twist pitch of the different layers is adjustedso that the layers have a similar impedance and the currentdistributes almost uniformly between them. This situation canbe simulated in 2-D models by imposing the same currentin each layer by means of integral constraints. The same canbe done in the case of multi-phase cables, where the currentsneed to have a specific phase shift.


A manuscript was submitted to IEEE Trans. Appl. Supercond. for possible publication.IEEE/CSC & ESAS SUPERCONDUCTIVITY NEWS FORUM (global edition), October 2015.

Invited keynote presentation 3A-LS-O1.2 given at EUCAS 2015; Lyon, France, September 6 – 10, 2015. A manuscript was submitted to IEEE Trans. Appl. Supercond. for possible publication.

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Fig. 2. Schematic views of analysis model for an HTS Roebel cable: (a)cross-sectional model and its analysis plane and (b) three-dimensional modeland its analysis region. Analysis results for a six-strand Roebel cable exposedto an ac magnetic field: (d) calculated current profile at peak phase and (d) ACloss density distribution. Adapted and reprinted with permission from [40].

Compact high-current cables such as Roebel [47],Conductor-On-Round-Core (CORC) [48] and Twisted-Stacked-Tape-Cables (TSTC) [49] have different geometriesand the adopted modeling approaches mirror those differences.Roebel cables are often simulated in 2-D as two stacks oftapes, where the effects of the tape transposition is accountedfor by imposing that the current flowing through them isthe same. A full 3-D model of a Roebel cable has beendeveloped [36], but at least for the time being does notseem to be a viable modeling option due to the complexityof building the geometry and the mesh and the excessivesize problem and computation time. A 2-D models basedon the infinitely-thin-tape approximation [40] has been usedmore extensively and also extended to coils made of suchcables [50]. CORC cables have an helicoidal geometry andare therefore topologically similar to HTS power cables.However, the pitch angle is much higher, in the range of 40◦,thus neglecting the axial component is a more questionableapproximation. However, in the case of magnetization AClosses, successive 2-D simulations made considering thechange of the angle between an external uniform field and thetapes of a cable provided good agreement with experimentaldata for a one-layer CORC cable [51]. A similar approachhas been followed for TSTC [52], although no experimentalverification exists yet.

Fig. 3. HTS-CC simulated with the 3-D electrothermal model (not to scale).Virtual probes are located along the length (z-axis) and across the width(x-axis) of the tape. The traces of the voltages during a quench are shown.Reprinted with permission from [56].

B. Fault Current Limiters

The modeling of the electromagnetic behavior of faultcurrent limiters in normal operating conditions usually reducesto simulating stack of tapes or coils, so it does not present anypeculiarity. Most of the modeling effort is focused on modelingthe current-limiting phase, for which the thermal aspects aremore important. A fully coupled 2-D electromagnetic-thermalmodel was proposed in [53]. However, if one is interestedin modeling quench, effects of non-uniformities, and normalzone propagation, simplified models assuming uniform currentdensity distribution in the superconducting layers are moresuitable. This assumption is justified by the very different timeconstants of the electromagnetic and thermal phenomena [54].These models neglect the magnetic flux dynamics insidethe superconductor and are capable of handling 3-D geome-tries [55], [56]. An example of the normal zone propagationvelocity (NZPV) is shown in Fig. 3, which shows the differentvoltage traces in different directions: due to the low NZPV, theelectrical field traces recorded along the length of the tape (z-axis) are perfectly distinct in the time domain, which is notthe case for the traces along the width (x-axis). This type ofmodeling has also been used to propose new tape architecturesaiming at accelerating the normal zone propagation [57]. Thequench propagation in coated conductors can also be studiedwith network model approaches [56], [58], [59].

C. Large Coils and Magnets

The modeling of large coil and magnets follows the tech-niques described in section II-A: one one side, simulation ofa few thousands tapes is possible thanks to the availabilityof cheap computing power and to the optimization of thecode [19]; on the other hand methods based on multi-scale



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Fig. 4. Current density in a large magnet coil at the end of the charg-ing curve. The homogenized bulk approximation is used. Negative currentdensity evidences magnetization currents. Reprinted with author’s permissionfrom [12].

or homogenization approaches are very attractive because ofthe much reduced size of the problem and computing times,without practical loss of accuracy in the results: in [26] theauthors report a 100-fold reduction of the computation time(30 minutes instead of 50 hours) with the homogenizationapproach with respect to simulating all tapes in a 500-turncoil configuration, and a 1% difference in the calculated AClosses.

An example of current distribution in a large coil systemwith the homogenization technique is shown in Fig. 4. Sim-ulated is a stack of 20 pancake coils made of 200 turnseach, with the current ramped up to certain level and thenlet constant. The reader can find the details of these calcula-tions on the relaxation effects in large magnets in [12]. Theimportant thing to note here is that, differently from Fig. 1, thepresence of the individual turns is neglected (since they arehomogenized into bulks), but the main features of the currentdistribution are maintained.

D. Electrical Machines

In electrical machines, HTS are employed in the forms oftapes (coils) or bulks (permanent magnets)

In the first case (where HTS are typically used in thecoils of the rotor), one does not usually need to simulatethe whole machine with the degree of resolution of theindividual HTS tape. Instead, the most convenient approachis a multi-scale one [60], [25], where the electromagneticcomputation for the machine is decoupled from that of theHTS coils. The concept is illustrated in Fig. 5, for the caseof a generator, whose rotor is made of superconducting coils(here schematically represented with four rectangles, each ofwhich is actually composed of several tens or hundreds ofsuperconducting tapes). If the coils of the rotor are tightlypacked, the magnetic field on the coupling boundary has aweak dependence upon the actual current distribution insideeach superconducting tape of the coils. This allows simulatingthe electromagnetic response of the machine without takinginto account the geometrical internal features of the rotor’scoils, or the peculiar E − J characteristics of the supercon-ductor. However, a reliable calculation of AC losses in thesuperconducting material requires the detailed knowledge ofthe current density distribution inside the superconductors. Thelatter can be recalculated from the machine’s simulation data,using the previously computed magnetic field on the couplingboundary as a Dirichlet condition for the simulation involvingsuperconductors.

In a fully superconducting rotating machine, due to thehigh electrical loading of the stator and the air-core magneticconfiguration, the magnetic field seen by a stator conductoris a combination of alternating and rotating fields [61], whichresults in different magnetic flux dynamics from those con-ventionally observed in numerical simulations of HTS, whereone usually simulates uniform fields of given orientation withrespect to HTS. An example of the current density over thecross-section of a round filament in the presence of a rotatingfield is given in Fig. 6.

As far as bulks are concerned, their modeling in electricalmachines has just started. An example is given in [62], wherethe field generated by an array of fully magnetized bulksuperconductors in an axial gap trapped-flux type electricmachine is computed with analytical approaches based on theBiot-Savart, the Fourier Transform Method, and Fast FourierTransform methods.

E. HTS as Permanent Magnets: Bulks and Tape Stacks

Finite-element-method-based models have recently beenused to calculate the maximum trapped field and the temper-ature evolution in HTS and MgB2 bulks used as permanentmagnets. The bulks are typically subjected to pulsed magneticfields, so that full time-dependent electromagnetic-thermalmodels are necessary. While the simulation of multiple pulsesis quite time-consuming, the bulk geometry is much simplerthan that of HTS tapes, and full 3-D simulations can beroutinely performed. One general challenge is represented bythe difficulty of having precise material properties over a widerange of temperatures and of applied fields, and by the factthat these properties are also non-uniform in the bulk volume.While inserting non-uniformities (such for example a position-dependent critical current density Jc) can be easily done [33],



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Fig. 5. Simulation strategy for computing the losses in the rotor of anelectrical machine, made of HTS coils. In the machine’s model, both electricload and rotation speed are known inputs. Using this model and assuminguniform current distribution in the coils, the magnetic field is calculated ona coupling boundary (red rectangle) around the rotor’s winding. The modelfor the coils (right) uses this calculated magnetic field as Dirichlet boundarycondition to compute the actual current distribution in the coils and thecorresponding AC losses. No feedback is considered from the coils’ modelto the machine’s model. Reprinted with author’s permission from [63].

Fig. 6. Distribution of the current density over the filament cross-sectionat the same time for different elliptical field configurations (parameter k)and different field penetrations (parameter αm). Reprinted with permissionfrom [61].

the correspondence between reality and model probably needsfurther verification. More details on modeling bulks can befound in a recently published topical review [3].

Stacks made of HTS tapes have also the potential to beused as permanent magnets [64]. The modeling approach isthe same as for bulks, with the difference that the geometry ismore complicated, due to the presence of layers of differentmaterials with high aspect-ratio. As a consequence, for thetime being simulations are usually limited to 2-D [65].

IV. CONCLUSION AND OUTLOOK

Numerical models of HTS have greatly evolved in recentyears, and it is now possible to simulate complex devices.Most situations of practical interest can be simulated with 2-Dapproaches, whereas full 3-D modeling is still rarely utilized.Among the current applications that require full 3-D modelsone can mention for example multi-layer CORC cables andstacks of HTS tapes, for which the challenges are represented

by layers twisted in different direction and the presence ofseveral thin layers of different materials, respectively. 2-Dmodels can simulate devices with large numbers of HTS tapesor turns, in the order of thousands. The size and computationtime of those problems can be considerably reduced withoutcompromising accuracy by utilizing multi-scale or homoge-nization approaches.

Despite a constant growth of a dedicated scientific commu-nity, numerical modeling of HTS is still a quite specializedfield, where the different groups have their own recipes tosolve particular problems. A more collective effort wouldintroduce several benefits, including an easier learning curvefor people who start modeling HTS devices without previousexperience (no need to reinvent the wheel each time) anda faster advance of the modeling capabilities [66]. In thisrespect, an attempt has been initiated, with the organization ofdedicated workshops [67] and the creation of a website [68],where one can find an up-to-date publication database, thedefinition of benchmark problems and – since a few months– publicly available codes, both commercial and open-source.

ACKNOWLEDGMENTS

The author would like to thank several colleagues forproviding useful information and reference material for thismanuscript: Prof. Frederic Sirois (Polytechnique Montreal,Canada), Dr. Victor Zermeno (Karlsruhe Institute of Technol-ogy, Germany), Dr. Mark Ainslie (University of Cambridge,UK), Dr. Enric Pardo and Dr. Michal Vojenciak (Institute ofElectrical Engineering, Bratislava, Slovakia), and Dr. AnttiStenvall (Tampere University of Technology, Finland).

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[37] D. Hu, M. D. Ainslie, J. P. Rush, J. H. Durrell, J. Zou, M. J. Raine,and D. P. Hampshire, “DC characterization and 3D modelling of atriangular, epoxy-impregnated high temperature superconducting coil,”Superconductor Science and Technology, vol. 28, no. 6, p. 065011, 2015.[Online]. Available: http://stacks.iop.org/0953-2048/28/i=6/a=065011

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[47] W. Goldacker, F. Grilli, E. Pardo, A. Kario, S. Schlachter, and M. Vojen-ciak, “Roebel cables from REBCO coated conductors: a one-century-oldconcept for the superconductivity of the future,” Superconductor Scienceand Technology, vol. 27, no. 9, p. 093001, 2014.

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8

[52] F. Grilli, V. M. R. Zermeno, and M. Takayasu, “Numerical modelingof twisted stacked tape cables for magnet applications,” Physica C(accepted), 2015.

[53] F. Roy, B. Dutoit, F. Grilli, and F. Sirois, “Magneto-Thermal Modelingof Second-Generation HTS for Resistive Fault Current Limiter DesignPurposes,” IEEE Transactions on Applied Superconductivity, vol. 18,no. 1, pp. 29–35, 2008.

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