Coupling Losses Algorithm for Superconducting Strands (CLASS)

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Superconductors (CICCs) and Fusion tokamaks

Approximatively at the same time of the discovery of superconductivity, A. Eddington explained that thermonuclear fusion is probably on going inside core of stars producing enough energy to sustain their gravitational pressure and shine across millions of kilometers. E. Rutherford demonstrates for the first time that fusion is accessible in laboratory by merging deuterium atoms. From this first attempt, the run to an experimental setup capable to achieve fusion at a larger scale (centimeter, meter) was started and in 1950, G.P. Thomson and M. Blackman patented a fusion reactor using a toroidal shape to confine the plasma. The first corresponding experimental setup was achieved in SSSR in 1969 and named tokamak for “toroidal chamber with magnetic system”. Experimental fusion machines have taken several forms since their first realisation: toroidal, spherical, and even more complex (stellarator).
Several confinement systems are used to maintain the plasma such as magnetic confinement based on conventional (copper based) or superconducting (NbTi or Nb3Sn) magnet system. The confinement system using toroidal field coils (superconducting or not) is placed as shown in Figure 3.Initially, fusion reactors were promoted under merits of its low production of nuclear wastes and of the great availability of its fuel (deuterium and tritium from lithium). Superconducting magnets systems are absolutely essential for their low energy cost during operation in order to produce in very large volumes the large magnetic field able to confine the plasma.
The need for superconducting magnets in large fusion devices was already recognized from the beginning. In the middle of the 1970’s, the initial major development programs were started based on the first definitions for the required magnet parameters. These programs followed two lines. Firstly was the development of conductor and magnet systems for small and medium size plasma devices such as T-7 and T-15 in the Soviet-Union [7], TRIAM in Japan [8] and most prominently TORE SUPRA (now WEST) at CEA in France [9]. Secondly, in the form of an international project, was initiated the development of a conductor and magnet arrangement with parameters pertinent to large fusion devices and their test in a dedicated special facility, the Large Coil Task (LTC) project (see [10]).
Following the successful execution of these projects, the design and construction of larger fusion devices with superconducting confinement magnets were started. Some of them are already in operation: the tokamaks EAST in China [11] and KSTAR in Korea [12], the stellarator LHD in Japan [13], the SST1 tokamak in India [14] and the stellarator Wendelstein W7-X in Germany [15]. Finally, the international fusion community felt brave enough to start the development, in a worldwide effort, of ITER, a tokamak-type reactor [16], [17] accompanied by a satellite tokamak JT-60SA, in Japan, with major contributions from Europe [18].
For the ITER coils, the requirements for high currents in the 70-80 kA range and for a very high voltages in operations, 10 to 20 kV to ground for the poloidal field and central solenoid systems inherent in the size of the magnetic systems, led to the selection of the cable-in-conduit-conductor (CICC, see [19]) as the best choice for the conductors in the present state of the superconducting technology. Moreover, this type of conductor is well adapted to support fast heat deposition. The principle of CICC is not recent: M. Hoenig at MIT (USA) introduced it in 1975 (see [20]) and all the superconducting tokamaks in operation except TORE SUPRA used this concept. The coil which pioneers this concept was the Westinghouse coil in the Large Task Coil described in [10], where Nb3Sn was the superconductor. The maximum performance of this magnet was unfortunately limited by some spreading out of a resistive phase in the magnet.
A modern CICC is basically made of several stages by cabling superconducting and copper strands and then by compacting the cable inside a conduit generally made of stainless steel. A CICC such as the one used for ITER is composed of several components, superconducting strands, copper strands, steel bandages (named wrapping), one or more helium channels and the steel encasing conduit, as shown in Figure 4 below. In a project like ITER the optimum composition of the conductor components is defined by the system design criteria.
Figure 4: Conductors of the two ITER model coils. On the left CS Model Coil (51mmx51mm, 40 kA), on the right an exploded view of the TF model Coil (40.7mm diameter, 80 kA) The CICC was invented to benefit from the very high volumetric heat capacity of helium, about 500 times the volumetric heat capacity of metallic materials, limiting by the way the temperature excursions in the case of fast energy deposition. This occurs in tokamaks after a very fast decrease to zero of the plasma current when plasma disruption occurs. In this case a fast magnetic field variation, with a time constant of the order of 100 ms for example, affects the whole coil over lengths of several meters, creating losses in the superconducting strands. This is rather similar to the kind of event which can occur in high field test facilities, affecting the outer superconducting magnet of a hybrid magnet when the central copper magnet disrupts. The CICC offers an adequate solution to this problem by providing:

A local helium reservoir

– A very long wetted perimeter. The diameter of the ITER TF cable is 39.7 mm. It is made of 900 superconducting strands 0.82 mm in diameter with a void fraction in the cable of 30 %. This fine subdivision of the strands can be translated in a total of 2.3 meters of wetted perimeter in the cable section, facilitating a large heat transfer to the helium reservoir.
– Small AC losses for the conductor by controlling its time constant through the contact resistance between strands (through cable void fraction/compaction).
The strands of CICC’s are magnetically transposed thanks to the cabling. This transposition is not perfect but it ensures an optimal sharing of the current among the strands in inductive mode.
During operation, the helium mass flow circulating in the conductor limits the temperature increase due to the residual nuclear heating and to the AC losses generated by the varying magnetic fields during a plasma discharge. The central channel in some cables as shown in Figure 4 helps to keep the pressure drop at an acceptable level.

Coupling losses in superconductors: the case of the strand

The elementary brick of CICC for fusion superconducting magnets is the composite strand. “Composite” means the strand is a mixed assembly of copper and superconducting filament. Coupling losses are generated in the copper matrix during field and current variations, however twisting the filaments is effective to limit these coupling losses.
There are hundreds of strands in a CICC. The composites strands (cylindrical) are consisting of hundreds superconducting filaments (NbTi, Nb3Sn) twisted in a copper matrix with a twist pitch 0 (as shown in Figure 5). Filament diameter usually lies in the range of microns and they are usually located in what is known as the filamentary zone. Strands used in fusion machines are often composed of several layers, alternating resistive and filamentary zones. A resistive zone is mainly composed of copper and filamentary zone is mainly composed of twisted filaments of superconducting materials embedded in a normal matrix (see Figure 5). In this filamentary zone, we also find a resistive matrix usually made of copper filling every space between the filaments.
Figure 5: Scheme of a superconducting strands on the left. Detailed architecture of JT-60SA TF conductor strand (0.81 mm diameter) on the right.
Architecture of the strand is different depending it is based on NbTi or Nb3Sn filaments. This is due to the fact that Nb3Sn is issued from a chemical reaction taking place in the strand once it is produced. Nb3Sn strands have to follow a peculiar thermal cycle to for the Nb3Sn phase to form in the strand. We can see the different architectures in Figure 7.
Figure 7: Different type of strands. NbTi (1st line) and Nb3Sn (2nd line). Extracted from [24].
The strand itself can generate losses even though it is in its superconducting state. Subject either to external magnetic field or to current variations, it will generate losses of two kinds: hysteresis losses and coupling losses. Hysteresis losses are due to the current flowing inside the superconducting filaments whereas coupling losses are due to current crossing normal zone to connect two superconducting filaments. These currents, crossing normal zone in a strand, as depicted in Figure 6, are due to current redistribution inside the strand. This current redistribution is due to the fact that, in order to shield its volume from the external magnetic field variation, the twist of filaments in the strand will generate current loops. All these current loops together, limit the external field penetration into the strand filamentary zone by creating a magnetic field inside the strand, which is opposed to the external one.
Where is the external field applied on the superconducting strand (or cable) and the internal field response to the applied one. is the time constant related to the current loop inside the strand.
This kind of currents generates, as described above, the so called coupling losses which are defined as:
where is the power per unit volume of strand with = 2 for a cylindrical composite. being a geometrical factor.
The relation between and can be illustrated for an applied sinusoidal field. If = sin( ) + , with = 2 the angular frequency, using the above first order differential equation (1), we obtain in complex notations:
The associated power density averaged over time (after a time long compared to ) will then be:
As it is widely used within the applied superconductivity community, we can also express the losses in terms of average losses per cycle per unit volume (of strand), this can be done very quickly multiplying by the period of the cycle. Using the expression of , we have:
Of course, these considerations and formulae only concern the coupling losses generated by a strand subjected to a transverse field (as shown in Figure 6) and assume that the outer edge filaments are not saturated and that the composite is not carrying any transport current. In case of saturation, we would need to add the penetration losses, which are well described in [21].

Considerations on losses in superconducting magnets for fusion

Presently, several experimental superconducting tokamaks are in operation. The largest superconducting tokamak ever built (JT-60SA) is in its final phase of assembly and commissioning and first operation is expected end in 2020. The first plasma discharges of the superconducting experimental reactor (ITER) are expected in 2025. Moreover, studies are devoted to the design of reactors producing electricity after ITER (DEMO, DTT). On the energetic plan, fusion is obviously very attractive. The energy consumption of, for example, 400 000 inhabitants is in the range of 500 MW of electric power. In order to produce this amount of energy we would only need to burn 175 kg of deuterium tritium in a fusion plant while we would have used either 1.35 megatons of coal, or 0.9 megaton of petrol, or 12.5 tons of uranium in a fission plant. With renewable energy, it corresponds to 35 2 of solar panels or around 25 2 of wind generator with a 20% load factor.
Using fusion plant instead of fission plant could offer nearly a carbon free electricity as fission but suppressing difficulties linked to fission as:
– Large available resources and suppression of the geopolitical tensions linked to fuel procurement.
– Simplification of the problems associated with long lived waste.
– No uncontrolled chain reaction or runaway.
The demonstration of the feasibility of fusion was initiated in the eighties by the production of fusion power in JET and TFTR (see [22] and [23]). The operation of existing superconducting tokamak (WEST, EAST, KSTAR, etc.) is preparing ITER even if it was not their initial goal. The main objective of the ITER project, which will deliver first plasmas in 2025, is to demonstrate at a representative scale the feasibility of energy production using a fusion reactor under the form of a tokamak. The objective is to produce thousands of plasma discharges (500 MW during 500 s).
Applied Superconductivity in MRI is mainly in DC field. During ITER scenarios, causing large field variations across the magnets, important losses are developed associated to temperature increase and reduction of temperature margins. This is a new very important challenge for applied superconductivity.
The ITER superconducting magnet system constitutes the largest ITER components, representing about 30% of the machine cost investment. This underlines the fact that designing the magnet system from specific design parameters in order to limit its AC losses remains a challenge. The objective is to keep the superconductors within their critical limits and to mitigate the power associated to the cryogenic system.
Since TORE SUPRA (CEA, Cadarache), now renamed WEST from its tungsten divertor upgrade, several other superconducting tokamaks are in operation. They are presented in Table 1 below:
Starting from the first small superconducting magnet in 1962, unprecedented advances have been accomplished in applied superconductivity in particular in MRI. Regarding fusion, important challenges have been mastered such as the development of CICC to support very high currents (50-70 kA), very high voltage (10 kV), the description of AC losses generated by field variation and eliminated by the helium flowing inside the CICC. Connections of several superconducting coils have also been managed with the development of various connections techniques as the twin box developed at CEA Cadarache.
In a CICC, strands are cabled and twisted in stages, and stages are also twisted and cabled as bigger stages. Regarding AC losses, the crucial role of the twist and transposition pitches is well known but the general impact of the geometry and in particular of the contact resistances between the cable strands has still to be explored. The cost investment of these large machines is dependent for a non-negligible part on the conductor itself. Progresses are needed to optimize and to analytically predict the behaviour of the CICCs under field variations.
Presently, the one time constant model is used in the fusion community to describe AC losses. Also numerical code (JackPot) or heuristic model (MPAS, for Multizone PArtial Shielding) exist but no analytical tools are capable of predicting the AC losses a CICC will generate with respect to its geometry, electrical contact distribution, and strand type. The objective of the thesis is to progress in this direction. Such a model presently exists at strand scale and it is named CLASS for Coupling Losses Algorithm for Superconducting Strand, developed by A. Louzguiti. This model (fully described in [24]) will be shortly introduced in Section II.1.2 for further uses. Based on this first step, he developed a two-stage cable model (see [24]) where two cabling stages of a CICC (example: the two first ones of JT-60SA TF conductor, a triplet of triplet) magnetically and electrically interact giving the amount of coupling losses generated by the coupling of two cabling stages.
The different versions of the conductors used in ITER are tested in the European SULTAN test facility (CRPP, see [25]). The test facility SULTAN inaugurated in 1992 in preparation of ITER has many objectives, among which:
– Critical properties characterisation of CICCs under representative magnetic field
– Losses measurement in CICCs in harmonic field variations ranging from 0 to 10 Hz
The experimental coupling losses are fitted using the heuristic MPAS model [26] (MPAS description will be done in Section II.1.1). The ITER scenarios are divided in small time steps into linear variations of magnetic fields, enabling to calculate the AC losses using the MPAS model and a model of hysteresis losses, then using a thermohydraulic code enables to check that the temperature margins are sufficient. This is the approach presently used in the ITER program to prepare the operation.

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Thesis content and objectives

This thesis work aims at progressing in the analytical description of coupling losses under time varying field inside large superconducting cable in conduit conductors used for the magnet system of a fusion reactor. To illustrate this approach, the discussion about the presented models will be confronted to the modelling of cables of type similar to the TF conductor of JT-60SA.
The MPAS model (see [26]), which relies essentially on the experimental data of coupling losses gathered by using test facility as SULTAN, is used for ITER to assess the magnetic parameters of the considered cable. This model is not able to predict AC losses but only to give a description on how there are distributed in the cable among each stage. However, the present thesis is in line with the work led in [24], where an analytical model capable to model any two-stage cable was developed. We name this model COLISEUM for COupling Losses analytIcal Staged cablEs Unified Model. This modelling of a two-stage cable is not sufficient to properly model or predict the coupling losses generated by a cable composed with five stages for example. Further development toward higher number of stages has therefore been carried out within the thesis work.
Both of the above models are based on analytical expression of coupling losses but COLISEUM is intrinsically predictive as needing cable-oriented inputs as cable geometry and conductances between the cable stages. Based on the geometrical description of the cable (strand radius, twist pitches, cabling stage radius) and on its electrical description, COLISEUM can compute and tentatively predict induced current and coupling losses of a two-stage CICC (see [24]).
In its original state, COLISEUM could describe one-stage or two-stage cables but it has not been tested in a wide variety of parameters in order to check its consistency (check for divergence, limits of the model). We will start presenting briefly the CLASS model and the two versions of COLISEUM in their actual state (one-stage and two-stage description) for further use in the present work. MPAS will also be presented in two derived versions from its original formulation (see [26]) named restricted and advanced. Of course, these versions of heuristic model (MPAS) are developed to improve the unique time constant model which has shown some weaknesses in coupling losses description of CICC especially in fast transient regimes (see section II.1.1.1).
Using both CLASS and COLISEUM, we will first demonstrate the capability of the one-stage COLISEUM to reproduce the magnetic behaviour (time constant and shielding coefficient) of a strand modelled by CLASS. By using COLISEUM to replicate the strand from CLASS, we will be able to integrate the strand in the modelling of a two-stage cable. This first step will allow us to show an original analytical model for a composite strand as if coupled in multiplets.
The consistency of the two-stage COLISEUM is also tested to consolidate the validity of its description. Range and value of coupling parameters (time constants and shielding coefficients) with respect to cabling parameters (twist pitch, conductances), absence of divergences in the coupling parameters, correspondence with the one-stage COLISEUM are points that will be checked in order to be perfectly confident in the COLISEUM for its iteration to a -stage description.
COLISEUM will be presented and discussed in its initial formulation and then simplified t its only relevant driving variables, already at the two-stage level, making it accessible to a fully explicit analytical approach and, further, prepare its extension to the description of cables with more than two stages. The analytical extension of the COLISEUM will be managed and presented as well as some applicative cases to start exploring its predictive capability. Several demonstrations strengthen our modelling and are also discussed in this thesis work.
Also, various collaborations have been pursued during this thesis work. The first one was with INFLPR (RO) and I. Tiseanu on the study of tomographic images of CICC. This study led us to the analysis of the contact distribution of the CICC (JT-60SA TF type). We also started to develop an approach to reconstruct the contact network in the section of cable from tomographic images. Collaboration with the University of Twente has also been continued in the confrontation of both our model (COLISEUM and JackPot). This collaboration aims at confronting both models (JackPot and COLISEUM) in order for COLISEUM to strengthen the validity of its description.
The hypothesis in MPAS will also be discussed in order to provide this model sufficient degrees of freedom regarding the variety of cabling patterns existing among all CICCs.
During this thesis, several experimental campaigns have been led using CICC samples of the same type as the JT-60SA TF production cable. These measurements are led to benchmark our models and quantify the effect of the void fraction on coupling losses. The choice of JT-60SA TF is related to the fact that we have the samples ready to be used due to another previous experiment [27]. The outputs of the experimental campaigns we led during this thesis are twofold: on one side, we have explored and quantified the effect of various void fractions (different compaction rate) on the CICC sample AC performances. On the other side, we have started to constitute an experimental database in order to confront COLISEUM (once upgraded to an -stage description) to the experimental data checking the validity of its predictions. A full description of the JOSEFA facility used to measure AC losses at CEA Cadarache will be given, with all the recent enhancements made on the facility to deliver a sinusoidal applied field. Several reminders will also be done regarding hysteresis and coupling losses modelling in order to describe the evaluation of AC losses from measured data.

Table of contents :

I. Introduction
I.1 Superconductivity
I.2 Superconductors (CICCs) and Fusion tokamaks
I.3 Coupling losses in superconductors: the case of the strand
I.4 Considerations on losses in superconducting magnets for fusion
I.5 Thesis content and objectives
II. Models
II.1 Presentation of the models
II.1.1 The MPAS model
II.1.2 Coupling Losses Algorithm for Superconducting Strands (CLASS)
II.1.3 One-stage COLISEUM
II.1.4 Two-stage COLISEUM
II.2 Applications of COLISEUM
II.2.1 Coupling between two stages: application of two-stage COLISEUM
II.2.2 Crosscheck of the one-stage COLISEUM and JackPot
II.2.3 Preliminary comments on the models
II.3 COLISEUM: Models overlap and cross-checks
II.3.1 CLASS vs one-stage COLISEUM
II.3.2 Reduction to a two time constant system for the two-stage COLISEUM
II.3.3 Two-stage COLISEUM vs one-stage COLISEUM
II.3.4 Global overlap: Application to a multiplets of Composite strands
III. Extension of the model
III.1 𝑛-stages COLISEUM iteration
III.1.1 First step
III.1.2 Numerical application
III.1.3 Iterative process
III.1.4 Numerical application
III.1.5 Discussion
IV. Experimental AC losses measurement
IV.1 The JOSEFA Facility
IV.1.1 Presentation of the facility
IV.1.2 Presentation of the method
IV.1.3 Short presentation of the test of the MAG42 conductor sample
IV.2 Hysteresis Losses
IV.3 Coupling Losses
IV.4 Experimental studies
IV.4.1 MAG 42 studies
IV.4.2 Complementary study: tomographic analysis
V Benchmark of the two models
V.1 COLISEUM application to experimental data
V.2 MPAS application to experimental data
V.2.1 Statistical study
V.2.2 Discussion and conclusion
V.3 Discussion between the two approaches
VI. Conclusions and Prospects
VI.1 Rationalization of MPAS
VI.2 Analytical extension of the model: 𝑛-stage COLISEUM
VI.3 Experimental coupling losses: JOSEFA facility
VI.4 Models benchmark on data
VI.5 Prospects
References

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