Stellarator Coils Research Articles

Optimization of finite-sized modular coils for advanced stellarators

To date, almost all coil-design codes, e.g. NESCOIL, COILOPT, FOCUS codes, etc, have been primarily attributed to the optimization of filament coils for stellarators. However, evolving to a practical/finite-sized coil from a filament coil, the finite-size effect of coils significantly constrains the fabrication tolerances of a coil system. This paper presents a novel approach that emphasizes the optimization of practical modular coils to reduce sensitivity to fabrication tolerances and to achieve the expected magnetic configurations precisely. A new evaluation parameter, surface twist, is defined in this paper and applied to the optimization sequence in addition to the practical coil line torsion and curvature. The approach has been applied to the framework of the filament coil scheme in the Chinese first quasi-axisymmetric stellarator. This practical coil system without surface twists has been accomplished. Compared to the original finite-sized coil design, the new result is a more considerable simplification of coil shapes, such that in a certain direction view each finite-sized coil becomes a planar-like one. Moreover, this method can also be implemented for the estimation of stochastic deviations of practical coils during the fabrication and assembly of the coil system.

Topology optimization of permanent magnets for stellarators

We introduce a topology optimization method to design permanent magnets for advanced stellarators. Recent research shows that permanent magnets have great potential to simplify stellarator coils. We adopt state-of-the-art numerical techniques to determine the presence of magnets in the entire design space. The FAMUS code is developed and it can design engineering-feasible permanent magnets for general stellarators satisfying the constraints of the maximum material magnetization and explicit forbidden regions. FAMUS has been successfully verified against the previously proposed linear method. Three different permanent magnet designs together with planar TF coils for a half-Tesla NCSX configuration have been obtained for demonstrations. The designs have good accuracy in generating the desired equilibrium and offer considerably large plasma access on the outboard side. The results show that FAMUS is a flexible, advanced numerical tool for future permanent magnet stellarator designs.

Optimization of finite-build stellarator coils

Finding coil sets with desirable physics and engineering properties is a crucial step in the design of modern stellarator devices. Existing stellarator coil optimization codes ultimately produce zero-thickness filament coils. However, stellarator coils have finite depth and thickness, which can make the single-filament model a poor approximation, particularly when coil build dimensions are relatively large compared to the coil–plasma distance. In this paper, we present a new method for designing coils with finite builds and present a mechanism to optimize the orientation of the winding pack. We approximate finite-build coils with a multi-filament model. A numerical implementation has been developed, and applications to the Helically Symmetric eXperiment stellarator and a new UW-Madison quasihelically symmetric configuration are shown.

Designing stellarators using perpendicular permanent magnets

We have developed a fast method to design perpendicular permanent magnets for simplifying stellarator coils based on existing codes. Coil complexity is one of the main challenges for stellarators. To date, only electromagnetic coils have been used to generate 3D fields for stellarators. Permanent magnets provide an alternative way to produce the desired magnetic field for optimized stellarators. In this paper, we revisit the concept of representing surface current using magnetic dipoles and carry out numerical validations. A surface magnetization is proven to be equivalent to the surface current that can be linearly solved by existing coil design codes. An incremental multi-layer method has been developed to obtain a practical solution that is attainable with present permanent magnets. With this method, we can reproduce a half-Tesla NCSX configuration using specially designed neodymium magnets together with simple planar coils. It shows that stellarator coils could be substantially simplified by adopting permanent magnets.

Improved performance of stellarator coil design optimization

Following up on earlier work which demonstrated an improved numerical stellarator coil design optimization performance by the use of stochastic optimization (Lobsien et al., Nucl. Fusion, vol. 58 (10), 2018, 106013), it is demonstrated here that significant further improvements can be made – lower field errors and improved robustness – for a Wendelstein 7-X test case. This is done by increasing the sample size and applying fully three-dimensional perturbations, but most importantly, by changing the design sequence in which the optimization targets are applied: optimization for field error is conducted first, with coil shape penalties only added to the objective function at a later step in the design process. A robust, feasible coil configuration with a local maximum field error of 3.66 % and an average field error of 0.95 % is achieved here, as compared to a maximum local field error of 6.08 % and average field error of 1.56 % found in our earlier work. These new results are compared to those found without stochastic optimization using the FOCUS and ONSET suites. The relationship between local minima in the optimization space and coil shape penalties is also discussed.

Physics analysis of results of stochastic and classic stellarator coil optimization

In this paper we follow up on the results from our previous publication (Lobsien et al 2018 Nucl. Fusion 58 106013), where it was found that stellarator coil design optimization can be substantially improved by using a stochastic optimisation approach. In that paper performance was quantified by lower (better) and more narrow (more robust) distributions of the penalty functions at the end of the optimisations. Here, we evaluate and compare the various coil sets of the previous paper but seek a verification and deeper understanding of the physics performance by replacing the relatively simple penalty function estimate with more accurate ones from state-of-the-art calculations of MHD stability, neoclassical transport in the -regime, fast particle confinement, and gyrokinetic behavior. The investigation shows that stochastic stellarator coil optimization generally outperforms the earlier non-stochastic stellarator optimization, also when using these more accurate metrics, generally confirming and quantifying the better performance. We do find some discrepancies, indicating that the penalty function does not represent a physics performance optimum perfectly. For example, as pointed out by others before us, the depth of the magnetic well is not a sufficiently good proxy for MHD stability, and the neoclassical transport can be significantly reduced in configurations that have relatively high field errors and therefore high penalty values. Thus, our work points to areas where better physics model inside the optimisation loop are needed, than what is currently represented by our penalty function.

Tuning of the rotational transform in Wendelstein 7-X**Notice: This manuscript has been authored by Princeton University under Contract Number DE-AC02-09CH11466 with the U.S. Department of Energy. The publisher, by accepting the article for publication acknowledges, that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others

The control of rotational transform in Wendelstein 7-X (W7-X) is key to both the island divertor operation and safety of plasma facing components. The island divertor concept in W7-X relies on an edge flux surface with rotational transform of resonating with an intrinsic n/m = 5/5 resonance to form a five lobed island chain. This island chain intersects with divertor plates to give rise to the island divertor. Changes in the relative position of the rational surface and the divertor plates can result in changes in divertor performance, thus the control of the rotational transform is essential to operation of the W7-X device. During the first divertor campaign electromagnetic loads resulted in elastic deformations of the shaped modular stellarator coils. Such deformations made these coils more planar, reducing the vacuum rotational transform, subsequently shifting the resonance outward. Unintended plasma wall interactions provided the first clear evidence of this effect during plasma operation. Flux surface measurements were utilized to estimate the correct level of current in the planar coils for correction of , and found to be around ∼. Scans of the planar coil current for iota correction were performed during plasma operation. These measurements suggest planar coil currents between −250 and ∼ would place the strike lines at the designed distance from the pumping gaps. Divertor Langmuir and upstream probe measurements confirm these estimates along with measurements of divertor neutral gas pressure.

Sawtooth oscillation behavior with varying amounts of applied stellarator rotational transform

Tokamak-like sawtooth oscillations are observed in the Compact Toroidal Hybrid (CTH), a current-carrying stellarator. CTH has the unique ability to change the amount of the applied vacuum rotational transform from external stellarator coils relative to the rotational transform generated by the internal plasma current to investigate the effects of strong three-dimensional magnetic shaping on sawtooth behavior. The observed sawteeth in CTH, for plasmas with monotonically decreasing rotational transform profiles dominated by the plasma current, have characteristics of those observed on tokamaks including (1) a central emissivity rise and then a sudden crash with a well-defined inversion radius, (2) the presence of an m = 1 emissivity fluctuation, and (3) the normalized inversion surface radius scales with the total edge rotational transform. We explore the properties of an ensemble of discharges in CTH in which the fractional rotational transform, defined as the vacuum rotational transform divided by the total rotational transform, is systematically varied from 0.04 to 0.42 to observe changes in sawtooth oscillation dynamics. Over this range of the fractional rotational transform, the measured sawtooth period decreased by a factor of two. At a high fractional rotational transform, the sawtooth amplitude is observed to consist of only low-amplitude oscillations while the measured crash time of the sawtooth oscillation does not appear to have a strong dependence on the amount of the fractional transform applied. Experimental results indicate that the low-amplitude sawteeth are accompanied by a decrease in the sawtooth period and predominantly correlated with the mean elongation (due to the increasing fractional rotational transform) of the non-axisymmetric plasmas within CTH rather than other global equilibrium parameters.

Differentiating the shape of stellarator coils with respect to the plasma boundary

The task of designing the geometry of a set of current-carrying coils that produce the magnetic field required to confine a given plasma equilibrium in stellarators is expressed as a minimization principle, namely that the coils minimize a suitably defined error expressed as a surface integral, which is recognized as the quadratic-flux. A penalty on the coil length is included to avoid pathological solutions. A simple expression for how the quadratic-flux and coil length vary as the coil geometry varies is derived, and an expression describing how this varies with variations in the surface geometry is derived. These expressions allow efficient coil-design algorithms to be implemented, and also enable efficient algorithms for varying the shape of the plasma surface in order to simplify the coil geometry, and a numerical illustration of this is given.

Stellarator coil optimization towards higher engineering tolerances

Recently designed optimized stellarator experiments have suffered from very tight construction tolerances, but some level of deviation of the coil system is unavoidable during fabrication of the coils and assembly of the coil system. In this paper, we present a new approach that incorporates reduced sensitivity to construction tolerances of the coil system into the optimization sequence. The approach was tested within the framework of the existing coil optimization scheme for Wendelstein 7-X. The results are compared with those of a coil set obtained by the original optimization. The result is a more optimal coil system with less stringent tolerances, such that small deviations cause reduced deterioration of the properties important for fusion performance.

An adjoint method for gradient-based optimization of stellarator coil shapes

We present a method for stellarator coil design via gradient-based optimization of the coil-winding surface. The REGCOIL (Landreman 2017 Nucl. Fusion 57 046003) approach is used to obtain the coil shapes on the winding surface using a continuous current potential. We apply the adjoint method to calculate derivatives of the objective function, allowing for efficient computation of analytic gradients while eliminating the numerical noise of approximate derivatives. We are able to improve engineering properties of the coils by targeting the root-mean-squared current density in the objective function. We obtain winding surfaces for W7-X and HSX which simultaneously decrease the normal magnetic field on the plasma surface and increase the surface-averaged distance between the coils and the plasma in comparison with the actual winding surfaces. The coils computed on the optimized surfaces feature a smaller toroidal extent and curvature and increased inter-coil spacing. A technique for computation of the local sensitivity of figures of merit to normal displacements of the winding surface is presented, with potential applications for understanding engineering tolerances.

Designing stellarator coils by a modified Newton method using FOCUS

To find the optimal coils for stellarators, nonlinear optimization algorithms are applied in existing coil design codes. However, none of these codes have used the information from the second-order derivatives. In this paper, we present a modified Newton method in the recently developed code FOCUS. The Hessian matrix is calculated with analytically derived equations. Its inverse is approximated by a modified Cholesky factorization and applied in the iterative scheme of a classical Newton method. Using this method, FOCUS is able to recover the W7-X modular coils starting from a simple initial guess. Results demonstrate significant advantages.

Hessian matrix approach for determining error field sensitivity to coil deviations

The presence of error fields has been shown to degrade plasma confinement and drive instabilities. Error fields can arise from many sources, but are predominantly attributed to deviations in the coil geometry. In this paper, we introduce a Hessian matrix approach for determining error field sensitivity to coil deviations. A primary cost function used for designing stellarator coils, the surface integral of normalized normal field errors, was adopted to evaluate the deviation of the generated magnetic field from the desired magnetic field. The FOCUS code (Zhu et al 2018 Nucl. Fusion 58 016008) is utilized to provide fast and accurate calculations of the Hessian. The sensitivities of error fields to coil displacements are then determined by the eigenvalues of the Hessian matrix. A proof-of-principle example is given on a CNT-like configuration. We anticipate that this new method could provide information to avoid dominant coil misalignments and simplify coil designs for stellarators.

New method to design stellarator coils without the winding surface

Finding an easy-to-build coils set has been a critical issue for stellarator design for decades. Conventional approaches assume a toroidal ‘winding’ surface, but a poorly chosen winding surface can unnecessarily constrain the coil optimization algorithm, This article presents a new method to design coils for stellarators. Each discrete coil is represented as an arbitrary, closed, one-dimensional curve embedded in three-dimensional space. A target function to be minimized that includes both physical requirements and engineering constraints is constructed. The derivatives of the target function with respect to the parameters describing the coil geometries and currents are calculated analytically. A numerical code, named flexible optimized coils using space curves (FOCUS), has been developed. Applications to a simple stellarator configuration, W7-X and LHD vacuum fields are presented.

An improved current potential method for fast computation of stellarator coil shapes

Several fast methods for computing stellarator coil shapes are compared, including the classical NESCOIL procedure (Merkel 1987 Nucl. Fusion 27 867), its generalization using truncated singular value decomposition, and a Tikhonov regularization approach we call REGCOIL in which the squared current density is included in the objective function. Considering W7-X and NCSX geometries, and for any desired level of regularization, we find the REGCOIL approach simultaneously achieves lower surface-averaged and maximum values of both current density (on the coil winding surface) and normal magnetic field (on the desired plasma surface). This approach therefore can simultaneously improve the free-boundary reconstruction of the target plasma shape while substantially increasing the minimum distances between coils, preventing collisions between coils while improving access for ports and maintenance. The REGCOIL method also allows finer control over the level of regularization, it preserves convexity to ensure the local optimum found is the global optimum, and it eliminates two pathologies of NESCOIL: the resulting coil shapes become independent of the arbitrary choice of angles used to parameterize the coil surface, and the resulting coil shapes converge rather than diverge as Fourier resolution is increased. We therefore contend that REGCOIL should be used instead of NESCOIL for applications in which a fast and robust method for coil calculation is needed, such as when targeting coil complexity in fixed-boundary plasma optimization, or for scoping new stellarator geometries.

Stabilizing turbulence in fusion stellarators

In the earliest days of the Cold War, physicists on both sides of the Iron Curtain raced to harness energy from nuclear fusion, which could, in principle, provide nearly limitless electricity. Innovative devices with names like pinch machines, levitrons, and superstators flourished, and for most of the 1950s it was unclear which would eventually prove most promising. But in 1968, at the Third International Conference on Plasma Physics and Controlled Nuclear Fusion Research, Soviet scientists announced that their tokamak—which confined a thermonuclear plasma within a donut-shaped magnetic field—had achieved temperatures 10 times higher than any other experiment (1). The interior of the Wendelstein 7-X stellarator. Visible are the plasma vessel, one of the stellarator coils, a planar coil, part of the support structure, and the cryostat together with a lot of cooling pipes and power supply lines. Photo by Wolfgang Filser and image courtesy of Max Planck Institute for Plasma Physics. Although the figure was initially met with shock and skepticism, it was quickly confirmed by British physicists (2). The tokamak has remained the dominant research avenue for pursuing fusion until today, with machines like the multibillion-euro International Thermonuclear Experimental Reactor (ITER), the world’s largest confined plasma physics experiment, currently under construction in France. But another invention, a close cousin of the tokamak called a stellarator, has lately begun to receive renewed attention. Despite their complexity, stellarators offer ways around many of the instabilities that plague tokamaks. “In simple words, the stellarator is just tamer,” says plasma physicist Thomas Klinger, codirector of the Wendelstein 7-X (W7-X) stellarator experiment in Greifswald, Germany. “It’s difficult to build, but easier to run.” Researchers now think the advantages of stellarators might extend to one more area that’s crucial to any real-world fusion reactor: the management of turbulence. Scientists have put enormous effort into …

Dimensional accuracy of additively manufactured structures for modular coil windings of stellarators

The paper studies the dimensional accuracy of additively manufactured (AM) coil winding structures for small experimental stellarators fabricated under industrial non-laboratory conditions. Insufficient accuracy is one of the main issues hindering the use of additive manufacturing (AM) for coil windings structures (coil casings and coil forms or frames) for stellarators.The dimensional accuracy of one complex modular coil frame and four planar coil casings is studied for different AM techniques and materials, in particular Selective Laser Sintering (SLS) in polyamide, Stereolithography (SLA) in resin, Fused Deposition Modelling (FDM) in ABS and PolyJet in resin. The measurements are performed by a Mitutoyo Coordinate Measuring Machine. The AM parts are hollow for subsequent internal casting with (fibre-reinforced) resin for strength and performance enhancement, method named 3Dformwork. The paper reports the features of the sample parts, the metrology methodology utilised, the performed measurements and the resulting dimensional deviations for each AM technique.Professional AM in FDM showed deviation between nominal dimensions and measurements of ±0.1% (one sigma, 68% of measurements) and PolyJet ±0.15%. Personal web-based AM in SLA exhibited deviation ±0.3% and SLS in polyamide ±0.4% (one sigma). The PolyJet part showed dimensional instability under harsh environment and would require immediate 3Dformwork. The FDM part presented the lower cost for the particular case study among the professional AM. Thus, high quality PolyJet and FDM additive manufacturing in plastics are at the verge of achieving the requirement of 0.1% minimum accuracy for stellarator coil windings. Therefore, AM may contribute to the fabrication of accurate winding structures for stellarators and other accurate components in fusion.

Low edge safety factor operation and passive disruption avoidance in current carrying plasmas by the addition of stellarator rotational transform

Low edge safety factor operation at a value less than two (q(a)=1/ι̷tot(a)<2) is routine on the Compact Toroidal Hybrid device with the addition of sufficient external rotational transform. Presently, the operational space of this current carrying stellarator extends down to q(a)=1.2 without significant n = 1 kink mode activity after the initial plasma current rise phase of the discharge. The disruption dynamics of these low edge safety factor plasmas depend upon the fraction of helical field rotational transform from external stellarator coils to that generated by the plasma current. We observe that with approximately 10% of the total rotational transform supplied by the stellarator coils, low edge q disruptions are passively suppressed and avoided even though q(a) < 2. When the plasma does disrupt, the instability precursors measured and implicated as the cause are internal tearing modes with poloidal, m, and toroidal, n, helical mode numbers of m/n=3/2 and 4/3 observed on external magnetic sensors and m/n=1/1 activity observed on core soft x-ray emissivity measurements. Even though the edge safety factor passes through and becomes much less than q(a) < 2, external n = 1 kink mode activity does not appear to play a significant role in the disruption phenomenology observed.

Suppression of vertical instability in elongated current-carrying plasmas by applying stellarator rotational transform

The passive stability of vertically elongated current-carrying toroidal plasmas has been investigated in the Compact Toroidal Hybrid, a stellarator/tokamak hybrid device. In this experiment, the fractional transform f, defined as the ratio of the imposed external rotational transform from stellarator coils to the total rotational transform, was varied from 0.04 to 0.50, and the elongation κ was varied from 1.4 to 2.2. Plasmas that were vertically unstable were evidenced by motion of the plasma in the vertical direction. Vertical drifts are measured with a set of poloidal field pickup coils. A three chord horizontally viewing interferometer and a soft X-ray diode array confirmed the drifts. Plasmas with low fractional transform and high elongation are the most susceptible to vertical instability, consistent with analytic predictions that the vertical mode in elongated plasmas can be stabilized by the poloidal field of a relatively weak stellarator equilibrium.

Major issues in the design and construction of the stellarator of Costa Rica: SCR-1

This paper aims at briefly describing the design and construction issues of the stellarator of Costa Rica 1 (SCR-1). The SCR-1 is a small modular Stellarator for magnetic confinement of plasma developed by the Plasma Physics Group of the Instituto Tecnológico de Costa Rica (ITCR). The SCR-1 is based on the small Spanish Stellarator UST_1 (Ultra Small Torus 1), created by Eng. Vicente Queral. These mains issues consist of the size of the Stellarator, closeness between coils, coupling of ECH to the vacuum chamber and the device for support. The size has become a problem because the vacuum chamber does not allow a lot of space to attach diagnosis devices, the heating system, the vacuum system and the very same support of the chamber. As a result of this lack of space, the Stellarator's coils are placed very close to each other; this means that two of the coils around of the vacuum chamber clash and cannot be placed as designed. The issue regarding the coupling of the ECH (electron cyclotron radio-frequency) to the vacuum chamber comprises the fact that the wave guide with rectangular shape does not match the CF port with circular shape on the vacuum chamber. In addition, the device for supporting the Stellarator has presented a challenge because of its size and the placement of the coils; in other words, there is not enough space between the ports and coils in the Stellarator to place appropriately the device for support.