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Quasi-isodynamic stellarator with one field period and eight coils per half-field period. Top: standard stellarator optimization approach where stage 1 and stage 2 optimizations were performed sequentially (left) and the single-stage optimization result (right). Lower Left: Superposition of magnetic surfaces at constant cylindrical toroidal angle ϕ of the QFM and the final single-stage equilibrium, as well as the Poincaré plot resulting from tracing magnetic field lines in the obtained coils. Middle Right: Contours of constant magnetic field strength on a surface at s = 0.495 in Boozer coordinates (θ, φ). Bottom Right: profile of rotational transform ι.
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We introduce a novel approach for the simultaneous optimization of plasma and coil engineering objectives in a fixed-boundary equilibrium that is computationally efficient and applicable to a broad range of vacuum and finite plasma pressure scenarios. Our approach treats plasma boundary and coil shapes as independently optimized variables, penalizi...
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... achieve this goal, we used eight independent coils with a total of N F = 16 Fourier modes, each coil with a maximum length of L max = 5.5, a maximum value of κ max = 10.0, and a minimum distance between coils of 0.12. The results of our optimization are displayed in figure 7 (top), where we show the configuration obtained from stage 1 and stage 2 independent optimizations. The residuals of the quasi-isodynamic objective function f QI in stage 1 were 2.4 × 10 −3 , while the residuals of the squared flux f QF in stage 2 were 2.9 × 10 −6 . ...
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... residuals of the quasi-isodynamic objective function f QI in stage 1 were 2.4 × 10 −3 , while the residuals of the squared flux f QF in stage 2 were 2.9 × 10 −6 . In the single-stage optimization shown in figure 7 (top right), the residuals of the quasi-isodynamic objective function f QI remained mostly unchanged at 2.9 × 10 −3 , while the squared flux improved to 4.4 × 10 −7 . We then run VMEC in fixedboundary mode using the QFM surfaces obtained from the stage 1 and stage 2 independent optimizations and the singlestage approach. ...
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... the traditional two-stage approach did not produce a usable result, whereas the single-stage method did. The figures in figure 7 provide further insight into the effectiveness of our optimization. On the lower left, we see that the reduction in the squared flux results in an agreement between the singlestage fixed boundary equilibrium, a fixed boundary equilibrium based on the QFM surface, and the Poincaré plots. ...
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Citations
... The first one starts from the plasma equilibrium and the parameters to describe fixed-boundary equilibria are varied. During the optimization, figures of merit to assess coil complexities are added [7][8][9]. The other type starts from coils. ...
Advanced stellarators are typically optimized in two stages. The plasma equilibrium is optimized first, followed by the design of coils/permanent magnets. However, the coils/permanent magnets in the second stage may become too complex to achieve the desired equilibrium. To address this problem, a quasi-single-stage optimization method has been proposed. In this paper, we introduce this method for designing permanent magnet (PM) stellarators. The new approach combines straightforward PM metrics to penalize the maximum required PM thickness and the mismatch between the fixed-boundary equilibrium and the free-boundary one, along with typical physical targets. Since the degrees of freedom of the PMs are not included and directly used to minimize the objective function in this method, so we call it ‘quasi-single-stage’ optimization. We apply this quasi-single-stage optimization method to find a new quasi-axisymmetric PM design. The new design starts from MUSE, which was initially designed using a two-stage optimization approach. The resulting design, MUSE++, exhibits an order of magnitude lower quasi-symmetric error and a one-order reduction in normal field error. We show that MUSE++ has approximately 30% fewer magnets compared to a proxy model ‘MUSE-0’ that uses the same FAMUS optimization without the benefit of a single-stage equilibrium optimization. These results demonstrate that the new single-stage optimization method can concurrently improve plasma properties and simplify permanent magnet complexity.
... Using such an approach, we note that Ref. [22] was able to find flexible stellarator equilibria that support multiple configurations by carefully placing planar coils and increasing the number of terms in the objective function. The methods employed here are the single-stage optimization method using fixed boundary equilibria introduced in Ref. [23], the direct coil optimiza-tion method introduced in Ref. [24], and a new guided coil method. ...
Single-stage optimization, also known as combined plasma-coil algorithms or direct coil optimization, has recently emerged as a possible method to expedite the design of stellarator devices by including, in a single step, confinement, stability, and engineering constraints. In this work, we show how such frameworks allow us to find new designs in a streamlined manner, yielding a broad range of new configurations. Examples are shown for stellarators with a small number of coils and quasisymmetric stellarators with only one to three coils per half field period, with external trim coils, helical coils, and a single set of coils generating both a quasi-axisymmetric and a quasi-helical equilibrium.
... However, recent work has shown that these difficulties can be surpassed. Improved coil optimisation techniques have shown that simple coil systems can be achieved (Wechsung et al. 2022;Jorge et al. 2023b;Kappel et al. 2023). Experimental results in W7-X have shown reduced levels of neoclassical transport (Beidler et al. 2021). ...
In this work, we propose a method of optimising stellarator devices to favour the presence of an electron root solution of the radial electric field. Such a solution can help avoid heavy impurity accumulation, improve neoclassical thermal ion confinement and helium ash exhaust and possibly reduce turbulence. This study shows that an optimisation for such a root is possible in quasi-isodynamic stellarators. Examples are shown for both vacuum and finite plasma pressure configurations.
... A partial solution to this issue may be to perform single-stage optimization, in which both the plasma shape and coil shapes are optimized together. [8][9][10][11] However, single-stage optimization can be computationally challenging. Single-stage optimization also strongly benefits from a good initial condition. ...
The separation between the last closed flux surface of a plasma and the external coils that magnetically confine it
is a limiting factor in the construction of fusion-capable plasma devices. This plasma-coil separation must be large
enough so that components such as a breeding blanket and neutron shielding can fit between the plasma and the coils.
Plasma-coil separation affects reactor size, engineering complexity, and particle loss due to field ripple. For some
plasmas it can be difficult to produce the desired flux surface shaping with distant coils, and for other plasmas it
is infeasible altogether. Here, we seek to understand the underlying physics that limits plasma-coil separation and
explain why some configurations require close external coils. In this paper, we explore the hypothesis that the limiting
plasma-coil separation is set by the shortest scale length of the magnetic field as expressed by the ∇B tensor. We tested
this hypothesis on a database of > 40 stellarator and tokamak configurations. Within this database, the coil-to-plasma
distance compared to the minor radius varies by over an order of magnitude. The magnetic scale length is well correlated
to the coil-to-plasma distance of actual coil designs generated using the REGCOIL method [Landreman, Nucl. Fusion 57,
046003 (2017)]. Additionally, this correlation reveals a general trend that larger plasma-coil separation is possible with a
small number of field periods.
... [24,25]), and (∂ α J) B = 0 guarantees excellent neoclassical transport properties. For this reason, historically (through reduction of the effective ripple ε ef f [26]), but also in recent works (by explicitly minimizing the distance to perfect quasisymmetry [17], to exact quasiisodynamicity [20,27], or to other types of perfect omnigeneity [28]) Orbits with ∂ s J ≈ 0, termed "superbananas", have an orbit-averaged drift that is directed in the radial direction and that leads to large radial fluxes. Generally speaking, ...
... Expressions (26) and (27) are equations, since L b 11 and δ b ij generally depend on E r . Neverthless, for most fusion-relevant scenarios (see e.g. ...
... When this is the case, flat ion temperature will cause E r to be close to zero, according to equation (26). This will make L D 11 large, which will keep the ion temperature gradient small, according to equation (27). In summary, a small E r and hence a small (∂ s J) Φ , can be very deleterious for the performance of a reactor, which would need more heating in order to access the desired temperatures close to the axis. ...
Stellarator magnetic configurations need to be optimized in order to meet all the required properties of a fusion reactor. In this work, it is shown that a flat-mirror quasi-isodynamic configuration (i.e. a quasi-isodynamic configuration with sufficiently small radial variation of the mirror term) can achieve small radial transport of energy and good confinement of bulk and fast ions even if it is not very close to perfect omnigeneity, and for a wide range of plasma scenarios, including low β and small radial electric field. This opens the door to constructing better stellarator reactors. On the one hand, they would be easier to design, as they would be robust against error fields. On the other hand, they would be easier to operate since, both during startup and steady-state operation, they would require less auxiliary power, and the heat loads on plasma-facing components caused by fast ion losses would be reduced to acceptable levels.
... Including both MHD and gyrokinetic stability metrics in the target function will be necessary when designing a viable fusion reactor. Finding a set of coils with reasonable properties would also be a valuable project, as was done in Jorge et al. (2023). The optimization approach in this paper also seems to be sensitive to initial conditions: we were unable to find any good configurations n fp > 1, except for those initialized from a solution found using NAE. ...
We present a novel method for numerically finding quasi-isodynamic stellarator magnetic fields with excellent fast-particle confinement and extremely small neoclassical transport. The method works particularly well in configurations with only one field period. We examine the properties of these newfound quasi-isodynamic configurations, including their transport coefficients, particle confinement and available energy for trapped-electron-instability-driven turbulence, as well as the degree to which they change when a finite pressure profile is added. We finally discuss the differences between the magnetic axes of the optimized solutions and their respective initial conditions, and conclude with the prospects for future quasi-isodynamic optimization.
... Part of the loss of quasi-symmetry is inherent to the two-stage approach to coil design: the coils are optimized to reproduce a target magnetic field, but not to generate a quasi-symmetric field. Single-stage optimization approaches (Giuliani et al. 2022a,b ;Jorge et al. 2023), however, directly optimize coils for quasi-symmetry, ensuring that they achieve optimal quasi-symmetry levels over the space of coils satisfying the coil length constraint. Single-stage approaches would hence improve the unfortunate trade-off between coil length and quasi-symmetry. ...
In designing stellarators, any design decision ultimately comes with a trade-off. Improvements in particle confinement, for instance, may increase the burden on engineers to build more complex coils, and the tightening of financial constraints may simplify the design and worsen some aspects of transport. Understanding trade-offs in stellarator designs is critical in designing high performance devices that satisfy the multitude of physical, engineering and financial criteria. In this study, we show how multi-objective optimization (MOO) can be used to investigate trade-offs and develop insight into the role of design parameters. We discuss the basics of MOO, as well as practical solution methods for solving MOO problems. We apply these methods to bring insight into the selection of two common design parameters: the aspect ratio of an ideal magnetohydrodynamic equilibrium and the total length of the electromagnetic coils.
... [24,25]), and (∂ α J) B = 0 guarantees excellent neoclassical transport properties. For this reason, historically (through reduction of the effective ripple ε ef f [26]), but also in recent works (by explicitly minimizing the distance to perfect quasisymmetry [17], to exact quasiisodynamicity [20,27], or to other types of perfect omnigeneity [28]) Orbits with ∂ s J ≈ 0, termed "superbananas", have an orbit-averaged drift that is directed in the radial direction and that leads to large radial fluxes. Generally speaking, ...
... When this is the case, flat ion temperature will cause E r to be close to zero, according to equation (26). This will make L D 11 large, which will keep the ion temperature gradient small, according to equation (27). In summary, a small E r and hence a small (∂ s J) Φ , can be very deleterious for the performance of a reactor, which would need more heating in order to access the desired temperatures close to the axis. ...
Stellarator magnetic configurations need to be optimized in order to meet all the required properties of a fusion reactor. In this work, it is shown that a flat-mirror quasi-isodynamic configuration (i.e. a quasi-isodynamic configuration with sufficiently small radial variation of the mirror term) can achieve small radial transport of energy and good confinement of bulk and fast ions even if it is not very close to perfect omnigeneity, and for a wide range of plasma scenarios, including low $\beta$ and small radial electric field. This opens the door to constructing better stellarator reactors. On the one hand, they would be easier to design, as they would be robust against error fields. On the other hand, they would be easier to operate since, both during startup and steady-state operation, they would require less auxiliary power, and the damage to plasma-facing components caused by fast ion losses would be reduced to acceptable levels.
Filament-based coil optimizations are performed for several quasi-helical stellarator configurations, beginning with the one from Landreman & Paul (Phys. Rev. Lett., vol. 128, 2022, 035001), demonstrating that precise quasi-helical symmetry can be achieved with realistic coils. Several constraints are placed on the shape and spacing of the coils, such as low curvature and sufficient plasma–coil distance for neutron shielding. The coils resulting from this optimization have a maximum curvature 0.8 times that of the coils of the Helically Symmetric eXperiment (HSX) and a mean squared curvature 0.4 times that of the HSX coils when scaled to the same plasma minor radius. When scaled up to reactor size and magnetic field strength, no fast particle losses were found in the free-boundary configuration when simulating 5000 alpha particles launched at $3.5\,\mathrm {MeV}$ on the flux surface with a normalized toroidal flux of $s=0.5$. An analysis of the tolerance of the coils to manufacturing errors is performed using a Gaussian process model, and the coils are found to maintain low particle losses for smooth, large-scale errors up to amplitudes of approximately $0.15\,\mathrm {m}$. Another coil optimization is performed for the Landreman–Paul configuration with the additional constraint that the coils are purely planar. Visual inspection of the Poincaré plot of the resulting magnetic field-lines reveal that the planar modular coils alone do a poor job of reproducing the target equilibrium. Additional non-planar coil optimizations are performed for the quasi-helical configuration with $5\,\%$ volume-averaged plasma beta from Landreman et al. (Phys. Plasma, vol. 29, issue 8, 2022, 082501), and a similar configuration also optimized to satisfy the Mercier criterion. The finite beta configurations had larger fast-particle losses, with the free-boundary Mercier-optimized configuration performing the worst, losing approximately $5.5\,\%$ of alpha particles launched at $s=0.5$.
