A flexible and differentiable coil proxy for stellarator equilibrium optimization

TL;DR

This paper tackles the coil-plasma balance in stellarator optimization by introducing a quasi-single-stage framework that couples equilibrium design with a differentiable coil complexity proxy, QUADCOIL. The key idea is to optimize the plasma boundary while implicitly solving a winding-surface subproblem to generate coil proxies, enabling gradients to flow without expanding the decision space. A differentiable winding-surface generator and adjoint differentiation through the coil subproblem are developed, with an augmented-Lagrangian approach used to solve the QCQP. Numerical results on MUSE-like equilibria demonstrate improved dipole-density metrics and maintained rotational-transform characteristics, though adjoint gradients encounter difficulties for problems with many inequality constraints. The work highlights practical pathways to scalable, differentiable quasi-single-stage optimization for stellarators and outlines future improvements to broaden applicability and robustness.

Abstract

Balancing plasma performance and coil cost is a significant challenge when designing a stellarator power plant. Most present stellarator designs are produced by two-stage optimization: the first for the equilibrium and the second for a coil design reproducing its magnetic configuration. It is challenging to find a compromise between plasma and coils with this approach. In recent years, single-stage approaches have gained popularity, which attempt to optimize both the plasma and coils simultaneously to improve the plasma-coil balance. In exchange, it can substantially increase the problem's dimensionality and introduce the ill-posedness of filamentary coil optimization to equilibrium optimization. This paper introduces a new ``quasi-single-stage'' method representing a flexible and differentiable coil proxy that directly predicts coil complexity during equilibrium optimization. The proxy is based on the adjoint differentiation of a winding surface coil subproblem. Our proxy can balance coil and plasma performance without introducing new degrees of freedom or ill-posedness. We present initial numerical results that demonstrate the proxy's effectiveness for single-stage optimization.

Figures (9)

• Figure 1: The values (left) and partial derivatives (right) of $f_K(x'_*)$ with respect to the plasma Fourier coefficient $R_{00}$ measured using the MUSE equilibrium. Note the close agreement between the finite difference and adjoint derivatives. The first row uses a fixed winding surface based on the location of PM holder in the MUSE device. The second row uses winding surfaces generated using Alg. \ref{['alg:ch:formulation:direct']} and demonstrates auto-differentiation through the winding surface generator.
• Figure 2: A comparison between the uniform-offset surface with our methods on NCSX, with $d_\text{cs}=0.5$m. Note that the new methods we present in this paper removes self-intersections, improves quadrature point uniformity, while preserving "bean-shaped" features on the inboard side.
• Figure 3: Limitation of Alg. \ref{['alg:ch:formulation:rule']}. This figure shows two self-intersecting planar curves produced by uniform offsets. The gray curve represents the plasma surface. The figure on the right shows an example with multiple self intersections. The black curve represents the winding surface. The red portion shows parts that Alg. \ref{['alg:ch:formulation:rule']} removes. Note that the routine does not work well for the complex offset curve shown on the right. Nevertheless, it works sufficiently well for the numerical studies in Section \ref{['sec: results']}.
• Figure 4: The values (left) and partial derivatives (right) of $f_c$ with respect to the plasma Fourier coefficient $R_{00}$ in problem \ref{['eq: quadcoil A']} (upper) and \ref{['eq: quadcoil B']}. The adjoint gradient is no longer accurate in \ref{['eq: quadcoil A']} because QUADCOIL solves \ref{['eq: quadcoil A']} as multi-constraint problem with $O(n_g'\times m_g')$ constraints. For a more detailed discussion, see Appendix \ref{['sec:appendix_limitations']}.
• Figure 5: Outer flux surface of new vacuum field \ref{['eq: quadcoil A']}