# Identification of important error fields in stellarators using Hessian matrix method

## TL;DR

This paper enhances a Hessian matrix method to quickly identify critical coil deviations affecting error fields in stellarators, aiming to optimize coil tolerances and reduce costs.

## Contribution

The paper introduces an improved analytical Hessian matrix approach to pinpoint sensitive coil deviations impacting stellarator magnetic properties.

## Key findings

- Successfully identified coil deviations affecting magnetic island size.
- Determined coil perturbations that deteriorate quasi-symmetry.
- Applied method to CFQS configuration with promising results.

## Abstract

Error fields are predominantly attributed to inevitable coil imperfections. Controlling error fields during coil fabrication and assembly is crucial for stellarators. Excessively tight coil tolerance increases time and cost, and, in part, led to the cancellation of NCSX and delay of W7-X. In this paper, we improve the recently proposed Hessian matrix method to rapidly identify important coil deviations. Two of the most common figures of merit, magnetic island size and quasi-symmetry, are analytically differentiated over coil parameters. By extracting the eigenvectors of the Hessian matrix, we can directly identify sensitive coil deviations in the order of the eigenvalues. The new method is applied to the upcoming CFQS configuration. Important perturbations that enlarge n/m=4/11 islands and deteriorate quasi-axisymmetry of the magnetic field are successfully determined. The results suggest each modular coil should have separate tolerance and some certain perturbation combinations will produce significant error fields. By relaxing unnecessary coil tolerance, this method will hopefully lead to a substantial reduction in time and cost.