# Modular Coil Design for the Ultra-low Aspect Ratio Quasi-axially Symmetric Stellarator MHH2 Page: 4 of 8

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essential harmonics of the magnetic field to yield needed

plasma properties.

Typically, we represent coils parametrically as two

dimensional Fourier series in terms of toroidal and poloidal

angles on a winding surface. The winding surface itself in turn

is represented as Fourier series in the toroidal and poloidal

angles. This double representation has the advantage in that it

allows one to choose the initial coil geometry in a more flexible

and intuitive way. It also allows a more efficient optimization

than by specifying directly the Cartesian coordinates of the

coils. The initial choice of the winding surface is important

since the optimization is highly non-linear and the

configuration space is complex with many valleys and hills.

The optimization is to find the "local" minimum of the penalty

function we specified. There is no unique solution in this multi-

dimensional optimization. An optimal solution is such that all

constraints are satisfied and the penalty function is minimized.

The initial choice of the winding surface is to make it

resemble the last closed magnetic surface of the fixed-boundary

plasma optimized with respect to the physics properties with an

offset large enough to meet the separation constraint between

the winding surface and coils and to set the outboard far

enough to minimize the ripple caused by the discrete coils. To

minimize the perturbation due to the discrete coils, we find that

the average minor radius of the outboard surface needs to be at

least twice as large as the average plasma minor radius.

For a DT reactor the tritium breeding and coil protection

from radiation damage typically require a blanket and shield to

have certain minimum thickness. We included the coil aspect

ratio R/Asi(C-P) as a constraint in the design optimization,

where R is the plasma major radius and Asi(C-P) is the

minimum separation between the coils and the LCMS. In

addition, we impose the constraints of coil separation ratio

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toroidal setosi*afapro o MHK 4R/A&i (C-C), where A&, (C-C) is the minimum separation

among coils, and the minimum radius of curvature in the coil

optimization. We allow coils to have different currents, but

they have to maintain stellarator symmetry. Typically we

search solutions for which the coil aspect ratio is <6, coil

separation ratio < 12, and major radius to minimum radius of

curvature <10. During the last stage of optimization in which

free boundary equilibrium is solved, we vary the coil geometry

as well as coil currents to minimize the non-axisymmetric

"noise" in the magnetic spectrum, the effective ripples and the

collisionless orbits of escaping a particles.

Typically, state variables consist of ~200 Fourier

coefficients describing the coil geometry and location, and the

penalty function consists of ~3000 physics and engineering

constraints imposing acceptance criteria for QA and coil

properties. The search of optimum in the design space is

carried out by the Levenberg-Marquardt non-linear

minimization technique [5]. VMEC [6] is used for the

calculation of plasma equilibrium and NEO [7] and ORBIT3D

[8] are used for the evaluation of effective helical ripples and

the loss of a particles, respectively.

III. A SIXTEEN MODULAR COIL DESIGN FOR MHH2

The configuration used as the basis of the coil design

discussed here is called MIHH2-K14 whose physics

characteristics are detailed in [1]. Fig. 1 shows the last closed

magnetic surface for which the coil design is intended to target.

A typical design using only modular coils is illustrated in Fig. 2

which was obtained by the three steps of optimization with the

increasing sophistication and complexity outlined in Section II.

There are four distinct types of coils in each of the half periods

with the coil aspect ratio 5.5 and coil separation ratio 10. The

ratio of the plasma major radius to the minimum radius of

curvature of these coils is about 13. It is seen in Fig. 2 that the

coils are reasonably smooth, but in the inboard region near the

crescent-shaped plasma at the beginning of a field period they

are twisted to provide the push along the ridges.

One of the most important coil design parameters is the

ratio of the maximum magnetic field in the coils to the field on

the magnetic axis, B. /Bo. The fusion power density is

Fig. 2. Top and perspective views of a modular coil design with coil aspect

ratio 5.5. The LCMS of the plasma is also shown. There are four distinctive

types of coils for a total of 16 coils in two field periods.

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Ku LP, the ARIES-CS Team. Modular Coil Design for the Ultra-low Aspect Ratio Quasi-axially Symmetric Stellarator MHH2, report, September 27, 2005; Princeton, New Jersey. (https://digital.library.unt.edu/ark:/67531/metadc884844/m1/4/: accessed March 20, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.