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
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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R/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 . VMEC  is used for the
calculation of plasma equilibrium and NEO  and ORBIT3D
 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 . 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.