Metric Definitions

The following metrics are used to evaluate coil optimization submissions. Notation: \(C_i\) coil curve, \(S\) plasma surface, \(\mathbf{r}_i\) point on coil, \(\kappa_i\) curvature, \(N\) number of coils, \(\mathbf{B}\) magnetic field, \(\mathbf{n}\) surface normal.

Definitions

\(\bar{B}_n\): Average of the absolute value of the normalized normal field component across the plasma surface. (dimensionless)
\(\max(B_n)\): Maximum value of the normalized normal field component across the plasma surface. (dimensionless)
\(J\): Variance of incremental arclength between coil segments. (\(\text{m}^2\))
\(\kappa_\text{max}\): Maximum curvature value across all coils. (\(\text{m}^{-1}\))
\(\text{MSC}\): Mean squared curvature per coil, averaged across all coils. (\(\text{m}^{-2}\))
\(L\): Total length of all coils. (\(\text{m}\))
FC: Sequence of Fourier orders used in continuation method. The optimization starts with a low-order representation,…
\(N\): Number of base coils before applying stellarator symmetry. (dimensionless)
\(L_{\text{SC}}\): Total length of superconducting tape required at reactor scale, accounting for the number of turns in each coil’s… (\(\text{km}\))
\(d_{cc}\): Minimum distance between any two coils. (\(\text{m}\))
\(d_{cs}\): Minimum distance between any coil and the plasma surface. (\(\text{m}\))
\(F_\text{max}\): Maximum force magnitude across all coils. (\(\text{N}/\text{m}\))
\(\tau_\text{max}\): Maximum torque magnitude across all coils. (\(\text{N}\))
\(\text{LN}\): Topological measure of how coils are linked together. (dimensionless)
\(t\): Total time required to complete the optimization. (\(\text{s}\))

Composite Score

The Score column summarises reactor-scale engineering feasibility.

Hard constraints (any violation → score = 0): coil-surface linkage, coil-coil linking, finite-build clearance. Soft constraints contribute via margin exponents \(m_i\):

\[\begin{split}m_i = \begin{cases} 1 - \text{value}_i\,/\,\text{bound}_i & \text{value} \leq \text{bound}\\[6pt] \text{value}_i\,/\,\text{bound}_i - 1 & \text{value} \geq \text{bound} \end{cases}\end{split}\]

Score \(= \exp\!\left(\frac{1}{n}\sum_{i=1}^{n} m_i\right)\). RMS curvature uses \(\sqrt{\text{MSC}}\); arclength variation uses \(\sqrt{\text{Var}}\).

Constraint	Direction	Bound	Margin \(m_i\)
avg \(\langle B{\cdot}n\rangle / \langle B\rangle\) (`avg_BdotN_over_B`)	\(\leq\)	0.01	\(1 - \text{value}\;/\;0.01\)
Min coil-surface distance (`reactor_scale_min_cs_separation`)	\(\geq\)	1.3 m	\(\text{value}\;/\;1.3 - 1\)
Min coil-coil distance (`reactor_scale_min_cc_separation`)	\(\geq\)	0.7 m	\(\text{value}\;/\;0.7 - 1\)
Total coil length (`reactor_scale_total_length`)	\(\leq\)	220 m	\(1 - L\;/\;220\)
Max curvature \(\kappa\) (`reactor_scale_max_curvature`)	\(\leq\)	1.0 m^-1	\(1 - \kappa_{\max}\;/\;1.0\)
RMS curvature \(\sqrt{\text{MSC}}\) (`reactor_scale_mean_squared_curvature`)	\(\leq\)	1.0 m^-1	\(1 - \sqrt{\text{MSC}}\;/\;1.0\)
Arclength variation \(\sqrt{\text{Var}}\) (`reactor_scale_arclength_variation`)	\(\leq\)	1.0 m	\(1 - \sqrt{\text{Var}}\;/\;1.0\)
Total superconductor length \(L_{\text{SC}}\) (`total_superconductor_length_km`)	\(\leq\)	100 km	\(1 - L_{\text{SC}}\;/\;100\)
Max turns per coil \(N_{\text{turns}}\) (`N_turns_per_coil`)	\(\leq\)	300 (turns)	\(1 - \max_i N_{\text{turns},i}\;/\;300\)

Interpretation: Score = 0 → hard infeasible; 0 < Score < 1 → soft violated; Score ≥ 1 → constraints met. Entries sorted by score descending.

Reactor-Scale Constraints

Scaled to ARIES-CS (\(a = 1.7\,\text{m}\), \(B_0 = 5.7\,\text{T}\)). Hard constraints (any violation → score = 0):

Constraint	Bound	Description
Coils linked to plasma surface	= True	Every base coil must topologically encircle the plasma.
Coil-coil linking number (\(\|\text{LN}\| \approx 0\))	≤ 0.5 (dimensionless)	Coils must not interlink with one another.
Finite-build coil-coil clearance (\(d_{\text{cc}} > w_{\text{WP}}\))	≥ 0.0 m	Centreline distance \(d_{\text{cc,min}}\) must exceed the largest winding-pack width \(w_{\text{WP,max}}\) to prevent physical overlap of finite-build coils.

Soft constraints — margin factors; violations lower score but do not set to 0:

Metric	Bound	Direction	Units
avg \(\bar{B}_n\)	\(\leq 0.01\)	max	(dimensionless)
Minimum coil-surface distance	\(\geq 1.3\)	min	m
Minimum coil-coil distance	\(\geq 0.7\)	min	m
Total coil length	\(\leq 220.0\)	max	m
Max curvature \(\kappa\)	\(\leq 1.0\)	max	m⁻¹
Max \(\sqrt{\text{MSC}}\) (RMS curvature)	\(\leq 1.0\)	max	m⁻¹
Arclength variation \(\sqrt{\text{Var}}\)	\(\leq 1.0\)	max	m
Total superconductor length \(L_{\text{SC}}\)	\(\leq 100.0\)	max	km
Max turns per coil \(N_{\text{turns}}\)	\(\leq 300\)	max	(turns)

Winding-Pack Model

The optimiser models coils as single filamentary turns carrying total current \(I\). A real reactor winding pack has \(N_{\text{turns}}\) turns per coil, each carrying \(I/N_{\text{turns}}\), to keep per-turn Lorentz forces and conductor field within limits. For each coil \(i\) we compute two turn counts and take the maximum:

\(N_{\text{turns},\,i} = \max\bigl(N^{(\text{force})}_i,\, N^{(J_c)}_i\bigr)\)

Force-based turns. With \(N\) turns, the force per unit length on each turn is \(F_{\text{turn}} = F_{\text{reactor}}/N\). To keep \(F_{\text{turn}} \leq 0.5\) MN/m (structural limit):

\(N^{(\text{force})}_i = \lceil F_{\text{reactor},i} / (0.5\,\text{MN/m}) \rceil\)

Jc-based turns. Ensures the HTS operates within its critical current envelope. Uses a Kim-like REBCO \(J_c(B,T)\) model (Stellaris params, Lion et al. 2025).

Required ampere-turns at reactor scale: \(NI_i = I_{\text{device},i} \times B_{\text{scale}} \times L_{\text{scale}}\)
Peak conductor field (with winding-pack self-field factor 1.3): \(B_{\text{peak},i} = f_{\text{WP}} \times B_{\text{ext},i}\)
Critical current of cable: \(I_{c,\text{cable}} = J_c(B_{\text{peak}}, T_{\text{op}}) \times A_{\text{HTS}}\)
Operating current per turn: \(I_{\text{turn}} = \min(I_{\text{lead,max}},\, \eta \times I_{c,\text{cable}})\)
Turns from Jc: \(N^{(J_c)}_i = \lceil NI_i / I_{\text{turn},i} \rceil\)

Soft constraint (bound 300). \(\max_i N_{\text{turns},i}\) contributes to the composite score via a margin: designs with \(\max_i N_{\text{turns},i} < 300\) are rewarded, designs with \(\max_i N_{\text{turns},i} > 300\) are penalized.

Finite-build width. Each turn occupies \(20\,\text{mm} \times 20\,\text{mm}\). Winding-pack side length: \(w_{\text{WP}} = \sqrt{N_{\text{turns}}} \times 20\,\text{mm}\). Clearance between coil packs: \(d_{\text{cc,min}} - w_{\text{WP,max}}\). Negative = infeasible.

Per-turn force/torque. \(F_{\text{turn}} = F_{\text{reactor}}/N_{\text{turns}}\), \(\tau_{\text{turn}} = \tau_{\text{reactor}}/N_{\text{turns}}\). Reported on leaderboard.