# Considerations on Stellarator’s Optimization from the Perspective of the Energy Confinement Time Scaling Laws

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{ren}, specific to each device and each level of optimisation for individual machines. The f

_{ren}coefficient is believed to account for higher order effects not ascribable to variations in the 0D quantities, the only ones included in the database used to derive ISS04, the International Stellarator Confinement database. This hypothesis is put to the test with symbolic regression, which allows relaxing the assumption that the scaling laws must be in power monomial form. Specific and more general scaling laws for the different magnetic configurations have been identified and perform better than ISS04, even without relying on any renormalisation factor. The proposed new scalings typically present a coefficient of determination R

^{2}around 0.9, which indicates that they basically exploit all the information included in the database. More importantly, the different optimisation levels are correctly reproduced and can be traced back to variations in the 0D quantities. These results indicate that f

_{ren}is not indispensable to interpret the data because the different levels of optimisation leave clear signatures in the 0D quantities. Moreover, the main mechanism dominating transport, in reasonably optimised configurations, is expected to be turbulence, confirmed by a comparative analysis of the Tokamak in L mode, which shows very similar values of the energy confinement time. Not resorting to any renormalisation factor, the new scaling laws can also be extrapolated to the parameter regions of the most important reactor designs available.

## 1. Stellarator Configurations: Optimisation and the Scaling of the Confinement Time

_{E}, the so called ISS04. This scaling is in power law monomial form and, in order to fit the experimental data acceptably, it employs a dimensionless renormalisation factor, called f

_{ren}, for each device and for each sufficiently different optimisation level. In the traditional interpretation, f

_{ren}is considered to reflect the different aspects of the optimisation, which cannot be accounted for by 0D quantities.

## 2. Brief Overview of Symbolic Regression via Genetic Programming for the Extraction of Scaling Laws from Empirical Databases

## 3. The International Stellarator Confinement Database and the ISS04 Scaling Law

_{e}the average plasma density, and $t$

_{2/3}the rotational transform at two thirds of the minor radius. It is worth mentioning that, since in the literature the uncertainties affecting f

_{ren}are not reported, it is impossible to calculate the confidence intervals of Equation (1).

_{E}on the rotational transform from the DB. This is mainly the unfortunate consequence of two factors: the design of certain devices and the experimental programme of others. The heliotron/torsatron family of devices, due to engineering constraints, presents a very strong collinearity between the rotational transform$\text{}{t}_{2/3}$and the aspect ratio. In their turn, W7-A and W7-AS machines do not scan a significant range of ${t}_{2/3}$. Consequently, it is difficult to isolate the effect of the rotational transform on τ

_{E}. To alleviate this deficiency, an approach, similar to the one devised in [22] to obtain ISS04, has also been implemented to derive the scalings reported in this work. TJ-II dependence of τ

_{E}on the rotational transform is used as first guess for symbolic regression. This approach works quite well for the devices with shear, whose scaling shows a power law dependence on ${t}_{2/3}$with an exponent not much different from the one of TJ-II. On the other hand, for the shearless configurations much weaker dependencies on ${t}_{2/3}$would be better supported by the International Stellarator Confinement Database, a subject discussed more extensively in the next section.

## 4. Scaling Laws for Different Magnetic Configurations

_{ren}of ISS04 ranges from 0.25 to 1. Moreover, the dependence from ${t}_{2/3}$ is quite difficult to determine, and indeed, as mentioned in [22], it was derived from a different set of data not included in the International Stellarator Confinement DB. It has therefore been decided to provide more freedom to the nonlinear fit of the model, selected by SR via GP, by implementing a non-linear least-square fit with constrained coefficients [23]. To perform the fit, a regularization term has been added to the loss function so that it becomes:

_{ren}ranges between 0.25 and 1, an interval quite substantial. The proposed scalings can improve in the ISS04 thanks to their flexibility.

^{2}) is very high, about 0.9 for all scaling laws obtained with SR via GP. Considering that R

^{2}represents the proportion of the dependent variable variance that is explained by the regressors, the obtained scalings basically account for all the information contained in the database. Around 10% of random errors in the estimates of the energy confinement times is indeed probably an overestimate of the accuracy of the entries in the DB. It is therefore not reasonable to expect R

^{2}values higher than the ones already achieved.

_{ren}had to be introduced to derive the ISS04. The new methodology of SR via GP allows relaxing the power law constraint, alleviating the rigidity of the scalings without relying on any renormalisation coefficient. The effects of the non-power law terms are shown graphically in Figure A1 and Figure A2. It is also worth mentioning, as supported again by the plots reported in Figure A1 and Figure A2, that not constraining the scalings to be power law monomials allows fitting quite well also the data of the individual devices. Normally the scaling laws for thermonuclear devices are designed to better fit the trends between devices but not the individual machines’ data (and this applies also to the Tokamak scalings [12,13]).

## 5. Scaling Laws and Optimisation

_{ren}is not indispensable to fit the different devices, a fundamental point remains to be clarified. In the DB there are devices, particularly LHD and W7-AS, which present different levels of optimisation. This is confirmed by the fact that four different values of f

_{ren}were necessary to fit the data of LHD with ISS04. In particular, LHD at R = 3.60 m is much better neoclassically optimized than LHD at R = 3.90. In the case of W7-AS, three different values of f

_{ren}had to be used to properly reproduce the entries of the database. Since the need for renormalisation was interpreted as a way to take into account higher order effects not reflected in the 0D quantities, the effectiveness of unified scaling laws, obtained without f

_{ren}, and valid for very different optimisation levels, needs explaining. The next two subsections are therefore devoted to investigating the relationship between the scalings obtained with SR via GP and f

_{ren}. The analysis is based on calculating first the ratio between the scaling laws obtained with SR via GP and the ISS04. These ratios reproduce almost perfectly the trends of τ

_{E}with f

_{ren}. The parts of the non-power law scaling reflecting the trends of f

_{ren}are then discussed in detail.

#### 5.1. Non-Power Law Scalings and f_{ren} for Different Optimisation Levels of LHD

_{LHD/ISS04}and reads as follows:

_{LHD/ISS04}. From these plots, it is easy to see how the scaling of Equation (2), obtained with SR via GP, can reproduce the effects of the various optimization levels on the 0D quantities equally well, if not better, than the renormalization factor f

_{ren}. The ratio R

_{LHD/ISS04}depends basically on major and minor radius, with a small residual effect on the magnetic field. The part of the ratio depending on the minor and major radii, R

_{LHD/ISS04}(a,R), is plotted vs. the major radius in Figure 4, while the other main trends are illustrated graphically in Figure A3, Figure A4, Figure A5 and Figure A6. For an easier visual comparison with f

_{ren}, the values of this ratio have been averaged for each level of optimisation.

_{LHD/ISS04}(a,R), fits quite well f

_{ren}and basically takes into account the effects of the changes in the configurations implemented to optimise confinement. Moreover, the good quality of the fits, testified by the high values of the indicator R

^{2}, suggest that practically all the effects of the optimisation efforts in LHD can be ascribed to the modifications in the 0D quantities.

#### 5.2. Non-Power Law Scalings and f_{ren} for Different Optimisation Levels of W7-AS

_{W-AS/ISS04}. For a ${t}_{2/3}$ exponent of 0.3, the functional dependence of R

_{W-AS/ISS04}is

_{W-AS/ISS04}is compared with f

_{ren}in Figure 3. The part depending on the minor and major radii, R

_{W7-AS/ISS04}(a,R), is shown versus R in the plots of Figure 5 (while the other major dependencies are reported in Figure A3, Figure A4, Figure A5 and Figure A6). Again, for an easier visual comparison, the values of this ratio have been averaged for each level of optimisation.

_{W7-AS/ISS04}depends strongly on plasma density and magnetic field, whereas in the case of LHD, the most important factors are geometrical, the minor and major radii. This evidence needs to be further investigated, but some points can be already emphasised. First, the completely different trends of R

_{LHD/ISS04}and R

_{W7-AS/ISS04}stress again the need for particularising the scaling laws for the different magnetic configurations, with and without shear. Second, the dubious and confusing nature of f

_{ren}becomes even more evident. This arbitrary factor tends to obscure more than clarify the underlying trends and the phenomenology. On the other hand, a significant effort should be devoted to understanding the causes of the different forms of Equations (5) and (6), for example, by studying whether they are due to dissimilar approaches to optimisation or to intrinsic specificities of the magnetic configuration physics.

## 6. Comparison with the Tokamak in the L Mode of Confinement and Extrapolation to Stellarator Reactors

#### 6.1. Comparison with the Tokamak in L Mode of Confinement

_{E}in the L mode. Overall, the DB contains 1140 entries, identified using the same selection criteria adopted in [26]. As customary, to compare the two configurations, the scaling laws of the Tokamak have been reformulated by writing the current in terms of the safety factor q

_{95}in the cylindrical approximation, corrected by the elongation [22]. For this confinement regime, the best credited scaling law in the literature is Equation (7), the so called ITER97 [27]. The methodology of symbolic regression via genetic programming has been applied also to the ITPA database of L-mode discharges [28]. The one obtained with SR is not a power law and is reported as Equation (8).

_{ren}= 1, which is a reasonable choice, since this is the value of renormalization factor typical of the largest devices and the best optimisation levels. A comparison of the predictions of the various models for a device of ITER engineering parameters is reported in Table 5. It is interesting to note that the estimates for the Stellarator configuration span practically the same interval than the two for the Tokamak. The extrapolation to ITER is probably still acceptable, since the gap in terms of operational parameters is not excessive. On the other hand, it should also be remembered that the assumption, underlying all these extrapolation exercises, is that the physics of the plasmas in devices of ITER dimensions will not be different from the ones of the present-day generation, an aspect that can be addressed only experimentally.

#### 6.2. Extrapolation to the Demonstrative Reactor Region of the Parameter Space

_{ren}= 1.

## 7. The Need to Move beyond f_{ren}: Concluding Remarks

_{ren}are very different for the two magnetic configurations. In the case of the devices with shear, the different optimisation levels can be accounted for by a dependence on the geometric factors, mainly minor and major radii. In the shearless devices, the optimisation process seems to leave a signature mainly in magnetic field and plasma density. In this context, it is worth stating again that, in any case, the results obtained with SR via GP are completely agnostic about higher order effects, not leaving clear traces in the 0D quantities. The proposed methodology is fully data driven and therefore can account only for the information contained in the data. Consequently, whether the aforementioned dissimilarities are the result of different optimisation procedures or are inherent to the physics of the two magnetic configurations remains to be established. On the other hand, the obtained results and the comparison with the L mode Tokamak support the opinion that turbulence transport is the dominant effect in determining the Stellarator energy confinement time, once the devices are reasonably optimised for neoclassical transport [32].

^{2}levels, achieved by the fits, indicate that the variance unaccounted for by the scalings is compatible with the random uncertainties to be associated with the entries of τ

_{E}. Consequently, SR via GP has converged on models, which exploit all the information available in the database (if anything some have probably a tendency to overfit). It seems therefore only reasonable to state that further progress in the understanding would require an upgrade of the database, which is not completely satisfactory given the complexities of the configuration. Moreover, to investigate higher order effects, additional entries, such as turbulence measurements, also will have to be included; current profiles and radiation patterns have also proved to be essential to understand the behaviour of Tokamaks [33,34]. Another quite high priority would be to particularise the models for the devices with metallic plasma facing components, given the effects of these materials on the operation and performance of JET with the new ITER Like Wall [35,36]. The dependence of the energy confinement time on the rotational transform should also be clarified experimentally, to understand its actual form and whether it is the same for both main magnetic configurations.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Effects of the Exponential Terms on the Scaling Laws

**Figure A1.**Comparison of the scaling laws with and without the exponential term for the devices with shear. In the legend of the plots, GAUSS indicates the corrective non-power law term in the scaling laws.

**Figure A2.**Comparison of the scaling laws with and without the exponential term for the devices without shear. In the legend of the plots, GAUSS indicates the corrective non-power law term in the scaling laws.

## Appendix B. Scaling Laws obtained with SR via GP and f_{ren}

**Figure A3.**Analysis of LHD data for various optimisation levels vs. the minor radius. (

**Left**) The part of the model Equation (2), which depends on major and minor radius f(a,R). (

**Centre**) The actual confinement time in LHD vs. the minor radius. (

**Right**) Trend of f

_{ren}in ISS04 vs. the minor radius.

**Figure A4.**Analysis of LHD data for various optimisation levels vs. iota. (

**Left**) The part of the model Equation (2), which depends on major and minor radius f(a,R). (

**Centre**) The actual confinement time in LHD vs. iota. (

**Right**) Trend of f

_{ren}in ISS04 vs. ${t}_{2/3}$

**Figure A5.**Analysis of W7-AS data for various optimisation levels vs. the magnetic field and comparison with f

_{ren}.

**Figure A6.**Analysis of W7-AS data for various optimisation levels vs. the plasma density and comparison with f

_{ren}.

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**Figure 1.**The log-log plots for the comparison of the models and the data; the experimental values of τ

_{E}are on the y axis and the estimates of the models on the x axis. (

**Left column**) the models obtained with SR via GP. (

**Right columns**) the corresponding estimates of ISS04.

**Figure 2.**The trends of the equations obtained with SR via GP compared to the ISS04. For the ISS04, two scalings are shown: one with a renormalization factor equal to 1 and one (indicated by <f

_{ren}>) with the renormalization factor averaged over the entries in the DB.

**Figure 3.**(

**Top**) comparison of the trend of the energy confinement time with R

_{LHD/ISS04}and f

_{ren}for LHD. (

**Bottom**) comparison of the trend of the energy confinement time with R

_{W-AS/ISS04}.and f

_{ren}for W7-AS (for ${t}_{2/3}$ = 0.3).

**Figure 4.**Analysis of LHD data for various optimisation levels vs. the major radius. (

**Top left**) The part of the model Equation (2), which depends on major and major radius R

_{LHD/ISS04}(a,R). (

**Top right**) the actual confinement time in LHD vs. the major radius. (

**Bottom**) trend of f

_{ren}in ISS04 vs. the major radius.

**Figure 5.**Analysis of W7-AS data for various optimisation levels vs. the major radius. (

**Top left**) The part of the model Equation (3), which depends on major and minor radius f(a,R). (

**Top right**) the actual confinement time in W7-AS vs. the major radius. (

**Bottom**) trend of f

_{ren}in ISS04 vs. the major radius.

**Table 1.**The main quantities in the International Stellarator Confinement database and their range of values. The meaning of the symbols is the usual: a is the minor radius, R the major radius, P the input power, n

_{e}the average electron density, B the on-axis magnetic field, and ${t}_{2/3}$ the rotational transform at two thirds of the minor radius.

Quantity | $\left[\mathbf{min}\left(\xb0\right),\mathbf{max}\left(\xb0\right)\right]$ | $\left[\mathit{\mu},\mathit{\sigma}\right]$ |
---|---|---|

$a\left[m\right]$ | $\left(\left[0.088,0.634\right]\right)$ | $\left(\left[0.23,0.12\right]\right)$ |

R [m] | $\left(\left[0.938,3.821\right]\right)$ | $\left(\left[1.94,0.68\right]\right)$ |

P [MW] | $\left(\left[0.04,6.52\right]\right)$ | $\left(\left[1.09,1.35\right]\right)$ |

${n}_{e}\left[{10}^{19}{m}^{-3}\right]$ | $\left(\left[0.22,34.31\right]\right)$ | $\left(\left[5.42,7.40\right]\right)$ |

$B\left[T\right]$ | $\left(\left[0.44,2.56\right]\right)$ | $\left(\left[1.37,0.63\right]\right)$ |

${t}_{2/3}$ | $\left(\left[0.092,1.607\right]\right)$ | $\left(\left[0.73,0.44\right]\right)$ |

Dataset | Devices | Entries |
---|---|---|

SHEAR | ATF | 229 |

CHS | 196 | |

HELE | 120 | |

HELJ | 54 | |

LHD | 162 | |

Total | 761 | |

SHEARLESS | W7-A | 13 |

W7-AS | 629 | |

TJ-II | 316 | |

Total | 958 |

**Table 3.**Comparison of ISS04 and the non-power law scalings for the machines with and without shear.

Device | Eq | $\mathbf{MSE}\text{}\left[{\mathit{s}}^{2}\right]$ | RMSE[s] | AIC | BIC | ${\mathit{R}}^{2}$ |
---|---|---|---|---|---|---|

SHEAR | ISS04 Equation (1) | $2.64\times {10}^{-5}$ | $5.14\times {10}^{-3}$ | $-8.0062\times {10}^{3}$ | $-7.9691\times {10}^{3}$ | 0.9767 |

Equation (2) | $2.16\times {10}^{-5}$ | $4.70\times {10}^{-3}$ | $-8.1539\times {10}^{3}$ | $-8.1075\times {10}^{3}$ | 0.9809 | |

SHEARLESS | ISS04 Equation (1) | $6.43\times {10}^{-6}$ | $2.53\times {10}^{-3}$ | $-1.1424\times {10}^{4}$ | $-1.1385\times {10}^{4}$ | $0.8974$ |

Equation (3) | $6.78\times {10}^{-6}$ | $2.60\times {10}^{-3}$ | $-1.1372\times {10}^{4}$ | $-1.1328\times {10}^{4}$ | $0.8920$ | |

Equation (4) | $6.27\times {10}^{-6}$ | $2.50\times {10}^{-3}$ | $-1.1445\times {10}^{4}$ | $-1.1402\times {10}^{4}$ | $0.9000$ |

**Table 4.**Comparison of the statistical indicators used to qualify the quality of the models: MSE is the mean square error, RMSE the root the mean square error, and k the number of parameters of the models.

Equation | k | $\mathbf{MSE}\text{}\left[{\mathit{s}}^{2}\right]$ | RMSE [s] | AIC | BIC |
---|---|---|---|---|---|

${\tau}_{E}^{TOKliterature}$ Equation (7) | 12 | $3.60\times {10}^{-3}$ | $6.00\times {10}^{-2}$ | $-6.40\times {10}^{-4}$ | $-6.36\times {10}^{-4}$ |

${\tau}_{E}^{TOKSRviaGP}$ Equation (8) | 11 | $2.29\times {10}^{-3}$ | $5.39\times {10}^{-2}$ | $-6.63\times {10}^{-4}$ | $-6.58\times {10}^{-4}$ |

Model | ${\mathit{\tau}}_{\mathit{E}}^{\mathbf{ITER}}\left[\mathit{s}\right]$ |
---|---|

ISS04 Equation (1) | $2.65$ |

SHEAR Equation (2) | ${2.34}_{1.45}^{3.29}$ |

SHEARLESS03 Equation (3) | ${2.76}_{1.02}^{8.11}$ |

SHEARLESS02 Equation (4) | ${2.08}_{0.71}^{5.10}$ |

L mode literature Equation (7) | ${2.12}_{2.00}^{2.23}$ |

L mode SR via GP. Equation (8) | ${2.78}_{2.18}^{3.53}$ |

ARIES-CS | HSR4/18 | HSR5/22 | |
---|---|---|---|

a[m] | 1.7 | 2 | 1.8 |

R[m] | 7.75 | 18 | 22 |

P [MW] | 462 | 486 | 594 |

$n\left[{10}^{19}{m}^{-3}\right]$ | 40 | 26 | 21 |

B[T] | 5.7 | 5.0 | 4.75 |

${t}_{\frac{2}{3}}$ | 0.7 | 0.917 | 0.947 |

**Table 7.**Estimates for the machines of Table 6.

Model | ${\mathit{\tau}}_{\mathit{E}}^{\mathbf{ARIES}-\mathbf{CS}}\left[\mathit{s}\right]$ | ${\mathit{\tau}}_{\mathit{E}}^{\mathbf{HSR}4/18}\left[\mathbf{s}\right]$ | ${\mathit{\tau}}_{\mathit{E}}^{\mathbf{HSR}5/22}\left[\mathit{s}\right]$ |
---|---|---|---|

ISS04 Equation (1) | $1.08$ | $2.06$ | $1.41$ |

SHEAR Equation (2) | ${0.83}_{0.58}^{1.73}$ | ${2.28}_{1.54}^{3.50}$ | ${1.67}_{1.12}^{2.58}$ |

SHEARLESS03 Equation (3) | ${1.74}_{0.57}^{5.70}$ | ${2.76}_{0.86}^{9.16}$ | ${1.78}_{0.57}^{5.79}$ |

SHEARLESS02 Equation (4) | ${1.39}_{0.42}^{3.81}$ | ${2.13}_{0.62}^{5.97}$ | ${1.38}_{0.41}^{3.79}$ |

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**MDPI and ACS Style**

Murari, A.; Peluso, E.; Spolladore, L.; Vega, J.; Gelfusa, M.
Considerations on Stellarator’s Optimization from the Perspective of the Energy Confinement Time Scaling Laws. *Appl. Sci.* **2022**, *12*, 2862.
https://doi.org/10.3390/app12062862

**AMA Style**

Murari A, Peluso E, Spolladore L, Vega J, Gelfusa M.
Considerations on Stellarator’s Optimization from the Perspective of the Energy Confinement Time Scaling Laws. *Applied Sciences*. 2022; 12(6):2862.
https://doi.org/10.3390/app12062862

**Chicago/Turabian Style**

Murari, Andrea, Emmanuele Peluso, Luca Spolladore, Jesus Vega, and Michela Gelfusa.
2022. "Considerations on Stellarator’s Optimization from the Perspective of the Energy Confinement Time Scaling Laws" *Applied Sciences* 12, no. 6: 2862.
https://doi.org/10.3390/app12062862