## 5 Scaling Laws and Optimisation
### Comparison with the Tokamak in L Mode of Confinement  
To compare the scaling laws obtained for the Stellarator with the Tokamak's, the International Tokamak Physics Activity (ITPA) International L-Mode Confinement (DB3v13f) Data Base has been considered [25]. The study therefore focusses on the scaling of \(\tau_{E}\) 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). \[\tau_{E}^{L\ mode\ literature}=\frac{0.103\cdot\mu_{0}}{\pi}\cdot\left(\frac{2 \pi}{\mu_{0}}\right)\left(\frac{Ba^{2}k_{a}}{q_{95}R}\right)\cdot q_{95}^{0.04}B^{-0.42}R^{0.51}n^{0.43}M^{-0.11}p^{-0.44}a^{1.95} \tag{7}\]  
\[\tau_{E}^{L\ mode\ SR}=\frac{0.011\cdot\mu_{0}}{\pi}\cdot\left(\frac{2\pi}{\mu _{0}}\right)\left(\frac{Ba^{2}k_{a}}{q_{95}R}\right)\cdot B^{-0.45}R^{2.12}k_ {a}^{0.70}p^{-0.53}a^{0.50}q_{95}^{1.34}n^{0.41}\cdot e^{(0.28\cdot\frac{\mu_{ 0}}{\pi}\cdot(\frac{2\pi}{\mu_{0}}\cdot\frac{2\pi^{2}k_{a}}{q_{95}R})\cdot M^{ -1.20}a^{0.93}(\cdot\frac{\mu_{0}}{q_{95}R})^{0.41})} \tag{8}\]  
The non-power law scaling outperforms the power law according to all the statistical indicators, as reported in Table 4. Moreover, the database is large enough to perform an extrapolation exercise; the scaling laws (7) and (8) have been fitted to the small devices, and then, their accuracy in modelling JET data has been tested [27]. The capability of the non-power law scaling to extrapolate to JET is qualitatively better than the equation in the literature: indeed, the residuals of (8) are an order of magnitude smaller than the ones of (7) [27]. As stated previously, since the models for the Stellarator confinement obtained with SR via GP do not include any renormalization factor, they can be compared with the scalings of the Tokamak in L mode. For the ISS04 it is assumed that \(f_{ren}=1\), 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. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Equation** & **k** & **MSE [\(s^{2}\)]** & **RMSE [\(s\)]** & **AIC** & **BIC** \\ \hline \(\tau_{E}^{TOK\ literature}\) & \multirow{2}{*}{12} & \multirow{2}{*}{3.60 \(\times\) 10\({}^{-3}\)} & \multirow{2}{*}{6.00 \(\times\) 10\({}^{-2}\)} & \multirow{2}{*}{\(-6.40\times\) 10\({}^{-4}\)} & \multirow{2}{*}{\(-6.36\times\) 10\({}^{-4}\)} \\ Equation (7) & & & & & \\ \hline \(\tau_{E}^{TOK\ SR\ via\ GP}\) & \multirow{2}{*}{11} & \multirow{2}{*}{2.29 \(\times\) 10\({}^{-3}\)} & \multirow{2}{*}{5.39 \(\times\) 10\({}^{-2}\)} & \multirow{2}{*}{\(-6.63\times\) 10\({}^{-4}\)} & \multirow{2}{*}{\(-6.58\times\) 10\({}^{-4}\)} \\ Equation (8) & & & & & \\ \hline \hline \end{tabular}
\end{table}