Optimizing magnetic diagnostics using the FP database

The existence of an equilibrium database enables sensitivity studies to be carried out for competing diagnostic designs before they are ever implemented experimentally [67,68]. The measurement setup can thus be optimized using the database as a testing ground and the extent of information in the measurements determined via parametric regressions. Here, we investigate the accuracy achievable for recovery of the flux surface geometry using a more realistic diagnostic setup consisting of fewer measurements. Particular emphasis is put on the effects of nuisance background magnetic fields. With the equivalence of geometric and $p(s)$ information in mind, we do not attempt to optimize the recovery of the geometry directly. Instead we use the previously defined RMS pressure profile recovery error $\langle \delta_\mathrm{p} \rangle$ as the figure of merit for a given diagnostic setup. This procedure is of particular interest in the context of designing a magnetic diagnostic system for the next generation stellarator W7-X, since the procedure we apply for W7-AS is equally applicable to the newer device. In fact, such sensitivity studies are not only limited to magnetic diagnostics, but can be of use any system for which the raw data can be adequately simulated in an equilibrium database.

A truly optimal diagnostic setup involves considering myriad constraints, from the structural considerations of the number and position of available mounting points for magnetic coils to the engineering considerations of expected accuracy. We do not attempt to present a final solution for this problem here, but rather outline the optimization method for a ``sensible'' set of parameters for the purposes of illustration. We continue with the assumption that the diagnostics available are probes of small extent which provide local measurements of magnetic field components.

We begin by assessing the precision demands on the measurements. Referring back to figure 5.1, we see that the steady state value of the PCA eigenvalues for the dense magnetic measurements with 1% noise is roughly $2.5\times10^{-10}$ mT$^2$, corresponding to an absolute error of 0.016 mT. This error must be placed in context: it is roughly the value of the Earth's magnetic field. It is unlikely that any conceivable magnetic diagnostic, regardless of how carefully compensated, would be capable of accurately measuring such a tiny variation against the large background magnetic field on a fusion experiment, since any stray fields present, e.g. due to minute variations in the main fields, would completely dominate the signal. Thus, while we were content to use a small level of stabilizing noise to examine the best-case recovery of $p(s)$ earlier, it is entirely inappropriate to assume the same when investigating a setup that must display sufficient robustness for use with experimental data. Furthermore, some coils (due to their orientation) may only ever measure miniscule fields, so the addition of relative noise to the measurements is also far from realistic here.

Figure 5.10: RMS $p(s)$ profile recovery RMSE $\langle \delta_\mathrm{p} \rangle$ versus retained PCs of the dense magnetic measurements for the levels of absolute noise (baseline error with 0 PCs = 2.63 kPa as before)

A more practical approach is to assume some fixed absolute error in the measurements, e.g. representing the level of nuisance fields present during discharges. Only exhaustive analysis of very accurate field measurements would allow a rigorous determination of such a noise level, which we are not in a position to perform. For an idea of the impact of nuisance fields on the previous results, we repeat the analysis of figure 5.2 using various absolute noise levels added to the measurements and show the trend in figure 5.10. The results are much less optimistic than the previous case, where the largest error of 10% relative noise is now equivalent to an absolute noise of only 0.06 mT. Also, the previously notable drop in $\langle \delta_\mathrm{p} \rangle$ for the 9 included PCs is absent for all but the lower noise levels here. In fact, the plot shows that $\langle \delta_\mathrm{p} \rangle$ actually rises when too many PCs are added for the higher noise levels, which is due to the use of the adjusted RMSE to account for the number of parameters in the model.

As well as the obvious dependence of the overall error on the noise level, it is evident from figure 5.10 that the number of independently measurable PCs is also strongly dependent on the background noise. For instance, if we could be guaranteed a background level of only 0.06 mT, then the 8$^\mathrm{th}$ PC brings an overall decrease in error, however for 0.20 mT, our 720 measurements yield only 3 PCs of any predictive value. This helps to identify a lower limit to the number of coils needed since in principle, a given number of PCs is measurable using the same number of correctly placed coils. However a large degree of redundancy is desirable in an experimental setup in order to facilitate correction of missing or failing signals and to improve the signal to noise level in the case of random, uncorrelated measurement noise.

We choose an intermediate error of 0.1 mT and show how to arrive at an optimal arrangement using an array of coils measuring just one component of the magnetic field in a single $\phi$-plane, much like the array of $B_{\mathrm{\theta}}$ coils on W7-AS. The figure of 0.1 mT may appear optimistic and merits further justification. On stellarators, the magnetic diagnostics purely measure signals due to the plasma, as sampling usually commences only when the currents in the external coils have reached steady values. It is therefore possible to successfully detect smaller fields than on tokamaks, where the detection system must be sensitive enough to measure small signals in tandem with the large poloidal field due to the plasma current. We remark that on the ASDEX-Upgrade tokamak, the magnetic diagnostic signals are accurate to roughly 1% [69]. The maximum and RMS values over the database of magnetic field measurements from our simulated probes in the $\phi =0$ plane are 8 mT and 1.35 mT respectively; if similar accuracy could be achieved on W7-AS this would correspond to an error level of 0.08 mT as a fraction of the maximum signal and much less as a fraction of the RMS signal.

We examine measurements from arrays with a variable number of probes sampling the components of the magnetic field normal and tangential to the vacuum vessel and in the toroidal direction in 10 $\phi$-planes with an absolute noise level of 0.1 mT in all cases. We remark that from an engineering standpoint, this is quite favourable since the orientation of each probe can be determined easily relative to the vacuum vessel. Recoveries with the toroidal component are consistently inferior to those using either the radial or poloidal component, and we thus exclude these from further discussion. A PCA is performed on the field measurements and the resulting PCs are used with the baseline predictors to recover $p(s)$, yielding a corresponding value of $\langle \delta_\mathrm{p} \rangle$.

Figure: $\langle \delta_\mathrm{p} \rangle$ behaviour versus numbers of PCs added to the baseline model for arrays comprising variable numbers of probes oriented tangentially to the vacuum vessel in the $\phi =0^\circ$ plane with 0.1 mT assumed absolute noise
Figure: $\langle \delta_\mathrm{p} \rangle$ behaviour versus numbers of PCs added to the baseline model for arrays comprising variable numbers of probes oriented tangentially to the vacuum vessel in the $\phi =36^\circ$ plane with 0.1 mT assumed absolute noise

At this noise level, only 2-4 PCs are of predictive value. We examine $\langle \delta_\mathrm{p} \rangle$ behaviour for arrays with a maximum of 72 to a minimum of just 4 coils, all equally spaced in the poloidal angle $\theta$. The value of $\langle \delta_\mathrm{p} \rangle$ is very similar for arrays measuring the normal component to those measuring the tangential component of the magnetic field, indicating that roughly the same degree of information is present in both.

We show results using the poloidal component from just the $\phi =0^\circ$ and $\phi =36^\circ$ planes in figures 5.11 and 5.12 to illustrate our finding that the most information-rich area to locate the coil arrays is near the $\phi =0^\circ$ and adjacent planes, where the plasma cross-section is most triangular. The recovery towards $\phi =36^\circ$ (the elliptical plane) is markedly poorer, which is in accordance with the tendency of magnetic measurements to give less information as the cross-section becomes more circular (i.e. where the shape is dominated by the $m=1$ harmonic). In all cases, there are at most 2 PCs with significant predictive value and we remark that this agrees with our findings in chapter 3, where the two leading PCs of the measurements were also responsible for essentially the entire decrease in error of the flux surface geometry.

Figure 5.13: Schematic of the reduced array of magnetic probes in the $\phi =0^\circ$ plane. Each is oriented to measure the component of the magnetic field tangential to the vacuum vessel

Figure 5.14: Leading PC eigenvectors for the 72 poloidal magnetic coil setup in the $\phi =0$ plane versus poloidal angle $\theta$

Having shown the $\phi =0^\circ$ plane to be the best location for the probe array, we see from figure 5.11 that the difference in $\langle \delta_\mathrm{p} \rangle$ for 24 and 18 coils is relatively small, however with 12 coils there is again a noticeable rise. Trading off the number of coils versus the error, an 18-coil setup as illustrated in figure 5.13 appears to be a good compromise between accuracy and compactness. The PCA eigenvectors are illustrated for the densest case of 72 coils in fig. 5.14 and are recognizable as cosine-like harmonics of the magnetic field. Note that there is no constant (zeroth order) term due to the absence of net toroidal current in the database; neither are sine-like harmonics present since the magnetic field is up-down symmetric in this poloidal plane. The deviation of the eigenvectors from pure cosine functions of $\theta$ is due to the ellipticity of the vacuum vessel: orienting the probes such that they measure the component of $B$ tangential to a circle centered on the vacuum vessel yields eigenvectors very close to $\cos(\theta), \cos(2\theta),$ etc. The distribution of eigenvector weights clearly shows where fewer coils are necessary (variable weights near zero are obviously less important), giving further scope for optimizing the number of coils if necessary.

The recovery of the usual set of parameters is shown under the heading ``18 coil'' in table 5.3 for the baseline model plus 2 PCs of the measurements above and 0.1 mT absolute noise level. For ease of comparison, we have reproduced the results of table 5.1 with 9 PCs of the 720 idealized magnetic measurements and a (relative) noise level of 1% of the spread. The loss in accuracy compared with the results in table 5.1 is approximately a factor of two for most Fourier coefficients, which is quite modest when we consider that here the number of independent measurements is a factor of 40 fewer (18 versus 720) and the noise level a factor of roughly 6 higher (0.1 mT versus an equivalent of 0.016 mT). Comparing these with the results of chapter 3, they are very close to the case using 2 PCs of the set of experimentally available signals, where we assumed a 5% relative noise level. We see this as a demonstration of the robustness of the procedure, since an assumed absolute error of 0.1 mT in all signals is quite large when compared to the RMS value of the 72 signals in the $\phi =0$ plane, which is just 1.35 mT.

Figure 5.11 shows that the accuracy with which $p(s)$ can be identified over the database is a function of the number of coils. This result is valid if the experimental noise is uncorrelated and normally distributed, where an increase in the number of probes gives more accurate measurements by exploiting the random nature of the errors. We chose an array of 18 coils for the example of an optimized setup above, however this is less than half the number of measurements available on ASDEX-Upgrade, a 2-D device. The absence of rapid feedback control demands on the magnetic diagnostic systems on stellarators is clear, however fast magnetic-based FP reconstructions can only be performed accurately with a suitably extensive suite of precise magnetic signals. We suggest that in this light, the array with 72 coils is perhaps not over-optimistic.

Table 5.3: Recovery accuracy for database parameters using magnetic data from the reduced probe array

General				Idealized		18 coil
Variable	Units	Mean	Spread	RMSE	$\varepsilon$(%)	RMSE	$\varepsilon$(%)
$V$	m$^3$	0.9684	0.1992	0.0031	1.57	0.0055	2.77
$\beta_{\mathrm{axis}}$	%	1.81	0.91	0.18	20.19	0.33	36.36
$W_{\mathrm{p}}$	kJ	6.40	3.60	0.010	0.26	0.13	3.65
${r_{\mathrm{eff}}}$	cm	15.511	1.598	0.026	1.66	0.045	2.81
${\Phi_{\mathrm{edge}}}$	mWb	96.57	22.38	0.31	1.40	0.60	2.76
${B_{\mathrm{0}}}$	T	1.2650	0.1004	0.0029	2.97	0.0056	5.59
$R_{0,0}$	cm	205.73	2.26	0.10	4.60	0.25	11.10
$R_{0,1}$	cm	-6.185	0.592	0.030	5.03	0.070	11.94
$R_{1,-2}$	cm	-0.243	0.266	0.005	1.90	0.011	3.99
$R_{1,-1}$	cm	-1.026	1.104	0.018	1.65	0.032	2.94
$R_{1,0}$	cm	7.872	8.453	0.026	0.31	0.052	0.61
$R_{1,1}$	cm	3.453	3.706	0.015	0.40	0.024	0.65
$R_{1,2}$	cm	0.568	0.617	0.010	1.67	0.017	2.68
$R_{2,-2}$	cm	-0.0085	0.0139	0.0012	8.50	0.0020	14.06
$R_{2,-1}$	cm	-0.2812	0.3310	0.0037	1.11	0.0077	2.31
$R_{2,0}$	cm	0.581	0.681	0.019	2.76	0.048	7.01
$R_{2,1}$	cm	0.3298	0.3836	0.0043	1.13	0.0091	2.37
$R_{2,2}$	cm	0.2142	0.2504	0.0028	1.11	0.0061	2.44
$Z_{0,1}$	cm	-1.068	0.997	0.036	3.59	0.081	8.11
$Z_{1,-2}$	cm	-0.119	0.161	0.008	4.85	0.012	7.28
$Z_{1,-1}$	cm	-0.732	0.872	0.025	2.93	0.033	3.79
$Z_{1,0}$	cm	14.83	15.86	0.060	0.38	0.13	0.83
$Z_{1,1}$	cm	-3.208	3.425	0.030	0.88	0.049	1.43
$Z_{1,2}$	cm	-0.511	0.561	0.007	1.28	0.011	1.90
$Z_{2,-2}$	cm	0.0243	0.0345	0.0016	4.71	0.0033	9.50
$Z_{2,-1}$	cm	-0.095	0.108	0.005	4.69	0.011	10.19
$Z_{2,0}$	cm	-0.481	0.559	0.010	1.75	0.023	4.14
$Z_{2,1}$	cm	-0.273	0.328	0.004	1.26	0.009	2.64
$Z_{2,2}$	cm	-0.1163	0.1401	0.0021	1.51	0.0046	3.29