Geiger, J.

Otte, M.

Sengupta, A.

Werner, A.

English

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Fast recovery of vacuum configuration ofWEGA stellarator with error field effects

MPG.PuRe

Fast recovery of vacuum configuration of WEGA stellarator with errorfield effectsA.Sengupta, A.Werner, M.Otte, J.GeigerMax-Planck-Institut f̈ur Plasmaphysik, Euratom Association, Greifswald, Germany1. INTRODUCTIONFast recovery of magnetic configurations is a crucial issue for all fusion devices. Being inherentlysteady state devices stellarators need rapid and reliable techniques, which would monitor the evolutionof different physics parameters in real time, to be implemented with them.The WEGA stellarator is in operation at the Max-Planck-Institut für Plasmaphysik in Greifswald,Germany, since July 2001. A classical stellarator meant primarily for educational purpose, it has amajor radiusR=72 cm, a plasma minor radiusaeff ∼ 11 cm and a planar magnetic axis. The naturalconfiguration, generated with 40 toroidal field (TF) coils and 2 pairs of helical field (HF) coils, has a five-fold toroidal periodicity and stellarator symmetry. Two pairs of vertical field (VF) coils of the Helmholtztype, one pair above and the other below the torus, provide the horizontal shift of the magnetic axis. Theplasma (resonant) start-up is by means of a 2.45 GHz ECRH system at a magnetic field of 87.5 mT,while with a recently implemented 28 GHz ECRH the machine is operated at 0.5 Tesla.However, real configurations do not enjoy the periodicity and the symmetry as mentioned above. Incase of WEGA, due to an assumed horizontal misalignment between the TF and the HF coils [1], themagnetic field has error fields superimposed on it. As a result the stellarator symmetry breaks down andthe toroidal periodicity becomesn=1, meaning unique features for the entire torus.Encouraging results were obtained with Function Parametrization (FP) for W7-X parameter recoverywhere cubic polynomials were used to model vacuum configurations [2] in terms of external coil currentratios (CCR), and the finite-beta case [3] in terms of CCR, plasma pressure and toroidal plasma currents.Since error field effects were always neglected, it was now decided to test the applicability of the methodon a smaller device including the perturbations due to these effects. The confining magnetic field of astellarator is generated entirely by currents in external coils, and since this does not require a plasmaa magnetic configuration exists in vacuum. We analyzed the vacuum configurations of WEGA withmagnetic islands having perturbed modes, and recovering the important islands formed an importantpart of this work.2. PRINCIPLES OF FPThe basic principle of FP consists in getting a simple representation of a certain (dependent orresponse) variableY as a function of a set of independent (or predictor) variablesx̄ (which are assumedto be knowna priori), resulting from a statistical analysis on a large dataset containing these variables.The ranges of variables over what is also called the training dataset should encompass those expectedin the experiment as these statistical models learn the trends of variation within the data and are usuallypoor extrapolators. A typical FP model has the formyi = F (x̄) + ǫ where the functionF isan estimate of the dependency of the true valuey of Y on x̄, andǫ is a random error term which showsthat this representation is only an approximate one. Also known as “regression” in statistics, the abovefunction is set up using the principles of least squares whereby the coefficients ofF are estimated fromthe minimisation of a mean squared error of the estimateỹ of Y . An error analysis shows the kind of35th EPS Conference on Plasma Phys. Hersonissos, 9 - 13 June 2008 ECA Vol.32D, P-2.051  (2008)function that best fits a particularyi to the data. The quality of the fit can then be tested by another erroranalysis on an independent subset of the training dataset. These time-consuming offline steps precedethe ultimate application of the above equation, when new data forx̄ are fed in to calculate unknownyi. This process is very fast as it simply involves evaluatingF . Finally, a regression is termed as linear(non-linear) depending on whether the coefficients ofF are linear (non-linear) in the equation foryi.3. THE DATABASE OF WEGA CONFIGURA-0.5 1 1.5 2 2.5 3−0.0500.05iHFi VFFigure 1: WEGA configuration space inCCR.TIONSAs described above, setting up of an FP model re-lies on analysing a dataset showing the trends of varia-tion of all relevant variables that the model is to approx-imate. Physical parameters describing the WEGA con-figurations, such as the rotational transform on the mag-netic axis (ι-ax) and at the boundary (ι-b), the axis positionRax, the on-axis magnetic field strengthBax, the positionr(n/m)is and the widthw(n/m)is of important magnetic is-lands (in particular, those with toroidal to poloidal mode number ratios ofn/m = 1/5, 1/4 and 1/3),and the profile ofι- as a function ofr2eff (reff = an effective minor radius for labelling flux surfaces)were listed as the response variables for the regression. Definitions ofris andwis were the same as thosein [2]. Since magnetic configurations essentially depend on CCR (except for a scaling of the magneticfield whose strength varies explicitly with the current), the predictor variables (components ofx̄) wereiHF = IHF /ITF andiV F = IV F /ITF , wherei andI denote CCR and the coil currents, respectively. TheTF coil is maintained at a constant current in the experiment, and so it was in the dataset and formed thenormalisation parameter. For CCR, the ranges 0.5≤ iHF ≤ 2.7, and -0.05≤ iV F ≤ 0.05 were used ingenerating the data.Using a field line tracing code [4] whose inputs were−0.05 0 0.050.20.30.40.50.60.70.80.91iVFι axiHF = 2.70iHF = 1.90Figure 2: Variation of ι-ax vs iV F at twofixed values ofiHF .the currents in the TF, HF and VF coils of WEGA, wegenerated a dataset of 250 configurations. Figure 1 showsthe configuration space for CCR. Three aspects of thescatter plot are worth mentioning. First, the data pointsare not random but were generated systematically in a 2-D grid defined byiHF andiV F . Second, there is a voidin the bottom-left corner. This region, at smalliHF , cor-responds to ultra-low values of rotational transformι- sothat flux surfaces are not well-formed due to insufficienttwist in the magnetic field lines. As seen from figure 2,ι- increases asiV F (from negative) tends to zero and positive values at fixediHF , so the void does notcontinue upwards. Third, a clustering of points is seen in the region with 1.4≤ iHF ≤ 2.0. These pointswere generated in order to get magnetic islands of the desired modes in the configuration.4. STATISTICAL ANALYSIS RESULTSAs explained in section 2, the purpose of statistical analysis on a dataset is to develop global repre-35th EPS 2008; A.Sengupta et al. : Fast recovery of vacuum configuration of WEGA stellarator with error field... 2 of 4sentations of physical quantities in terms of the predictors, and a study of the error statistics leads to adecision on the best fit. These will now be described for the physical parameters of WEGA. From thedatabase, 150 configurations were used for setting up the FP models, making sure the parameter valuescovered all relevant experimental cases, and another 75 to test the quality of fit.a) Scalar parameter recoveryTable 1: Recovery statistics of scalar parameters of WEGA vacuum configurations using linear regression models.Parameter mean σ R2 measure of fit from models 100 (RMS error)/σ Remarksq-FP c-FP 4-FP 5-FP q-FP c-FP 4-FP 5-FPι-ax 0.31 0.22 0.9991 0.9997 0.9999 0.9999 2.79 1.63 0.93 0.80 1) For islandι-b 0.33 0.22 0.9927 0.9967 0.9973 0.9976 7.34 5.96 5.50 5.05 parametersaeff 8.91 cm 4.40 cm 0.8534 0.9020 0.9448 0.9544 38.29 30.00 23.50 21.40 q-FP outputsRax 70.58 cm 3.95 cm 0.8230 0.9360 0.9785 0.9931 42.00 25.30 14.94 7.59 anticipated toBax 0.30 T 0.018 T 0.8311 0.9509 0.9752 0.9889 41.10 22.16 15.75 11.11 be poor.r(1/3)is 8.09 cm 2.44 cm — 0.9606 0.9766 0.9934 — 19.85 15.30 8.12 2) For 1/4, 1/5w(1/3)is 2.05 cm 1.15 cm — 0.5207 0.8572 0.9099 — 69.23 37.79 30.00 regressingr2isr(1/4)is 8.58 cm 2.06 cm — 0.9527 0.9924 0.9974 — 21.74 8.74 5.10 improved fit.w(1/4)is 1.08 cm 0.45 cm — 0.8690 0.9323 0.9790 — 36.19 26.03 14.50 3) Fit of ι-br(1/5)is 7.64 cm 2.57 cm — 0.9979 0.9991 0.9991 — 4.28 2.61 2.49 saturated withw(1/5)is 0.59 cm 0.41 cm — 0.9353 0.9431 0.9449 — 24.39 21.95 19.51 c-FP model.The regressions were done and tested with a quadratic (q-FP),a cubic (c-FP), a 4th order (4-FP)and a 5th order (5-FP) polynomial in (iHF , iV F ) including interaction terms, that involved 6, 10, 15and 21 model coefficients, respectively. In our previous studies [3, 4] on W7-X configurations, a c-FPmodel was always found to be necessary and sufficient. For WEGA a significant improvement in theregression accuracy was observed, for all the configuration parameters regressed, when a 4-FP modelwas used. The error statistics are tabulated in Table 1, from which it is also clear that forRax, Bax, andthe parameters of at least two island modes there was a further enhancement in the quality of regressionwith a 5-FP model. The necessity of higher order regression terms in the models suggests quite acomplex dependence of the physical parameters of WEGA configuration on the coil currents. Thismay be attributed to the error field effects, when the lack of symmetry and a reduction of the naturalperiodicity introduced the non-linearities.However, for some of the parameters other causes0.5 1 1.5 2 2.5 3606570758085iHFRax (cm)  iVF = 0.03iVF = 0.0iVF = −0.03Figure 3: Variation of magnetic axis posi-tion with iHF at three fixed values ofiV F .were also likely. For example, from figure 3 we findthe behaviour ofRax with iHF for positive, negative andzero vertical field as being quite complicated in an over-all sense, so a single model that approximates this vari-ation is also expected to be complex. For this and theother parameters which showed progressively better fitseven up to 5-FP, we then decided to test the feasibility ofnon-linear regression using rational functions. The pur-pose was to study the usefulness of these functions fromthe viewpoint of accuracy as compared to, e.g., the 5-FPmodel. Non-linear regressions involve iterative convergence of the error function to its minimum inthe hyperspace of the model coefficients, and the Levenberg-Marquardt scheme was used for that. Theresults were very encouraging, and the rational functions were finally settled with as the best fit.DenotingiHF by x1, iV F by x2, root-mean-squared error byǫrms and standard deviation byσ, the35th EPS 2008; A.Sengupta et al. : Fast recovery of vacuum configuration of WEGA stellarator with error field... 3 of 4best fit rational functions for some of the relevant configuration parameters wereRax =(a0+a1x1+a2x2+a3x21+a4x22)(a5+a6x1+a7x2+a8x21+a9x22+a10x1x2+a11x31+a12x32)with 13 coefficients, givingǫrms = 1.1 mm,anR2 measure of fit of 0.9992 (defined as the fraction of the total variance in the regressed parameterexplained by the chosen model) and a percentage spread error (ǫperc = 100ǫrms/σ) of 2.84,Bax =(b0+b1x1+b2x2+b3x21+b4x22)(b5+b6x1+b7x2+b8x21+b9x22+b10x31)with 11 coefficients, givingǫrms = 0.0007 T, anR2 measureof fit of 0.9988 andǫperc = 3.48,r(1/3)is =(c0+c1x1+c2x2+c3x21+c4x22)(c5+c6x1+c7x2+c8x21+c9x22)with 10 coefficients, givingǫrms = 1.8 mm, anR2 measure offit of 0.9932 andǫperc = 8.27, andw(1/3)is =(d0+d1x1+d2x2+d3x21+d4x22+d5x31+d6x32)(d7+d8x1+d9x2+d10x21+d11x22+d12x32)with 13 coefficients, givingǫrms = 2.1 mm, anR2measure of fit of 0.9515 andǫperc = 22.For Rax andBax there was a remarkable improvement in accuracy over 5-FP, while for the islandparameters the 5-FP results were reproduced, by the use of rational functions with 8-10 coefficients less.b) Profile parameter recoveryAs already listed, the only relevant profile quantity for WEGA is that ofι- which was consideredto be a function ofr2eff (apart from CCR) because physically the profile quantities are functions of thetoroidal flux which varies asr2eff . The ι--profile was best fitted with a radial polynomial of the formι-(iHF , iV F , r2eff ) = p0 + p1r2eff + p2(r2eff )2 where CCR (iHF , iV F ) are contained withinthe coefficientspi. When CCR were combined in a “mixed” cubic form, the uncertainty ofι- ver theentire profile was±0.0025; when the combination was in a “mixed” 4th order form, the uncertaintywas±0.0015. So the latter was chosen as the best fit, consistent with the scalar parameter results. Itmay be noted that the radial function is smooth and so cannot represent the discontinuity in theι--profildue to islands. Also, it allows extrapolations beyond the minor radius of an island chain in a givenconfiguration and will return a consistent value ofι- for anreff even if no good flux surfaces exist in thereal configuration for thatreff . To avoid this problem the regression ofι--profile must be combined withthat ofris and/oraeff in order to restrict the valid range ofreff for the regression functions.5. CONCLUSIONSThe FP on WEGA, performed on a small device with a very modest number of independently vari-able predictors, has shown the inherent non-linearities in the dependencies of the physical state of theWEGA magnetic configurations, with error field perturbations, on the external coil currents. The best-fit statistical models have been either 4th order polynomials with linear regression, or rational functionswith non-linear regression. That the latter improved upon the fit further demonstrated the complexi-ties of some of the dependencies, even though the model sizes remained reasonably small. Our resultswere very encouraging, including the challenging issue of the island width whose determination in thedatabase may itself be erroneous due to problems in detecting the outer separatrix. The FP functions arenow in the process of being implemented in software to run with the control system.REFERENCES:[1] M.Otte, J.Lingertat, 29th EPS Conference on PPCF, Montreaux, ECA, Vol.26B, P-5.036 (2002)[2] A.Sengupta, et al,Nuclear Fusion44 (2004) 1176[3] A.Sengupta, et al,Plasma Phys. Contr. Fusion49 (2007) 649[4] A.Werner, IPP-Greifswald, Germany: private communications35th EPS 2008; A.Sengupta et al. : Fast recovery of vacuum configuration of WEGA stellarator with error field... 4 of 4

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Fast recovery of vacuum configuration ofWEGA stellarator with error field effects

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