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\label{sec:conclusions}  Two new codes---FREEBIE and VacTH---have been successfully set up on COMPASS, which enabled to perform an extensive cross-benchmarking and validation of free-boundary equilibrium tools. We show that FREEBIE can predict equilibria that are consistent with EFIT++ reconstructions from experimental data. FREEBIE model equilibria, either with linear $p'$ and $FF'$ profiles or with pressure profiles from Thomson scattering diagnostic, then served to assess the credibility of EFIT++ reconstructions.   % This has not been up to now possible.  We show that magnetic reconstruction EFIT++ with linear $p'$ and $FF'$ features a relatively good accuracy of 1 -- 2 cm in the plasma shape reconstruction but introduces systematic errors both in the shape and in internal plasma parameters, such as $W$, $l_{\mathrm i}$, $\beta_{\mathrm p}$ or $q_0$. The reconstruction properties can be significantly improved by using quadratic $p'$ for elongated and divertor plasmas, which removes the systematic error and also improves the LCFS reconstruction. EFIT++ converges in 100~\% cases in this regime.  diff --git a/JUrban_SOFT2014.pdf b/JUrban_SOFT2014.pdf  index 461659d..44bf19e 100644  Binary files a/JUrban_SOFT2014.pdf and b/JUrban_SOFT2014.pdf differ diff --git a/JUrban_SOFT2014.tex b/JUrban_SOFT2014.tex  index ad95ba0..26b8141 100644  --- a/JUrban_SOFT2014.tex  +++ b/JUrban_SOFT2014.tex 
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\usepackage{booktabs}  % \usepackage[firstpage]{draftwatermark}  %\journal{Nuclear Physics B}  \journal{Fusion Engineering and Design}  \begin{document} 
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\author[label1]{M.~Komm}  \author[label2]{I.~Lupelli}   \author[label1,label5]{M.~Peterka}   % \address[label1]{Institute of Plasma Physics AS CR, v.v.i., Association EURATOM / IPP.CR, Za~Slovankou 3, 182 00 Praha 8, Czech Republic}  \address[label1]{Institute of Plasma Physics ASCR, Za~Slovankou Za Slovankou  3, 182 00 Praha 8, Czech Republic} \address[label2]{CCFE, Culham Science Centre, Abingdon, Oxfordshire, UK}  \address[label3]{CEA, IRFM, F-13108 Saint Paul Lez Durance, France}  \address[label4]{Laboratoire J.A. Dieudonné, UMR 7351, Université de Nice Sophia-Antipolis, Parc Valrose, 06108 
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\address[label5]{Department of Surface and Plasma Science, Faculty of Mathematics and Physics, Charles University in Prague, V Hole\v{s}ovi\v{c}k\'ach 2, 180~00 Praha 8, Czech Republic}  \input{Abstract.tex}  % \begin{abstract} %% Text of abstract   % Various MHD (magnetohydrodynamic) equilibrium tools, some of which being recently developed or considerably updated, are used on the medium-size COMPASS tokamak [R. Pánek et al., Czech J Phys 56, B125, 2006]. MHD equilibrium is a fundamental property of the tokamak plasma, whose knowledge is required for many diagnostics and modelling tools. Proper benchmarking and validation of equilibrium tools is thus key for interpreting and planning tokamak experiments. We present here benchmarks and comparisons to experimental data of the EFIT++ reconstruction code [L.C. Appel et al., to be submitted to Nucl. Fusion], the free-boundary equilibrium code FREEBIE [J.-F. Artaud, S.H. Kim, EPS 2012, P4.023], and a rapid plasma boundary reconstruction code VacTH [B. Faugeras et al., PPCF 2014, accepted]. We demonstrate that FREEBIE can calculate the equilibrium and corresponding poloidal field (PF) coils currents for given plasma parameters. Both EFIT++ and VacTH can reconstruct equilibria generated by FREEBIE from synthetic diagnostic data (including an artificial noise) and hence might be suitable for real-time control. Optimum reconstruction parameters are estimated; in addition, possible enhancements using more diagnostics are discussed and simulated using synthetic diagnostics. FREEBIE can also calculate the temporal evolution of the poloidal field coils currents for a whole plasma scenario.  % \end{abstract}    \begin{keyword}  %% keywords here, in the form: keyword \sep keyword  tokamak \sep equilibrium \sep COMPASS %% PACS codes here, in the form: \PACS code \sep code  \PACS 52.55.Fa \sep 52.65.Kj  %% MSC codes here, in the form: \MSC code \sep code 
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The work of J. Urban was supported by Czech Science Foundation grant 13-38121P.  The work at IPP ASCR was supported by MSMT \#LM2011021.   % \appendix  % \input{Appendix.tex}  % \section*{References} % (fold)  \bibliographystyle{model1-num-names}  \bibliography{bibliography/biblio}{}  % \bibliographystyle{unsrt}  % \bibliographystyle{abbrvnat}  % \usepackage{natbib}  % \bibliographystyle{chicago} \bibliographystyle{elsarticle-num}  \bibliography{biblio}{}  \end{document}  \endinput  diff --git a/RZstats.pdf b/RZstats.pdf  new file mode 100644  index 0000000..255c8e0  Binary files /dev/null and b/RZstats.pdf differ diff --git a/RZstats_BW.pdf b/RZstats_BW.pdf  new file mode 100644  index 0000000..868da3d  Binary files /dev/null and b/RZstats_BW.pdf differ diff --git a/Results.tex b/Results.tex  index b79b0f3..4c8faac 100644  --- a/Results.tex  +++ b/Results.tex 
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\begin{figure*}[!htb]  \centering %\begin{center}  \hfill{}  \includegraphics[width=18cm]{figures/example_6962.pdf} \includegraphics[width=18cm]{example_6962.pdf}  \hfill{}  %\end{center}  \caption{EFIT++ reconstructed pressure profiles and contours of $\bar\psi=\left(0.5,1\right)$ and VacTH LCFS from FREEBIE data, shot 6962 with Thomson scattering pressure profiles. EFIT++ parameters: $n_\mathrm{mp} = 16$, $n_\mathrm{fl} = 4$, $n_{FF'} = 1$. VacTH parameters: $n_\mathrm{mp} = 8$, $n_\mathrm{fl} = 16$, $n_P = n_Q = 4$. 3~\% random input data noise is used in the case of VacTH and EFIT++ with $n_{p'} = 2$, zero noise otherwise.} 
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VacTH reconstructs the LCFS reasonably, even with noisy inputs. Although a bending artefact emerge on the inboard side. Similar artefacts can be observed in other VacTH results as well. This is probably a result of the specific COMPASS configuration as such a behaviour was not observed in the case of WEST \cite{vacthref}. $n_P = n_Q = 4$ is used in this case as these values are minimum for reasonable VacTH results, while higher values are too sensitive to the input noise.  % A comparison of plasma shapes for shot 4275 is shown in Fig. \ref{fig:ex4275}. Numerical values of reconstruction errors are presented in Table \ref{table:ex4275}. We can observe a very good agreement between the original equilibrium and the reconstructed shapes. In this case, FREEBIE was using linear $p'$ and $FF'$ polynomials so that the EFIT++ model agrees with the target data. VacTH uses 8 magnetic probes and 16 flux loops. As we discuss later, flux loops are essential for reliable VacTH results. Even global kinetic properties are well reconstructed in EFIT++; the largest error around 10~\% is in $l_{\mathrm i}$ (i.e. basically in the toroidal current density profile).  % \begin{table*}  % \centering  % \small  % \begin{tabular}{lrrrrrrrrrrrrr}  % \toprule  % code & time & $\left| \Delta R_{\mathrm in} \right|$ & $\left| \Delta R_{\mathrm out} \right|$ & $\left| \Delta Z_{\mathrm min} \right|$ & $\left| \Delta Z_{\mathrm max} \right|$ & $\delta \kappa$ & $E_\mathrm{mp}$ & $E_\mathrm{fl}$ & $\delta W$ & $\delta l_{\mathrm i}$ & $\delta \beta_{\mathrm p}$ & $\delta q_0$ & $\delta q_{95}$ \\  % \midrule  % EFIT++ & 0.97 & 0 & 0.002 & 0.001 & 0.001 & 0.003 & 0.001 & 0.0009 & 0.04 & 0.09 & 0.03 & 0.03 & 0.007 \\  % EFIT++ & 0.99 & 4e-05 & 9e-05 & 0.001 & 0.001 & 0.004 & 0.001 & 0.0006 & 0.04 & 0.09 & 0.04 & 0.02 & 0.005 \\  % EFIT++ & 1.02 & 0.0005 & 0.0002 & 0.002 & 0.002 & 0.008 & 0.002 & 0.002 & 0.02 & 0.1 & 0.02 & 0.02 & 0.009 \\  % EFIT++ & 1.05 & 0.001 & 0.0008 & 4e-05 & 0.0003 & 0.004 & 0.003 & 0.005 & 0.01 & 0.07 & 0.02 & 0.01 & 0.009 \\  % EFIT++ & 1.1 & 0.001 & 0.0005 & 0.005 & 0.0002 & 0.005 & 0.004 & 0.002 & 0.04 & 0.09 & 0.03 & 0.02 & 0.05 \\  % VacTH & 0.97 & 0 & 0.001 & 0.0006 & 0.0002 & 0.002 & 8e-07 & 7e-05 & -- & -- & -- & -- & -- \\  % VacTH & 0.99 & 4e-05 & 0.0004 & 0.001 & 0.0008 & 0.004 & 4e-07 & 0.0001 & -- & -- & -- & -- & -- \\  % VacTH & 1.02 & 0.002 & 0.0008 & 0.005 & 0.002 & 0.01 & 2e-06 & 0.002 & -- & -- & -- & -- & -- \\  % VacTH & 1.05 & 0.006 & 0.002 & 5e-05 & 0.002 & 0.01 & 3e-06 & 0.02 & -- & -- & -- & -- & -- \\  % VacTH & 1.1 & 0.005 & 0.002 & 0.002 & 0.003 & 0.02 & 2e-06 & 0.003 & -- & -- & -- & -- & -- \\  % \bottomrule  % \end{tabular}  % \caption{Errors for the same cases as in Fig. \ref{fig:ex4275}.}  % \label{table:ex4275}  % \end{table*}  % \begin{figure*}  % \centering %\begin{center}  % \hfill{}  % \includegraphics[width=18cm]{figures/example_4275.pdf}  % \hfill{}  % %\end{center}  % \caption{Contours of $\bar\psi=\left(0.25,0.5,0.75,1\right)$, reconstruction from FREEBIE data, shot 4275. EFIT++ parameters: $n_\mathrm{mp} = 16$, $n_\mathrm{fl} = 4$, $n_{p'} = n_{FF'} = 1$. VacTH parameters: $n_\mathrm{mp} = 8$, $n_\mathrm{fl} = 16$, $n_P = n_Q = 5$. $E_\mathrm{mp, fl}$ is the relative error of the reconstructed magnetic probes and flux loops values, respectively.}  % \label{fig:ex4275}  % \end{figure*}  % The second shot for the comparison is 6962, which has been chosen because Thomson scattering (TS) profiles are available. In Fig. \ref{fig:ex6962} we show reconstructions for more realistic equilibria and an artificial noise in the synthetic diagnostic signals. TS pressures were used in FREEBIE equilibria, which course no longer feature linear $p'$ and $FF'$ profiles.   % It is apparent that the reconstruction is not as good as in the previous case. Nevertheless, the plasma shape is well reconstructed with EFIT++ and VacTH, except for the last time slice in which VacTH yields too large plasma in the upper part.  % Expectedly, EFIT++ reconstruction with magnetic data only cannot reliably reconstruct kinetic plasma parameters, such as the stored energy or $q_0$. Rather unexpected is a relatively good agreement of $l_\mathrm{i}$. However, this agreement is compensated by a large error of $\beta_{\mathrm p}$ so that the quantity $\beta_{\mathrm p} + l_{\mathrm i}/2$ remain within a 10~\% error bar. Notable are large errors (up to 40~\%) in the stored energy, which are mainly caused by incompatible pressure profiles.  % \begin{table*}  % \centering  % \small  % \begin{tabular}{lrrrrrrrrrrrrr}  % \toprule  % % code & time & $\Delta R_{\mathrm in}$ & $\Delta R_{\mathrm out}$ & $\Delta Z_{\mathrm min}$ & $\Delta Z_{\mathrm max}$ & $\delta \kappa$ & $E_\mathrm{mp}$ & $E_\mathrm{fl}$ & $\delta W$ & $\delta \left(\beta_{\mathrm p} + l_{\mathrm i}/2 \right)$ & $\delta q_0$ & $\delta q_{95}$ \\  % code & time & $\left| \Delta R_{\mathrm in} \right|$ & $\left| \Delta R_{\mathrm out} \right|$ & $\left| \Delta Z_{\mathrm min} \right|$ & $\left| \Delta Z_{\mathrm max} \right|$ & $\delta \kappa$ & $E_\mathrm{mp}$ & $E_\mathrm{fl}$ & $\delta W$ & $\delta l_{\mathrm i}$ & $\delta \beta_{\mathrm p}$ & $\delta q_0$ & $\delta q_{95}$ \\  % \midrule  % EFIT++ & 980 & 0 & 0.009 & -0.003 & 0.003 & 0.01 & 0.04 & 0.01 & 0.1 & 0.3 & 0.1 & 0.5 & 0.02 \\  % EFIT++ & 1014 & 0.0004 & -0.0005 & -0.001 & 0.001 & 0.002 & 0.01 & 0.01 & 0.006 & 0.1 & 0.003 & 0.07 & 0.003 \\  % EFIT++ & 1080 & -0.002 & -0.002 & -0.002 & 0.002 & 0.003 & 0.01 & 0.02 & 0.2 & 0.1 & 0.2 & 0.06 & 0.02 \\  % EFIT++ & 1147 & -0.002 & -0.0006 & -0.007 & -0.002 & 0.006 & 0.01 & 0.02 & 0.1 & 0.2 & 0.1 & 0.2 & 0.01 \\  % EFIT++ & 1214 & -0.004 & 0.007 & 0.006 & -0.004 & 0.04 & 0.03 & 0.01 & 0.2 & 0.3 & 0.2 & 0.03 & 0.001 \\  % VacTH & 980 & 0 & -0.01 & 0.01 & -0.02 & 0.04 & 0.0002 & 0.01 & -- & -- & -- & -- & -- \\  % VacTH & 1014 & 0.006 & -0.003 & -0.01 & -0.001 & 0.02 & 7e-05 & 0.02 & -- & -- & -- & -- & -- \\  % VacTH & 1080 & 0.01 & -0.003 & 0.004 & -0.003 & 0.01 & 3e-05 & 0.009 & -- & -- & -- & -- & -- \\  % VacTH & 1147 & 0.02 & -0.003 & -0.002 & 0.002 & 0.04 & 5e-05 & 0.01 & -- & -- & -- & -- & -- \\  % VacTH & 1214 & 0.01 & 0.001 & -0.02 & 0.02 & 0.07 & 0.0001 & 0.01 & -- & -- & -- & -- & -- \\  % \bottomrule  % \end{tabular}  % \caption{Errors for the same cases as in Fig. \ref{fig:ex6962}.}  % \label{table:ex6962}  % \end{table*}  \begin{figure*}[!htb]  \centering %\begin{center}  \hfill{}  \includegraphics[width=18cm]{figures/RZstats.pdf} \includegraphics[width=18cm]{RZstats.pdf}  \hfill{}  %\end{center}  \caption{Absolute errors in LCFS extents for convergent cases. EFIT++ results in the first row, VacTH in the second row. S1 denotes linear $p'$ and $FF'$ in EFIT++ as well as FREEBIE, s2 denotes TS pressure profiles in FREEBIE and $n_{p'} = n_{FF'} = 1$ in EFIT++, s3 denotes TS pressure profiles in FREEBIE and $n_{p'} = 2$ in EFIT++. Full lines show the means. VacTH parameters are $n_P=n_Q=4$, ``8 mp, 16 fl'' denotes $n_\mathrm{mp}=8$, $n_\mathrm{fl}=16$. Input error is calculated as an average of $I_{\rm{p}}$, magnetic probes and flux loops values. Zero input error data are scattered for a better visibility.} 
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\begin{figure*}[!htb]  \centering %\begin{center}  \hfill{}  \includegraphics[width=18cm]{figures/kinetic_stats_opt.pdf} \includegraphics[width=18cm]{kinetic_stats_opt.pdf}  \hfill{}  %\end{center}  \caption{Internal plasma parameters relative errors for EFIT++ reconstructions using $n_{p'}=n_{FF'}=1$ in the top row and more optimized $n_{p'}$ in the bottom row. Zero input error data are scattered for a better visibility.} 
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Absolute errors of the reconstructed LCFS extents for convergent cases from the scan are shown in Fig. \ref{fig:RZstats}. We can observer that the LCFS reconstructed with EFIT++ for target linear $p'$ and $FF'$ profiles (selection 1) are within 1~cm errors. There are, however, cases with up to 3 cm errors in $Z_{\rm{min}}$ for the more realistic TS pressure profiles (selection 2) if $n_{p'}=1$ is used. This error can be reduced by using $n_{p'}=2$. Input errors do not pose major difficulties for EFIT++.   VacTH is performing reasonably well for its most favourable diagnostic set of 8 magnetic probes and 16 flux loops and $n_P = n_Q = 4$. With a higher number of harmonics or with less flux loops, VacTH becomes unreliable and yields significant errors. Unfortunately, only 4 flux loops are currently available on COMPASS. In fact, it is easier for VacTH to fit magnetic probes than flux loops  % while magnetic probes do not fully determine the magnetic flux.   % This is also apparent in $E_\mathrm{mp}$ and $E_\mathrm{fl}$ values in Tables \ref{table:ex4275} and \ref{table:ex6962}.   An additional optimization of the fitting weights or algorithm is probably needed. The current behaviour might be quite anti-intuitive as VacTH performs significantly worse with 16 flux loops and 16 or 64 magnetic probes in comparison to 16 flux loops and only 8 magnetic probes.   EFIT++ internal plasma parameters reconstruction results are shown in Fig. \ref{fig:kinetic_stats}. It shows that purely magnetic reconstruction with $n_{p'}=n_{FF'}=1$ introduces (except for $I_{\rm{p}}$) a systematic error for realistic pressure profiles, i.e. for plasmas that do not have the same profile parametrization.  
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\begin{figure}  \centering %\begin{center}  \hfill{}  \includegraphics[width=8cm]{figures/convergence_ratio_6962.pdf} \includegraphics[width=8cm]{convergence_ratio_6962.pdf}  \hfill{}  %\end{center}  \caption{Ratio of converged cases to all cases, shot 6962, TS profiles. Opt refers to optimized code parameters.} 
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