Free-boundary Equilibrium Code Research Articles

Free boundary equilibrium determination of HL-3 discharges

In 2022–2023, HL-3 (previously known as HL-2M) realized the divertor discharge after its first plasma campaign. To analyze equilibrium configurations of HL-3, a free boundary equilibrium code FEEQS, based on the finite element method, has been used in recent experiments. Free boundary equilibrium determinations are carried out for discharges performed in both limiter and divertor configurations. Compared with the experimental diagnostic data from fast camera photos and saturated ion current signals.etc, various time evolution of plasma parameters are well reproduced, such as plasma shape, currents in the poloidal field coils, strike-point displacement. The use of FEEQS successfully supported the operations in HL-3, and physics study as well, which laid a foundation for the application of free boundary equilibrium integration simulation in the next step.

Numerical analysis for the free-boundary current reversal equilibrium in the AC plasma current operation in a tokamak

Abstract In recent decades, tokamak discharges with zero total toroidal current have been reported in tokamak experiments, and this is one of the key problems in alternating current (AC) operations. An efficient free-boundary equilibrium code is developed to investigate such advanced tokamak discharges with current reversal equilibrium configuration. The calculation results show that the reversal current equilibrium can maintain finite pressure and also has considerable effects on the position of the X-point and the magnetic separatrix shape, and hence also on the position of the strike point on the divertor plates, which is extremely useful for magnetic design, MHD stability analysis, and experimental data analysis etc. for the AC plasma current operation on tokamaks.

Magnetic equilibrium optimisation and divertor integration in spherical tokamak reactors

Managing the heat exhaust presents one of the main challenges in the design of a spherical tokamak reactor. The development of a robust exhaust solution is intrinsically linked to the optimisation of the plasma equilibrium, together with a compatible poloidal field (PF) system. A multidisciplinary design framework should balance a host of often conflicting physics and engineering requirements and identify a set of self-consistent constraints. This work employs an iterative approach which integrates the core plasma scenario and divertor magnetic topology, ensuring compatibility with the available technology. The aim of this paper is to develop and assess a number of optimal magnetic configurations consistent with the UKAEA STEP Prototype Plant designs. Firstly, we use the Fiesta free-boundary equilibrium code to optimise the global configuration for a chosen double-null scenario, employing both standard and alternative exhaust-handling schemes. We then analyse the local scrape-off layer (SOL) magnetic topology in order to preliminarily assess the anticipated divertor performance. We discuss the advantages and disadvantages of each configuration in terms of the selected divertor metrics, the effects on the core plasma scenario, engineering feasibility, and the implications for whole reactor design.The inner divertor proves to be the most challenging issue in a compact device such as STEP. The small target area and reduced SOL width at the small strike-point radius lead to excessive heat loads beyond the power-handling capacity of standard concepts. We propose an alternative inner X-divertor configuration and compare it with the standard divertor optimised for maximal connection length and poloidal flux expansion. The objective is to achieve the optimal global configuration with a realistic coil set, taking into account space constraints and material limitations. We discuss the feasibility of each considered exhaust solution in STEP, where the PF shaping coils can be placed either in unfavourable locations outside of the toroidal field (TF) coil or mounted inside the TF and operating near their engineering limits.

3-D Electromagnetic Model of the WEST Tokamak Sector During Plasma Disruptions

WEST is an acronym for tungsten (W) Environment in Steady-state Tokamak. WEST derives from the upgrade of the Tore Supra tokamak, moving from a limiter configuration with carbon plasma-facing components (PFCs) to a divertor configuration with W PFCs. It is located in the south of France (CEA Cadarache) and it assumes a key role as a testing facility for ITER. A computational analysis campaign has been carried out to investigate electromagnetic (EM) loads during plasma disruptions in WEST, the first one with the actual WEST experimental configuration. The applied methodologies as well as the obtained results have been benchmarked and could be employed to assess the behavior of WEST and analogous tokamak machines during plasma disruptions. Specifically, the research campaign has focused on a 20° sector of the WEST tokamak and it has been carried out by means of the ANSYS APDL v19.2 code. The input disruptions have been simulated with the free-boundary plasma equilibrium code FEEQS, while the obtained results mainly relate to the evaluation of eddy currents as well as the magnetic field distribution inside the tokamak. Furthermore, two different configurations of the WEST sector have been analyzed. The analyses in Configuration A are characterized by the same axisymmetric geometrical domain and electrical resistances as in the FEEQS analysis. As a consequence, they allow making a direct comparison between FEEQS and APDL results, benchmarking the model. On the other hand, the analyses of Configuration B are based on an improved configuration, characterized by more realistic 3-D geometry, electrical properties, and loads. Therefore, these last calculations can be employed to evaluate the magnetic field distribution inside the tokamak as well as the EM loads of in-vessel components.

A Galerkin method for the weak formulation of current diffusion and force balance in tokamak plasmas

This paper presents a new numerical approach to the simulation of plasma equilibrium evolution in tokamaks. It is based on an unusual coupling formulation of current diffusion and axi-symmetric equilibrium equations that does not require the use of flux-surface averages and allows for fast Newton-type iterations schemes. A detailed derivation of the coupled system of non-linear equations and its discretization is provided. Numerical examples, based on an implementation in the free-boundary equilibrium code FEEQS, highlight feasibility and applicability.

Tokamak discharge simulation coupling free-boundary equilibrium and plasma model with application to JT-60SA

In order to simulate a full tokamak discharge, a fast integrated tokamak modeling tool for scenario design, METIS, is coupled with a quasi-static free-boundary magnetic equilibrium code, FEEQS. The first code is able to compute at each time the plasma equilibrium, the current density and plasma pressure profiles. The generated outputs are iteratively used by the second code that computes the poloidal field coils currents needed in order to obtain a given plasma shape. This is used to assess the feasibility of scenarios designed by integrated modeling simulations. This approach is much faster with respect to full simulators, equipped with specific feedback controllers, and can be regarded as complementary or preparatory to much more time-consuming scenario control optimisation. The results obtained with the equilibrium code are benchmarked with data already available and computed using the TOSCA code. To illustrate how the coupled codes work, the optimization of one of the reference scenarios of the JT-60SA tokamak is carried out. The use of this new simulation tool for tokamak scenario design is discussed.

First demonstration of real-time kinetic equilibrium reconstruction on TCV by coupling LIUQE and RAPTOR

Kinetic equilibrium reconstruction (KER) in tokamaks is the solution of the free-boundary MHD equilibrium that best fits the external magnetic measurements and the internal plasma profiles as imposed from modeling and/or available internal measurements. In this paper, for the first time, the KER has been performed in real-time by coupling the free-boundary equilibrium code LIUQE to the 1.5D transport code RAPTOR, during a TCV discharge. The transport code is used as a dynamic state observer evolving the current diffusion equation, the electron heat and particle diffusion equations and correcting the prediction with the measurements available in real-time, making use of the extended Kalman filter (EKF) method. The simple coupling strategy, based on a matching of the free functions ( and ) on the right-hand-side of the Grad Shafranov equation, is shown to be effective in providing consistency of pressure and current density profiles between the two codes. We also show that, despite this highly non-linear system, this technique does not require significant additional computational time with respect to the running times of the two independent codes and does not require a tight coupling of the two codes allowing therefore their independent development. The KER is compared to standard real-time equilibrium reconstruction, performed routinely in TCV, where only the external magnetic measurements are taken into account. The following cases are considered: flat central ne, current density broadening due to applied electron cyclotron current drive and neoclassical tearing mode causing a temperature drop. The KER reproduces the expected changes of the internal profiles, tracking their dynamic evolution in their physical timescales.

“PROCESS”: Systems studies of spherical tokamaks

With a reduced aspect ratio, spherical tokamaks have a number of attractive features for a fusion power plant. This can be studied using systems codes, which allow for the rapid conceptual study of power plants covering everything, from the plasma through to electricity generation. In this paper, we describe models in the systems code process that have been added specifically for spherical tokamaks. Within process an alternative relation for the plasma current is included which accounts for the increased ratio of Ip/aB. We have tested this against a series of equilibria created with the free boundary equilibrium code fiesta, and additionally performed our own fit. We also outline the engineering changes that can be made to the device and describe a water-cooled copper centrepost model. To test our models we recreate the published designs for the Fusion Nuclear Science Facility (FNSF) and a High Temperature Superconducting Pilot Plant (HTS-PP) and find good agreement. We conclude by highlighting the efficiencies needed to produce net electricity from small fusion devices.

Automatic identification of the plasma equilibrium operating space in tokamaks

In order to identify the plasma equilibrium operating space for future tokamaks, a new objective function is introduced in the inverse static free-boundary equilibrium code FEEQS.M. This function comprises terms which penalize the violation of the central solenoid and poloidal field coils limitations (currents and forces). The penalization terms do not require any weight tuning. Hence, this new approach automatizes to a large extent the identification of the operating space. As an illustration, the new method is applied on the ITER 15 and 17 MA inductive scenarios, and similar operating spaces compared to previous works are found. These operating spaces are obtained within a few (∼ 3) hours of computing time on a single standard CPU.

Investigation of the n = 1 resistive wall modes in the ITER high-mode confinement

The n = 1 resistive wall mode (RWM) stability of ITER high-mode confinement is investigated with bootstrap current included for equilibrium, together with the rotation and diamagnetic drift effects for stability. Here, n is the toroidal mode number. We use the CORSICA code for computing the free boundary equilibrium and AEGIS code for stability. We find that the inclusion of bootstrap current for equilibrium is critical. It can reduce the local magnetic shear in the pedestal, so that the infernal mode branches can develop. Consequently, the n = 1 modes become unstable without a stabilizing wall at a considerably lower beta limit, driven by the steep pressure gradient in the pedestal. Typical values of the wall position stabilize the ideal mode, but give rise to the ‘pedestal’ resistive wall modes. We find that the rotation can contribute a stabilizing effect on RWMs and the diamagnetic drift effects can further improve the stability in the co-current rotation case. But, generally speaking, the rotation stabilization effects are not as effective as the case without including the bootstrap current effects on equilibrium. We also find that the diamagnetic drift effects are actually destabilizing when there is a counter-current rotation.

Identification of the plasma boundary shape and position in the Damavand tokamak

A series of experiments and numerical calculations have been done on the Damavand tokamak for accurate determination of equilibrium parameters, such as the plasma boundary position and shape. For this work, the pickup coils of the Damavand tokamak were recalibrated and after that a plasma boundary shape identification code was developed for analyzing the experimental data, such as magnetic probes and coils currents data. The plasma boundary position, shape and other parameters are determined by the plasma shape identification code. A free-boundary equilibrium code was also generated for comparison with the plasma boundary shape identification results and determination of required fields to obtain elongated plasma in the Damavand tokamak.

Towards Automated Magnetic Divertor Design for Optimal Heat Exhaust

Avoiding excessive structure heat loads in future fusion tokamaks is regarded as one of the greatest design challenges. In this paper, we aim at developing a tool to study how the severe divertor heat loads can be mitigated by reconfiguring the magnetic confinement. For this purpose, a free boundary equilibrium code is integrated with a plasma edge transport code to work in an automated fashion. Next, a practical and efficient adjoint based sensitivity calculation is proposed to evaluate the sensitivities of the integrated code. The sensitivity calculation is finally applied to a realistic test case and compared with finite difference sensitivity calculations.

Validation of equilibrium tools on the COMPASS tokamak

Various MHD (magnetohydrodynamic) equilibrium tools, some of which being recently developed or considerably updated, are used on the COMPASS tokamak at IPP Prague. 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 (Appel et al., 2006), the free-boundary equilibrium code FREEBIE (Artaud and Kim, 2012), and a rapid plasma boundary reconstruction code VacTH (Faugeras et al., 2014). We demonstrate that FREEBIE can calculate the equilibrium and corresponding poloidal field (PF) coils currents consistently with EFIT++ reconstructions from experimental data. Both EFIT++ and VacTH can reconstruct equilibria generated by FREEBIE from synthetic, optionally noisy diagnostic data. Hence, VacTH is suitable for real-time control. Optimum reconstruction parameters are estimated.

Integrated modelling of DEMO-FNS current ramp-up scenario and steady-state regime

An approach to the integrated modelling of plasma regimes in the projected neutron source DEMO-FNS based on different codes is developed. The consistency check of the steady-state regime is carried out, namely, the possibility of the plasma current ramp-up, acceptance of growth rates of MHD modes in the steady-state regime, heat loads to the wall and divertor plates and neutron yield value. The following codes are employed for the integrated modelling. ASTRA transport code for calculation of plasma parameters in the steady-state regime, NUBEAM Monte Carlo code for NBI incorporated into the ASTRA code, DINA free boundary equilibrium and evolution code, SPIDER free boundary equilibrium and equilibrium reconstruction code, KINX ideal MHD stability code, TOKSTAB rigid shift vertical stability code, edge and divertor plasma B2SOLPS5.2 code and Semi-analytic Hybrid Model (SHM) code for self-consistent description of the core, edge and divertor plasmas based on the experimental scaling laws. The consistent steady-state regime for the DEMO-FNS plasma and the plasma current ramp-up scenario are developed using the integrated modelling approach. Passive copper coils are suggested to reduce the plasma vertical instability growth rate to below ∼30 s−1.The outer divertor operation in the ‘high-recycling’ regime is numerically demonstrated with a maximal heat flux density of 7–9 MW m−2 that is technically acceptable.

Novel free-boundary equilibrium and transport solver with theory-based models and its validation against ASDEX Upgrade current ramp scenarios

Tokamak scenario development requires an understanding of the properties that determine the kinetic profiles in non-steady plasma phases and of the self-consistent evolution of the magnetic equilibrium. Current ramps are of particular interest since many transport-relevant parameters explore a large range of values and their impact on transport mechanisms has to be assessed. To this purpose, a novel full-discharge modelling tool has been developed, which couples the transport code ASTRA (Pereverzev et al 1991 IPP Report 5/42) and the free boundary equilibrium code SPIDER (Ivanov et al 2005 32nd EPS Conf. on Plasma Physics vol 29C (ECA) P-5.063 and http://epsppd.epfl.ch/Tarragona/pdf/P5_063.pdf), utilizing a specifically designed coupling scheme. The current ramp-up phase can be accurately and reliably simulated using this scheme, where a plasma shape, position and current controller is applied, which mimics the one of ASDEX Upgrade. Transport of energy is provided by theory-based models (e.g. TGLF (Staebler et al 2007 Phys. Plasmas 14 055909)). A recipe based on edge-relevant parameters (Scott 2000 Phys. Plasmas 7 1845) is proposed to resolve the low current phase of the current ramps, where the impact of the safety factor on micro-instabilities could make quasi-linear approaches questionable in the plasma outer region. Current ramp scenarios, selected from ASDEX Upgrade discharges, are then simulated to validate both the coupling with the free-boundary evolution and the prediction of profiles. Analysis of the underlying transport mechanisms is presented, to clarify the possible physics origin of the observed L-mode empirical energy confinement scaling. The role of toroidal micro-instabilities (ITG, TEM) and of non-linear effects is discussed.

Self-consistent simulation of plasma scenarios for ITER using a combination of 1.5D transport codes and free-boundary equilibrium codes

Self-consistent transport simulation of ITER scenarios is a very important tool for the exploration of the operational space and for scenario optimization. It also provides an assessment of the compatibility of developed scenarios (which include fast transient events) with machine constraints, in particular with the poloidal field coil system, heating and current drive, fuelling and particle and energy exhaust systems. This paper discusses results of predictive modelling of all reference ITER scenarios and variants using two suites of linked transport and equilibrium codes. The first suite consisting of the 1.5D core/2D SOL code JINTRAC (Wiesen S. et al 2008 JINTRAC-JET modelling suite JET ITC-Report) and the free-boundary equilibrium evolution code CREATE-NL (Albanese R. et al 2003 ISEM 2003 (Versailles, France); Albanese R. et al 2004 Nucl. Fusion 44 999), was mainly used to simulate the inductive D–T reference Scenario-2 with fusion gain Q = 10 and its variants in H, D and He (including ITER scenarios with reduced current and toroidal field). The second suite of codes was used mainly for the modelling of hybrid and steady-state ITER scenarios. It combines the 1.5D core transport code CRONOS (Artaud J.F. et al 2010 Nucl. Fusion 50 043001) and the free-boundary equilibrium evolution code DINA-CH (Kim S.H. et al 2009 Plasma Phys. Control. Fusion 51 105007).

A New Simulation Framework Based on the Kepler and Scicos Open-Source Software for the Design and Qualification of Tokamak Control Algorithms: First Test Case Results

Plasma control is recognized to be a crucial issue for the achievement of ITER objectives. One of the most challenging tasks for the preparation of the ITER operation will therefore be the design and qualification of a variety of control algorithms. This highlights the need for a simulation platform capable of supporting the design, integration and test of advanced control algorithms on complex physics models. With this aim, a generic multi-purpose “flight” simulator (GMFS) is being developed at IRFM (Institut de Recherche sur la Fusion par confinement Magnétique), CEA Cadarache, France.The GMFS is based on Kepler, a free interdisciplinary open-source Java software. Kepler will be used as a simulation platform to test and improve control algorithms before their actual use in the real control system. The physics and engineering codes complementary to the control algorithms will be supplied by the EFDA Integrated Tokamak Modelling Task Force (ITM-TF). The GMFS will be benchmarked, at the beginning, on the Tore Supra Tokamak.In this paper we will report on a test case suitable to demonstrate the feasibility of a part of GMFS, namely the development of workflows where to create and verify ITER plasma boundary feedback control algorithms. It consists of: a) derivation of a linear plasma response model; b) design of a control diagram under the ScicosLab/Scicos open-source software; c) porting of the diagram under Kepler; d) substitution of the Kepler controller with a controller generated by a special Scicos extension; e) substitution of the simplified static linear model with the free-boundary equilibrium code CEDRES++.The test case demonstrated the feasibility of employing Kepler, ScicosLab/Scicos and other expressly-made codes in view of the conception of valuable instruments for the active control of ITER and it can be considered as a first step in this direction.

The CEDRES++ equilibrium code and its application to ITER, JT-60SA and Tore Supra

Free-boundary equilibrium codes are an essential tool for tokamak design studies and scenario development. In this contribution, the CEDRES++ code is presented and applications to ITER, JT-60SA and Tore Supra are reported on. A satisfactory benchmark of CEDRES++ with the DINA code was performed on an ITER Scenario 2 case at 32 different times across the discharge, both in direct and inverse mode. Ongoing design studies for the poloidal field coils system of JT-60SA, performed in the frame of the Broader Approach, were cross-checked with CEDRES++. Finally, a dimensioning study was carried out for the coils and power supplies for plasma vertical control in the frame of a feasibility study for the implementation of a divertor in Tore Supra.

Validation of the thermal transport model used for ITER startup scenario predictions with DIII-D experimental data

Control simulations of ITER startup using 2D free-boundary equilibrium and 1D transport codes rely on accurate predictions of the electron and ion temperature profiles that determine the electrical conductivity and pressure profiles during the current rise. We present results of validation studies that apply the transport model used by the ITER team to DIII-D discharge evolution and compare predictions with data from similarity experiments. Results presented here detail difficulties and sensitivities associated with the modelling of time-dependent current profile evolution required to asses performance of the poloidal-field coil system and controllers on ITER.

Simulating ITER plasma startup and rampdown scenarios in the DIII-D tokamak

DIII-D experiments have investigated ITER startup scenarios, including an initial phase where the plasma was limited on low field side poloidal bumper limiters. In addition, li feedback control has been tested with the goal of producing discharges in ITER within the capabilities of the poloidal field coil set and favourable to the intended mode of operations in the subsequent constant current (flattop) phase. These discharges have been modelled using the Corsica free boundary equilibrium code. High performance hybrid scenario discharges (βN = 2.8, H98,y2 = 1.4) and ITER H-mode baseline discharges (βN > 1.6, H98,y2 = 1–1.2) have been obtained experimentally in an ITER similar shape after the ITER-relevant startup. Studies have been initiated to develop a reliable scenario for exiting the burn phase and ramping down the plasma current in ITER without disruptions.