Magnetic axis safety factor of finite β spheromaks and transition from spheromaks to toroidal magnetic bubbles
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Bellan, Paul M.↗
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Results for “SAFETY FACTOR”
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The effect of safety factor profile on transport in steady-state, high-performance scenarios
In this study, an analysis of the dependence of transport on the safety factor profile in high-performance, steady-state scenario discharges is presented. This is based on experimental scans of q 95 and q min taken with fixed β N , toroidal field, double-null plasma shape, divertor pumping, and electron cyclotron current drive input. The temperature and thermal diffusivity profiles were found to vary considerably with the q-profile, and these variations were significantly different for electrons and ions. With fixed q 95 , both temperature profiles increase and broaden as q min is increased and the magnetic shear becomes low or negative in the inner half radius, but these temperature profile changes are stronger for the electrons. Power balance calculations show the peak in the ion thermal diffusivity (χ i ) at ρ – 0.6 – 0.8 increases with q 95 or q min .
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Computational Fluid Dynamics (CFD) and Safety Factors: Computer modeling of complex processes needs old-fashioned experiments to stay in touch with reality
Computational fluid dynamics (CFD) is recognized as a powerful engineering tool. That is, CFD has advanced over the years to the point where it can now give us deep insight into the analysis of very complex processes. There is a danger, though, that an engineer can place too much confidence in a simulation. If a user is not careful, it is easy to believe that if you plug in the numbers, the answer comes out, and you are done. This assumption can lead to significant errors. As we discovered in the course of a study on behalf of the Department of Energy's Savannah River Site in South Carolina, CFD models fail to capture some of the large variations inherent in complex processes. These variations, or scatter, in experimental data emerge from physical tests and are inadequately captured or expressed by calculated mean values for a process. This anomaly between experiment and theory can lead to serious errors in engineering analysis and design unless a correction factor, or safety factor, is experimentally validated. For this study, blending times for the mixing of salt solutions in large storage tanks were the process of concern under investigation. This study focused on the blending processes needed to mix salt solutions to ensure homogeneity within waste tanks, where homogeneity is required to control radioactivity levels during subsequent processing. Two of the requirements for this task were to determine the minimum number of submerged, centrifugal pumps required to blend the salt mixtures in a full-scale tank in half a day or less, and to recommend reasonable blending times to achieve nearly homogeneous salt mixtures. A full-scale, low-flow pump with a total discharge flow rate of 500 to 800 gpm was recommended with two opposing 2.27-inch diameter nozzles. To make this recommendation, both experimental and CFD modeling were performed. Lab researchers found that, although CFD provided good estimates of an average blending time, experimental blending times varied significantly from the average.
12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗
Nonlinear control of the minimum safety factor in tokamaks by optimal allocation of spatially moving electron cyclotron current drive
The minimum value of the safety factor profile is related to the magnetohydrodynamic (MHD) stability of the plasma confined in a tokamak. Therefore, active control of the minimum safety factor may mitigate MHD instabilities that can degrade or even terminate plasma confinement. Typically, in most tokamak scenarios, the minimum safety factor evolves spatially with time, i.e., the location at which the safety factor achieves the minimum value changes with time. In addition to the inherent nonlinearities in the minimum safety factor evolution, its spatial variation makes the control design challenging. In particular, complexity in control design may arise from the need for time-dependent nonlinear models that account for spatial variation of the minimum safety factor. Furthermore, the minimum safety factor may drift to locations where the actuator authority is low. The problem of minimum safety factor control with target location tracking and moving electron cyclotron current drive (ECCD) is addressed in this work. A nonlinear time-dependent model that incorporates the spatial variation of the minimum safety factor is presented. A nonlinear controller based on optimal feedback linearization is developed to track a target minimum safety factor. The proposed controller treats the ECCD position as a controllable variable. In other words, the controller prescribes the ECCD position (in addition to the non-inductive powers) in real time based on an optimal criterion that is defined a priori. This work also presents the steps necessary to integrate the minimum safety factor controller with a total energy controller to achieve multiple control objectives simultaneously. In conclusion, the proposed integrated control algorithm is tested using nonlinear simulations in the Control Oriented Transport SIMulator (COTSIM) for a DIII-D tokamak scenario.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Nonlinear control of safety factor gradient in tokamaks using spatially variable electron cyclotron current drives
Active control of plasma properties may be necessary to achieve stable operation of next-generation tokamaks over large time scales. Such control algorithms can regulate the plasma properties to avoid the onset of magnetohydrodynamic (MHD) instabilities. For instance, the global and local properties of the safety factor profile are linked to the onset of neoclassical tearing modes (NTMs). This work proposes a model-based control approach for deterring/delaying NTMs through active modulation of one of the safety factor properties - the gradient of the safety factor profile at a particular rational safety factor surface. In particular, a novel control-oriented model for the local safety factor gradient is developed. The nonlinear control model is governed by a nonautonomous ordinary differential equation that accounts for a given rational safety factor surface’s spatial variation over time. Further, to improve the controllability of the spatially evolving parameter, the control model treats ECH&CD positions, along with noninductive powers, as controllable variables. A nonlinear control algorithm based on feedback linearization with optimization is synthesized to achieve the objective of regulating the safety factor gradient around a given target. The proposed algorithm allocates optimal ECH&CD positions, in addition to auxiliary powers, at each time instant as the rational safety factor surface drifts to locations with a low control authority. Stability guarantees of the proposed control law are also discussed in this work. The proposed algorithm is tested for a DIII-D tokamak scenario in nonlinear simulations carried out using the Control Oriented Transport SIMulator (COTSIM). Both fixed and moving ECH&CD cases are studied, and their outcomes are compared. Simulation results demonstrate that enthusiastic regulation of the safety factor gradient can be achieved during the ramp-up and flat-top phases of tokamak operation in both fixed and moving ECH&CD cases. However, real-time updates of ECH&CD positions can prevent the saturation of auxiliary powers.
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Test-Based Calibration of Safety Factors for Capacity Models
In this paper, a simple procedure to calibrate the safety factor of capacity models is presented. The calibration can be carried out based on any available database of experimental tests, even of limited size. The procedure aims to assess the model capability of predicting the test results and to calibrate the safety factor so that the capacity equation meets the target reliability level required by the code or sought by the calibrator. After predicting each test of the database with the capacity equation under consideration, the test-prediction pairs are checked for the property of linearity, and the relative error for the properties of homoscedasticity and normality. Once these properties are fulfilled—which may require a nonlinear transformation of the test values and/or the predictions—the closed-form equation proposed in this paper is employed to compute a target design value. The model safety factor is finally obtained by comparing such target design value with the design value obtained from the code. The paper also proposes two approximate analytical equations to compute the tolerance factor, used to attain any given fractile, as a function of the (even small) number of tests, with any assigned confidence level. A fundamental outcome of the procedure is that it yields an objective indicator of the model accuracy, measured by the standard deviation of its error, which may be regarded as a parameter useful for selecting the most reliable model among different competing ones. In the long run, the application of the proposed procedure will allow achieving a uniform reliability level throughout all capacity models used in codes and guidelines. A further advantage is that the partial safety factors so derived can be straightforwardly updated when more experiments become available. Finally, as an example, the proposed procedure is herein applied to the ACI 318 shear design capacity equation for concrete members unreinforced in shear.
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Self-organized magnetic equilibria in tokamak plasmas with very low edge safety factor
Tokamak plasmas often exhibit self-organizing behavior in which internal modes shape the toroidal current density profile, a common example being the sawtooth instability. However, such behavior has not been studied in detail for edge safety factor below 2 due to disruptive kink instabilities that typically prevent operation in this regime. Now, steady tokamak plasmas with an edge safety factor down to 0.8 have been created in the Madison Symmetric Torus, where disruptions are prevented due to a thick, conductive wall and a feedback power supply that sustains the plasma current. Internal measurements and nonlinear magnetohydrodynamic modeling reveal a family of safety factor profiles with a central value clamped near unity as the edge safety factor decreases, indicating current profile broadening through a relaxation process. Furthermore, as the safety factor decreases, the magnetic fluctuations become irregular, and the electron energy confinement time decreases.
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X-point effects on the ideal MHD modes in tokamaks in the description of dual-poloidal-region safety factor
The flux coordinates with dual-region safety factor (q) in the poloidal direction are developed in this work. The X-point effects on the ideal MHD modes in tokamaks are then analyzed using this coordinate system. Since the X-point effects mainly affect the edge region, the modes localized at the tokamak edge are particularly examined. Two types of modes are studied. The first is related to the conventional peeling or peeling-ballooning modes. The mode existence aligned with the local magnetic field in the poloidally core region as observed experimentally is confirmed. The X points are shown to contribute to a stabilizing effect for the conventionally treated modes with the surface-averaged q and with the tokamak edge portion truncated. The other is the axisymmetric modes localized in the vicinity of X points, which can affect the cross-field-line transport near the X points. The existence of axisymmetric modes points to the possibility of applying a toroidally axisymmetric resonant magnetic perturbation (RMP) in the X-point area for mitigating the edge localized modes, which can be an alternative to the current RMP design. The dual q description also has important implications for the existing non-axisymmetric RMP concept. It helps to understand why the RMP suppression of edge localized modes is difficult to achieve in the double-null tokamak configurations and points to the possibility of further improving the current RMP concept by considering the alignment to the local q.
Fourier analysis↗
Tokamak Operation with Safety Factor q 95 < 2 via Control of MHD Stability
Magnetic feedback control of the resistive-wall mode has enabled DIII-D to access stable operation at safety factor q95 = 1:9 in divertor plasmas for 150 instability growth times. Magnetohydrodynamic stability sets a hard, disruptive limit on the minimum edge safety factor achievable in a tokamak, or on the maximum plasma current at given toroidal magnetic eld. In tokamaks with a divertor, the limit occurs at q95 = 2, as con rmed in DIII-D. Since the energy con cement time scales linearly with current, this also bounds the performance of a fusion reactor. DIII-D has overcome this limit, opening a whole new high-current regime not accessible before. This result brings signi cant possible bene ts in terms of fusion performance, but it also extends resistive wall mode physics and its control to conditions never explored before. In present experiments, q95 < 2 operation is eventually halted by voltage limits reached in the feedback power supplies, not by intrinsic physics issues. Improvements to power supplies and to control algorithms have the potential to further extend this regime.
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Simultaneous control of the electron temperature and safety factor profiles in DIII-D using model-based optimal control techniques
Future tokamak power plants will likely operate using a single, well-defined plasma scenario, either in steady state or for very long pulse lengths. In order to enhance the robustness of the scenario, feedback controllers for a variety of plasma properties will be necessary to counteract any disturbances and ensure safe operation. However, only a limited set of actuators will be available to control many different quantities. Because of this, it is necessary to develop controllers that are able to regulate multiple plasma properties using a limited set of actuators. To this end, a controller has been developed for the simultaneous regulation of both the electron temperature and safety factor profiles in DIII-D. This algorithm uses a linear quadratic integral control synthesis approach based on a linearized model of the dynamics of the two profiles. Two neural network surrogate models, NubeamNet and MMMnet, are included to improve the fidelity of the model. Furthermore, the controller has been tested in simulation using COTSIM, and has demonstrated the ability to simultaneously track changes in both the electron temperature and safety factor targets, including changes in both the magnitude and the shape of the profiles.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Regulation of the central safety factor and normalized beta under low NBI torque in DIII-D
An algorithm has been designed to simultaneously control the central safety factor (q 0 ) and normalized beta (β N ) while ensuring near-zero torque from the neutral beam injection in DIII-D. Feedback control of q 0 and β N in tokamaks can be beneficial due to the close relationship that these variables have with plasma performance and magneto-hydrodynamic stability. In addition, low neutral-beam-torque conditions are of special interest in present devices because future burning-plasma tokamaks such as ITER will most likely operate at very low plasma rotation. The control synthesis of the algorithm presented in this work is based on a linearized, one-dimensional (1D) model of the current-profile dynamics coupled with a zero-dimensional (0D) plasma-energy balance. The actuators considered are neutral beam injection and electron-cyclotron heating and current drive, and discrete logic determines the neutral-beam injection powers that deliver near-zero torque. Here, the algorithm has been tested in nonlinear, 1D simulations using COTSIM (Control-Oriented Transport SIMulator) and in DIII-D experiments, demonstrating satisfactory performance.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Steady state scenario development with elevated minimum safety factor on DIII-D
On DIII-D, a high β scenario with minimum safety factor (q min ) near 1.4 has been optimized with new tools and shown to be a favourable candidate for long pulse or steady state operation in future devices. Furthermore, the new capability to redirect up to 5 MW of neutral beam injection (NBI) from on- to off-axis improves the ability to sustain elevated q min with a less peaked pressure profile. The observed changes increase the ideal magnetohydrodynamics (MHD) n = 1 mode β N limit thus providing a path forward for increasing the noninductive current drive fraction by operating at high β N . Quasi-stationary discharges free of tearing modes have been sustained at βN = 3.5 and β T = 3.6% for two current profile diffusion timescales (about 3 s) limited by neutral beam duration. The discharge performance has normalized fusion performance expected to give fusion gain Q ≈ 5 in a device the size of ITER. Analysis of the poloidal flux evolution and current drive balance show that the loop voltage profile is almost relaxed even with 25% of the current driven inductively, and q min remains elevated near 1.4. Our observations increase confidence that the current profile will not evolve to one unstable to a tearing mode. In preliminary tests a divertor heat flux reduction technique based on producing a radiating mantle with neon injection appears compatible with this operating scenario. 0D model extrapolations suggest it may be possible to push this scenario up to 100% noninductive current drive by raising β N . Similar discharges with q min = 1.5–2 were susceptible to tearing modes and off-axis fishbones, and with q min > 2 lower normalized global energy confinement time is observed.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Feedback-assisted extension of the tokamak operating space to low safety factor
Recent DIII-D and RFX-mod experiments have demonstrated stable tokamak operation at very low values of the edge safety factor q(a) near and below 2. The onset of n = 1 resistive wall mode (RWM) kink instabilities leads to a disruptive stability limit, encountered at q(a) = 2 (limiter plasmas) and q{sub 95} = 2 (divertor plasmas). However, passively stable operation can be attained for q(a) and q{sub 95} values as low as 2.2. RWM damping in the q(a) = 2 regime was measured using active MHD spectroscopy. Although consistent with theoretical predictions, the amplitude of the damped response does not increase significantly as the q(a) = 2 limit is approached, in contrast with damping measurements made approaching the pressure-driven RWM limit. Applying proportional gain magnetic feedback control of the n = 1 modes has resulted in stabilized operation with q{sub 95} values reaching as low as 1.9 in DIII-D and q(a) reaching 1.55 in RFX-mod. In addition to being consistent with the q(a) = 2 external kink mode stability limit, the unstable modes have growth rates on the order of the characteristic wall eddy-current decay timescale in both devices, and a dominant m = 2 poloidal structure that is consistent with ideal MHD predictions. The experiments contribute to validating MHD stability theory and demonstrate that a key tokamak stability limit can be overcome with feedback.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Hybrid model predictive control techniques for safety factor profile and stored energy regulation while incorporating NBI constraints
Abstract A novel hybrid Model Predictive Control (MPC) algorithm has been designed for simultaneous safety factor ( q ) profile and stored energy ( w ) control while incorporating the pulse-width-modulation constraints associated with the neutral beam injection (NBI) system. Regulation of the q -profile has been extensively shown to be a key factor for improved confinement as well as non-inductive sustainment of the plasma current. Simultaneous control of w is necessary to prevent the triggering of pressure-driven magnetohydrodynamic instabilities as the controller shapes the q profile. Conventional MPC schemes proposed for q -profile control have considered the NBI powers as continuous-time signals, ignoring the discrete-time nature of these actuators and leading in some cases to performance loss. The hybrid MPC scheme in this work has the capability of incorporating the discrete-time actuator dynamics as additional constraints. In nonlinear simulations, the proposed hybrid MPC scheme demonstrates improved q -profile+ w control performance for NSTX-U operating scenarios.
Physics↗
The motional Stark effect diagnostic reliably measures significant deviations in safety factor profile during DIII-D sawteeth
Motional Stark effect (MSE) data acquired during large fast-ion stabilized sawteeth are critically reexamined. Here, the safety factor at the sawtooth crash changes by Δq ≃ 0.15, much more than any likely errors.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Reconstruction of tokamak plasma safety factor profile using deep learning
Abstract The motional Stark effect (MSE) diagnostic has been a standard measurement for the magnetic field line pitch angle in tokamaks that are equipped with neutral beams. However, the MSE data are not always available due to experimental constraints, especially in future devices without neutral beams. Here we develop a deep-learning based model (SGTC-QR) that can reconstruct the safety factor profile without the MSE diagnostic to mimic the traditional equilibrium reconstruction with the MSE constraint. The model demonstrates promising performance, and the sub-millisecond inference time is compatible with the real-time plasma control system.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Mathematical Simulation of Humidification of Earth on a Slope and Calculation of Its Safety Factor
This paper presents a mathematical model of the simultaneous processes of moisture, heat, and salt transfer in a porous medium, as well as a pattern of calculation of earth stability on a slope depending on the above factors and nonlinear dependences on them of the strength characteristics of the earth. The corresponding boundary-value problem has been solved by the method of radial basis functions. A comparison has been made between the results of numerical experiments showing the relationship between the change in the value of the safety factor and the position of the slip curve depending on the heat, salt, and moisture transfer.
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