Time-dependent SOLPS-ITER simulations of the tokamak plasma boundary for model predictive control using SINDy ^*

Abstract

Time-dependent SOLPS-ITER simulations have been used to identify reduced models with the Sparse Identification of Nonlinear Dynamics (SINDy) method and develop model-predictive control of the boundary plasma state using main ion gas puff actuation. A series of gas actuation sequences are input into SOLPS-ITER to produce a dynamic response in upstream and divertor plasma quantities. The SINDy method is applied to identify reduced linear and nonlinear models for the electron density at the outboard midplane $\nesepm$ and the electron temperature at the outer divertor $\tesepa$. Note that $\tesepa$ is not necessarily the peak value of $T_e$ along the divertor. The identified reduced models are interpretable by construction (i.e., not black box), and have the form of coupled ordinary differential equations (ODEs). Despite significant noise in $\tesepa$, the reduced models can be used to predict the response over a range of actuation levels to a maximum deviation of 0.5$\%$ in $\nesepm$ and 5 - 10$\%$ in $\tesepa$ for the cases considered. Model retraining using time history data triggered by a preset error threshold is also demonstrated. A Model Predictive Control (MPC) strategy for nonlinear models is developed and used to perform feedback control of a SOLPS-ITER simulation to produce a setpoint trajectory in $\nesepm$ using the Integrated Plasma Simulator (IPS) framework. The developed techniques are general and can be applied to time-dependent data from other boundary simulations or experimental data. Ongoing work is extending the approach to model identification and control for divertor detachment, which will present transient nonlinear behavior from impurity seeding, including realistic latency and synthetic diagnostic signals derived from the full SOLPS-ITER output.

Bifurcation-like transition of divertor conditions induced by X-point radiation in KSTAR L-mode plasmas *

“…In order to ensure that the time-dependent solution at each time is sufficiently quasi-steady state (QSS), 1 s was used for each ramp up and down. There are various methods to guarantee that each moment is sufficiently QSS (or converged), for example, by adding fixed gas throughput sections in between ramp up stages, forming a stair-like progression, and checking if the solution change is small enough in the fixed sections, or by comparing the time-dependent trace with a few steady-state solutions plotted in the phase space of n e,sep and T et [35]. In Lmode cases, simulations were performed with EIRENE time steps of either 1 × 10 −3 s or 1 s. For the H-mode-like condition, only the heating power was changed to 6 MW, and the rest of the conditions were kept the same, without including a pedestal or other H-mode specific features.…”

“…Instead, timedependent gas throughput scans for a long enough time compared to the relaxation timescale gives continuous curve rather than the discrete points, providing more insight into existence of the discontinuity or abrupt changes in the operation space. In addition, time-dependent gas throughput scans can be utilized for system identification and for training reduced models [35,47].…”

Review: Machine learning for advancing low-temperature plasma modeling and simulation

“…Moreover, MPC based on such ROMs has been suggested with limited availability of training data 174 . An aspect that has been applied using time-dependent simulations of the tokamak plasma boundary 173 , 175 . It may be specifically relevant also in the context of LTP processing with its diverse requirements for precise process control.…”

Full time-dependent SOLPS-ITER simulation of the SPARC tokamak: actuator design for particle and divertor condition control *

“…This approach can address problems which are dominated by plasma dynamics and the influence of neutral propagation time can be neglected. Such applications include ELM simulations [7,8], system identification of the plasma response to slow gas puff modulation [9], and the bifurcation of divertor conditions due to density ramps in KSTAR [10]. Alternatively, the neutral particle distribution can be advanced by EIRENE using a fixed plasma background.…”