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

Abstract

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mi>n</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">e</mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal">s</mml:mi> <mml:mi mathvariant="normal">e</mml:mi> <mml:mi mathvariant="normal">p</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">O</mml:mi> <mml:mi mathvariant="normal">M</mml:mi> <mml:mi mathvariant="normal">P</mml:mi> </mml:mrow> </mml:msubsup> </mml:math> and the electron temperature at the outer divertor <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">e</mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal">s</mml:mi> <mml:mi mathvariant="normal">e</mml:mi> <mml:mi mathvariant="normal">p</mml:mi> </mml:mrow> <mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">d</mml:mi> <mml:mi mathvariant="normal">i</mml:mi> <mml:mi mathvariant="normal">v</mml:mi> </mml:mrow> </mml:mrow> </mml:msubsup> </mml:math> . Note that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">e</mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal">s</mml:mi> <mml:mi mathvariant="normal">e</mml:mi> <mml:mi mathvariant="normal">p</mml:mi> </mml:mrow> <mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">d</mml:mi> <mml:mi mathvariant="normal">i</mml:mi> <mml:mi mathvariant="normal">v</mml:mi> </mml:mrow> </mml:mrow> </mml:msubsup> </mml:math> 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. Despite significant noise in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">e</mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal">s</mml:mi> <mml:mi mathvariant="normal">e</mml:mi> <mml:mi mathvariant="normal">p</mml:mi> </mml:mrow> <mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">d</mml:mi> <mml:mi mathvariant="normal">i</mml:mi> <mml:mi mathvariant="normal">v</mml:mi> </mml:mrow> </mml:mrow> </mml:msubsup> </mml:math> , the reduced models can be used to predict the response over a range of actuation levels to a maximum deviation of 0.5% in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mi>n</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">e</mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal">s</mml:mi> <mml:mi mathvariant="normal">e</mml:mi> <mml:mi mathvariant="normal">p</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">O</mml:mi> <mml:mi mathvariant="normal">M</mml:mi> <mml:mi mathvariant="normal">P</mml:mi> </mml:mrow> </mml:msubsup> </mml:math> and 5%–10% in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mi>T</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">e</mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal">s</mml:mi> <mml:mi mathvariant="normal">e</mml:mi> <mml:mi mathvariant="normal">p</mml:mi> </mml:mrow> <mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">d</mml:mi> <mml:mi mathvariant="normal">i</mml:mi> <mml:mi mathvariant="normal">v</mml:mi> </mml:mrow> </mml:mrow> </mml:msubsup> </mml:math> for the cases considered. Model retraining using time history data triggered by a preset error threshold is also demonstrated. A model predictive control strategy for nonlinear models is developed and used to perform feedback control of a SOLPS-ITER simulation to produce a setpoint trajectory in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msubsup> <mml:mi>n</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">e</mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal">s</mml:mi> <mml:mi mathvariant="normal">e</mml:mi> <mml:mi mathvariant="normal">p</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">O</mml:mi> <mml:mi mathvariant="normal">M</mml:mi> <mml:mi mathvariant="normal">P</mml:mi> </mml:mrow> </mml:msubsup> </mml:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xl