MPC strategies for density profile control with pellet fueling in nuclear fusion tokamaks under uncertainty

TL;DR

This paper tackles real-time control of the electron density profile in ITER-like tokamaks using discrete pellet fueling, a problem complicated by actuator delays and safety-critical edge-density constraints under parametric uncertainty. It introduces a reduced-order, LPV plant model derived via dynamic mode decomposition with control (DMDc) and formulates three MPC strategies: a baseline mixed-integer MPC (MI-MPC), a multi-stage scenario MI-MPC (msMI-MPC) that accounts for uncertainty with a scenario tree, and a computationally efficient multi-stage scenario PTH-MPC (msPTH-MPC) that combines PCA-based scenario reduction with a penalty-term homotopy approach to solve online QPs. Key results show MI-MPC can track well but may violate safety constraints; msMI-MPC and especially msPTH-MPC achieve constraint satisfaction under uncertainty, with msPTH-MPC attaining real-time computational performance close to the limit. The work demonstrates a promising uncertainty-aware, real-time density-control strategy for pellet fueling in ITER, pending validation in nonlinear JINTRAC simulations.

Abstract

Control of the density profile based on pellet fueling for the ITER nuclear fusion tokamak involves a multi-rate nonlinear system with safety-critical constraints, input delays, and discrete actuators with parametric uncertainty. To address this challenging problem, we propose a multi-stage MPC (msMPC) approach to handle uncertainty in the presence of mixed-integer inputs. While the scenario tree of msMPC accounts for uncertainty, it also adds complexity to an already computationally intensive mixed-integer MPC (MI-MPC) problem. To achieve real-time density profile controller with discrete pellets and uncertainty handling, we systematically reduce the problem complexity by (1) reducing the identified prediction model size through dynamic mode decomposition with control, (2) applying principal component analysis to reduce the number of scenarios needed to capture the parametric uncertainty in msMPC, and (3) utilizing the penalty term homotopy for MPC (PTH-MPC) algorithm to reduce the computational burden caused by the presence of mixed-integer inputs. We compare the performance and safety of the msMPC strategy against a nominal MI-MPC in plant simulations, demonstrating the first predictive density control strategy with uncertainty handling, viable for real-time pellet fueling in ITER.

Figures (5)

• Figure 1: (a) Diagram of a poloidal cross-section of the ITER tokamak with the plasma region in pink. Image courtesy of the ITER organization with modifications ITER_VV. The trajectory of the pellet (purple) into the plasma core is overlaid on a sketch of the 1D $n_{e}(\rho)$ profile (blue). $n_{e}(\rho)$ is constant on each magnetic flux surface, which are denoted in grey in (a) and labeled in (b). (b) The 3D plasma, sketched as a cylindrical approximation labeled with the flux label $\rho$, where $\rho = 0$ at the plasma center and $\rho = 1$ at the plasma edge. (c) Realization of the 1D $n_e(t,\rho)$ profile immediately before ($t = 5.135$ s) and after ($t = 5.136$ s) a pellet enters the plasma (c.f Fig. \ref{['fig:simulation']}). The region above $n_{e,lim}$ that $n_{e}(t,\rho)$ cannot enter is shown in red.
• Figure 2: Example scenario tree for msMI-MPC with $N_R = 2$, $n_p=2$, and $\tau_c = 5\tau_s$, branching at control nodes (red).
• Figure 3: Data-driven parameterization of $p_j$. (a) Score plot using the first two principal components of each data point (blue) computed with PCA. The red x's indicate the chosen scenario realizations. (b) The $\%$ of the total variance explained by each principal component.
• Figure 4: Simulation results with MI-MPC (black), msPTH-MPC (blue), and msMI-MPC (green). (a) $\bar{n}_{e,core}^{ref}$ compared to $\bar{n}_{e,core}(t)$$[10^{20}$ m$^{-3}]$. (b) The RRMSE of $n_e(t,\rho)$, averaged over $\rho = [0,0.4]$. (c) The constraint satisfaction at $n_{e,edge}(t)$$[10^{20}$ m$^{-3}]$ w.r.t $n_{e,lim}$ (red). MI-MPC violations of $n_{e,lim}$ are denoted with red x's (note that consecutive violations overlap). (d) Control decisions for each controller. (e) $t_{cpu}$ [s] required to compute each MPC strategy, compared to $t_{cpu}^{lim}$ (red).
• Figure 5: $n_e(t,\rho)$ over the simulation with MI-MPC. Jumps in $n_e(t,\rho)$ occurring near $\rho = 0.8$ each time a pellet is fired exhibits the variation in the ablation profile varies captured in $p(t)$. Each $n_{e,lim}$ violation is labeled as a red x. The black frame region corresponds to (b). (b) The plant $n_e(t,\rho)$ (black) compared to the $n_e(t,\rho)$ predicted by MI-MPC at $t = 5.136$ s. The zoomed frame shows the true $n_e(t,\rho)$ exceeding $n_{e,lim}$, while the MI-MPC-predicted $n_e(t,\rho)$ does not. The msMI-MPC scenarios predicted from the same $y(t)$ and decision $u(t)$ as MI-MPC (green) do predict that $n_e(t,\rho)$ will violate $n_{e,lim}$.