#### Research project ConFuNuc

#### Development of theoretical approaches and numerical methods to control the evolution of the safety profile in a nuclear fusion plasma

*Team:* **Dynamic and discrete events and Optimization**

*Labeling:* none

*Term:* 3 years (2017 - 2020)

*Funding:* Atlanstic RFI 2020

*LARIS staff involved:* Laurent Autrique, Thérèse Azar, Marie-Françoise Gérard, Jean-Claude Jolly

*Project partners:* Sylvain Brémond (CEA - Cadarache), Emmanuel Moulay (XLIM - Poitiers), Rémy Nouailletas (CEA - Cadarache), Laetitia Perez (LTéN - Nantes), Christophe Prieur (GIPSA-Lab - Grenoble),

*Abstract and objectives*

The general context of this research project concerns the modeled systems control by partial differential equations (PDEs). In the specific context of magnetic confinement modeling in thermonuclear fusion, it will be necessary to control the evolution of the safety profile in the fusion plasma in order to ensure its stability (in the presence of disturbances). This profile, distributed spatially, depends on the magnetic flux and the thermal state of the plasma. These two states are linked by nonlinear and coupled PDE systems. The stability of the fusion plasmas is an important lock which conditions the duration of the plasma and thus represents a major stake to satisfy the Lawson's criterion (aimed at establishing the energy profitability of the fusion reaction).

The interest of mathematical models based on PDEs over approaches based on finite-dimensional models is that they allow the control of infinitely many dynamics at the same time, so it is a non-limited approach of the number of modes envisaged in the synthesis of a control. There are now many techniques for controlling such systems. Let us mention the Lyapunov function approaches for hyperbolic systems, or backstepping approaches, or the robust command synthesis by solving Riccati equations in infinite dimension. The interest of these controls being growing in the context of nuclear fusion, it is necessary to have more flexibility and to take into account constraints on the inputs of the systems. So we must now consider new paradigms for the synthesis of non-linear control laws of systems modeled by PDEs. This is a recent subject and only a few results are available in the literature for specific equations such as the wave equation or the Korteweg-de Vries equation. In general, other nonlinear commands, such as Lured commands, or commands with a memory effect, or those calculated as outputs of ordinary nonlinear differential equations, will be considered.

In order to develop new approaches to control the evolution of the safety profile in the fusion plasma, we have decided to associate the scientific partners specialized in the following fields:

- Control and Identification of systems described by PDEs
- Nuclear fusion
- High temperature thermal systems

In order to evaluate the scientific results, different test campaigns will be carried out jointly:

- Digital implementation and validation on dedicated simulators, developed by the fusion community. These include Metis (developed by CEA) and Raptor (RApid Plasma Transport Simulator developed by EPFL - Lausanne).
- Experimental campaigns on Tokamak to test the control laws during different scenarios. Contacts will be continued in order to be able to experiment with TCV (Tokamak with variable configuration) at EPFL and as part of CEA's WEST project (Tungsten (W) Environment (E) Steady-state (S) Tokamak (T)).

The results will be disseminated by scientific articles, at international conferences and within the international network involved in research on nuclear fusion (ITER, ...)

*Context*

Fossil fuels (oil, gas, coal) still account today for about 85% of primary energy sources worldwide. But they are expected to be exhausted by typically a century, and are largely responsible for the ongoing climate change due to the emission of greenhouse gases generated by their combustion. Magnetic confinement nuclear fusion is a potential alternative energy source with many advantages in terms of fuel abundance, safety and the absence of long-lived radioactive waste. But its industrial exploitation requires carrying at high temperature (about 100 million degrees), in a physical state called plasma, a mixture of hydrogen varieties, for example confined by powerful magnetic fields (several Teslas) in so-called tokamak reactors. . The flagship project in the field, ITER: International Thermonuclear Experimental Reactor (www.iter.org), brings together partners representing about half of the world's population, to build and operate a facility under construction on the French site of Cadarache.

The control of the macroscopic physical quantities of such reactors is well controlled, but the control of the internal spatial profiles of these quantities remains an emerging issue with strong stakes in performance and stability of operation.

In order to propose appropriate and effective control laws, it is essential to take into account systems of Partial Differential Equations (PDE) which describe on the one hand the evolution of the spatial distribution of the magnetic flow within the plasma and on the other hand heat transfers (high temperatures) in the environment considered. These coupled systems have strong nonlinearities and new approaches need to be specifically developed.

The approach considered so far by most theoretical and experimental work on the control of internal profiles of nuclear fusion plasmas is a linear approach in finite dimension. In this approach, the dynamic evolution of these profiles around a point of equilibrium is represented by a system of linear equations coupled to a reduced set of spatial distribution parameters of the physical quantities in question (electrical current density, temperature and density of the plasma environment). This approach, which makes it possible to use a large corpus of theoretical results of the automatic (science of feedback systems) classical, encountered in practice problems of robustness of the experimental implementation against the variable conditions of the reactive environment.