Development of free-boundary equilibrium and transport solvers for simulation and real-time interpretation of tokamak experiments

To exploit fusion as a source of energy, a hot and dense confined plasma is needed. This is achieved in tokamaks by actively controlling the plasma state meaning shape, position and internal kinetic profiles. This thesis addresses two topics: the development of a tokamak simulator to design and test real-time controllers; the reconstruction of the plasma state from diagnostic measurements. A flexible control oriented simulator is needed to design the feedforward trace of the current evolution in the external conductors and to test/design and verify the controllers that maintain the desired shape, position and vertical stability. A fullyMatlab-based suite of routines called "LIUQE-suite", sharing the same optimized low level routines, has been developed at SPCEPFL addressing multiple problems related to the free-boundary equilibrium. It covers the full range of free-boundary magnetic equilibrium codes from computing the coil currents for a sequence of desired plasma equilibria (FBT code), simulating the plasma magnetic evolution coupled to controller dynamics (FGE code), performing magnetic equilibrium reconstruction (MER) from synthetic/experimental data both in post discharge and for real-time analysis (LIUQE code). In this thesis, two new codes have been added to this code suite. FGS (Forward Grad-Shafranov Static) which solves the forward static free-boundary equilibrium problem. FGE (Forward Grad-Shafranov Evolutive) which solves the dynamics of conductor current evolution, coupled to the resistive plasma current decay on subsequent states of free-boundary equilibria. Both codes uses the newly implemented Jacobian Free Newton Krylov (JFNK) method, which combines the stability property of the Newton-like algorithm without requiring to formexplicitly the Jacobian of the problem. This enables the implementation of a monolithic approach and avoids iterative coupling between the free-boundary solution and 0D current diffusion equation (CDE). A particular form of the CDE, derived in this thesis, allows to avoid the expensive computation of the geometrical coefficients in the coupling. A linearized version of FGE in state-space formis presented, consistently including deformable plasmas and current diffusion. A novel contribution of this thesis is also the derivation of an analytic Jacobian for the free-boundary forward equilibriumproblem discretized in space with finite differences, which can enable the implementation of a standard Newton method. FGS was used to investigate the impact of some simplification hypotheses in the LIUQE algorithm, related to the functional dependencies of the synthetic diagnostics on the plasma flux map, finding a small impact to the resulting magnetic equilibrium reconstruction (MER), confirming therobustness of LIUQE code. A proof of principle application of Bayesian analysis to obtainuncertainty quantification for MER is shown. To validate the implementation of FGE, thegrowth rates of VDEs for several TCV plasmas were compared to the rigid displacement RZIPmodel. After coupling the real vertical stability and position controller of TCV, the request ofthe central solenoid in the simulation was compared to the experiment, in order to verify thecorrectness of the CDE implemented. Good agreement was found in all cases,allowing FGSand FGE to be used for the development of TCV controllers in future.Kinetic equilibrium reconstruction (KER) is the consistent reconstruction of the plasma fluxsurface shapes and int