Development and validation of a tokamak skin effect transformer model

Abstract

A lumped parameter, state space model for the tokamak transformer including the slow flux penetration in the plasma (skin effect transformer model) is presented. The model does not require detailed or explicit information about plasma profiles or geometry. Instead, this information is lumped in system variables, parameters and inputs. The model has an exact mathematical structure built from energy and flux conservation theorems, predicting the evolution and non linear interaction of the plasma current and internal inductance as functions of the primary coil currents, plasma resistance, non-inductive current drive and the loop voltage at an specific location inside the plasma (equilibrium loop voltage). Loop voltage profile in the plasma is substituted by a three-point discretization, and ordinary differential equations are used to predict the equilibrium loop voltage as function of the boundary and resistive loop voltages. This provides a model for equilibrium loop voltage evolution, which is reminiscent of the skin effect. The order and parameters of this differential equation are determined empirically using system identification techniques. Fast plasma current modulation experiments with Random Binary Signals (RBS) have been conducted in the TCV tokamak to generate the required data for the analysis. Plasma current was modulated in Ohmic conditions between 200kA and 300kA with 30ms rise time, several times faster than its time constant L/R≈200ms. A second order linear differential equation for the equilibrium loop voltage is sufficient to describe the plasma current and internal inductance modulation with 70% and 38% fit parameters respectively. The model explains the most salient features of the plasma current transients, such as the inverse correlation between plasma current ramp rates and internal inductance changes, without requiring detailed or explicit information about resistivity profiles. This proves that lumped parameter modeling approach can be used to predict the time evolution of bulk plasma properties such as plasma inductance or current with reasonable accuracy; at least in Ohmic conditions without external heating and current drive sources.

State-space Coil Modelling in Plasma Magnetic Confinement Devices

“…It has been determined that the internal circuit of the stellarator can be represented by the following lumped parameter electrical system (Figure 2). This lumped parameter approach is widely employed when in fusion devices [7][8][9]:…”

Internal inductance predictive control for Tokamaks

“…From an electrical point of view the tokamaks are modeled as a toroidal transformer primary coupled with the plasma ring by a mutual inductance and the plasma acts as the secondary circuit, where and denote the plasma resistance and inductance respectively [31][32][33][34][35]. The plasma is maintained using poloidal field discharges with the particularity that the total inductance consists of the sum of a constant term (external inductance) due to the inductance of transmission lines and wiring which are geometrical factors, plus a variable term corresponding to the internal inductance of the system.…”

Sliding mode control of a tokamak transformer

“…However, for control systems design a lumped parameter formulation is generally preferable [52], [53]. To develop the various control system designs presented in this paper we have used a transformer model that includes a lumped parameter formulation for the skin effect [54]. We call this model a skin effect transformer model to differentiate it from the standard transformer model where secondary inductance is a fixed parameter.…”

“…A second order approximation for the equilibrium voltage (13) as a function of the boundary (4) and resistive drop (8) voltages is used [54]. The weight 0 1 k ≤ ≤ regulates the relative contributions from boundary and resistive voltages to the equilibrium voltage.…”