Carl-Magnus Fransson

Feedback stabilization of nonaxisymmetric resistive wall modes in tokamaks. I. Electromagnetic model

Active feedback stabilization of pressure-driven modes in tokamaks is studied computationally in toroidal geometry. The stability problem is formulated in terms of open-loop transfer functions for fluxes in sensor coils resulting from currents in feedback coils. The transfer functions are computed by an extended version of the MARS stability code [A. Bondeson et al., Phys. Fluids B 4, 1889 (1992)] and can be accurately modeled by low order rational functions. In the present paper stability is analyzed for a system with an ideal amplifier (current control). It is shown that feedback with modest gain, and a single coil array poloidally, gives substantial stabilization for a range of coil shapes. Optimum design uses sensors for the poloidal field, located inside the resistive wall, in combination with rather wide feedback coils outside the wall. Typically, the feedback does not strongly modify the plasma-generated magnetic field perturbation. A future companion paper [C. M. Fransson et al., Phys. Plasmas (ac...

Physics and control of resistive wall modes

Active feedback stabilization of resistive wall modes in tokamaks is studied both analytically, using large aspect ratio theory, and by means of toroidal computations. Extensive studies show that robust stabilization, with respect to variations in plasma current, pressure and flow velocity, can be achieved with a simple control system using poloidal sensors inside the first wall. The required coil voltages are modest, even for the two-wall structure of a tokamak reactor.

Feedback stabilization of nonaxisymmetric resistive wall modes in tokamaks. II. Control analysis

Active feedback of nonaxisymmetric resistive wall modes in tokamaks is investigated using control theory. Control systems are designed to stabilize the resistive wall mode of toroidal mode number n=1 and meet certain performance specifications for a set of test equilibria. The response of the plasma and resistive wall is described by low order rational functions from an electromagnetic model [Y. Q. Liu et al., Phys. Plasmas 7, 3681 (2000)]. Simple coil arrangements are assumed, both for the sensor and feedback coil arrays, and the sensors detect the perturbed poloidal field. The active coils are modeled both as broad strips and as thin wires, and several different controllers: P (proportional), PD (proportional plus derivative) and H∞ are investigated. An important parameter is the ratio, τ, of control system response time to the resistive wall time, and the analysis shows the restrictions on this ratio for acceptable performance. For an equilibrium that exceeds the no-wall beta limit by 63%, good control...

Active feedback stabilization of high beta modes in advanced tokamaks

Active feedback of non-axisymmetric external modes in tokamaks is studied by means of combined MHD and control analysis. The MARS code for toroidal MHD stability analysis has been extended to compute transfer functions for the electromagnetic response of the plasma, the coils and the resistive wall. These transfer functions are used for controller design. The controller is designed to allow for the longest possible response time of the amplifier-feedback coil circuit, such that certain predetermined performance criteria are satisfied. Calculations are presented for high beta advanced tokamak equilibria with ITER-FEAT shape. With an array of feedback coils that has only one coil in the poloidal direction, and assuming a single resistive wall, control of modes with toroidal mode number n = 1 is found to be possible for βN up to 5, or twice the no-wall limit. Sensors for the poloidal field are superior to sensors for the radial field. Feedback coils with a rather broad cross-section have significant advantages over thin-wire coils. The controller needs to have significant derivative action, but the requirements on the time constant of the amplifier-feedback coil circuit are moderate. This time needs to be less than a few resistive wall times, and broad strips allow an even longer time constant.

Model validation, dynamic edge localized mode discrimination, and high confidence resistive wall mode control in DIII-D

A linear model for feedback stabilization of n=1 resistive wall modes (RWMs) in the DIII-D [T.C. Simonen, J. Fusion Energy 11, 79 (1992)] tokamak is presented and validated with recent experimental data. The model uses a toroidal current sheet to represent the plasma surface and “picture frame” currents to represent the conducting structure. Since the model does not account for plasma rotation, recent low rotation DIII-D discharges are vital for validation. It is shown that edge localized modes (ELMs) cause the system to become unstable in DIII-D by affecting the magnetic field sensor measurements, and thus, exciting the active coils even though the RWM is already stabilized. Two procedures for discriminating ELMs from the sensor signals are suggested and by combining the two approaches, the ELM contributions in the closed loop can be removed almost completely. Filtered sensor signals and a validated closed loop model facilitates high confidence RWM feedback stabilization. Controllers with high stability ...

From PI to H/sub /spl infin// control in a unified framework

Based on a recently presented evaluation method, tuning rules are introduced and compared for a series of PID related controllers. For all of them, the integral gain and the high-frequency gain (the parameters in a PI controller), are used as the main tuning parameters. As these directly correspond to well known properties of a control system, they are easy to handle for an operator. It is also shown that a well-tuned PID controller augmented by a low-pass filter, to increase the roll-off rate, is quite competitive to a PLD weighted H/sub /spl infin// controller, and superior to the common PI weighted H/sub /spl infin// controller.

Feedforward feedback controller design for uncertain systems

We describe an optimization method for design of combined feedforward and feedback controllers when the plant model is uncertain. It is demonstrated that the feedback design and the feedforward design have to be made jointly for the performance to be optimal. The uncertainties used in the synthesis are given as intervals for the parameters with corresponding probability density functions. These are used in the evaluation of the objective function, which is the expected value of the effect of load disturbances. The minimization is subject to constraints on the sensitivity function and the controller response to reference signals and measurement noise. By changing the constraints the trade-off between performance, robustness and actuation is elucidated in the same manner as for plants with no explicit uncertainties. Depending on the problem character, i.e. SISO or MIMO, number of uncertain parameters and size of nominal closed loop in the uncertainty formulation, three different methods to guarantee the constraints are suggested: A direct evaluation, Horowitz-Sidi bounds with a Horowitz-Sidi test, or use of the structured singular value for robust performance. We also derive a second order approximation of the objective function to use when the number of uncertain parameters is high. The methods are illustrated on a nonlinear and uncertain control problem, namely the external carbon addition in predenitrification wastewater treatment plants.

Global Controller Optimization Using Horowitz Bounds

Abstract A procedure for global optimization of PID type controller parameters for SISO plants with model uncertainty is presented. Robustness to the uncertainties is guaranteed by the use of Horowitz bounds, which are used as constraints when low frequency performance is optimized. The basic idea of both the optimization and the parameter tuning is to formulate separate criteria for low, mid and high frequency closed loop properties. The trade-off between stability margins, high frequency robustness and low frequency performance is then elucidated and, hence, the final choice of parameters is facilitated. The optimization problems are non-convex and ill-conditioned and we use a combination of new global and standard local optimization algorithms available in the TOMLAB optimization environment to solve the problem. The method does not rely on a good initial guess and converges fast and robustly. It is applied to a controller structure comparison for a plant with an uncertain mechanical resonance. For a given control activity and stability margin as well as identical tuning parameters it is shown that a PID controller achieves slightly improved low frequency performance compared to an ℋ ∞ controller based on loop-shaping. The reason for this somewhat surprising result is the roll-off in the ℋ ∞ controller, which adds additional high frequency robustness compared to the PID controller. Computationally, a factor of 10–20 has been gained compared to an earlier, less general, version of the procedure.

Physics and stabilization of resistive wall modes in tokamaks

The theory of resistive wall modes (RWMs) is discussed and compared with experimental results. Special attention is given to the possibilities of stabilizing the RWM by plasma rotation and active feedback. A simple cylindrical model is used to illustrate various aspects of active control. Fully toroidal computations are also presented, including predictions for RWM stabilization in ITER. According to ideal MHD, robust control of RWM should be straightforward. Theory for stabilization by rotation is discussed, and a semikinetic model is introduced, which compares favourably with experiment. The semikinetic model produces somewhat lower rotation thresholds than previous models.

Feedback control of resistive wall modes in toroidal devices

Feedback control of nonaxisymmetric resistive wall modes is studied analytically for cylindrical plasmas and computationally for high beta tokamaks. Internal poloidal sensors give superior performance to radial sensors, for instance in terms of the highest achievable plasma pressure. A single poloidal array of feedback coils allows robust control with respect to variations in plasma pressure, current and rotation velocity. The control analysis is applied to advanced scenarios for ITER. Configurations with multiple poloidal coils and feedback systems for nonresonant MHD instabilities in reversed field pinches are also studied. The control study was carried out using the assumption of ideal amplifiers.