Feedback control and injection locking of the sawtooth oscillation in fusion plasmas

In this thesis various different control strategies have been developed to control the period of the sawtooth oscillation inside a tokamak. This sawtooth instability arises at the q = 1 surface in nuclear fusion plasmas and manifests itself as a repetitive slow rise and sudden crash of the plasma core temperature and pressure. The mixing effect of the sawtooth crash can regulate the exhaust of helium ash from and the fuelling of deuterium and tritium to the plasma core. At the same time, the crash can trigger other instabilities, such as neoclassical tearing modes, which is generally undesirable. These processes are affected by the periodicity of the sawtooth oscillation, hence, control over the sawtooth period is essential to enable optimization of the plasma performance. Based on the dominant dynamics of the underlying magnetics of the sawtooth, a control-relevant sawtooth model has been developed and numerically implemented. This model is actuated via an electron cyclotron current drive (ECCD) term, characterized by two inputs: the amount of driven current and its deposition location. The output of the model is the sawtooth period. This sawtooth model mimics the static input–output behaviour observed on tokamaks, and has therefore been used as a casestudy for the controller designs. Two different control strategies have been investigated: feedback control (closed loop) and injection locking (open loop). Feedback control In the closed loop approach, first the sawtooth period dynamics has been determined. Via dedicated step response simulations and the application of approximate realization techniques, the dynamic behaviour around various operating points has been identified and represented in the frequency domain. Next, three different feedback control approaches have been based upon these identified systems: Low complexity - First, a standard linear controller has been designed, considering only the deposition location as a control variable. The parameters of the chosen controller structure (PII) have been optimized based on frequency domain specifications. The resulting closed loop is relatively robust, but its performance can sometimes be unsatisfactory whenever the ECCD mirror launcher, i.e. the actuator for the deposition location, is slow compared to the sawtooth period. High performance - In a second approach the closed-loop performance has been improved by the employment of gain-scheduling, feedforward and anti-windup techniques. Alternatively, the amount of driven current has been used as a control variable, which yields a closed-loop performance improvement as well. Moreover, a multivariable controller design has been proposed to combine fast settling times of the sawtooth period with power efficiency. These high-performance control designs are particularly interesting for future fusion reactors. High robustness - On experimental devices plasma uncertainties and disturbances are relatively large. For such applications a very robust feedback controller has been designed, based on extremum seeking. This adaptive controller makes online estimations of the sawtooth input–output behaviour via an external perturbation, based upon which the deposition location is adjusted to steer towards a desired sawtooth period. Various simulations have demonstrated the large robustness of the approach. Injection locking In addition, open-loop injection locking has been presented as an alternative sawtooth period control strategy. In this strategy, the deposition location is kept constant while the driven current (or gyrotron power) is modulated with a certain period and duty cycle. Extensive simulations have revealed that the sawtooth period can lock to the modulation period. All combinations of modulation period and duty cycle for which locking occurs define the locking range, which is particularly large for depositions close to the q = 1 surface. Similar sawtooth periods as with continuous wave ECCD can be achieved, while consuming time-averaged less power. Sawtooth period locking can be both fast and robust; simulations have demonstrated convergence speeds within only a handful of crashes, while locking is maintained even in the presence of plasma variations or disturbances. These predictions have been validated by experimental results; injection locking of the sawtooth period has been demonstrated on TCV plasmas. The experimentally obtained time responses and locking range show strong resemblances with the predictive simulations. Based on the identified locking range, an open-loop controller has been designed and implemented. Successful application to a TCV discharge has demonstrated that this controller can force a desired sawtooth period unto the plasma, even if this setpoint is slowly changing over time. The results in this thesis form a basis for further research on sawtooth control. Future work includes the application of the feedback control strategies in tokamak experiments, and further investigation of the locking phenomenon.