To this day, many work has been done in the topic of sawtooth control, always with the goal of controlling the sawtooth period by affecting both quantities in the trigger criterion (1.6).
1.3.1 Actuator
In first place an actuator is needed to effectively influence the sawtooth oscillation. Several have been proposed including the use of Ion Cyclotron waves (IC), neutral beam injection (NBI) and Electron Cy- clotron Current Drive (ECCD). Choosing the best actuator for this purpose has been a subject of many debates, as for example in [13]. These actuators usually work by injecting current in certain zones of the tokamak, and this way altering the local magnetic shear which is responsible for the triggering of the crash. Another way to control the sawtooth period is through modification of the fast particle orbits in the vicinity of theq= 1surface which has a strong effect on the stability of the internal kink mode as will be explained later. It has been shown that IC waves are able to affect both these aspects [14], and the NBI has an effect mainly in the fast particle orbits [15,16]. ECCD has proven to be the most effective, partly because of it’s capability of doing highly localized deposition of the current near theq= 1surface as has been shown experimentally in a variety of tokamaks like ASDEX Upgrade [17] or TCV [18]. In ITER both ICRH and ECCD actuators are envisaged for sawtooth period control.
1.3.2 Model and experiments for sawtooth instability with fast particles
The first experimental demonstration of efficient real time feedback control of the sawtooth instability was achieved by Lennholm, in several experiments performed in Tore Supra [19,20] and JET tokamaks [21]. In order to influence the sawtooth behaviour, it was used in Tore Supra the Electron Cyclotron Resonance Heating (ECRH) and in JET the Ion Cyclotron Resonant Heating (ICRH). Both of this actuator influence the sawtooth activity by manipulating the absorption location of the injected waves. For the first case the absorption location was controlled by changing the ECRH beam direction through mobile mirrors situated inside the tokamak. For ICRH the absorption location was controlled by modifying the injected wave frequency. It was verified in these experiments that the dynamic behaviour of the sawtooth period in plasmas with fast ions in the core diverges a lot from plasmas without. In plasmas with fast ion pressure intentionally created on Tore Supra, the sawtooth was seen to jump between long and short periods when a sweep over a range of values was requested to the controller. These observations lead to the creation of a search and maintain controller that will be explained next. It works by firstly sweep the absorption location of the injected waves until the short period sawtooth regime is observed. At this point it registries the position of both the absorption location and current inversion radius, and from there on tries to maintain the distance between the two constant by continuously updating the former.
An explanation for the observed behaviour is also proposed based on the triggering criteria (1.6) as a
result from the presence of fast ions. It states that due to the known stabilizing effect of these particles on the internal kink mode, there is a different critical shear, sw, below which the crash will not be triggered.
This new effect can be described by the same criterion (1.6), by stating that the critical shear scritis now given by the maximum between the previous value and sw. This results in much longer sawteeth, as
Figure 1.5: Illustrative example of the reasoning suggested by Lennholm in [20], where the magnetic shear is represented by a capital letter. S1 is the shear atq = 1, Sw the critical shear given by the fast ions criterion and Scritthe typical critical shear. The new critical value is given by the maximum of S1and Scrit. Thus in this example the internal kink mode becomes unstable att =t2. However, if the shear at q= 1is slightly increased the crash is triggered att=t1.
illustrated in figure 1.5, unless the shear at theq= 1surface (s1) is increased rapidly enough to cause a crash before the fast ion criterion becomes dominant. This appears to replicate the duality in the sawtooth behaviour between long and short period.
Lennholm’s search and maintain controller relied on knowledge of the ECRH absorption location as well as real time sawtooth inversion radius determination, which are very difficult to estimate. Addition- ally, the slow speed of the mirror angle in the ECRH actuator and of the matching mechanism of the ICRH actuator limit the effectiveness of the controller.
1.3.3 Period locking
An open loop controller for the sawtooth instability based on injection locking has been developed by Lauret et al. and demonstrated in TCV plasmas [22]. This is a well known technique in electronic circuits and system theory [23]. Instead of using the deposition location of the waves at a constant power, this controller manipulates the sawtooth period by modulating the power of injected waves from an ECCD source. It was shown that, by keeping the deposition location constant while using a pulsed ECCD power with a certain period and duty cycle, a period locking is achieved i.e., the sawtooth oscillation synchronizes with the modulated input signal, and thus acquires the same period as the injected signal.
Since this is an open loop control method it does not rely on real time sawtooth period detection and therefore is not affected by the measurement noise, nor is limited by the speed of the actuator since the power of the ECCD launcher can be modified very quickly. Furthermore, when compared to the change of deposition location approach, this method converges must faster and is less sensitive to variations in the location. In a variation of period locking, called pacing, the modulation of the power is feedback
controlled making it very robust against variations in plasma parameters, at the cost of having to rely on a crash detection system. Lauret was the first to analyse this method from a theoretical perspective [24].
He was able to show mathematically that pacing control will lead to stabilization for a rather general class of the input output map (sawtooth period versus power modulation period).
1.3.4 Adaptive control and hysteresis
Figure 1.6: Hysteresis effect on the sawtooth period ob- served by Payley and reported in [25]. When the angle of the ECCD launcher is increased the observed saw- tooth period is different from the one observed when it is decreased.
Further work on the control of the sawtooth os- cillation was done by Payley [25] where he also demonstrates that it is possible to control saw- teeth using real time steerable EC launchers.
In his experiments he verified the existence of an hysteresis when sweeping the EC wave de- position location from the outer plasma to the inner plasma and back again, as can be seen in figure 1.6. This effect is explained by the re- distribution of the global plasma current that occurs at a slower time scale than the mirror angle movement, resulting in the displacement of the q = 1 surface. He concluded that it is important to build control algorithms based in models that take this hysteresis into account.
He then predicted that a closed loop controller that determines the position of the q = 1sur- face would yield much better performance.
1.3.5 Velocity distribution of the fast ions
In [26] Graves et al., through an analytical approach proved that both trapped and co-passing fast ions have a stabilizing effect on the internal kink mode and therefore, their presence can lengthen its period, while counter-passing fast ions have a destabilizing effect. With these results he stated that it is possible to control the sawtooth period by manipulating the velocity distribution of an auxiliary fast ion population generated with ICRH. The sawtooth can be lengthened or shortened either by increasing the number of co or counter-passing fast ions respectively, controlled wit the application of ICRH. This was demonstrated experimentally in several shots carried out in JET [27], where NBI was used to create co-passing ions that have an effect on the sawtooth similar to the predicted αparticles that will be released by fusion reactions in ITER. Even in the presence of stabilizing co-passing fast ions, the use of ICRH showed to be efficient in the destabilization of the internal kink mode and the consequent shortening of the sawtooth period.
1.3.6 Physics oriented modelling
In addition to investigate the feasibility of the already mentioned actuators [15,16,13], Chapman also studied the influence that the Graves’ effect can have in the sawtooth instability in ITER through a series of simulations [28]. In this work he assumed the conditions predicted for that tokamak, as well as the characteristics of the envisaged actuators.
He concluded that the contribution from the Graves’ effect can completely cancel the stabilizing effect of the fast particle population when the deposition location of the ICRH waves is just inside the q = 1 surface. The contribution from this effect then vanishes as the two locations are further apart.
1.3.7 Control oriented modelling
Later, Witvoet developed a control-oriented model for the sawtooth based on the crash trigger condition of Porcelli and the reconnection model of Kadomtsev, using only the magnetic diffusion equation [29].
This model describes the general behaviour of the sawtooth as a response to the injected power and location of an ECCD source, with a relatively low complexity. Then, using this model he derived several closed and open loop model-based controllers. In first place, a linear controller was developed consisting in a PI-controller that used as actuator the ECCD deposition location through variation of the angle of the mirror reflecting the ECCD beam. Although this method showed some robustness against small distur- bances, its performance was limited by the slow dynamics of the actuator. In a second approach, both the deposition location of the ECCD as the power of the injected beam were used as actuators [30]. Unlike the mirror angle, the dynamics of the second actuator is very fast and does not limit the performance of the controller. As a consequence with this design was achieved extremely small steady-state errors (the difference from the requested sawtooth period and its actual value after stabilization) with very good per- formance. However these controllers had a limited robustness against plasma perturbations, which are typically encountered in every hot plasmas. Hence, a third controller was developed based on a control approach called extremum seeking control [31]. It operates by constantly scanning the current operating point and adapt accordingly. To do so, the manipulated variable (in this case the mirror angle of the ECCD launcher) keeps oscillating with a small amplitude, and the response of the system is evaluated in order to calculate the gradient of a pre-determined cost function. Then it tries to reach the minimum of this function that corresponds to the desired sawtooth period. One of the advantages of this controller is that it does not rely on the model used to describe the sawtooth and thus can be used in many different situations. Using the aforementioned sawtooth model this controller was tested in many conditions and in all of them it managed to track the reference value of the sawtooth period as expected, demonstrating its high robustness. However, a non-zero steady-state error is inherent to this kind of controller, because the period is always oscillating around the reference value, which is needed for the on-line estimation of the gradient of the cost function. Finally Witvoet used the developed model in order to numerically verify the effectiveness of the period locking controller, already discussed, and showed that it can operate over a large range of parameters proving its robustness as well as good performance [32].