reduce the sawtooth period below the50s threshold, and later control it.
-1.0 -0.5 0.0 0.5 1.0 1.5
Magneticshear
s1 scrit sfp sKO sBu sGraves sw
0 10 20 30 40 50 60
0.0 0.1 0.2 0.3 0.4
Time after crash(s)
ρ ρ1
ρICRH
Figure 3.5: Evolution of the magnetic shear at the q = 1 surface and critical shear components for ρICRH = 0.46and no current driven with ECCD. The Kruskal-Oberman and fast particle terms saturate after only 20 s and its the remaining terms that contribute to the evolution ofsw.
the Graves’ terms. The largest of the two, the Graves’ term is only relevant when theq= 1surface is near the deposition location of the ICRH, and so it keeps increasing until theρ1reachesρICRHand the crash is triggered. On the other hand, whenρICRH is located at0.44, theq= 1surface grows beyondρICRHbefore
-0.5 0.0 0.5 1.0 1.5
Magneticshear
s1 scrit sfp sKO sBu sGraves sw
0 50 100 150 200 250
0.0 0.1 0.2 0.3 0.4 0.5
Time after crash(s)
ρ ρ1
ρICRH
Figure 3.6: Evolution of the magnetic shear at the q = 1 surface and critical shear components for ρICRH = 0.44and no current driven with ECCD. Notice that there is a minimum inswatt≈40s which corresponds to the instant whenρICRH=ρ1.
the crash is triggered. Consequently, as now the bottom plot of figure 3.6 depicts, after 35s theq = 1 surface is moving away fromρICRH and the Graves’ term is constantly decreasing, as the top plot in the same figure shows. In order for the crash to be triggered it is necessary to wait until the Bussac term grows to compensate it, which takes250s.
3.3.2 Response of the extended Witvoet model to both actuators
Although we showed that the sawtooth period can be greatly reduced, it is still above the50s mark we had established. So it seems clear that the two actuators must be used in simultaneous for that purpose.
This was done by adding a single ECCD with a total driven current of −400 kA. We opted to use the maximum driven current since it is the one that maximizes the destabilizing effect.
In figure 3.7 is depicted the 2D steady state map of the sawtooth period in respect toρICRH and to ρECCD. Each contour line represents a sawtooth period multiple of10s, from30s to320s. We see that in
Figure 3.7: Steady state map in a small region around the global minimum. The minimum sawtooth period observed is 30s.
the area coloured with the darkest blue, the sawtooth has successfully been shortened to periods smaller than50s. Thus this method represents a suitable solution for the problem of reducing and controlling the sawtooth period, and it is the one that we chose to investigate in this work. However the area of the map for whichτs<50represents only0.3%of the total map, which means that the combination of actuators has to be precisely controlled in order to obtain such short periods. Additionally, note that the extended Witvoet model shows very non-linear response to the actuators location which increases the complexity of the problem from a control engineering point of view.
In some locations of the map the period has abrupt variations, just like happened with only the ICRH actuator, explained in section 3.3. Notice for example that for ρICRH = 0.45, if we decreaseρECCD from 0.6to0.3, which means, if we follow the vertical line at thisρICRH we observe that the period initially is almost constant at τs = 300s. Then, atρECCD = 0.5 the period rapidly decreases to∼ 50s, and keeps
there untilρECCD = 0.42, when it instantly jumps to250s. It must be emphasized here that this is a very similar behaviour to that reported by Lennohlm in [20], and introduced in section 1.3.
The actual minimum period obtained with this method is29.9 s which is observed forρICRH = 0.45 andρECCD= 0.445. Notice that for both actuators the location of the minimum is different from the ones obtained individually, which proves that the destabilizing effect from each one is not cumulative. This is an important result that will later, in chapter 4, have implications when designing of the controller.
One important characteristic of the steady map for designing a controller is the DC-gain, i.e. the variation of the steady state output of the model due to a small variation in the actuators which in our case is equivalent to the gradient of the map. In common controllers like P and PI-controllers the steady state map is required to be monotonic, .i.e the DC-gain must not change its sign. Essentially these controllers work by applying a change in the actuator location which is proportional to the error, i.e. to the difference between the observed output and the requested one, until the two converge. This method is based in the assumption that a positive variation of the actuator always leads to a variation of the output in the same direction. If that is not the case, and at some point the DC-gain changes sign, the variation in the actuator will make the output to move away from the requested value, increasing the error. This leads to a further increase in the variation of the actuator location in the wrong direction and thus, the error would increase indefinitely or in other words, the controller becomes unstable.
In the steady state map of figure 3.7 this requirement is not fulfilled, since in the left half the gradient with respect toρICRHpoints in the positive direction while in the right half occurs the opposite. The same behaviour is observed in the case of the other actuator. This aspect of the model carries an additional limitation for the controller as will be discussed in chapter 4. As has been observed using solely the ICRH actuator (figure 3.4), when ρICRH is far from the minimum, the Graves’ effect ceases to have an impact in the triggering of the crash. Consequently forρICRH < 0.4∨ρICRH >0.7 the change in the sawtooth period due to a small variation inρICRHis negligible and the gradient points vertically. Also for the ECCD actuator, if the deposition location is far from the minimum, its effect on the sawtooth instability vanishes and a change in ρECCD has little or no impact in its period. As a consequence, in these locations the gradient points horizontally as can be seen in the 2D steady state map for ρICRH = 0.5, ρECCD = 0.3.
Further away from the minimum, the steady state map becomes completely flat which means that the DC-gain is null, and so these zones are unusable for control purposes.
3.3.2a) Bifurcation
Finally, a particularity of the model must be mentioned here, related with the discontinuity in its steady state map marked with black, that can be found in the lower left vicinity of the τs = 50 s contour line. In these rare cases, the sawtooth period does not converge to a single value. Instead what we observe is that after the crash of a long sawtooth cycle, the next cycle is very short, followed again by a long one, a behaviour that is also known as a bifurcation of the map. The periods of the long and short cycles are constant and close to the ones in surrounding locations. To understand the reason for this behaviour we show in figure 3.8 the evolution of the quantities involved in the triggering criteria, alongside the plot of the normalized radius of the q = 1 surface. When the first crash happens at
-0.5 0.0 0.5 1.0 1.5
Magneticshear
s1 scrit sfp sKO sBu sGraves sw
0 50 100 150 200 250
0.0 0.1 0.2 0.3 0.4 0.5
Time after crash(s)
ρ ρ1
ρICRH ρECCD
Figure 3.8: Evolution of the quantities responsible for the triggering of the crash for ρICRH = 0.43and ρECCD = 0.4355, showing the long sawtooth cycle followed by a short one. In the top plot are shown the magnetic shear at theq= 1surface as well as the critical shear components. In the bottom plot is drawn the normalized radius of theq= 1surface alongside the location of the ICRH and ECCD actuators.
t = 53 s, the configuration of the magnetic field did not have time to evolve towards a state close to the equilibrium. As a consequence, the magnetic field configuration after the crash, generated with the Kadomtsev reconnection model, varies slightly from the one obtained after a short sawtooth. Therefore, the magnetic field evolves differently from one case to the other. Specifically, in the second, longer cycle, theq= 1surface grows faster than in the first. This can be better seen in figure 3.9 where we show the first80seconds of each cycle overlapped, where the dashed lines represent the long one.
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Magneticshear
30 40 50 60 70
0.90 0.95 1.00 1.05
s1 scrit sfp sKO sBu sGraves sw
0 20 40 60 80
0.0 0.1 0.2 0.3 0.4 0.5
Time after crash(s)
ρ ρ1
ρICRH
ρECCD
Figure 3.9: Overlap of two cycles shown in figure 3.8. The dashed curves represent the second, longer cycle. In the top plot, is depicted also an amplification ofs1andswat the end of the short cycle, showing that at the same instant on the long cycle sw is slightly larger thans1, which is why the crash is not triggered. Additionallyswis growing faster thans1which ensures that the crash is not triggered on short term.
Due to the slightly faster growth of theq= 1surface after a short cycle, the maximum destabilizing contribution from the Graves’ effect is reached earlier. Additionally all the terms that composeswdepend on r1. Consequently, after the same53 s from the beginning of the second cycle, the critical shear is slightly aboves1and so the crash is not triggered at this instant. Sinceρ1(which is continuously growing) has already passed byρICRH at this point, the contribution from Graves’ effect decreases during the rest of the cycle and thus sw increases. In fact, it grows faster than s1 and so only when the Bussac term becomes relevant, the loss of the Graves’ destabilizing effect is compensated and the crash is triggered.
In the current case, it takes approximately230s for this to happen as can be seen in figure 3.9. After the crash of this long cycle, due to the new configuration of the magnetic field, it evolves at a slower pace and the crash is triggered again at53s. This feature is a direct consequence of the inherent memory of Witvoet model introduced by the Kadomtsev reconnection. It presents another limitation to the controller, since in fact, this phenomenon encompasses a discontinuity in the steady state map and so the DC-gain in this region is infinite.