Page 16 - Positional stability of field-reversed-configurations in the presence of resistive walls

instead of direct “measurements” of the plasma position, the simulations provide some useful information for the de- sign of a control system for use in experiments.
Firstly (and most obviously), we find that the simulated control system (coil locations, turns, voltage steps) is suf- ficient to control the instability for displacements of up to 20 cm with considerable margin. For the largest displace- ments, the current in any coil is still below 300 A.
Furthermore, we find that an increase in latency has a much smaller performance impact than an increase in cy- cle time. For example, the control performance with a cycle time of 60 μs and a single cycle (i.e., 60 μs) latency is much worse than with a cycle time of 10 μs and 6 cycles (i.e, 60 μs) latency. In other words, even when discretized to steps of 400 V, the smoothness of the actuator commands matters. The control hardware in an experiment should therefore be designed to support pipelined operation, so that sufficient time can be spent on the computation of each sample with- out compromising on the cycle time.
Analysis with the linear model revealed the importance of 3-D effects from the conducting wall. Instability growth times may be increased by a factor of 4 or more if the wall has cutouts at the right places, a difference that is not ap- parent when looking only at the characteristic L/R time of the wall itself.
Finally, we note that the same control algorithm was able to control a wide range of instabilities, including cases with wall resistivity scaled by a factor of two, and different unper- turbed equilibria (not shown in detail in this paper). Given that the control algorithm contains no parameters that were derived from the plasma equilibrium, there is thus a good chance that control of the plasma position will not require any additional input from a (slower running) equilibrium reconstruction algorithm.
VII. SUMMARY
We have presented a linear model for rigid displacement of a toroidally symmetric plasma. The model has a single free parameter, which is the (shape of the) volume that is considered rigid. We find that that, in the absence of super- conductors, any axisymmetric system is unstable to either a radial or an axial displacement (but not both). In the pres- ence of a conducting wall, the growth rate of the instability becomes proportional to the wall resistivity. As wall resis- tivity decreases, plasma inertia becomes insignificant and feedback control becomes feasible.
We have simulated the unperturbed evolution of an ax- isymmetric, beam-driven FRC over a relatively quiescent period using the Q2D code. At three different points in time, we perturbed the plasma and compared the non- linear evolution of the instability with the linear predictions. We found that predictions had a strong dependence on the choice of rigid volume, and used this to fix the free parame- ter. There does not seem to be a way to determine this pa- rameter without running non-linear simulations, so the pri- mary use of the linear model is not the determination of un-
controlled growth rates but the design of control algorithms. We tested the usefulness of the linear model for control algorithm design using a simple control algorithm devel- oped in Simulink. In order to perform non-linear Simula- tions without replicating the control logic in the Q2D code, we devised a method to integrate Simulink models with ar- bitrary plasma simulation codes that support custom initial conditions. The Simulink model is compiled into a shared object, and a supervisor process runs plasma simulation code and control algorithm in lockstep by appropriate mod- ification of the initial conditions. The advantage of this method is that it enables the use of a single codebase for linear design, non-linear testing, and experimental applica-
tion under real-time constraints.
When used for control algorithm design, we find the lin-
ear model to perform satisfactory. The global characteristics of the controlled non-linear system are reproduced over a wide range of control parameters. Linear and non-linear re- sults also indicate the same potential improvements to the control algorithm (taking into account the resistivity of the wall to avoid a breakdown of performance at very fast cy- cle times and low latencies), and similar design goals for an experimental control system (favoring longer latencies over longer cycle times). In addition, the linear model allowed us to examine the effects of a three dimensional wall on the in- stability growth times. Here we found that walls with similar L/R time may nevertheless cause significant changes to the instability growth time if e.g. there are cutouts in the wall region that best couples to the plasma.
VIII. ACKNOWLEDGEMENTS
The authors would like to thank Jim Bialek for his assis- tance with the VALEN code, and Richard Milroy for his assi- tance with the NIMROD code. The control algorithm used for this paper was proposed by Jesus Romero. Data analy- sis for this work was done with Python in Jupyter, using the SciPy, Matplotlib, and Pandas modules.
IX. REFERENCES
1E. V. Belova et al., Numerical study of tilt stability of prolate field-reversed configurations, Physics of Plasmas, 7(12):4996–5006, 2000.
2E. V. Belova et al., Advances in the numerical modeling of field-reversed configurations, Physics of Plasmas, 13(5):056115, 2006.
3J. Bialek et al., Modeling of active control of external magnetohydrody- namic instabilities, Physics of Plasmas, 8(5):2170, 2001.
4M. Bindermauer et al., A high performance field-reversed configuration, Physics of Plasmas, 22:056110, 2015.
5A. H. Boozer, Simplified multimode calculations of resistive wall modes, Physics of Plasmas, 17(7):072503, 2010.
6A. H. Boozer, Non-axisymmetric magnetic fields and toroidal plasma confinement, Nuclear Fusion, 55(2):025001, 2015.
7M. Chu and M. Okabayashi, Stabilization of the external kink and the re- sistive wall mode, Plasma Physics and Controlled Fusion, 52(12):123001, December 2010.
8 N. Iwasawa, A. Ishida, and L. C. Steinhauer, Linear gyroviscous stability of field-reversed configurations with static equilibrium, Physics of Plasmas, 8(4):1240, 2001.
16