The work presented here is part of the final masters project conducted on a single semester. As such it was limited in time and can be improved in many aspects.
Further developments can be done in the model to make it more reliable and accurate. Specifically, the evolution of the bulk plasma density and temperature can be modelled assuming diffusion of those quantities during a cycle followed by a reset of their profiles after a sawtooth crash. This reset would make the profiles flat inside the mixing radius, as is observed in experiments. Also, the equation that governs the evolution of fast particles can be modified to take into account effects like diffusion of these particles, losses to the wall and a better description of the slowing down process. Finally the model could be improved by removing the assumption of circular cross section, although this would implicate a great change in the code, as well as large increase in its complexity since it would require an extra coordinate to describe the magnetic field.
On the other side, the controller can be improved by using better algorithms of optimization that cover shorter trajectories. One of such algorithms is the Newton based scheme, which in addition to the gradient of the cost function the optimization rate is proportional to the inverse of the Hessian, i.e. the matrix of second derivatives. In the current work we tried to apply this scheme but with no success due to difficulties in the estimation of the Hessian and also lack of available time. However we believe that if such scheme was successfully implemented, it would yield large improvements in the performance of the controller, since the optimization rate would not be constant as is the case with our optimizer. Instead,
the vast changes in the gradient seen in the steady state map could be compensated. For example, when the controller reaches an area where the gradient is low, for which it had not been tuned to, the gain would be increased by the inverse of the second derivative and so the convergence time of the controller would decrease. However, note that the saturation used in our controller would still be necessary in order to cope with the infinite DC-gain observed at the bifurcation of the steady state map.
Also the mechanism implemented to make sure that the controller will not remain static indefinitely at a local minimum can be improved in two ways. Firstly by further tuning the parameters we chose and in the ultimate case by using a different algorithm. For example one that keeps the large perturbation amplitude constant until a new minimum is found instead of the fixed50sawtooth cycles.
Furthermore, the hysteresis problem must be addressed, and a study must be done to understand if it can threaten the operation of our controller. In the affirmative case, a solution may also consist in increasing the perturbation amplitude as Bolder suggests in [31].
The long sawtooth periods predicted in future tokamaks pose an obstacle to their operation in an efficient manner. This fact in combination with the many advantages inherent to sawtooth period control justify future investigation of this topic.
Bibliography
[1] Fusion for Energy Website, 2013. URLhttp://fusionforenergy.europa.eu.
[2] J. Wesson.Tokamaks. Oxford University Press, 3 edition, 2004. ISBN 0198509227.
[3] Euro Fusion Website. URL https://www.euro-fusion.org/wpcms/wp-content/uploads/2011/
09/jg05-537-1c-720x260.jpg.
[4] B. Reiter. Radiative Response on Massive Noble Gas Injection For Runaway Suppression in Disruptive Plasmas. PhD thesis, 2010.
[5] P. H. Rutherford. Nonlinear growth of the tearing mode. Phys. Fluids, 16(11):1903–1908, 1973.
ISSN 1070664X. doi: http://dx.doi.org/10.1063/1.1694232. URL http://ipaps.ucsd.edu/
students/courses/spring2015/physics218b/Rutherford.pdf.
[6] G. Witvoet. Feedback control and injection locking of the sawtooth oscillation in fusion plasmas. PhD thesis, Eindhoven University of Technology, 2011.
[7] F. Porcelli, D. Boucher, and M. N. Rosenbluth. Model for the sawtooth period and amplitude.Plasma Phys. Control. Fusion, 38(12):2163–2186, 1999. ISSN 0741-3335. doi: 10.1088/0741-3335/38/
12/010.
[8] M. Nave, N. Gorelenkov, K. McClements, S. Allfrey, B. Balet, D. Borba, P. Lomas, J. Manickam, T. Jones, and P. Thomas. Fast particle effects on the sawtooth stability of JET DT discharges*.Nucl.
Fusion, 42(3):281–289, 2002. ISSN 0029-5515. doi: 10.1088/0029-5515/42/3/308.
[9] I. Chapman, R. Buttery, S. Coda, S. Gerhardt, J. Graves, D. Howell, A. Isayama, R. La Haye, Y. Liu, P. Maget, M. Maraschek, S. Sabbagh, and O. Sauter. Empirical scaling of sawtooth period for onset of neoclassical tearing modes, 2010. ISSN 0029-5515.
[10] O. Sauter, E. Westerhof, M. L. Mayoral, B. Alper, P. A. Belo, R. J. Buttery, A. Gondhalekar, T. Hellsten, T. C. Hender, D. F. Howell, T. Johnson, P. Lamalle, M. J. Mantsinen, F. Milani, M. F. F. Nave, F. Nguyen, A. L. Pecquet, S. D. Pinches, S. Podda, and J. Rapp. Control of neoclassical tearing modes by sawtooth control.Phys. Rev. Lett., 88(10):105001, 2002. ISSN 0031-9007. doi: 10.1103/
PhysRevLett.88.105001.
[11] M. Maraschek. Control of neoclassical tearing modes. Nucl. Fusion, 52(7):074007, 2012. ISSN 0029-5515. doi: 10.1088/0029-5515/52/7/074007.
[12] M. Nave, J. Rapp, T. Bolzonella, R. Dux, M. Mantsinen, R. Budny, P. Dumortier, M. von Hellermann, S. Jachmich, H. Koslowski, G. Maddison, A. Messiaen, P. Monier-Garbet, J. Ongena, M. Puiatti, J. Strachan, G. Telesca, B. Unterberg, M. Valisa, P. de Vries, and c. t. t. J.-E. Workprogramme. Role of sawtooth in avoiding impurity accumulation and maintaining good confinement in JET radiative mantle discharges, 2003.
[13] I. T. Chapman. Controlling sawtooth oscillations in tokamak plasmas.Plasma Phys. Control. Fusion, 53(1):013001, jan 2010. ISSN 0741-3335. doi: 10.1088/0741-3335/53/1/013001. URLhttp:
//iopscience.iop.org/article/10.1088/0741-3335/53/1/013001/meta.
[14] L.-G. Eriksson, T. Johnson, M.-L. Mayoral, S. Coda, O. Sauter, R. Buttery, D. McDonald, T. Hell- sten, M. Mantsinen, A. Mueck, J.-M. Noterdaeme, M. Santala, E. Westerhof, P. de Vries, and J.-E. Contributors. On ion cyclotron current drive for sawtooth control. Nucl. Fusion, 46(10):
S951–S964, oct 2006. ISSN 0029-5515. doi: 10.1088/0029-5515/46/10/S12. URL http:
//iopscience.iop.org/article/10.1088/0029-5515/46/10/S12/meta.
[15] I. T. Chapman, I. Jenkins, R. V. Budny, J. P. Graves, S. D. Pinches, and S. Saarelma. Sawtooth control using off-axis NBI.Plasma Phys. Control. Fusion, 50(4):045006, apr 2008. ISSN 0741-3335.
doi: 10.1088/0741-3335/50/4/045006. URL http://iopscience.iop.org/article/10.1088/
0741-3335/50/4/045006/meta.
[16] I. T. Chapman, V. Igochine, J. Graves, S. Pinches, A. Gude, I. Jenkins, M. Maraschek, and G. Tardini.
Sawtooth control and the interaction of energetic particles. Nucl. Fusion, 49(3):035006, 2009.
ISSN 0029-5515. doi: 10.1088/0029-5515/49/3/035006. URL http://iopscience.iop.org/
article/10.1088/0029-5515/49/3/035006/meta.
[17] A. M¨uck, T. P. Goodman, M. Maraschek, G. Pereverzev, F. Ryter, H. Zohm, and A. U. Team. Sawtooth control experiments on ASDEX Upgrade, 2005. ISSN 0741-3335.
[18] C. Angioni, T. Goodman, M. Henderson, and O. Sauter. Effects of localized electron heating and current drive on the sawtooth period. Nucl. Fusion, 43(6):455–468, jun 2003. ISSN 0029- 5515. doi: 10.1088/0029-5515/43/6/308. URL http://iopscience.iop.org/article/10.
1088/0029-5515/43/6/308/meta.
[19] M. Lennholm, L. G. Eriksson, F. Turco, and F. Bouquey. Closed loop sawtooth period control using variable ECCD injection angles on tore supra. Fusion Sci. Technol., 55(1):45–55, 2009.
[20] M. Lennholm, L. G. Eriksson, F. Turco, F. Bouquey, C. Darbos, R. Dumont, G. Giruzzi, M. Jung, R. Lambert, R. Magne, D. Molina, P. Moreau, F. Rimini, J. L. Segui, S. Song, and E. Traisnel. Demon- stration of effective control of fast-ion-stabilized Sawteeth by electron-cyclotron current drive.Phys.
Rev. Lett., 102(11):100–103, 2009. ISSN 00319007. doi: 10.1103/PhysRevLett.102.115004.
[21] M. Lennholm, T. Blackman, I. T. Chapman, L. G. Eriksson, J. P. Graves, D. F. Howell, M. de Baar, G. Calabro, R. Dumont, M. Graham, S. Jachmich, M. L. Mayoral, C. Sozzi, M. Stamp, M. Tsalas,
and P. de Vries. Feedback control of the sawtooth period through real time control of the ion cyclotron resonance frequency. Nucl. Fusion, 51(7):073032+, 2011. ISSN 0029-5515. doi: 10.
1088/0029-5515/51/7/073032. URLhttp://dx.doi.org/10.1088/0029-5515/51/7/073032.
[22] M. Lauret, F. Felici, G. Witvoet, T. T. Goodman, G. Vandersteen, O. Sauter, and M. M. de Baar.
Demonstration of sawtooth period locking with power modulation in TCV plasmas. Nucl. Fusion, 52(6):062002, jun 2012. ISSN 0029-5515. doi: 10.1088/0029-5515/52/6/062002. URLhttp:
//iopscience.iop.org/article/10.1088/0029-5515/52/6/062002/meta.
[23] R. Adler. Study of locking phenomena in oscillators, 1973. ISSN 00189219.
[24] M. Lauret.Control of Mixing and Oscillations in Plasmas and Fluids. PhD thesis, Eindhoven University of Technology, 2014.
[25] J. I. Paley, F. Felici, S. Coda, T. P. Goodman, and F. Piras. Real time control of the sawtooth period using EC launchers. Plasma Phys. Control. Fusion, 51(5):055010, 2009. ISSN 0741-3335. doi:
10.1088/0741-3335/51/5/055010.
[26] J. P. Graves, I. Chapman, S. Coda, L. G. Eriksson, and T. Johnson. Sawtooth-control mechanism using toroidally propagating ion-cyclotron-resonance waves in tokamaks. Phys. Rev. Lett., 102(6):
1–4, 2009. ISSN 00319007. doi: 10.1103/PhysRevLett.102.065005. URLhttp://journals.aps.
org/prl/abstract/10.1103/PhysRevLett.102.065005.
[27] J. Graves, I. Chapman, S. Coda, M. Lennholm, M. Albergante, and M. Jucker. Control of magne- tohydrodynamic stability by phase space engineering of energetic ions in tokamak plasmas. Nat.
Commun., 3:624, 2012. ISSN 2041-1723. doi: 10.1038/ncomms1622. URLhttp://dx.doi.org/
10.1038/ncomms1622.
[28] I. T. Chapman, J. P. Graves, O. Sauter, C. Zucca, O. Asunta, R. J. Buttery, S. Coda, T. Goodman, V. Igochine, T. Johnson, M. Jucker, R. J. L. Haye, M. Lennholm, and J.-E. Contributors. Power requirements for electron cyclotron current drive and ion cyclotron resonance heating for sawtooth control in ITER. Nucl. Fusion, 53(6):066001, 2013. ISSN 0029-5515. doi: 10.1088/0029-5515/
53/6/066001. URLhttp://iopscience.iop.org/0029-5515/53/6/066001.
[29] G. Witvoet, M. de Baar, E. Westerhof, M. Steinbuch, and N. Doelman. Systematic design of a sawtooth period feedback controller using a Kadomtsev–Porcelli sawtooth model. Nucl. Fusion, 51 (7):073024, 2011. ISSN 0029-5515. doi: 10.1088/0029-5515/51/7/073024.
[30] G. Witvoet, M. Steinbuch, M. de Baar, N. Doelman, and E. Westerhof. Sawtooth period control strategies and designs for improved performance.Nucl. Fusion, 52(7):074005, jul 2012. ISSN 0029- 5515. doi: 10.1088/0029-5515/52/7/074005. URLhttp://iopscience.iop.org/article/10.
1088/0029-5515/52/7/074005/meta.
[31] J. Bolder, G. Witvoet, M. de Baar, N. van de Wouw, M. Haring, E. Westerhof, N. Doelman, and M. Steinbuch. Robust sawtooth period control based on adaptive online optimization.Nucl. Fusion,
52(7):074006, jul 2012. ISSN 0029-5515. doi: 10.1088/0029-5515/52/7/074006. URLhttp:
//iopscience.iop.org/article/10.1088/0029-5515/52/7/074006/meta.
[32] G. Witvoet, M. Lauret, M. de Baar, E. Westerhof, and M. Steinbuch. Numerical demonstration of injection locking of the sawtooth period by means of modulated EC current drive. Nucl. Fusion, 51 (10):103043, 2011. ISSN 0029-5515. doi: 10.1088/0029-5515/51/10/103043.
[33] M. Lennholm. Real time control of the sawtooth instability in fusion plasmas with large fast ion populations. PhD thesis, Eindhoven University of Technology, 2014.
[34] B. B. Kadomtsev. On disruptive instability in tokamaks.Sov. J. Plasma Phys., 1(389):710–715, 1975.
[35] T. Casper, Y. Gribov, A. Kavin, V. Lukash, R. Khayrutdinov, H. Fujieda, and C. Kessel. Development of the ITER baseline inductive scenario. Nucl. Fusion, 54(1):013005, 2014. ISSN 0029-5515.
doi: 10.1088/0029-5515/54/1/013005. URL http://iopscience.iop.org/article/10.1088/
0029-5515/54/1/013005/meta.
[36] a. Zocco, J. W. Connor, C. G. Gimblett, and R. J. Hastie. Improved criterion for sawtooth trigger and modelling.Plasma Phys. Control. Fusion, 55(7):074005, 2013. ISSN 0741-3335. doi: 10.1088/
0741-3335/55/7/074005. URLhttp://iopscience.iop.org/article/10.1088/0741-3335/55/
7/074005/meta.
[37] T. Hender, J. Wesley, J. Bialek, A. Bondeson, A. Boozer, R. Buttery, A. Garofalo, T. Goodman, R. Granetz, Y. Gribov, O. Gruber, M. Gryaznevich, G. Giruzzi, S. G¨unter, N. Hayashi, P. He- lander, C. Hegna, D. Howell, D. Humphreys, G. Huysmans, A. Hyatt, A. Isayama, S. Jardin, Y. Kawano, A. Kellman, C. Kessel, H. Koslowski, R. L. Haye, E. Lazzaro, Y. Liu, V. Lukash, J. Manickam, S. Medvedev, V. Mertens, S. Mirnov, Y. Nakamura, G. Navratil, M. Okabayashi, T. Ozeki, R. Paccagnella, G. Pautasso, F. Porcelli, V. Pustovitov, V. Riccardo, M. Sato, O. Sauter, M. Schaffer, M. Shimada, P. Sonato, E. Strait, M. Sugihara, M. Takechi, A. Turnbull, E. West- erhof, D. Whyte, R. Yoshino, H. Zohm, D. Group, the ITPA MHD, Magnet, D. Group, the Itpa Mhd, and Magnet. Chapter 3: MHD stability, operational limits and disruptions. Nucl. Fu- sion, 47(6):S128–S202, 2007. ISSN 0029-5515. doi: 10.1088/0029-5515/47/6/S03. URL http://iopscience.iop.org/article/10.1088/0029-5515/47/6/S03/meta.
[38] J. Huba. NRL plasma formulary. Technical report, Naval Research Laboratory, 2004. URLhttp:
//oai.dtic.mil/oai/oai?verb=getRecord&metadataPrefix=html&identifier=ADA429448.
[39] S. J. Zweben, S. Putvinski, M. P. Petrov, G. Sadler, K. Tobita, and K. M. Young. Alpha-Physics and Measurement Requirements for ITER. In Diagnostics Exp. Thermonucl. Fusion React., pages 467–
476. Springer US, Boston, MA, 1996. doi: 10.1007/978-1-4613-0369-5 58. URLhttp://link.
springer.com/10.1007/978-1-4613-0369-5_58.
[40] M. N. Bussac, R. Pellat, D. Edery, and J. L. Soule. Internal Kink Modes in Toroidal Plasmas with Circular Cross Sections. Phys. Rev. Lett., 35(24):1638–1641, dec 1975. ISSN 0031-9007. doi:
10.1103/PhysRevLett.35.1638. URLhttp://link.aps.org/doi/10.1103/PhysRevLett.35.1638.
[41] H. Zohm. The concept of ECRH/ECCD for ITER. In13th Jt. Work. Electron Cyclotr. Emiss. Electron Cyclotr. Reson. Heating. Russ., 2004. URL http://www.ec13.iapras.ru/papers/Zohm_Invited.
pdf.
[42] Y. Tan, W. Moase, C. Manzie, D. Nessic, and I. Mareels. Extremum seeking from 1922 to 2010.
Control Conf. (CCC), 2010 29th Chinese, 2010.
[43] M. Krsti´c and H.-H. Wang. Stability of extremum seeking feedback for general nonlinear dynamic systems. Automatica, 36(4):595–601, apr 2000. ISSN 00051098. doi: 10.1016/S0005-1098(99) 00183-1. URLhttp://linkinghub.elsevier.com/retrieve/pii/S0005109899001831.
[44] Y. Tan, D. Neˇsi´c, and I. Mareels. On non-local stability properties of extremum seeking control.
Automatica, 42(6):889–903, jun 2006. ISSN 00051098. doi: 10.1016/j.automatica.2006.01.014.
URLhttp://linkinghub.elsevier.com/retrieve/pii/S0005109806000690.
[45] A. Ghaffari, M. Krsti´c, and D. Neˇsi´c. Multivariable Newton-based extremum seeking. Automatica, 48(8):1759–1767, 2012. ISSN 00051098. doi: 10.1016/j.automatica.2012.05.059. URLhttp:
//dx.doi.org/10.1016/j.automatica.2012.05.059.
[46] W. H. Moase, C. Manzie, and M. J. Brear. Newton-like extremum-seeking part II: Simulations and experiments. Proc. IEEE Conf. Decis. Control, pages 3845–3850, 2009. ISSN 01912216. doi:
10.1109/CDC.2009.5400195.
[47] K. B. Ariyur, M. Krstic, and M. Krsti´c. Analysis and design of multivariable extremum seeking. In Proc. Am. Control Conf., volume 4, pages 2903–2908, 2002. ISBN 0780372980. doi: 10.1109/ACC.
2002.1025231.