# Mathematical model of the adaptable discharge control system in a tokamak with an iron core

( Pp. 11-20)

More about authors

Andreev Valery F.
doktor fiziko-matematicheskih nauk; fakultet vychislitelnoy matematiki i kibernetiki

Lomonosov Moscow State University Popov Alexander M. doktor fiziko-matematicheskih nauk, professor; fakultet vychislitelnoy matematiki i kibernetiki

Lomonosov Moscow State University

Lomonosov Moscow State University Popov Alexander M. doktor fiziko-matematicheskih nauk, professor; fakultet vychislitelnoy matematiki i kibernetiki

Lomonosov Moscow State University

Abstract:

The previously developed adaptation models of the program regime used the so-called “local” adaptation model, i.e. when the correction of the control currents was carried out at each moment of time regardless of their change at other times. The main element was that the adaptation algorithms used were linear, i.e. are based on the linearization of the Kirchhoff system of equations describing the evolution of the control currents during the discharge. In this paper, an algorithm for “global” adaptation of control currents is proposed. It is based on a mathematical model for controlling the discharge in an iron-core tokamak, taking into account the nonlinear behavior of a ferromagnetic. In this model, the evolution of currents is described by the Kirchhoff equations with nonlinear mutual induction coefficients, and the restrictions on the control currents, voltages, and the equilibrium and stability conditions for the plasma are included in the corresponding discrepancy functional. To find the discharge scenario, the optimal control problem is formulated and solved. The algorithm of “global” adaptation consists in the fact that information about the previous discharges is included in the minimized functional, and then the problem of optimal control is solved and a new program scenario of the discharge and the corresponding control currents and voltages are found. As a result, the model of adaptation of the program regime, first, is nonlinear, i.e. nonlinearized Kirchhoff equations are used to describe the evolution of the control currents, and, secondly, the “global” one, since the correction of all currents is carried out interdependently and consistently during the entire discharge scenario. This approach allows us to redistribute the currents in advance in the critical situation (for example, additional heating) and provide the required program regime. This is especially important in an iron-core tokamak, when the connections between the control currents become highly nonlinear. In this work, using the example of the problem of additional heating for the T-15 tokamak, the work of the “global” adaptation algorithm is demonstrated.

How to Cite:

Andreev V.F., Popov A.M., (2020), MATHEMATICAL MODEL OF THE ADAPTABLE DISCHARGE CONTROL SYSTEM IN A TOKAMAK WITH AN IRON CORE. Computational Nanotechnology, 4 => 11-20.

Reference list:

Blum J., Dei Cas R. Static and dynamic control of plasma equilibrium in a tokamak // CH1441-5. 1979. Pp. 1873-1878.

Ward D.J., Jardin S.J., Chen C.Z. Calculation of axisymmetric stability of tokamak plasmas with active and passive feedback // Journal of Comp. Physics. 1993. No. 104. Pp. 221-233.

Isoz P.F., Lister J.B., Marmillod Ph. A hybrid matrix multiplier for control of the TCV tokamak // Proc. 16th Symp. On Fusion Technology. 1264 Oxford, 1990 (1991).

Sharma A.S., Limebeer D.J.N., Jaimoukha L.J.B. Modeling and control of TCV // IEEE Trans. Control Systems Tech. 2005. Vol. 13. P. 356.

Gruber O., Genhardt J., McCarthy P. et al. Position and shape control on ASDEX-UPGRADE // Fusion Technology. 1992. Pp. 1042-1046.

Albanese R., Villone F. The linearized CREATE-L plasma response model for the control of current, position, and shape in tokamaks // Nucl. Fusion. 1998. Vol. 38. P. 723.

Walker M.L., Humphreys D.A., Leuer J. et al. Practical control issues on DIII-D and their relevance for ITER // General Atomics, Engineering Physics Memo, EPM111803a. 2003.

Walker M.L., Humphreys D.A. Valid coordinate systems for linearized plasma shape response models in tokamaks // Fusion Science and Technology. 2006. Vol. 50. Pp. 473-489.

Mitrishkin Y.V., Zenckov S.M., Kartsev N.M. et al. Linear and impulse control systems for plasma unstable vertical position in elongated tokamak // Proc. the 51st IEEE Conference on Decision and Control. Maui. Hawaii. USA. December 10-13, 2012. Pp. 1697-1702.

Zenkov S.M., Mitrishkin YU.V., Fokina E.K. Mnogosvyaznye sistemy upravleniya polozheniem, tokom i formoy plazmy v tokamake T-15 // Problemy upravleniya. 2013. № 4. S. 2-10.

Andreev V.F., Popov A.M. Redutsirovannaya model upravleniya plazmennym razryadom v tokamake s zheleznym serdechnikom // Computational nanotechnogy. 2020. T. 7. № 3. S. 17-28. DOI: 10.33693/2313-223X-2020-7-3-17-28.

SHafranov V.D. Ravnovesie plazmy v magnitnom pole. V kn.: Voprosy teorii plazmy / pod red. M.A. Leontovicha. M.: Gosatomizdat. 1963. Vyp. 2. S. 92-131.

Zakharov L.E., SHafranov V.D. Ravnovesie plazmy s tokom v toroidal nykh sistemakh. V sb.: Voprosy teorii plazmy / pod red. M.A. Leontovicha, B.B. Kadomtseva. M.: Energoizdat. 1982. Vyp. 11. S. 118-235.

Dnestrovskiy YU.N., Kostomarov D.P. V kn.: Vychislitel nye metody v fizike plazmy. M.: Mir. 1974. S. 483-506.

CHernous ko F.L., Banichuk N.V. Variatsionnye zadachi mekhaniki i upravleniya. M.: Nauka. 1973. 238 c.

Pontryagin L.S., Boltyanskiy V.G., Gamkrelidze R.V., Mishchenko E.F. Matematicheskaya teoriya optimal nykh protsessov. M.: Nauka. 1976. 392 c.

Tikhonov A.N., Arsenin V.YA. Metody resheniya nekorrektnykh zadach. M.: Nauka. 1979. 288 c.

Ward D.J., Jardin S.J., Chen C.Z. Calculation of axisymmetric stability of tokamak plasmas with active and passive feedback // Journal of Comp. Physics. 1993. No. 104. Pp. 221-233.

Isoz P.F., Lister J.B., Marmillod Ph. A hybrid matrix multiplier for control of the TCV tokamak // Proc. 16th Symp. On Fusion Technology. 1264 Oxford, 1990 (1991).

Sharma A.S., Limebeer D.J.N., Jaimoukha L.J.B. Modeling and control of TCV // IEEE Trans. Control Systems Tech. 2005. Vol. 13. P. 356.

Gruber O., Genhardt J., McCarthy P. et al. Position and shape control on ASDEX-UPGRADE // Fusion Technology. 1992. Pp. 1042-1046.

Albanese R., Villone F. The linearized CREATE-L plasma response model for the control of current, position, and shape in tokamaks // Nucl. Fusion. 1998. Vol. 38. P. 723.

Walker M.L., Humphreys D.A., Leuer J. et al. Practical control issues on DIII-D and their relevance for ITER // General Atomics, Engineering Physics Memo, EPM111803a. 2003.

Walker M.L., Humphreys D.A. Valid coordinate systems for linearized plasma shape response models in tokamaks // Fusion Science and Technology. 2006. Vol. 50. Pp. 473-489.

Mitrishkin Y.V., Zenckov S.M., Kartsev N.M. et al. Linear and impulse control systems for plasma unstable vertical position in elongated tokamak // Proc. the 51st IEEE Conference on Decision and Control. Maui. Hawaii. USA. December 10-13, 2012. Pp. 1697-1702.

Zenkov S.M., Mitrishkin YU.V., Fokina E.K. Mnogosvyaznye sistemy upravleniya polozheniem, tokom i formoy plazmy v tokamake T-15 // Problemy upravleniya. 2013. № 4. S. 2-10.

Andreev V.F., Popov A.M. Redutsirovannaya model upravleniya plazmennym razryadom v tokamake s zheleznym serdechnikom // Computational nanotechnogy. 2020. T. 7. № 3. S. 17-28. DOI: 10.33693/2313-223X-2020-7-3-17-28.

SHafranov V.D. Ravnovesie plazmy v magnitnom pole. V kn.: Voprosy teorii plazmy / pod red. M.A. Leontovicha. M.: Gosatomizdat. 1963. Vyp. 2. S. 92-131.

Zakharov L.E., SHafranov V.D. Ravnovesie plazmy s tokom v toroidal nykh sistemakh. V sb.: Voprosy teorii plazmy / pod red. M.A. Leontovicha, B.B. Kadomtseva. M.: Energoizdat. 1982. Vyp. 11. S. 118-235.

Dnestrovskiy YU.N., Kostomarov D.P. V kn.: Vychislitel nye metody v fizike plazmy. M.: Mir. 1974. S. 483-506.

CHernous ko F.L., Banichuk N.V. Variatsionnye zadachi mekhaniki i upravleniya. M.: Nauka. 1973. 238 c.

Pontryagin L.S., Boltyanskiy V.G., Gamkrelidze R.V., Mishchenko E.F. Matematicheskaya teoriya optimal nykh protsessov. M.: Nauka. 1976. 392 c.

Tikhonov A.N., Arsenin V.YA. Metody resheniya nekorrektnykh zadach. M.: Nauka. 1979. 288 c.

Keywords:

adaptation of the program regime, iron core, tokamak, inverse problem, optimal control problem.

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