RU2702137C1 - Method of forming magnetic model of shape and current of plasma with feedback in tokamak - Google Patents

Abstract

FIELD: manufacturing technology.

SUBSTANCE: invention relates to a method of forming magnetic control systems of shape and current of plasma with feedback in tokamaks and can be used for stabilization of plasma separatrix position at reflection of action of small detachment disturbances in tokamaks. When generating feedback, variable input and output plasma parameters are used in tokamak. Input parameters are values of voltage vectors on windings of poloidal field and central solenoid of tokamak, which provide magnetic coupling with plasma. Output parameters at each moment in time represent a one-dimensional array of values of change of plasma current vectors, currents in windings of poloidal field and central solenoid, signals on magnetic loops and probes, signals proportional to vertical and horizontal displacement of magnetic axis of plasma or center of current of plasma cord. Based on said parameters, a complete plasma shape control model is obtained at each moment in time.

EFFECT: design of a plasma control model with a recovery code, shorter time for debugging a control system on tokamak, a thermonuclear reactor-tokamak or a thermonuclear power plant based on tokamak, as well as shorter experiment time.

3 cl, 16 dwg

Description

Technical field

The invention relates to methods for forming magnetic systems for controlling the shape and current of a plasma with feedback in tokamaks, and in particular to methods for simulating closed-loop control systems for the position, current and shape of a plasma with a code for restoring plasma equilibrium in feedback based on the results of experimental data, and can be It is used to stabilize the position of the plasma separatrix when reflecting the action of disturbances such as small stall in tokamaks.

State of the art

The design of the tokamak (toroidal chamber with magnetic coils) is a transformer, in the primary winding of which a current pulse is supplied from an external energy source, and the plasma coil is a secondary coil of the transformer in which the current heating the plasma flows (see Fig. 1). Also, for additional heating of the plasma, injectors of neutral atoms and microwave electromagnetic waves are used. Different types of plasma instabilities are suppressed by the strong toroidal field created by the winding of the toroidal field.

It has been established that upon reaching the Lawson criterion [2], when the product of the plasma confinement time τ and its density n and temperature T reaches a certain level:

the reaction of fusion of the nuclei of light elements begins with the release of a large amount of energy. When the fusion of deuterium and tritium nuclei occurs:

then a helium-4 nucleus and a neutron are formed, which carries 80% of the synthesis energy. Helium-4 nuclei remain in the plasma and contribute their energy to maintain the thermonuclear reaction. Neutrons are absorbed by the outer shell (the first wall in the tokamak reactor), heat it, and then heat is removed to further convert it into electricity. Deuterium-tritium mixture (DT) has an extremely high energy intensity: 1 gram of DT mixture is equivalent to 10,000 liters of oil.

The most popular and progressive are tokamaks having toroidal axially symmetric magnetic configurations. Tokamaki were proposed in the USSR, at the Institute of Atomic Energy. I.V. Kurchatov [1], then spread throughout the world [2]. The first tokamak was built in 1954.

In 2009, the international community (Europe, Japan, Russia, the USA, South Korea, China and India) began the construction of the world's first ITER tokamak reactor (ITER, International Thermonuclear Experimental Reactor) in France (Cadarache).

To control the shape of the plasma, i.e. the location of the plasma boundary in space, it is necessary to reconstruct the distribution of the plasma toroidal current density, poloidal magnetic flux and the plasma separatrix from magnetic measurements outside the plasma. Moreover, the current situation is such that there is no definite standard for plasma control systems, although these systems have been developing over the course of fifty years. This is due to the fact that all tokamaks have different poloidal systems, different parameters and different teams developing plasma control systems in specific tokamaks with different ideas and degrees of qualification in this area. Plasma (ionized gas) is a multiply connected nonlinear object with distributed and time-varying parameters, subject to various types of uncontrolled disturbances, the dynamics of which have not been fully studied in view of its extreme complexity. For this reason, this object can be attributed to extremely complex objects, since it is necessary to control a super-large number of charged particles (ions and electrons) in a limited volume with a density of about 10 ¹⁴ particles / cm ³ .

Moreover, in vertically elongated tokamaks, in particular, in ITER, because of the vertical instability of the plasma and the limited control actions, the controllability region in the state space of the plasma model is limited and relatively small. Because of this, uncontrolled disturbances can at any time take the system out of the control domain, which will lead to the loss of the stability control system.

In an ITER type tokamak reactor (France) or a DEMO type thermonuclear power station (Japan), the loss of stability by a plasma control system leads to an accident: a tokamak chamber burns out and high-temperature plasma energy is released to the outside. The way out of this situation is the development of such plasma control systems that can reliably withstand plasma disturbances and are guaranteed to hold the plasma in the control region, providing the necessary control accuracy.

Based on the foregoing, the study of plasma in tokamaks as an object of control of a complex physical nature, the construction of its models according to experimental data and using the laws of plasma physics, the development of methods for reliable plasma diagnostics by magnetic measurements outside the plasma, the development, study and application of effective automatic systems in a physical experiment plasma control in the presence of uncontrolled external influences, the presence of incomplete possible information, limited management resources, and The multiplicity of criteria for evaluating control, the development of a methodology for studying plasma, and methods for constructing and modeling diagnostic and plasma control systems is an important and urgent task.

All the above tokamaks have a common similarity, consisting in the fact that they are elongated vertically, but they all differ from each other by poloidal systems, i.e. the implementation of the designs of winding systems that create poloidal fields. Moreover, generalized magnetic control schemes for the position, current, and shape of the plasma in tokamaks are known.

To simulate such systems, complete tokamak plasma models are used, based on nonlinear plasma-physical codes such as TSC, CORSICA (USA), TOSCA (Japan), MAXFEA (Italy), PET, DINA (Russia). These codes for obtaining plasma equilibrium solve the Grad – Shafranov equation [6, 7]. This is a second-order partial differential equation in cylindrical coordinates taking into account axial plasma symmetry, relating the distribution of the plasma toroidal current density J _ϕ with the distribution of the poloidal flux ψ created by it, which looks like this:

Here, the plasma toroidal current density J _ϕ is expressed in terms of the gas kinetic plasma pressure p and the function F proportional to the poloidal plasma current:

On current tokamaks, a fairly simple magnetic diagnosis is used to determine the position of the plasma in its stabilization circuit, based on a change in the external measured magnetic flux when the plasma cord is displaced. To determine the signals of the plasma shape, codes are used to restore the equilibrium of the plasma, which can determine the distribution of the poloidal flow, toroidal current, and the position of the separatrix.

The plasma equilibrium in a tokamak is described by the distribution of the poloidal magnetic flux ψ (r, z) in a cylindrical coordinate system (r, ϕ, z) taking into account axial plasma symmetry. The poloidal flux ψ at point P is defined as the magnetic flux per radian of a horizontal circle centered on the axis of the tokamak passing through point P and bounding the area S (Fig. 2) [8]:

- вектор индукции магнитного поля,

- скалярное произведение индукции и нормали элемента поверхности. Из определения полоидального потока следует, что он должен обращаться в ноль на аксиальной оси и на бесконечности (граничные условия).Where

is the magnetic field induction vector,

is the scalar product of induction and the normal of a surface element. It follows from the definition of a poloidal flow that it should vanish on the axial axis and at infinity (boundary conditions).

The plasma boundary (separatrix) in the poloidal plane (the vertical plane passing through the z axis) can be found as the largest closed line of the poloidal flow level. If the separatrix touches the tokamak chamber, the plasma configuration is called the limiting one. If the plasma does not touch the walls of the tokamak, then they say that the plasma is in the divertor phase. The divertor phase is also characterized by the presence on the separatrix of an X-point at which the magnetic field vanishes. The separatrix has branches that “enter” the first wall of the tokamak or divertor plates. Along these branches, impurities exit the plasma and are absorbed by the divertor plates. Thus, to determine the shape of the plasma, it is necessary to find the distribution of the poloidal flow.

A method is known from the prior art for modeling magnetic plasma control systems in tokamaks, in which the signals for controlling the plasma shape are taken directly from complete nonlinear plasma models in tokamaks, since plasma modeling codes allow this to be done when modeling plasma equilibrium using the Grad-Shafranov equation (Dokuk V. N., Kavin A.A., V.E. Lukash, M.M. Sokolov, PP Khairutdinov, V.A. Khairutdinova, "Numerical simulation of plasma control in the modernized T-15 tokamak", "VANT", Ser. " Thermonuclear Fusion ”, 2014, vol. 3 7, issue 3, pp. 56-70). To hold the plasma in the magnetic field of the tokamak, poloidal field windings are used, each of which is connected to an actuator, which is either a controlled multiphase thyristor rectifier, or a voltage inverter, or a current inverter [3] (Fig. 3). The inputs of the actuators are connected to the regulators P _ZR , P _PF , through which they close the feedback to control the vertical Z and R horizontal position of the plasma or currents in the windings of the poloidal field and the central solenoid (internal control cascade). Feedback is understood in terms of the present application as the input of the output signal of the control object to its input with some conversion carried out by the controller and actuator. An external feedback cascade forms a regulator for controlling the plasma current I _p and the gaps g between the plasma separatrix (the magnetic surface inside which the plasma is held) and the first tokamak wall.

This method does not use reconstruction of the plasma shape, namely, the restoration of the distribution of the poloidal plasma flow, the density of the toroidal plasma current and the location of the separatrix. This method is implemented in the absence of a feedback plasma restoration code, which leads to insufficient completeness of the systems for modeling magnetic plasma control in a tokamak and to the lack of full accuracy in the recovery of plasma separatrix.

A known method of using the recovery code directly on the tokamak itself, reflected in the flowchart shown in FIG. 4. (QP Yuan, BJ Xiao, “Plasma current, position and shape feedback control on EAST”, “Nuclear Fusion”, No. 53, 2013, http://dx.doi.org/10.1088/0029-5515/53/ 4/043009).

When controlling current and plasma shape in real time on existing tokamaks with vertically elongated plasma, codes for restoring plasma equilibrium in real time in closed control systems are used to determine the position of the plasma separatrix (Fig. 4) (see, for example, [9-11] ) In these systems, the input of the recovery code receives signals from the magnetic diagnosis of the tokamak: integrated signals from magnetic loops proportional to the magnetic flux at given points Ψ, signals about the magnetic field B from magnetic probes (Mirnov sensors), signals from Rogowski belts about the plasma current I _p , currents in the central solenoid I _CS and in the windings of the poloidal field I _PF . From the equilibrium restoration code, the current and plasma forms are supplied with signals about the field components in the assumed location of the X-point B _z and B _r , the poloidal flow Ψ and the separatrix shift δg.

Based on equation (1), the problem of stationary identification (restoration) of plasma equilibrium can be formulated, in particular, as follows, provided that the tokamak has only magnetic loops [8].

It is required to find the plasma region S, the distribution of the toroidal current J _ϕ in it, and the distribution of the poloidal flow created by it, satisfying equation (1) and constraints of the form:

where I _p and Ψ _i are the measured values of the total plasma current and poloidal flux on magnetic loops, M is the number of loops measuring the flux.

The functions p and F in (1) are unknown, but depend only on the poloidal flow. In view of this, the components of the density of the plasma toroidal current J _ϕ can be approximated by the functions of the poloidal flow:

To solve the problem of identifying equilibrium, the Picard iterative method is used. At each iteration, the plasma current density is approximated by a linear combination of the basis functions ƒ (ψ) from the poloidal flow obtained at the previous iteration. The poloidal flow in this case is calculated as the sum determined by the selected basic functions:

Where

, создаваемого единичным бесконечно тонким кольцевым током с координатами

, распределение плотности тока которого имеет вид дельта функции

, и выражаемая через эллиптические интегралы первого K и второго рода Е [12]:Here G is the Green's function for equation (1), which has the physical meaning of a poloidal flow at a point

created by a single infinitely thin ring current with coordinates

whose current density distribution has the form of a delta function

, and expressed through elliptic integrals of the first K and second kind E [12]:

,

и токи в катушках I_m подбираются на каждой итерации через минимизацию суммы квадратов разностей между рассчитанными и измеренными токами и потоками:Odds

,

and the currents in the coils I _{m are} selected at each iteration through minimizing the sum of the squared differences between the calculated and measured currents and flows:

и

- рассчитанный и измеренный полные токи плазмы,

- измеренные токи в обмотках токамака,

- погрешности измерений полоидального потока, тока плазмы и тока в обмотках соответственно.Here

and

- calculated and measured total plasma currents,

- measured currents in the tokamak windings,

- measurement errors of poloidal flow, plasma current and current in the windings, respectively.

The closest way to the claimed invention is a method of creating a model of plasma magnetic control systems in tokamaks (Fig. 3) (Mitrishkin Yu.V., Korostelev A.Ya., Dokuka VN, Khairutdinov PP “Synthesis and modeling of a two-level magnetic control system plasma of a tokamak reactor ”,“ Plasma Physics ”, 2011, Volume 37. No. 4. P. 307-349). The model obtained in accordance with the indicated method does not use the feedback plasma restoration code. The signals for controlling the shape of the plasma are taken directly from the full nonlinear plasma models in tokamaks, since the plasma modeling codes allow this to be done when modeling the plasma equilibrium using the Grad-Shafranov equation.

The disadvantage of this method is the insufficient completeness of the simulation due to the lack of a recovery code for the distribution of the poloidal plasma flow, the density of the toroidal plasma current and the location of the separatrix, as a result of which the plasma shape diagnostics are not taken into account in the simulation. This makes it impossible to obtain a detailed idea of the operation of a complete system for controlling the shape and current of a plasma with a code for restoring plasma equilibrium in feedback.

Disclosure of invention

The technical problem of the claimed invention is to overcome the technical drawbacks inherent in analogues, which leads to the need to create a method for creating a model of magnetic plasma control in a tokamak feedback, implemented using a recovery code that provides the most reliable distribution of the poloidal plasma flow.

The technical result of the claimed invention is to increase the reliability of the simulation of magnetic control processes in a tokamak, which allows, by conducting experiments on the created model, to avoid the need for unjustified adjustment of the plasma position in real tokamaks and, as a result, improves the reliability of operation of real tokamaks by using the results of the invention.

The inventive method can be applied to stabilize the position of the plasma separatrix when reflecting the action of disturbances such as small stall in tokamaks.

The problem is solved in that the method of forming a model of magnetic control of the plasma shape and current with feedback in a tokamak, including simulating plasma in a tokamak, with actuators, control circuits for the position of the plasma and currents in the windings of the poloidal field, modeling the change in the position of the plasma separatrix in the divertor phase a plasma discharge using a multidimensional regulator of the shape and plasma current included in the feedback, during which the output values are changed with signals of the regulator of the shape and current of the plasma at each moment of time within the limits corresponding to a relatively small deviation of the position of the plasma separatrix from the given one, and also measure the current vectors in the windings of the poloidal field and the central solenoid of the tokamak, signals on magnetic loops and probes, plasma current, according to the obtained values which form the restoration of plasma equilibrium in a tokamak using the recovery code, for use in feedback, according to the technical solution, when forming feedback using use variable input and output plasma parameters in a tokamak,

the input parameters are the values of the voltage vectors on the windings of the poloidal field and the central tokamak solenoid, which provide magnetic coupling with the plasma, and

output parameters at each moment of time are a one-dimensional array of values of changes in the plasma current vectors, currents in the windings of the poloidal field and the central solenoid, signals on magnetic loops and probes, signals proportional to the vertical and horizontal displacement of the magnetic axis of the plasma or the center of current of the plasma cord,

the resulting one-dimensional array of values is summed up with similar values of the array of scenario signals corresponding to the given location of the plasma separatrix in the divertor phase of the plasma discharge at each time instant, as a result of which the input signals of the plasma equilibrium restoration code are generated, based on the solution of the nonlinear inverse boundary value problem, and at the output of the code restoration receive signals proportional to the input signals of the regulator of the form and plasma current, which are fed to the input of the regulator and thereby the circuit ie feedback, resulting in a complete management model of plasma form at any given time. The output signals of the current regulator and plasma shape correspond to the currents in the windings of the poloidal field and the central solenoid. The output signals of the regulator of the form and plasma current are fed to the input of the current control cascades in the windings of the poloidal field and the central tokamak solenoid.

The claimed invention, as well as technical solutions that are analogues, are illustrated by the following drawings. Wherein

In FIG. 1 is a tokamak diagram with control coils of poloidal magnetic fields and a central solenoid (primary sectioned winding),

in FIG. 2 - arrangement of poloidal flow in a tokamak,

in FIG. 3 is a block diagram known from the analogue (Dokuka V.N., Kavin A.A., V.E. Lukash, M.M. Sokolov, PP Khairutdinov, V.A. Khairutdinova, “Numerical simulation of plasma control in a modernized Tokamake T-15 "," VANT ", Ser." Thermonuclear Fusion ", 2014, v. 37, issue 3, p. 56-70),

in FIG. 4 is a block diagram of a magnetic plasma control system in tokamaks, known from the closest analogue associated with a real physical experiment,

in FIG. 5 is a flowchart of a method according to the claimed invention; FIG. 6 - vertical sections and cross sections of foreign tokamaks most adapted in solving plasma control problems, where a) DIII-D (1.66 m, 0.67 m, 1986, USA), b) JET (3.00 m, 1.25-2.1 m, 1992 , England), c) - JT-60U (3.40 m, 1.00 m, 1991, Japan), d) - TCV (0.88 m, 0.25-0.7 m, 1992, Switzerland), e) - ASDEX Upgrade (1.65 m, 0.5 -0.8 m, 1991, Germany), f) Globus-M (0.36 m, 0.24 m, 1999, Russia), g) T-15 (1.48 m, 0.67 m, 2018, Russia). The tokamak markings indicate the year of their launch (or the planned year of launch), large and small radii.

In FIG. 7 - vertical sections of existing tokamaks, where a) - section of the EAST tokamak design (copy of ITER, 1.75 m, 0.43 m, 2006, China), b) - section of the ITER tokamak-reactor being constructed (6.20 m, 2.00 m, planned launch year 2025 , France). The tokamak markings indicate the year of their launch, large and small radii.

In FIG. 8 - level lines of the restored distribution of poloidal flow in the GLOBUS-M tokamak and measurement points: a - poloidal flow on the separatrix; 6 - gaps between the first wall and the separatrix,

In FIG. 9 - evolution of the shape and position of the plasma, restored for discharge No. 31648 tokamak Globus-M,

In FIG. 10 - The restored distribution of the poloidal flow for time t = 180 ms discharge No. 31648 tokamak Globus-M,

In FIG. 11 is a reconstructed distribution of the density of the toroidal plasma current for a time t = 180 ms of discharge No. 31648 of the Globus-M tokamak,

In FIG. 12 - Plasma boundary for discharge No. 31488 (t = 0.19),

In FIG. 13 - comparison of separatrices obtained by the filament method and the DINA code on a model discharge,

In FIG. 14 is a block diagram of a poloidal flow control system for a divertor discharge phase,

In FIG. 15 - The result of the control system: a - deviation of the magnetic field components at the X point from zero; 6 - the difference of the poloidal flow at the X-point from the poloidal flow at the points of the desired location of the plasma boundary,

In FIG. 16 — evolution of the plasma boundary: a — 0.18 s; b - 0.185 s; c - 0.19 s; g - 0.195 s and the desired location of the plasma boundary.

With reference to FIG. 1, where a known tokamak design is shown, are indicated:

1 - primary winding;

2 - magnetic circuit,

3 - winding of a toroidal field;

4 - toroidal field;

5 - winding of the poloidal field;

6 - poloidal field;

7 - helical magnetic field;

8 - plasma coil.

The claimed invention is based on the use of a dynamic plasma model with variable input parameters, while appropriately processed output signals of this model are used as input signals for the equilibrium restoration code.

All calculations to justify the proposed method were performed using the MatLab system of numerical and mathematical modeling and the Simulink system of modeling dynamic systems.

In the absence of plasma in the tokamak, the dynamics of the current vector I in the coils of the poloidal field and the elements of the tokamak vacuum chamber (horizontal rings) is described by the Kirchhoff vector-matrix differential equation relating the change in the column vector of the magnetic flux Ψ _s through the circuit of conductors with the diagonal matrix of active resistances of conductors R and the vector of applied stresses U:

In this case, the flux vector Ψ _c is determined by the product of the constant matrix of inductances of the conductors

and current vector I, and therefore the Kirchhoff differential equation is linear:

и формы плазмы, определяемых распределениями полоидального потока ψ и плотности тока плазмы J_ϕ, реконструкции которых посвящены предыдущие разделы.When a plasma circuit with current I _{p is} added to the system, the magnetic flux vector Ψ _s nonlinearly depends on the position of the plasma magnetic axis

and plasma shapes determined by the distributions of the poloidal flux ψ and the plasma current density J _ϕ , the reconstruction of which was discussed in the previous sections.

The magnetic flux also depends on the shape and position of the plasma

и токами через эти катушки. Матрицы индуктивностей, связанных с плазмой, выражаются из восстановленных распределения плотности тока, формы и положения плазмы при помощи функций Грина, например, для M_pc так:through the plasma circuit, determined by the intrinsic inductance of the plasma M _pp and the current through the plasma, as well as the matrix of mutual inductances between the plasma and the control coils (coils of the poloidal field)

and currents through these coils. Plasma inductances associated with plasma are expressed from the reconstructed distribution of current density, plasma shape and position using Green's functions, for example, for M _pc like this:

These formulas determine the relationship of linear models with reconstructed plasma flow and current distributions and lead to the solution of the complete consistent plasma identification problem.

To obtain a linear model, the following physical assumptions are made:

масса и сопротивление плазмы пренебрежимо малы, поэтому они полагаются равными нулю,

the plasma mass and resistance are negligible, therefore they are assumed to be zero,

считается, что плазма не меняет свою форму при смещении по вертикали и горизонтали (концепция «жесткой модели» [15]),

it is believed that the plasma does not change its shape when shifted vertically and horizontally (the concept of a “rigid model” [15]),

изменение полного тока плазмы не сопровождается изменением профиля распределения плотности тока.

a change in the total plasma current is not accompanied by a change in the current density distribution profile.

We now write the linearized Kirchhoff equations in deviations from the equilibrium distributions ψ and J _ϕ . The first equation describes the contours of the coils and elements of the tokamak chamber, and the second describes the plasma circuit:

Since the plasma mass is taken equal to zero, the force acting on the plasma in the poloidal plane

or in coordinate form

at any moment should be equal to zero. This equality must also be satisfied for deviations from the equilibrium distributions ψ and J along the position of the magnetic axis of the plasma and currents in the conductive elements:

через вектор изменения токов в катушках и элементах камеры

, подставим

в уравнение (2), выразим из второго уравнения полученной системы скорость изменения тока плазмы

и подставим ее в первое уравнение. В итоге уравнение линейной модели примет вид:We express from (3) the deviation

through the vector of current changes in the coils and camera elements

, substitute

in equation (2), we express from the second equation of the resulting system the rate of change of the plasma current

and substitute it in the first equation. As a result, the equation of the linear model takes the form:

в этой формуле зависит от равновесия плазмы и находится по формулеEffective Inductance Matrix

in this formula depends on the equilibrium of the plasma and is found by the formula

In the final form, the linear plasma model in the state space can be written as

Matrices A and B are expressed through matrices of inductances and resistances according to the formulas

where E is the identity matrix, N is the number of tokamak windings, V is the number of horizontal ring elements into which the vacuum chamber is divided.

в наборе из 10-ти точек Р1-Р10 на границе плазмы (фиг. 8а) и приращения δg 6-ти зазоров между границей плазмы и вакуумной камерой (фиг. 8б). Выходы модели как отклонения от восстановленного равновесия токов в активных и пассивных структурах, положения магнитной оси плазмы, тока плазмы, потоков и расстояний сепаратрисы и первой стенки представляются в одном векторе:The model’s output signals include current deviations in 8 control coils from equilibrium values, horizontal and vertical displacements of the plasma magnetic axis position, change in plasma current, poloidal flow increment

in a set of 10 points P1-P10 at the plasma boundary (Fig. 8a) and an increment δg of 6 gaps between the plasma boundary and the vacuum chamber (Fig. 8b). The model outputs as deviations from the restored equilibrium of currents in active and passive structures, the positions of the plasma magnetic axis, plasma current, flows and distances of the separatrix and the first wall are presented in one vector:

and are related to the states δI by the matrix C:

- вектор-столбец магнитного потока, создаваемого в горизонтальных контурах, проходящих через точки Р1-Р10 (фиг. 8а),

- матрица взаимных индуктивностей между данными контурами, катушками полоидального поля и элементами камеры, а

- вектор-столбец взаимных индуктивностей между рассматриваемыми контурами и плазмой.Here

- a column vector of the magnetic flux generated in horizontal contours passing through points P1-P10 (Fig. 8A),

- a matrix of mutual inductances between these circuits, coils of the poloidal field and camera elements, and

- a column vector of mutual inductances between the contours under consideration and the plasma.

The essence of the invention consists in using a code for restoring plasma equilibrium in the feedback of a current and plasma control system of a tokamak.

соответствующие малым отклонениям от сценарных сигналов, складывать с соответствующими сценарными сигналами

то получится полная имитация работы объекта управления в эксперименте.To obtain a discharge in a tokamak vacuum chamber, a scenario is defined in advance, which is a programmatic change in currents in the central solenoid I _CS and poloidal current windings I _PS so that they give the desired magnetic plasma configuration in the tokamak. When plasma equilibrium is restored from experimental data, signals from magnetic loops and probes that were in the discharge and which can also be attributed to scenario signals are also used (Fig. 5). Since the linear model, as described above, is obtained for small deviations from the scenario signals, then, if the input signals are input to the linear model, as in the experiment, generated by the same feedback cascades (Fig. 5), and the model output signals

corresponding to small deviations from the scenario signals, add to the corresponding scenario signals

this will result in a complete imitation of the operation of the control object in the experiment.

. Таким образом происходит замыкание обратной связи в системе управления током и формой плазмы с кодом восстановления равновесия плазмы внутри замкнутого контура управления. Это является новым предложением по сравнению с известными техническими решениями систем магнитного управления плазмой в токамаках при их компьютерном моделировании (фиг. 3).The resulting sum of the model output signals and scenario signals will correspond to the output signals of the experiment, by which it is possible to restore the plasma equilibrium in the tokamak. The equilibrium restoration code calculates the distribution of the poloidal flow, from which it is possible to obtain variations of the flow signals at the desired points δΨ, deviations of the gaps δg at given points on the separatrix, the horizontal and vertical field components B _r , B _z at the X-point. These signals are fed to the input of the regulator of the form and plasma current

. Thus, feedback closes in the current and plasma control system with a code for restoring the plasma equilibrium inside a closed control loop. This is a new proposal in comparison with the known technical solutions of plasma magnetic control systems in tokamaks during their computer simulation (Fig. 3).

In the particular case of the implementation of the claimed invention, the method is used in tokamaks elongated vertically.

The output signals of the controller can correspond to the current values in the windings of the poloidal field and the central solenoid.

The output signals of the regulator of the shape and current of the plasma can be fed to the input of the cascades of current control in the windings of the poloidal field and the central solenoid of the tokamak.

As the output variables of the linear model, for example, the plasma current, the current in the windings of the poloidal field and the central solenoid, signals on magnetic loops and probes are set.

A plasma equilibrium restoration code is used, based, for example, on the Grad-Shafranov equation

According to FIG. 1, the tokamak contains a primary winding 1, inside of which there is a magnetic circuit 2, a winding (coil) 3 of a toroidal field 4, windings (coils) 5 of a poloidal field 6 for controlling the position and shape of the plasma cord. A current pulse from an external energy source enters the primary winding 1, the resulting helical magnetic field 7 is formed, and the plasma coil 8 is the secondary coil of the transformer, in which the current flowing through the plasma flows. For additional plasma heating, injectors of neutral atoms and microwave electromagnetic oscillations are used. Plasma instabilities are suppressed by a strong toroidal field 4 created by the winding 3.

In FIG. 3-4 are block diagrams illustrating analogues of the claimed invention (described in the "prior art" description section).

для управления током плазмы I_p и зазорами g между сепаратрисой плазмы (магнитной поверхностью, внутри которой удерживается плазма) и первой стенкой токамака, соединенный с регулятором токов в обмотках полоидального поля и центрального соленоида токамака

, выход которого соединен с исполнительным устройством, охваченным отрицательной обратной связью, и регулятор положения магнитной оси плазмы P_Z-R (для управления вертикальным Z и R горизонтальным положением плазмы), выход которого соединен с исполнительным устройством ИУ_z-r, также охваченными отрицательной обратной связью. Выходы исполнительных устройств соединены со входами линейной модели.In FIG. 5 is a flowchart illustrating the claimed method. The block diagram according to the claimed invention contains a multidimensional regulator

for controlling the plasma current I _p and the gaps between the plasma separatrix (the magnetic surface inside which the plasma is held) and the first tokamak wall connected to the current regulator in the windings of the poloidal field and the central tokamak solenoid

the output of which is connected to an actuator covered by negative feedback, and a regulator of the position of the magnetic axis of the plasma P _ZR (to control the vertical Z and R horizontal position of the plasma), the output of which is connected to an actuator of the DUT _zr , also covered by negative feedback. The outputs of the actuators are connected to the inputs of the linear model.

Namely, the outputs of the actuators are connected to the inputs of the linear model calculation unit, one vector output of which is connected in series with the regulator P _ZR through an adder with a negative input, while the input of the adder is supplied with setting actions Z and R that determine the required plasma position.

another vector output is connected in series with the input of the recovery code block,

, выходы которого являются задающими воздействиями для каскада управления токами в обмотках полоидального поля и центрального соленоида,one scalar output is connected through the negative input of the adder with a current and plasma regulator

the outputs of which are the driving actions for the cascade of current control in the windings of the poloidal field and the central solenoid,

, выход которого подключен ко входу исполнительного устройства ИУ_PF, а на другой вход данного сумматора подаются задающие воздействия по токам в обмотках полоидального поля и центрального соленоида,the vector output, which is a sub-vector of the overall vector output of the linear model, is connected in series through the negative input of the adder to the input of the multidimensional controller

the output of which is connected to the input of the actuator IU _PF , and to the other input of this adder are given the driving currents in the windings of the poloidal field and the central solenoid,

the full input of the equilibrium restoration unit consists of the sum of the output vector signal of the linear model and the corresponding scenario signals, which are the vector of currents in the windings of the poloidal field and the central solenoid, plasma current, signals from magnetic loops and probes.

.and the third output is connected in series with the regulator

.

тока и формы плазмы через отрицательный вход сумматора, на другой вход которого подаются задающие воздействия по зазорам между сепаратрисой плазмы и первой стенкой.The outputs of the recovery code block are connected to the input of the multidimensional controller

current and plasma forms through the negative input of the adder, to the other input of which the driving actions are given along the gaps between the plasma separatrix and the first wall.

, являющийся внутренним каскадом управления током и формой плазмы, контур управления формой плазмы, включающий в себя код восстановления равновесия плазмы и регулятор управления током и формой плазмы

, причем все перечисленные контуры включают в себя линейную модель плазмы в токамаке с переменными выходными сигналами, что представляет взаимодействие данных замкнутых контуров между собой через линейную модель плазмы.The block diagram includes a plasma position control circuit that contains the actuator DI _ZR and a plasma position regulator P _ZR , a current control circuit in the windings of the poloidal field that contains the actuator IU _PF and the regulator

, which is the internal cascade for controlling the current and shape of the plasma, the control loop for the shape of the plasma, which includes the code for restoring the plasma equilibrium and the regulator for controlling the current and shape of the plasma

moreover, all of these circuits include a linear plasma model in a tokamak with variable output signals, which represents the interaction of these closed circuits with each other through a linear plasma model.

To measure the signals of plasma current, currents in the windings of the poloidal field and the central solenoid, Rogowski belts are used in the experiment, the signals on magnetic loops are integrated to obtain magnetic flux values at their locations.

The implementation of the invention

подают значения задающих референсных сигналов

и

, заранее определенных в соответствии с выбранным сценарием плазменного разряда.At the input of a multidimensional controller

give values of reference reference signals

and

predetermined according to the selected plasma discharge scenario.

, выходные сигналы с которого подают на исполнительное устройство, где вычисляют векторы напряжения на обмотках полоидального поля и центрального соленоида токамака в каждый момент времени и подают полученные сигналы на вход линейной модели плазмы, причем указанные обмотки входят в линейную модель плазмы в токамаке, поскольку они магнитно связаны с плазмой и воздействуют на ее размещение в пространстве путем изменения магнитной конфигурации.As the output signals of the controller receive current values in the windings of the poloidal field and the central solenoid, which are sent to the controller

, the output signals from which are fed to the actuator, where the voltage vectors on the windings of the poloidal field and the central tokamak solenoid are calculated at each moment of time, and the received signals are fed to the input of the linear plasma model, these windings being included in the linear plasma model in the tokamak, since they are magnetically associated with plasma and affect its placement in space by changing the magnetic configuration.

в пределах малого отклонения положения сепаратрисы плазмы от заданного положения, соответствующего плазменному разряду диверторной фазы в каждый момент времени.At the same time, the values of the output and output signals of the controller are set

within a small deviation of the position of the plasma separatrix from a predetermined position corresponding to the plasma discharge of the divertor phase at each moment of time.

сигналов на магнитных петлях, пропорциональных изменению магнитного потока в заданных точках δΨ, сигналы об изменении магнитного поля δB с магнитных зондов, значение тока плазмы I_p. Затем полученный на выходе линейной модели одномерный массив значений сигналов в данный момент времени

суммируют с массивом аналогичных значений сценарных сигналов

соответствующих заданному расположению сепаратрисы плазмы на диверторной фазе плазменного разряда в каждый момент времени. После чего, полученный массив сигналов, соответствующий суммированным значениям, подают на вход кода восстановления равновесия плазмы. На выходе кода восстановления получают значения сигналов, пропорциональных входным сигналам регулятора формы и тока плазмы и соответствующие положению равновесия плазмы, которые подают на вход многомерного регулятора

, цикл замыкается.Based on the received input signals using the dynamic linear plasma model, the values of the current vectors in the windings of the poloidal field and the central tokamak solenoid are calculated

signals on magnetic loops proportional to the change in magnetic flux at given points δΨ, signals on a change in the magnetic field δB from magnetic probes, the value of the plasma current I _p . Then, the one-dimensional array of signal values at the given time obtained at the output of the linear model

summarize with an array of similar values of scenario signals

corresponding to a given location of the plasma separatrix in the divertor phase of the plasma discharge at each time point. After that, the resulting array of signals corresponding to the summed values is fed to the input of the plasma balance recovery code. The output of the recovery code gives the values of signals proportional to the input signals of the regulator of the form and plasma current and corresponding to the plasma equilibrium position, which are fed to the input of the multidimensional regulator

, the cycle closes.

In this way, feedback is implemented with the plasma equilibrium restoration code embedded in it according to the values of the signals received at each instant of time at the output of the linear plasma model, summed with the corresponding scenario signals, and the resulting sum is the input multidimensional signal of the plasma equilibrium restoration code, the output of which is the shift of the selected points of the separatrix relative to the given values, closes the feedback, coming to the input of the current regulator and the shape of the plasma.

Case Studies

Example 1

In FIG. Figure 9 shows images of the position and magnetic configuration of the plasma at different instants of time, illustrating the change in the shape and position of the plasma reconstructed for discharge No. 31648 of the Globus-M tokamak. Magnetic configurations are obtained every 10 ms, starting from the 150th sec of discharge. At time points 150, 160, 170 sec, there is a limited configuration in the plasma, and at time points 180, 190, 200 ms there is a divertor configuration with one upper X-point. Thus, the use of a feedback code for restoring plasma equilibrium in feedback allows us to trace the evolution of the magnetic configuration during the entire discharge when it is restored at discrete time instants.

Example 2

и

. При численном моделировании использовалась квадратная сетка размером 64×64. Эти трехмерные изображения восстановленных равновесий иллюстрируют работоспособность созданного авторами изобретения кода восстановления равновесия плазмы методом итераций Пикара по экспериментальным данным токамака Глобус-М. Код использовался в обратной связи для моделирования системы управления формой и током плазмы.In FIG. 10 and 11 show the reconstructed distribution of the poloidal flow and the corresponding distribution of the plasma current density on a three-dimensional scale in coordinates

and

. In numerical simulation, a 64 × 64 square grid was used. These three-dimensional images of restored equilibria illustrate the operability of the code for restoring plasma equilibrium created by the inventors by the Picard iteration method based on experimental data from the Globus-M tokamak. The code was used in feedback to simulate a plasma shape and current control system.

Example 3

The following is a specific example of creating a plasma shape and current control system with a code for restoring plasma equilibrium using the movable filament method using the claimed invention.

To describe physical processes in a tokamak, a cylindrical coordinate system (r, ϕ, z) is usually used under the assumption that any physical quantity does not depend on the angle ϕ with axial symmetry of the tokamak. To describe electromagnetic phenomena in a tokamak chamber, the function of the poloidal flow is introduced as a poloidal flow per radian passing through a circle at the boundary of which lies the point P:

The main idea of the filament method (English filament - thread, hair, fiber) is to approximate the distributed plasma current by the sum of the currents from point sources (infinitely thin circular turns lying in a horizontal plane) [16]:

where J _φ (r) - toroidal density plasma current, r - the radius vector on the plane _{(r, z), I m} - the current in the m-th filament, r _m - coordinate of the m-th filament, M - the number of filaments, δ (rr _m ) is the delta function.

As a result of such a replacement, the distribution of the poloidal flux from the plasma on the (r, z) plane can be found as the sum of the distributions of the poloidal flux from point sources:

- распределение полоидального потока от плазмы, I_m - ток в m-ом филаменте, G(r_m, r) - функция Грина, [7] явный вид которой имеет вид:Where

- distribution of the poloidal plasma flow, I _m - the current in the m-th _{filament, G (r m, r)} - Green's function, [7] the explicit form which has the form:

.where μ ₀ is the magnetic constant, K, E are elliptic integrals of the first and second kind,

.

Similarly, the distributions of the poloidal flow created by the control coils and passive structures - elements of the vacuum chamber are written out:

,

- токи в катушках и элементах камеры, L - число катушек, V - число элементов вакуумной камеры.Where

,

- currents in the coils and elements of the chamber, L is the number of coils, V is the number of elements of the vacuum chamber.

As a result, the total distribution of the poloidal flow on the (r, z) plane includes three sums:

The filament method involves determining the location of the filaments and the currents flowing in them, so that the deviation rate of the magnetic signal created by the filaments (magnetic induction or magnetic flux) and the current value in the filaments from the measured values is minimal.

At the current Globus-M installation, during the scenario, the magnetic flux is measured at the 21st point of the location of the magnetic loops fixed along the perimeter of the vacuum chamber. Also, during a plasma discharge, the plasma current and plasma current are determined in total with the current through the vacuum chamber using Rogowski belts.

Based on the data listed, the following functionality is compiled:

- измеренные значения тока плазмы и тока плазмы в сумме с током по вакуумной камере. Все измеряемые величины зависят от времени и в эксперименте регистрируются с некоторым дискретным шагом, поэтому равновесие плазмы восстанавливается для какого-то выбранного момента времени, т.к. для каждого момента времени будет иметься свое восстановленное равновесие.which must be minimized by the values of the coordinates of the filaments, the currents flowing in them, as well as the currents in the elements of the vacuum chamber, because when the camera is arbitrarily divided into ring elements, the currents flowing in these elements are unknown. In the functional F, the quantities c _i , p ₁ , p ₂ are normalizing coefficients, N = 21 is the number of magnetic loops, Ψ _i is the measured value of the flux of the i-th magnetic loop,

- the measured values of the plasma current and plasma current in total with the current through the vacuum chamber. All measured values depend on time and in the experiment are recorded with some discrete step, therefore, the plasma equilibrium is restored for some chosen time moment, because for each moment in time there will be its own restored equilibrium.

To simplify the minimization procedure, the currents and Green's function are linearized:

- начальные значения токов в филаментах и элементах камеры, определяемые как соответствующий полный ток, деленный на количество элементов,

- начальные положения филаментов. Минимизация функционала теперь проводится по отклонениям токов

и

и отклонениям координат

.Where

- the initial values of the currents in the filaments and camera elements, defined as the corresponding total current divided by the number of elements,

- the initial position of the filaments. Functionality minimization is now carried out by current deviations

and

and coordinate deviations

.

When using (5), formula (4) when discarding second-order quantities of smallness can be reduced to the form:

Where

The normal pseudosolution of the system kx = p is found by means of a singular decomposition. The result of solving the system is shown in FIG. 12. The correctness of the restoration of equilibrium was confirmed on the plasma-physical code DINA (developed by the SSC RF TRINITY) (Lukash VE, Dokuka VN, Khayrutdinov PP Software and computer complex DINA in the MATLAB system for solving tokamak plasma control problems. , series: Thermonuclear Fusion, vol. 1, 2004, pp. 40-49). In FIG. Figure 13 shows a comparison of the plasma boundaries obtained using the filament method and the DINA code.

By analogy with the well-known method of obtaining linear models (known, for example, from sources Mitrishkin YV, Korenev PS, Kartsev NM, and Patrov MI Plasma shape control with a linear model for Globus-M tokamak. 41 ^st Conference on Plasma Physics, Berlin, Germany , June 23-27, 2014, P4.054. Walker M., Humphreys D. Valid coordinate systems for linearized plasma shape response models in tokamaks, Fusion Science and Technology, 2006, vol. 50, no. 4, pp. 473- 489) linear dynamic models were obtained taking into account plasma and a vacuum chamber of the following form:

Where

L, V - the number of coils and elements of the vacuum chamber,

- вектор-столбец, состоящий из приращений токов в катушках и элементах камеры,

- a column vector consisting of increments of currents in coils and camera elements,

is a column vector consisting of voltage increments in the control coils.

Exit vector

It includes plasma vertical and horizontal shifts, increment of the total plasma current and current increments in the L control coils, as well as 21 values of the poloidal current increments at the locations of the magnetic loops. In accordance with this, the matrix C has the dimension (2 + 1 + L + 21) × (L + V).

The existing Globus-M installation already has a developed and implemented system for stabilizing the plasma position of the Globus-M tokamak. For plasma position control, HFC (Horizontal Field Coils), VFC (Vertical Field Coils) coils are reserved. The blocks under development are highlighted in FIG. 14 grayed out. First of all, this is the plasma current I-regulator, which is necessary to maintain the plasma current at a certain level during the quasistationary stage of the discharge in a tokamak. This regulator receives the measured data on the plasma current and, through negative feedback, is connected to the cascade of the multiphase thyristor rectifier CS (Central Solenoid), as this coil actually controls the plasma current according to the transformer principle.

The following algorithm is proposed for controlling the shape of the plasma. Based on the output from the linear model (the values of the poloidal flux at the 21st magnetic loop location point, plasma current, vacuum chamber current, coil current), the reconstruction algorithm calculates the r-, z-components of the magnetic field at the location of the desired X-point location, and also the difference in the poloidal flow between the X-point and the points of the desired location of the plasma boundary. The task of the multidimensional controller K _MIMO (s) is to reduce these values to zero, since it is known that the magnetic field at the X-point for any discharge is zero, and the plasma boundary is determined by a line of equal level of poloidal flux. This regulator is synthesized by the H _∞ optimization method (loop shaping), and provides a margin of robust stability ε = 0.485. The simulation result is shown in FIG. 15, 16.

References

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2. Wesson J. Tokamaks. Oxford: Clarendon Press, 1997.

3. Ed. Chizhenko I.M. Handbook of conversion technology. - Kiev: "Technique", 1978. - 447 p.

4. Mitrishkin Yu.V., Korostelev A.Ya., Dokuka V.N., Khayrutdinov P.P. Synthesis and modeling of a two-level system of magnetic control of a plasma of a tokamak reactor // Plasma Physics. 2011. Volume 37. No. 4. S. 307-349 (prototype).

5. Dokuka V.N., Kavin A.A., V.E. Lukash, M.M. Sokolov, P.P. Khairutdinov, V.A. Khayrutdinova. Numerical simulation of plasma control in the modernized T-15 tokamak. VANT. Ser. Thermonuclear Fusion, 2014, v. 37, no. 3, p. 56-70 (analog).

6. Shafranov V.D. The equilibrium of plasma in a magnetic field // Problems of plasma theory. 1963. Issue 2. S. 92-131.

7. Ariola M., Pironti A. Magnetic control of tokamak plasmas. Springer-Verlag, 2008.

8. Korenev P.S., Mitrishkin Yu.V., Patrov M.I. Reconstruction of the equilibrium distribution of tokamak plasma parameters by external magnetic measurements and the construction of linear plasma models. Mechatronics, automation and control. Volume 17, No 4, 2016, p. 254-265. DOI: 10.17587 / mau.17.254-266.

9. M.L. WALKER, D.A. HUMPHREYS, J.A. LEUER, J.R. FERRON, and B.G. PENAFLOR. Implementation of model-based multivariable control on DIII-D, GA-A23468, 2000 (analogue).

10. V. Mertens, G. Raupp, and W. Treutterer. Chapter 3: Plasma Control in ASDEX Upgrade. Fusion Science and Technology, Vol. 44, Nov. 2003, pp. 593-604 (analogue).

11. Q.P. Yuan, B.J. Xiao, Z.P. Luo, M.L. Walker, A.S. Welander, A. Hyatt, J.P. Qian1, R.R. Zhang, D.A. Humphreys, J.A. Leuer, R. D. Johnson, B.G. Penaflor and D. Mueller. Plasma current, position and shape feedback control on EAST. Nucl. Fusion 53, 2013, 043009, 10pp (equivalent).

12. Abramowitz M., Stigan I. Handbook of special functions. - M.: Science, 1979. 832 p.

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14. Mitrishkin YV, Korenev PS, Kartsev NM, and Patrov MI Plasma shape control with a linear model for Globus-M tokamak. 41 ^st Conference on Plasma Physics, Berlin, Germany, June 23-27, 2014, P4.054.

15. Walker M., Humphreys D. Valid coordinate systems for linearized plasma shape response models in tokamaks, Fusion Science and Technology, 2006, vol. 50, no. 4, pp. 473-489.

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Claims (11)

1. The method of forming a model of magnetic control of the shape and current of the plasma with feedback in a tokamak, including plasma simulation in a tokamak with actuators, control circuits for the position of the plasma and currents in the windings of the poloidal field, modeling the change in the position of the plasma separatrix at the divertor phase of the plasma discharge using a multidimensional regulator of the shape and current of the plasma included in the feedback, during which carry out a change in the values of the output signals of the regulator of the form and plasma current at each moment of time within the limits corresponding to a relatively small deviation of the position of the plasma separatrix from the given, and measure the current vectors in the windings of the poloidal field and the central solenoid of the tokamak, signals on magnetic loops and probes, plasma current, from the obtained values of which form the restoration of plasma equilibrium in the tokamak using the recovery code, for use in feedback, characterized in that when forming the feedback using variable input and output plasma parameters in a tokamak, the input parameters are the values of the voltage vectors on the windings of the poloidal field and the central tokamak solenoid, which provide magnetic coupling with the plasma, and output parameters at each moment of time are a one-dimensional array of values of changes in the plasma current vectors, currents in the windings of the poloidal field and the central solenoid, signals on magnetic loops and probes, signals proportional to the vertical and horizontal displacement of the magnetic axis of the plasma or the center of current of the plasma cord, the resulting one-dimensional array of values is summed up with similar values of the array of scenario signals corresponding to the given location of the plasma separatrix in the divertor phase of the plasma discharge at each time instant, as a result of which the input signals of the plasma equilibrium restoration code are generated, based on the solution of the nonlinear inverse boundary value problem, and at the output of the code restoration receive signals proportional to the input signals of the regulator of the form and plasma current, which are fed to the input of the regulator and thereby the circuit ie feedback, resulting in a complete management model of plasma form at any given time. 2. The modeling method according to claim 1, characterized in that the output signals of the current regulator and plasma shape correspond to the currents in the windings of the poloidal field and the central solenoid. 3. The modeling method according to claim 2, characterized in that the output signals of the shape and current plasma regulator are fed to the input of the current control cascades in the windings of the poloidal field and the central tokamak solenoid.

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