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Multivariable non-interacting control of plasma configuration in JT-60U
Fusion Engineen.'ng Fusion Engineeringand Design 24 (1994) 375-388
ELSEVIER
andDesign
Multivariable non-interacting control of plasma configuration in JT-60U R. Yoshino Department of Fusion Plasma Research, Naka Fusion Research Establishment, Japan Atomic Energy Research Institute, Naka-machi, Naka-gun, lbaraki-ken 311-01, Japan
Received 14 September 1993; revised 2 March 1994 Handling Editor: M. Ohta
Abstract
A multivariable non-interacting control technique has been applied to the configuration control of single-null divertor plasmas in JT-60U. Strong interactions between the controls of the plasma current and the plasma horizontal position--which inevitably occur during their independent control, owing to large inductive coupling between the ohmic heating coil and vertical magnetic field coil--have been drastically suppressed by adopting the non-interacting control. The multivariable non-interacting control technique is a general method and can be applied to any tokamak machine, including tokamak fusion reactors such as ITER. Poloidal field coil currents and voltages required for configuration control in ITER-CDA have been investigated using this control algorithm.
1. Introduction
Control of the plasma configuration is essential to obtain required plasma performances. The height of a null point from the divertor plate should be controlled to optimize the divertor effect; clearances between the first wall and the plasma's outermost surface must be controlled to minimize the plasma interaction with the first wall or to optimize the r.f. heating efficiency, etc. Several control algorithms of the plasma shape have been applied in tokamak machines [1], and these can be classified into the following two methods. The first method is the control o f the plasma's outermost flux surface as with ASDEX [2], JET
[3] and Tore Supra [4], in which feedback controls are performed to reduce the flux differences at some desired points. In PBX-M, with the highly controlled plasma shape, flux differences are decomposed into three modes of a singular value [5]. The shortcoming of this first method is the requirement of well-predicted preprogrammed currents of poloidal field (PF) coils with small real-time correction during a discharge [6]. To get such preprogrammed currents, predictive calculations with high accuracy and/or many trial shots must be carried out. The second method is the direct control of the plasma shape parameter (e.g. horizontal and vertical positions of the plasma center, clearances
0920-3796/94/$07.00 © 1994 Elsevier Science S.A. All rights reserved SSD1 0920-3796(94)00080-Q
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R. Yoshino / Fusion Engineering and Design 24 (1994) 375 388
between the outermost flux surface and the first wall) calculated from signals of magnetic probes by using some functions. Mathematical formulas of the horizontal and vertical positions of the plasma center can be obtained for circular crosssections [7] and have been applied in TFTR [8]. For the divertor plasma configuration, theoretical formulation of the plasma shape parameter (including the height of the null point) for the realtime control is difficult. However, regression analysis of fitting functions for the plasma shape parameter enables their real-time calculation, as obtained in JT-60 [9], DIII-D [10], ASDEX-U [11], and JT-60U [12]. The merit of the second control method is that preprogrammed currents of PF coils are not required, in principle. From the viewpoint of designing a feedback control system for reactor plasmas, general control algorithms are desirable. Multivarible non-interacting control is one such control scheme, and can be adopted in the above second control method. In TFTR, an ohmic heating (OH) coil and an equilibrium field coil are used to control both the plasma current and the vertical plasma position [8]. However, the non-interacting control method is not clearly explained by the values of the gains chosen for the plasma current and the vertical plasma position in this paper. In JT-60, the non-interacting control method was proposed to swing the separatrix line on the divertor plate to reduce the heat flux onto the plate while maintaining the configuration of the main plasma [13]. Such numerical simulations clarified the usefulness of non-interacting control to reduce interactions between the control variables and to decrease the response time. However, non-interacting control was not examined in the experiments owing to the limit on the computational speed of the feedback control computer. In DIII-D, perfect non-interacting control of the plasma shape (or gaps between the outermost flux surface and the first wall) is difficult because of the constraint that the sum of the PF coil currents must equal zero [10]. In ASDEX-U, the non-interacting control method has not been tried yet [11]. The applicability of the multivariable non-interacting control method to configuration control
has been examined experimentally in JT-60U for the first time, with an improved control system [14], as presented in this paper. JT-60U is a large tokamak machine with a major radius of about 3.3 m, a minor plasma radius of about 1 m with a plasma elongation of 1.3-- 1.8, a maximum plasma volume of l l 0 m -~, and a designed maximum plasma current of 6 MA [ 15]. Here, 5.0 MA is the maximum achieved plasma current so far. JT-60U can be used to investigate control algorithms relevant to tokamak fusion reactors. In non-interacting control, a linear relationship between small perturbations of the PF coil currents (6/) and small displacements of control variables (rX) from an equilibrium state, which can be presented as 6I : ArX, is essential to obtain a matrix gain. This linear relationship (or matrix A) can be obtained from some theoretical formulas, from an equilibrium database, or from small perturbations of an equilibrium configuration. These derivation methods of matrix A are presented in this paper. This paper is organized as follows, In Section 2. the matrix gain of the proportional feedback control in the non-interacting control is derived in a general format. In Section 3, a highly improved performance of the non-interacting control method experimentally obtained in JT-60U is presented. In Section 4, the non-interacting control method is applied to an [TER-CDA plasma to investigate the PF coil currents and voltages required for the plasma configuration control.
2. Non-interacting control method The matrix gain of the non-interacting control for the plasma configuration is derived generally in this section. Then, methods to obtain the linear relationship between the control variables and currents of PF coils, which is essential to obtain the matrix gain of the non-interacting control, are presented,
2.1. Derivation of matrix gain Basic equations necessary to obtain the proportional feedback matrix gain of the multivariable
R. Yoshino / Fusion Engineering and Design 24 (1994) 375-388
non-interacting control are presented in this section. Differential and integral control gains are not discussed in this paper, because they are not essential in non-interacting control. Integral control can reduce the offsets of control variables from the reference values, but deteriorates the quick response to sudden disturbance. Furthermore, offsets of control variables are not serious problems in the plasma shape control, because they can be adjusted easily by modifying the reference values. Therefore, integral control is not adopted in this paper. The characteristic frequency of proportional feedback control is usually selected to be lower than the characteristic frequency of eddy currents which flow in the machine components (e.g. vacuum vessel, cryostat). The effect of eddy currents is not included explicitly in the following investigation of proportional matrix gain. Of course, the time delay of the control system itself (e.g. about 2 ms in JT-60U) and part of the delay effect caused by eddy currents can be compensated for by differential control, as presented in Section 3.2. The following equations are necessary to obtain the non-interacting matrix gain. The circuit equation of the plasma current lp is Lp]p +
Mtpcri+ Ip(OLpfi?x).f¢
+ r/pip = 0
(1)
where I is an n vector of PF coil currents, Mp¢ is an n vector of the mutual inductance between the plasma current and PF coil currents, rip is the resistivity of the plasma current, and x is an m vector of control variables. Here, n is the number of PF coil currents. The circuit equation of PF coil currents I is Mi +
M~ip + ((~Mpe/Ox)Ip~ --I- R I =
g
(2)
where E is an n vector of the control voltages of the PF coil power supplies, and R is an (n x n) diagonal matrix of the resistances of the PF coils. When proportional control with the compensation of the resistivity drop is adopted, E becomes E = R1 q'- RGxlp(x ref - x) -.~ R a p ( I r y - Ip)
(3)
Here, x r~f is the reference vector of the control variables. In JT-60U, (ARp, AZp, Xp) is selected as x, as is explained in Section 3. I;ff is a reference value of the plasma current. Gx is the (n x m)
377
matrix gain of the plasma shape, and Gp is the n vector gain of the plasma current. The relationship between the displacements of the control variables 6Ip and 6x from an equilibrium state and that of the PF coil currents d l is given by (Eq. (1) is partially used) 61 = Apfi/p q- AxlprX
(4)
where Ap is an n vector of PF coil currents, and Ax is an (n x m) matrix of PF coil currents. Then, Eq. (4) can be rewritten as 6I = ArX
(5)
where A is defined as an [n x (m + 1)] matrix of (Ap, Ax), and dX is an (m + 1) vector of (d/p, lpdX) t. Each control variable must be controlled by each vector presented in matrix A (i.e. vectors of Ap, Axl, Ax2, • • • ) without modifying the other control variables. The non-interacting matrix gain can be obtained from Ap and Ax as follows. The state equation of 6Ip can be obtained from Eqs. (2) and (3) using definitions of x = x ref and .~ = 0: (MAp + Mr~)f/:p = - RGp lip
(6)
If we select a characteristic control frequency fp for the control of the plasma currents, the feedback gain vector Gp multiplied by R becomes RGp = (MAp + Mpc)2nfp
(7)
The matrix gain of the plasma configuration also can be derived from E.qs. (2) and (3) using definitions o f l p = _p 7~ec and Ip = 0. Then, the state equation of dx becomes (MA x + ~Mpc/Ox)lp6Yc = - R G x r x
(8)
When we select a diagonal matrix fx as the characteristic frequency of the control variables, the matrix gain multiplied by a diagonal matrix of resistance R becomes RGx = (MAx + 8MpJSX)2nfx
(9)
Then, we define an [n x (m + 1)] matrix Mp and an (m + 1) vector A, such that Mp = (Mr, c, 8Mr, c/Ox )
(10)
ft = (fp, f~)
(11)
378
R. Yoshino / Fusion Engineering and Design 24 (1994) 375 388
Then, the [n x (m + 1)] matrix gain G = (Gp, G x) of the plasma current and the plasma shape is obtained as RG = (MA + Mp)2~f
(12)
The same relationship for the plasma current and PF coil currents can be obtained from a circuit equation of the plasma current, and an equation of force balance to the horizontal direction as (3Ip = BpM
2.2. Derivation of matrix A Matrix A is necessary to obtain the non-interacting matrix gain as presented in Eq. (12). Matrix A can be derived from small displacements from an equilibrium configuration as follows. To obtain Ap, a n equilibrium plasma code can be used to obtain two different coil current vectors of 11 and 12, which make the same plasma configuration. When d~P is the difference in the influx in volt seconds supplied by these PF coil currents, the relationship between 6Iv and 6I is
,SI = (Lp/hqJ)(ll - I2)3Ip = Apflp
(13)
The matrix A,. also can be obtained by using an equilibrium code of the plasma configuration. Each control variable is displaced while keeping the other variables fixed, and with the same plasma current and the same influx in volt seconds. When the number of control variables is smaller than the number of PF coils, some restrictions on the coil currents (such as the same current values for some PF coil currents) is required to obtain a unique solution of A,- When Axi is the displacement of one control variable and A!,.i is the displacement of the coil currents, the required 61,.~ for Ip6Xi is M,, = ( A!,.,/(Iw Ax,))Ip6xi
(14)
This can be obtained for each control variable. Then, A X can be estimated as A,. = [AI,.,/(Ip~Axl), A!,.2/(Ip2Ax2) . . . . ]
(15)
The relationship between the control variables and PF coil currents also can be obtained statistically using a least-squares method for an equilibrium database [13], as presented by
x = B,.(I/Iv) + C,.
From Eqs. (17) and (18), the matrix A can be obtained as
F"pl'
A=LB~j
(19)
The procedure for determining the matrix A decides the applicability of the matrix. If A is obtained from a large equilibrium database, quite a wide range of plasma configurations can be controlled by one matrix gain. If matrix A is obtained from Eqs. (13) and (15) with small changes in the plasma parameter, as investigated for a plasma of I T E R - C D A (see Section 4), the range of applicability is limited to small displacements of the plasma parameters. However, in tokamak fusion reactors, it will be possible to calculate matrix A in real time for each equilibrium state, owing to the expected slow control speed caused by the long time constant of the vacuum vessel and the limit on the excitation voltages of superconductors.
3. Application of non-interacting control of JT-60U The non-interacting control has been applied to the configuration control of divertor plasmas in JT-60U. The control system of JT-60U is first briefly presented, and then the matrix gain for non-interacting control of the plasma current and the plasma horizontal position is derived according to the general method presented in the previous section. Finally, an experimental result obtained by the non-interacting control method is presented for comparison with that obtained by an independent control method.
(16)
where B~ is an (m x n) constant matrix and C~ is an (m + 1) constant vector. Then, the following relationship can be easily obtained.
6x = B,.(61/1p)
(18)
(17)
3. I. Control system of JT-60U Control of the plasma current and the plasma shape in JT-60U is performed by four types of PF coil. These are an OH coil, a vertical magnetic
R. Yoshino I Fusion Engineering and Design 24 (1994) 375-388
~ F , r [ ~
~
~Ij~Bl~r~..,~l~ ~
/ I XP~. "/~D_IAGONAL _ PORT
Fig. 1. Locations of the PF coils and the control variables. Types o f PF coil are as follows: F, OH coil; V, vertical magnetic field coil; H, horizontal magnetic field coil; D, divertor coil; D C W , disruption control winding, with a dominant mode of m/n = 3•2. D C W is not used for the configuration control. Control variables are as follows: Ip, plasma current; ARp and AZp, displacements of the vertical and horizontal positions of the plasma geometrical center from the center of the vacuum vessels; Xp, height o f null point from the divertor plate.
field coil (V coil), a horizontal magnetic field coil (H coil) and a divertor coil (D coil), as presented in Fig. 1 [15]. The connection of the V coil can be changed to produce three modes of divertor configuration. The elongation of the divertor plasma is 1.3-1.5 in the standard mode, and 1.5-1.8 in the highly elongated mode. In the third mode, continuous controls of the elongation and the triangularity are possible. However, the third mode requires an additional power supply, and
379
has not been tried yet. The mutual inductants of the PF coils in the highly elongated mode are presented in Table 1. Here, the mutual inductance between the OH coil and the V coil is extremely high, and is almost the same as the self-inductance of the OH coil. This high mutual inductance, which is also high in the standard mode of the V coil connection, causes a large interaction between the control of the plasma current and the horizontal plasma position. The selected control variables of the divertor plasma in JT-60U are the plasma current (Ip), the displacements of the vertical and horizontal positions of the plasma geometrical center from the center of the vacuum vessel (ARp, AZp), and the height of the null point from the divertor plate (Xp), as presented in Fig. 1. The control variables are calculated in real time, using formulas statistically derived from the predicted equilibrium database of about 3000 cases by regression analysis [9, 12]. These formulas use 6-17 sets of predictor variables calculated from magnetic probes, the plasma current and the D coil current. The power supplies of the PF coils are controlled by a plasma feedback computer consisting of three 32-bit reduced instruction set computers (RISCs) MC88100 [14], and by a direct digital controller (DDC) with a 32-bit microcomputer (MC68030) for each power supply of PF coil, as presented in Fig. 2. The pipeline computer system organized by the RISCs executes the following tasks: (1) collecting data of magnetic probes, PF coil currents and reference values of control variables; (2) calculating control variables using statistically derived formulas; (3) calculating command values to PF coil power supplies following programmed control algorithms; (4) sending com-
Table 1 Inductance matrix of PF coils in JT-60U (units of henries)
F Va H D
F
Va
H
D
1.24 x 10 2
- 1 . 0 3 x 10 -2 3.37 x 10 -2
- 9 . 5 2 x 10 -5 6.21 x 10 -5 1.91 x 10 -3
8.92 -9.74 -2.43 2.10
a The connection o f the V coil is in the highly elongated mode.
× x x x
10 -4 10 -4 10 -4 10 -3
380
R. Yoshino / Fusion Engineering and Design 24 (1994) 375-388
Plosmo Feedbock Computer ,r-. . . . . . . . . . . . . . 7 II
- - - - -
1;' i.!
v~"
,
i /hlp~p"q
DDC
,
•
q
i
MATRIX I I~"EaI,÷BP"~ 'it . . . . . . . . . . G A 1N
9.~
~-"~!'"L~J__ ~,"_F-_O~C~'~'__!i'I,.,g,, V,L2 C, IF . . . . . . . . . . .
~
~v
co~
~
Lv
I
(eGch10ms }
~AR~,,AzJf,,X~, ~ pre- calculation . . . . . . . . . .
io
~
I
L ~ - IRogowski,BN,T probes IF J
Fig. 2. Control system of the plasma current and configuration in JT-60U. A plasma feedback computer controls the plasma current and the plasma configuration, and receives reference values from a supervisory computer each 10 ms. It calculates Ip, ARp, AZp and Xp from raw data of a Rogowski coil and BT and By magnetic probes, then calculates commands and sends them to the DDC of each PF coil power supply. In the figure, voltage control is performed in the OH(F) coil power supply, and current control is performed in the other power supplies. This control diagram is also used for the case presented in Fig. 3.
mand values to DDCs. The control of AZp is performed with a cycle time of 250 ps and a delay time of 500 gs, and the other control variables are executed with a cycle time of 500 ps and a delay time of 1 ms. A faster control loop is executed for the control of AZp. The vertical position is unstable, because of the plasma elongation of 1.31.8. The control algorithms of each PF coil (such as current control, voltage control, proportional control to the plasma current, and feedback control) can be changed individually according to a preset program. Here, non-interacting control is one of the control algorithms executed in a discharge. Furthermore, the set of the matrix gain can be changed five times during a discharge, according to a preset-time schedule. DDCs of V, H and D coil power supplies have local gains of Kv, Kn and KD. These local gains are 0.05(Tdecay/"Ccl ). Here, "('decayis the current decay time constant (=L/R) of each PF coil (in seconds), and Zc~ is the cycle time of each D D C (e.g. 500 gs). In contrast, the OH coil power supply is controlled by the voltage command from the plasma feedback computer.
3.2. Matrix gain Two 2 x 2 matrix gains are installed in a central feedback control computer for the configuration control in JT-60U. Here, a 4 x 4 matrix gain cannot be used, owing to the limit on the computation speed. Therefore, two sets of non-interacting controls, i.e. (lp and A R p ) and ( A Z p and X p ) , have been selected. In particular, a high mutual inductance between the O H coil and V coil (see Table 1) causes a strong interaction between the controls of Ip and a R p , so that their non-interacting control must be adopted. However, independent control with a diagonal matrix is enough for the control of (AZp and Xp), owing to the relatively low mutual inductance between the H and D coils. Differential control can be performed for the following four cases: Iv control by the F coil: •Rp control by the V coil; AZp control by the H coil; and Xp control by the D coil. Here the differential time of ARp control is selected to be 6 ms and that of AZp control is 6 - 1 0 ms. Differential times of other control variables are set to be zero. The time delay of the control system itself is
R. Yoshino / Fusion Engineering and Design 24 (1994) 375-388
about 2 ms and the shielding time of the vacuum vessel to the dipole magnetic field is 8.5 ms. The matrix A for the non-interacting control of (Ip and ARp) can be obtained from a circuit equation of Ip and Shafranov equation, with assumptions of tip = 0, /Tp = 0, ~ = 0 and A,~p = 0. Here, ~IpARp is used instead of ARp in a real system. Then, the relationship between the control variables of (lp, #olpARp) and the PF coil currents of (IoH and Iv) can be presented as
?,o.) = ~2 F -M"°" ~MpoH
-M"l-'r L. t3MpvI ] R, J
xr
]:AF
L#oIp~ARp]
",
]
L#oIp,$ARvJ
(20)
where Ao
:
In(8Rj'] \Up )
--]- t i p + - -
.°=['o :v}['o,+,
Here, r/v is the resistance of the V coil. The derivation of Eq. (22) is presented in Appendix B. Then, GFB can be written as 1 Gva =
0
l~-3 2
aL f0= 1 -R,>
Yv--
RG
(23)
Noninteracting Gain Matrix
Vv ~ R j
7 -1
1
A comparison of independent control and noninteracting control has been executed in a divertor plasma with the highly elongated mode of the V coil connection, as presented in Fig. 3. The reference ARp was swung by + 2 cm at 10 Hz. The matrix gain for Ip and ARp is changed from a non-interacting gain to a diagonal matrix gain at t = 8.8 s. The matrix gain GFB for the non-interacting control from 8.3 to 8.8 s is selected with the
Rj OVv
\ap/
1
~/v 1 + K v
fl = A o - 1 + A 0 - - - -
g = ln(8Rj~ + 4
0
3.3. Non-interacting control of lp and ARp
t3R
L
381
Diagonal Gain Matrix
2.1
Ao 7
I
T~
ref
2.0
1 OMpv eL.
27rRj
0R
Here, the shift of the plasma current center from the plasma geometrical center is assumed to be given by
1.9 2
# iJllllllllllllllllllll,
lllll,l,,llllJJrlllltll
ARrer p ~- +-2cm,JOHz
,i
-3
,:~ , ! ~ ,'~
<3
The derivation of Eq. (20) is presented in Appendix A. The matrix gain G of non-interacting control can be obtained according to Eq. (12). In JT-60U, feedback control gains are installed in two types of computer, which are a plasma feedback computer and DDCs, as presented in Fig. 2. Then, the matrix gain GFBinstalled in the plasma feedback computer for the control of (Ip and/~olpARp) c a n be presented with the gain (Kv) for a minor feedback loop in the DDC of a V coil power supply (see Fig. 2) as
-8
8.3
' " ' .... ' .... ' .... , .... , .... , .... ,,,,',,8,,37,m.... 8.8
TIME
9.3
(sec)
Fig. 3. Swing of the horizontal plasma position ARp. A reference command of ARn, denoted by a broken line, is swung by _+2 cm at 10 Hz. The matrix gain is changed from the non-interacting gain to a diagonal matrix gain at 8.8 s. Large fluctuations of 144 kA are observed in the plasma current under the independent control method (t = 8.8-9.3 s), but this is suppressed drastically to 8 kA by the non-interacting control method (t = 8.3-8.8 s).
382
R. Yoshino / Fusion Engineering and Design 24 (1994) 375 388
characteristic control frequencies of 6 Hz for Ip control and 30 Hz for ARp control, such that
10 ,04 Mv J
+8.0 x lO 3 × [ kt°~/p
- 1.o x 104]
]
Lgolp6ARp]
(24)
However, the matrix gain for independent control from 8.8 to 9.3 s is a diagonal matrix gain, such that 6Iv
J
0.0 Lpolp6ARpJ
- 1.0 x 104
(25)
An OH coil power supply is controlled by the voltage command, and a V coil power supply is controlled by the current command. The units of the parameters are as follows: Von in volts, Iv and I v in amperes, and Rp in meters. In contrast, the gain matrix of AZp and Xp is a constant diagonal matrix. ARp is overswung at 10 Hz, but this frequency is selected to demonstrate the effect of the non-interacting control. A large Iv fluctuation of 144 kA, which occurs inevitably in independent control from t = 8.8 to 9.3 s, is drastically suppressed to about 8 kA (5% of 144kA) by non-interacting control from t = 8.3 to 8.8 s. This result shows clearly the usefulness of non-interacting control. The frequency analysis comparing the non-interacting control method and the independent control method was performed for shape control in JT-60, as presented in Fig. 6 of Ref. [13]. According to this frequency analysis, at low swing frequencies of the control variables (e.g. 1 Hz), non-interacting control has no benefit and independent control is sufficient. At higher frequencies of about 50 Hz or more, both control methods cannot be used, owing to the delay effect of eddy currents. However, at medium frequency of 1020 Hz, the interaction between the control variables (inevitably caused by independent control) can be reduced drastically by non-interacting control. The same performance of the non-interacting control should be observed in JT-60U, so ARp is swung at 10 Hz.
4. Application to ITER-CDA
In ITER-CDA, two control loops were designed for plasma shape control [6]. A slow control loop is performed by 14 PF superconductor coils with the characteristic frequency up to about 0.3 Hz to control the plasma current and the plasma shape. A fast control loop is performed by in-vessel coils of normal conductors, with a characteristic control frequency of 5 10 Hz, which controls the vertical plasma position, stabilizing the positional instability resulting from the highly elongated plasma (~: ~ 2.0), The selected control variables in the slow control loop are Ip, R .... R~,,, Zp, R,, and Z,, as presented in Fig. 4, in which the locations of the 14 PF superconductor coils with u p - d o w n symmetry are also presented. The plasma current lp must be controlled at the required value according to the operational scenario. Rout and Ri. are the horizontal locations of the outside and inside outermost plasma surfaces at the same vertical height with the plasma current axis. Each of Ro., and R~,, determines the clearance between the plasma outermost surface and the first wall, which should be a reasonable value to achieve the divertot performance and to decrease the erosion of the first wall caused by the charge exchange.
E su 4U 5U
Dzu
2U 1U IL 2L
[] 7L
5L 4L
(Rx, Zx)
[~ 6L
[~5L Fig. 4. Control variables of the plasma configuration in tTERCDA, and locations of PF coils.
R. Yoshino / Fusion Engineering and Design 24 (1994) 375-388 R o u t should be controlled to obtain the good coupling of the lower hybrid wave with the plasma. Ri, should be controlled to obtain a clearance of about 10 cm or more, in order to avoid the plasma touching the inner first wall owing to an inward horizontal shift caused by a minor disruption. Control of Zp is necessary to equalize the heat load on to two divertor plates, settled at the top and bottom of the first wall. The low frequency component of the in-vessel coil current Iiu, which will be used for fast control of Zp, should be reduced to achieve the control margin of Zp. Thus, the reduction of the low frequency mode in /in is also the control objective of the slow control loop of Zp. Rn and Z, are the horizontal and vertical locations of the null point, which should be controlled to guide the separatrix lines onto the divertor plate. ITER-CDA has a double-null divertor configuration, and the locations of the two null points should be controlled. Swing of the separatrix lines on the divertor plate is required to decrease the heat flux effectively. It takes too much time to obtain several thousand equilibrium values for a database for proFurthermore,
Table 2 M a t r i x A in I T E R - C D A Ip
383
ducing a linear relationship as presented by Eq. (16). Therefore, matrix A was investigated according to Eq. (15), by moving each control variable while keeping the other variables in an equilibrium state. Here, the merit of using Eq. (15) is that the matrix A can be obtained without knowledge of the theoretical formulations of the control variables (such as force balance equations in the horizontal direction and the vertical direction). The reference equilibrium configuration is presented in Fig. 5(g). Then, a 14 × 6 matrix A has been obtained, as presented in Table 2 for ITER-CDA with 22 MA, R p / a p = 6.0/2.0 m, and bp/l i = 0.6/0.65. In the case of Zp, the plasma was moved rigidly in a vertical direction. The units of matrix A are kiloampere teslas. Each magnetic field produced by each vector of matrix A is presented in Fig. 5. The voltages of the coil power supplies RG are then estimated by (MA)2ztf, as presented in Table 3, by selecting each control speed at 400 kA s(22 MA per 75 s) for Ip, 2 cm s -1 for Rout, Rin, Zp and Zn, and 3 cm per 1.25 s (swing of 3 cm per 0.2Hz) for R o. As presented in Table 3, the matrix A is also useful for estimating the coil voltages required for configuration control.
( u n i t s o f k i l o a m p e r e teslas) Rout
Rin
Rn
Zn
Zp
PF1U PF2U PF3U PF4U PF5U PF6U PF7U
1.453 1.453 1.533 1.559 7.860 6.480 5.700
× × × × × × ×
10 ° 10° 10° 10 ° 10 -1 10 - 2 10 -3
-7.159 -7.159 3.477 5.193 -1.287 1.664 -1.039
× x × × × × ×
101 101 102 102 102 102 102
2.475 2.475 -4.604 2.178 5.673 -1.266 9.119
× × × × × x ×
101 101 101 101 101 102 101
2.800 2.800 -2.010 -9.200 1.412 4.400 -1.040
× × × × x × ×
101 101 102 10° 102 10 ° 101
-4.590 -4.590 - 1.530 -7.776 -1.485 1.451 -7.929
x × × × × × ×
10° 10 ° 10 ° 101 102 102 101
-2.300 -2.300 -5.530 -6.220 -1.190 -3.360 3.360
× × × × × × ×
101 10 I 10 l 10 l 101 10 l 10 l
PF1L PF2L PF3L PF4L PF5L PF6L PF7L
1.453 1.453 1.533 1.559 7.860 6.480 5.700
× × × × × × ×
10° 10 ° 10° 10° 10 -1 10 -2 10 3
-7.159 -7.159 3.477 5.193 -1.287 1.664 -1.039
× × × × × × ×
101 101 102 102 102 102 102
2.475 2.475 -4.604 2.178 5.673 -1.266 9.119
× × × × × × ×
101 101 101 101 101 102 101
2.800 2.800 -2.010 -9.200 1.412 4.400 -1.040
× × × × × × x
101 101 102 10° 102 10 ° 101
-4.590 -4.590 -1.530 -7.776 -1.485 1.451 -7.929
× × × × × × x
10° 10° 10° 101 102 102 101
2.300 2.300 5.530 6.220 1.190 3.360 -3.360
× × × × × × x
101 l0 t 10 ~ 10 ~ 10 j 101 101
C o n t r o l l e d p l a s m a is Ip = 22 M A , is in k i l o a m p e r e s .
Rp/ap = 6.0/2.0 m , a n d flp/li = 0.6/0.65. U n i t s o f Rout, Rin ,
Zp a n d
Z , a r e c e n t i m e t e r s , a n d Ip
R. Yoshino / Fusion Engineering and Design 24 (/994) 375.-388
384
[]
(b)
0
E3
i , i , 1 ,~1,
1 ,.,J.,~.
t i I I [ i r i I i
I0
E~
(c)
[]
(d
[]
(g)
5
v
0
N
-5
[] ,
E~
,
,
-I0
5
10 R
(m)
l
Fig. 5. Magnetic field produced by each current vector presented in matrix A: (a) Rm; (b) Rout; (c) Rn; (d) Z,,; (e) Zv; (1) lp; (g) A reference plasma configuration used to investigate each current vector in matrix A. Controlled plasma is 1o = 22 Ma, Rr/a r, = 6.0,,' 2.0 m, and ~p/lj = 0.6/0.65.
R. Yoshino / Fusion Engineering and Design 24 (1994) 375-388 Table 3 Required coil voltage RG for I T E R - C D A (units of kilovolts) 10
Rout
Rin
Rn
Z.
Zp
PF1U PF2U PF3U PF4U PF5U PF6U PF7U
1.48 1.40 1.02 0.36 0.46 2.63 1.59
0.00 -0.04 -0.19 -0.03 0.43 -3.13 1.23
-0.66 -0.34 0.53 0.07 - 1.37 3.70 -0.93
0.07 -0.12 -0.72 -0.12 1.44 -0.18 -0.06
0.08 0.06 -0.02 -0.12 -1.35 3.87 -0.57
-0.3 0.82 -1.3 -0.4 0.57 -3.2 0.79
PFIL PF2L PF3L PF4L PF5L PF6L PF7L
1.48 1.40 1.02 0.36 0.46 2.63 1.59
0.00 -0.04 -0.19 -0.03 0.43 -3.13 1.23
-0.66 -0.34 0.53 0.07 -1.37 3.70 -0.93
0.07 -0.12 -0.72 -0.12 1.44 -0.18 -0.06
0.08 0.06 -0.02 -0.12 -1.35 3.87 -0.57
-0.17 0.12 0.06 -0.004 0.24 -0.33 0.048
Equilibrium plasma and units are the same for Table 2.
5. Conclusions The deviation of the matrix gain of non-interacting control for the plasma configuration is presented in this paper. The relationship between small perturbations of the poloidal coil currents (6/) and small displacements of control variables from an equilibrium state (fiX), which can be presented as 61 = A f X , is of essential importance to determine the matrix gain of non-interacting control. The matrix A can be obtained from some theoretical formulas for some plasma parameters, from an equilibrium database by using regression analysis, or from small perturbations of an equilibrium plasma configuration, as presented in this paper. The non-interacting control technique has been applied to the control of the plasma configuration with single-null divertors in JT-60U. Comparisons of the independent control method and the noninteracting control method have been experimentally performed in the control of the plasma current Ip and the vertical positions of a plasma's geometrical center from the center of a vacuum vessel (ARp), swinging a reference of ARp by +2 cm at 10 Hz. When independent control is adopted, a large mutual inductance between the OH and V coils inevitably causes strong interac-
385
tion between the controls of Ip and ARp. However, the fluctuation of Ip is drastically suppressed by the non-interacting control method by about 5% of that in the independent control method. The multivariable non-interacting control method can be used generally, and can be applied to any tokamak machines and tokamak fusion reactors. As an example, the matrix A for an ITER-CDA plasma has been investigated, and the PF coil currents and voltages required for control of the plasma configuration have been calculated for selected control speed of the control variables, using the non-interacting control algorithm.
Acknowledgments Tha author wishes to thank the Japan Atomic Energy Research Institute, who have contributed to the JT-60U project, and he especially would like to express his gratitude to Mrs. M. Matsukawa and M. Takahasi, and Drs. M. Sugihara, K. Shinya, L. Zaharov, T. Kimura and Y. Shimomura for their fruitful discussions. He would like to express his gratitude to Dr. H. Ninomiya for his continuous support.
Appendix A: Derivation of Eq. (20) A 1. Relationship between Rp and P F coil currents
The Shafranov equation is 2
\
-"}- tip -+ap
-1- 2 1 r R j B v = 0
(A1)
When Ion and I v are the coil currents of the OH and V coils, the equilibrium vertical magnetic field can be presented as Bv = VOHIo, + Vvlv
(A2)
Here, vi (i = OH, V) can be presented as 1 dMpi vi - 27zRj d R j
(A3)
In JT-60U, VoH is much smaller than Vv. However, to achieve the generality of the equation, the vertical magnetic field produced by the OH coil is
386
R. Yoshino / Fusion Engineering and Design 24 (1994) 375- 388
not neglected. Here, Rj is the horizontal position of the plasma current center, ap is the averaged plasma minor radius. In JT-60U, control of the plasma horizontal position is performed for the geometrical center of the plasma column Rp. The relationship between Rp and Rj can be approximated by
tip + ~
Rj = Rp -}-
- 1
(A4)
From Eq. (A4), with assumptions of rip = 0,/~p 0 and Ii. = 0, the following relationship can be obtained: ~--
ap2{B li 2Rp\ '~p+ 2 - 1 ) ] / ~ p
/~j__[1
Then, with the definition
1
(15,
offo as
(A6)
The derivative of the Shafranov Eq. (A1) with Eq. (A6) can lead to
Af0dgp A0dRp A0( R, dt Ip
dG.
dlv5 (A7)
Here, f~ and Ao are defined as Rj 8Vv5 j] --- A0 - 1 + Ao( -RJ 8Vo. vo. OR,
R--7
(All)
Here, Lp is the plasma inductance, such that Lp=/~oRj(ln 8Rj a - ~ + ~l,.- 2)
(112)
Furthermore, from Eq. (A3), we can obtain 2~RjBz =
Ao
(')Mpi
8g-~l,=-Itolp~ -
Z
/ = OH,V
(113)
The derivative of Eq, (A12), with assumptions of rip = 0 and /] = 0, leads to \
ap
L dip
J d-t-
(A14)
dRp
(A15, In Eq. (A15), the relationship between Ip, Rp and the PF coil currents is presented. Here, g is defined as
ln(8Rj~ +/,. \ap] 2 -1
(A9)
Lp
dRp
gfo][
- #oAo
0
(A16)
A modified Shafranov Eq. (AI0) and a modified circuit equation of the plasma current (Eq. (A15)) can be combined as
- olo d7
-- 47zRj VOH d-~- -1- VV ~ - / / =
Ao 2
A3. Derivation of matrix A
2
d/v']
d/,.)
(A8)
li - 3 - -
d/oH
(
p--~- + g 0 g 0 # 0 I p ~ - -]-~plp-t-i=O~H, V M p i - d - T j = 0
g=
From Eqs. (AI), (A7) and (A9), the relationship between Rp and the OH and V coil currents can be presented as flf0/2oi p dRp
( dI, ?~MpidRj ) i=o~H.v M p i ~ - + ?Rj dt li =-0
Then, Eq. (AI 1) can be modified using Eqs, (A13) and (A14) as
l~j = foRp
+ tip +
+
dt
Eq. (A5) becomes
ap
dL v
dLp=#o(lnSRj+~_ 1~dRJ
ap2(flp+ li l )
Ao - In -8Rj -
dip
G¥
dlv
I . dRy,
(A10)
A2. Relationship between Ip, Rp and PF coil currents From the circuit equation of the plasma current, we have
__ M p o H
2 3MpoH
-
-
mpv
OMpv
d/o.
d,v +[-;'q
57
(A17)
R. Yoshino / Fusion Engineering and Design 24 (1994) 375-388 The control speed is much faster than the timescale of the plasma current decay time (Lp/?lp), so the resistivity drop of the plasma current can be ignored to investigate the feedback control gain. Then, the relationship between the PF coil currents and control variables, i.e. as in Eq. (20), can be obtained from Eq. (A17) as
387
Here, the mutual inductance matrix M is defined as
[Lo. Mo.v] M=
Monv
Lv ]
(B2)
and the voltages of the OH and V coils are controlled as
."on -~- =/2 63MpoH dlv
L ~7-
L
c~R
OMpvl
l / / 2 ~ R ~ j L- p ° A °
fofll
RjJ
[-l
dt dRp
r --MpoH
--Mpv 1-1 r 0Mpv / 2 ~ ]
Lp
/
Rj 1
L-#oA o
II
MpOH /~0---~'1 OMov
MA +
M~v
V
=
RG[
The circuit ec uations of the OH and V coil currents can be ~resented as
dlv
+
?
[M'°" #0-TY-f0/ 1 OMpv l M.v ;; ~:oj
d~ X
dt dRp
~oI.-~-
:rE°H--q°nl°nl
L e,,-rtv/v J
l |
(B4)
The matrix gain G can be obtained from Eq. (B4) for selected characteristic control frequencies offlp and fRp for Iv and Rp as
MA+
B1. Matrix gain G in the general form
~-
~o~p
b,0Ip(Rp - Rp)d
RG=
1 ~3Mpon
at dRp
1 ~Mvoufo] M"o.;o~-I
Appendix B: Derivation of Eq. (22)
[
i l
~ ~ f0 ~
fief [ P r~f O
Then, Eq. (A20) is the same as Eq. (5).
Lon Monv 1 Monv Lv ]
(B3)
gfO 1
/of, /
J = LPoIp6RpJ
-,iV
1
The state equation of the plasma current and the horizontal plasma position can then be obtained from Eqs. (A20) and (B1)-(B3) as
(AI9) the relationship between the displacements of the control variables (Mp and #0IprRp) from an equilibrium state and those of the PF coil currents O/OH and 6Iv) can be presented as
"l/OH
[ref p __];ef
RGL~Ip (R [ff - RDe~)J
(A18)
By defining the matrix A as
A--|2 OMpon L aR
t/vJLg21 g22_J [ r e f /ref 1 xI --P --P I kklolp(Rpef RIoOj V
d/~ X
6Iv
L Ev - qvlv ] =
(B1)
x
-
10Mvv
I
}
27ZflP 0 0 2~zfRp
(B5)
Thus, Eq. (B5) is the same as Eq. (12). B2. Matrix gain GFBinstalled in JT-6OU' s plasma feedback computer In JT-60U, a matrix gain feedback control is installed in two types of computer, which are a plasma feedback computer and a DDC for each thyristor bank of PF coil power supplies, as pre-
R. Yoshino / Fusion Engineering and Design 24 (1994) 375 388
388
sented in Fig. 2. E~%m and -~vr are calculated in the plasma feedback computer as
Here, GFB is defined as GvB--
A/r~yfJ =
lJkg[, ~
;
g=V~"Jk~,oldR;"~-R~) (B6)
H e r e , gVB (i,j = 1, 2) is the f e e d b a c k g a i n i n s t a l l e d in the p l a s m a f e e d b a c k c o m p u t e r . T h e c o m m a n d v o l t a g e E~'~ is sent d i r e c t l y to t h e p o w e r s u p p l y o f the O H coil as (see Fig. 2)
Eb~ '
=
rloHIoa + AEb'~
(BY)
The voltage command E~°m to the thyristor bank of the V coil is calculated in the DDC as E~,° ' ' = r]v{I~°m -+- K v ( I ~ ° " - Iv)}
(B8)
Here, Kv is the local feedback gain in the DDC (see Fig. 2) and "v'refis the reference coil current sent from the plasma feedback computer to the V coil DDC, such that
I~°m = Iv + A/'ref ~-v
(B9)
From Eqs. (B8) and (B9), E~¢°m becomes E~°~ = q v l v + r/v( 1 + Kv)AIF r Then Eq. (B6) can be presented as
[
(B10)
EbO~' - qonlou 1 E~/°m -- r/V/v ]
Lqv(1 + K v ) A I F r J = ~r/;
.
0 1F r/v( 1 + Kv)jLgVB g~;Sj
[
× Lmlp(RF' - np) =[~
0][1 qv 0 [
×
]
0 Ilion 1 +Kv
Ipf--IP °'-
l
r )j
°l[
l JLgl;" g~gJ (ml)
",,
(B12)
1jLg~,B g2~2BJ
T h e n , E q . (22) c a n be o b t a i n e d f r o m E q s . (B3), ( B l l ) a n d (B12) as
RG=
0 t/v
0
l+KvJ
v,3
(B13)
References [1] F.C. Schfiller and A.A.M. Oomens, Plasma engineering aspects of large tokamak operation, Fus. Eng. Des. 22 (1993) 35 55. [2] F. Schneider, Proc. l lth Symp, in Fusion Technology, Oxford, Vol. 2, 1978, pp. 1027 1032. [3] D. Ciscato, L. de Kock and P. Noll, Proc. llth Symp. in Fusion Technology, Oxford, Vol. 2, 1978, pp. 1033 1039. [4] J.M. Ane et al., Proc. 15th Symp. on Fusion Technology, Oxford, Vol. 1, 1988, pp. 657 661. [5] R.E. Bell et al., Proc. 13th Symp. in Fusion Engineering, VoL 1, 1990, pp. 467 470. [6] F. Hofmann et al., Nucl. Fus. 32 (1992) 897 902. [7] V.S. Mukhovatov and V.D. Schafranov, Nuc. Fus. 11 (197t) 605. [8] R.D. Wooley el al., Fus. Technol. 8 (1985) 1807 1812. [9] N. Hosogane et al., Nucl. Fus. 26 (1986) 657. M. Matsukawa et al., Plasma Phys. Control. Fus. 34 (1992) 907 921. [10] S. Kinoshita et al., General Atomics Rep. GA-A 19584, 1990. [l I] P.J. McCarthy et al., Proc. 19th EPS Conf. on Controlled Fusion and Plasma Physics, lnnsbruck, Vol. 1, 1992, pp. 459 462. [ 12] M. Matsukawa et at., Detecting method of plasma position by regression analysis, JAERI Rep., Japan Atomic Energy Research Institute, JAERI-M 92-073, 1992, pp. 16 19. [13] R.Yoshino et al., Control of divertor configuration in JT-60, Proc. 13th Symp. on Fusion Technology, Vol. I, 1984, pp. 1141 1146. [14] T. Kimura et al., VME multiprocessor system ['or control al the JT-60 upgrade, tEEE Trans. Nucl. Sci. 36 (1989) 1554 1558. [ 15] tt. Nimomiya et al., JT-60 upgrade device for confinement and steady state studies, Plasma Dev. Oper. I (1990) 43 65. [16] Y. Shimomura et al., ITER Poloidal Field System, ITER Doc. Ser. 27, IAEA, 1990, pp. 100 .114.
ELSEVIER
andDesign
Multivariable non-interacting control of plasma configuration in JT-60U R. Yoshino Department of Fusion Plasma Research, Naka Fusion Research Establishment, Japan Atomic Energy Research Institute, Naka-machi, Naka-gun, lbaraki-ken 311-01, Japan
Received 14 September 1993; revised 2 March 1994 Handling Editor: M. Ohta
Abstract
A multivariable non-interacting control technique has been applied to the configuration control of single-null divertor plasmas in JT-60U. Strong interactions between the controls of the plasma current and the plasma horizontal position--which inevitably occur during their independent control, owing to large inductive coupling between the ohmic heating coil and vertical magnetic field coil--have been drastically suppressed by adopting the non-interacting control. The multivariable non-interacting control technique is a general method and can be applied to any tokamak machine, including tokamak fusion reactors such as ITER. Poloidal field coil currents and voltages required for configuration control in ITER-CDA have been investigated using this control algorithm.
1. Introduction
Control of the plasma configuration is essential to obtain required plasma performances. The height of a null point from the divertor plate should be controlled to optimize the divertor effect; clearances between the first wall and the plasma's outermost surface must be controlled to minimize the plasma interaction with the first wall or to optimize the r.f. heating efficiency, etc. Several control algorithms of the plasma shape have been applied in tokamak machines [1], and these can be classified into the following two methods. The first method is the control o f the plasma's outermost flux surface as with ASDEX [2], JET
[3] and Tore Supra [4], in which feedback controls are performed to reduce the flux differences at some desired points. In PBX-M, with the highly controlled plasma shape, flux differences are decomposed into three modes of a singular value [5]. The shortcoming of this first method is the requirement of well-predicted preprogrammed currents of poloidal field (PF) coils with small real-time correction during a discharge [6]. To get such preprogrammed currents, predictive calculations with high accuracy and/or many trial shots must be carried out. The second method is the direct control of the plasma shape parameter (e.g. horizontal and vertical positions of the plasma center, clearances
0920-3796/94/$07.00 © 1994 Elsevier Science S.A. All rights reserved SSD1 0920-3796(94)00080-Q
376
R. Yoshino / Fusion Engineering and Design 24 (1994) 375 388
between the outermost flux surface and the first wall) calculated from signals of magnetic probes by using some functions. Mathematical formulas of the horizontal and vertical positions of the plasma center can be obtained for circular crosssections [7] and have been applied in TFTR [8]. For the divertor plasma configuration, theoretical formulation of the plasma shape parameter (including the height of the null point) for the realtime control is difficult. However, regression analysis of fitting functions for the plasma shape parameter enables their real-time calculation, as obtained in JT-60 [9], DIII-D [10], ASDEX-U [11], and JT-60U [12]. The merit of the second control method is that preprogrammed currents of PF coils are not required, in principle. From the viewpoint of designing a feedback control system for reactor plasmas, general control algorithms are desirable. Multivarible non-interacting control is one such control scheme, and can be adopted in the above second control method. In TFTR, an ohmic heating (OH) coil and an equilibrium field coil are used to control both the plasma current and the vertical plasma position [8]. However, the non-interacting control method is not clearly explained by the values of the gains chosen for the plasma current and the vertical plasma position in this paper. In JT-60, the non-interacting control method was proposed to swing the separatrix line on the divertor plate to reduce the heat flux onto the plate while maintaining the configuration of the main plasma [13]. Such numerical simulations clarified the usefulness of non-interacting control to reduce interactions between the control variables and to decrease the response time. However, non-interacting control was not examined in the experiments owing to the limit on the computational speed of the feedback control computer. In DIII-D, perfect non-interacting control of the plasma shape (or gaps between the outermost flux surface and the first wall) is difficult because of the constraint that the sum of the PF coil currents must equal zero [10]. In ASDEX-U, the non-interacting control method has not been tried yet [11]. The applicability of the multivariable non-interacting control method to configuration control
has been examined experimentally in JT-60U for the first time, with an improved control system [14], as presented in this paper. JT-60U is a large tokamak machine with a major radius of about 3.3 m, a minor plasma radius of about 1 m with a plasma elongation of 1.3-- 1.8, a maximum plasma volume of l l 0 m -~, and a designed maximum plasma current of 6 MA [ 15]. Here, 5.0 MA is the maximum achieved plasma current so far. JT-60U can be used to investigate control algorithms relevant to tokamak fusion reactors. In non-interacting control, a linear relationship between small perturbations of the PF coil currents (6/) and small displacements of control variables (rX) from an equilibrium state, which can be presented as 6I : ArX, is essential to obtain a matrix gain. This linear relationship (or matrix A) can be obtained from some theoretical formulas, from an equilibrium database, or from small perturbations of an equilibrium configuration. These derivation methods of matrix A are presented in this paper. This paper is organized as follows, In Section 2. the matrix gain of the proportional feedback control in the non-interacting control is derived in a general format. In Section 3, a highly improved performance of the non-interacting control method experimentally obtained in JT-60U is presented. In Section 4, the non-interacting control method is applied to an [TER-CDA plasma to investigate the PF coil currents and voltages required for the plasma configuration control.
2. Non-interacting control method The matrix gain of the non-interacting control for the plasma configuration is derived generally in this section. Then, methods to obtain the linear relationship between the control variables and currents of PF coils, which is essential to obtain the matrix gain of the non-interacting control, are presented,
2.1. Derivation of matrix gain Basic equations necessary to obtain the proportional feedback matrix gain of the multivariable
R. Yoshino / Fusion Engineering and Design 24 (1994) 375-388
non-interacting control are presented in this section. Differential and integral control gains are not discussed in this paper, because they are not essential in non-interacting control. Integral control can reduce the offsets of control variables from the reference values, but deteriorates the quick response to sudden disturbance. Furthermore, offsets of control variables are not serious problems in the plasma shape control, because they can be adjusted easily by modifying the reference values. Therefore, integral control is not adopted in this paper. The characteristic frequency of proportional feedback control is usually selected to be lower than the characteristic frequency of eddy currents which flow in the machine components (e.g. vacuum vessel, cryostat). The effect of eddy currents is not included explicitly in the following investigation of proportional matrix gain. Of course, the time delay of the control system itself (e.g. about 2 ms in JT-60U) and part of the delay effect caused by eddy currents can be compensated for by differential control, as presented in Section 3.2. The following equations are necessary to obtain the non-interacting matrix gain. The circuit equation of the plasma current lp is Lp]p +
Mtpcri+ Ip(OLpfi?x).f¢
+ r/pip = 0
(1)
where I is an n vector of PF coil currents, Mp¢ is an n vector of the mutual inductance between the plasma current and PF coil currents, rip is the resistivity of the plasma current, and x is an m vector of control variables. Here, n is the number of PF coil currents. The circuit equation of PF coil currents I is Mi +
M~ip + ((~Mpe/Ox)Ip~ --I- R I =
g
(2)
where E is an n vector of the control voltages of the PF coil power supplies, and R is an (n x n) diagonal matrix of the resistances of the PF coils. When proportional control with the compensation of the resistivity drop is adopted, E becomes E = R1 q'- RGxlp(x ref - x) -.~ R a p ( I r y - Ip)
(3)
Here, x r~f is the reference vector of the control variables. In JT-60U, (ARp, AZp, Xp) is selected as x, as is explained in Section 3. I;ff is a reference value of the plasma current. Gx is the (n x m)
377
matrix gain of the plasma shape, and Gp is the n vector gain of the plasma current. The relationship between the displacements of the control variables 6Ip and 6x from an equilibrium state and that of the PF coil currents d l is given by (Eq. (1) is partially used) 61 = Apfi/p q- AxlprX
(4)
where Ap is an n vector of PF coil currents, and Ax is an (n x m) matrix of PF coil currents. Then, Eq. (4) can be rewritten as 6I = ArX
(5)
where A is defined as an [n x (m + 1)] matrix of (Ap, Ax), and dX is an (m + 1) vector of (d/p, lpdX) t. Each control variable must be controlled by each vector presented in matrix A (i.e. vectors of Ap, Axl, Ax2, • • • ) without modifying the other control variables. The non-interacting matrix gain can be obtained from Ap and Ax as follows. The state equation of 6Ip can be obtained from Eqs. (2) and (3) using definitions of x = x ref and .~ = 0: (MAp + Mr~)f/:p = - RGp lip
(6)
If we select a characteristic control frequency fp for the control of the plasma currents, the feedback gain vector Gp multiplied by R becomes RGp = (MAp + Mpc)2nfp
(7)
The matrix gain of the plasma configuration also can be derived from E.qs. (2) and (3) using definitions o f l p = _p 7~ec and Ip = 0. Then, the state equation of dx becomes (MA x + ~Mpc/Ox)lp6Yc = - R G x r x
(8)
When we select a diagonal matrix fx as the characteristic frequency of the control variables, the matrix gain multiplied by a diagonal matrix of resistance R becomes RGx = (MAx + 8MpJSX)2nfx
(9)
Then, we define an [n x (m + 1)] matrix Mp and an (m + 1) vector A, such that Mp = (Mr, c, 8Mr, c/Ox )
(10)
ft = (fp, f~)
(11)
378
R. Yoshino / Fusion Engineering and Design 24 (1994) 375 388
Then, the [n x (m + 1)] matrix gain G = (Gp, G x) of the plasma current and the plasma shape is obtained as RG = (MA + Mp)2~f
(12)
The same relationship for the plasma current and PF coil currents can be obtained from a circuit equation of the plasma current, and an equation of force balance to the horizontal direction as (3Ip = BpM
2.2. Derivation of matrix A Matrix A is necessary to obtain the non-interacting matrix gain as presented in Eq. (12). Matrix A can be derived from small displacements from an equilibrium configuration as follows. To obtain Ap, a n equilibrium plasma code can be used to obtain two different coil current vectors of 11 and 12, which make the same plasma configuration. When d~P is the difference in the influx in volt seconds supplied by these PF coil currents, the relationship between 6Iv and 6I is
,SI = (Lp/hqJ)(ll - I2)3Ip = Apflp
(13)
The matrix A,. also can be obtained by using an equilibrium code of the plasma configuration. Each control variable is displaced while keeping the other variables fixed, and with the same plasma current and the same influx in volt seconds. When the number of control variables is smaller than the number of PF coils, some restrictions on the coil currents (such as the same current values for some PF coil currents) is required to obtain a unique solution of A,- When Axi is the displacement of one control variable and A!,.i is the displacement of the coil currents, the required 61,.~ for Ip6Xi is M,, = ( A!,.,/(Iw Ax,))Ip6xi
(14)
This can be obtained for each control variable. Then, A X can be estimated as A,. = [AI,.,/(Ip~Axl), A!,.2/(Ip2Ax2) . . . . ]
(15)
The relationship between the control variables and PF coil currents also can be obtained statistically using a least-squares method for an equilibrium database [13], as presented by
x = B,.(I/Iv) + C,.
From Eqs. (17) and (18), the matrix A can be obtained as
F"pl'
A=LB~j
(19)
The procedure for determining the matrix A decides the applicability of the matrix. If A is obtained from a large equilibrium database, quite a wide range of plasma configurations can be controlled by one matrix gain. If matrix A is obtained from Eqs. (13) and (15) with small changes in the plasma parameter, as investigated for a plasma of I T E R - C D A (see Section 4), the range of applicability is limited to small displacements of the plasma parameters. However, in tokamak fusion reactors, it will be possible to calculate matrix A in real time for each equilibrium state, owing to the expected slow control speed caused by the long time constant of the vacuum vessel and the limit on the excitation voltages of superconductors.
3. Application of non-interacting control of JT-60U The non-interacting control has been applied to the configuration control of divertor plasmas in JT-60U. The control system of JT-60U is first briefly presented, and then the matrix gain for non-interacting control of the plasma current and the plasma horizontal position is derived according to the general method presented in the previous section. Finally, an experimental result obtained by the non-interacting control method is presented for comparison with that obtained by an independent control method.
(16)
where B~ is an (m x n) constant matrix and C~ is an (m + 1) constant vector. Then, the following relationship can be easily obtained.
6x = B,.(61/1p)
(18)
(17)
3. I. Control system of JT-60U Control of the plasma current and the plasma shape in JT-60U is performed by four types of PF coil. These are an OH coil, a vertical magnetic
R. Yoshino I Fusion Engineering and Design 24 (1994) 375-388
~ F , r [ ~
~
~Ij~Bl~r~..,~l~ ~
/ I XP~. "/~D_IAGONAL _ PORT
Fig. 1. Locations of the PF coils and the control variables. Types o f PF coil are as follows: F, OH coil; V, vertical magnetic field coil; H, horizontal magnetic field coil; D, divertor coil; D C W , disruption control winding, with a dominant mode of m/n = 3•2. D C W is not used for the configuration control. Control variables are as follows: Ip, plasma current; ARp and AZp, displacements of the vertical and horizontal positions of the plasma geometrical center from the center of the vacuum vessels; Xp, height o f null point from the divertor plate.
field coil (V coil), a horizontal magnetic field coil (H coil) and a divertor coil (D coil), as presented in Fig. 1 [15]. The connection of the V coil can be changed to produce three modes of divertor configuration. The elongation of the divertor plasma is 1.3-1.5 in the standard mode, and 1.5-1.8 in the highly elongated mode. In the third mode, continuous controls of the elongation and the triangularity are possible. However, the third mode requires an additional power supply, and
379
has not been tried yet. The mutual inductants of the PF coils in the highly elongated mode are presented in Table 1. Here, the mutual inductance between the OH coil and the V coil is extremely high, and is almost the same as the self-inductance of the OH coil. This high mutual inductance, which is also high in the standard mode of the V coil connection, causes a large interaction between the control of the plasma current and the horizontal plasma position. The selected control variables of the divertor plasma in JT-60U are the plasma current (Ip), the displacements of the vertical and horizontal positions of the plasma geometrical center from the center of the vacuum vessel (ARp, AZp), and the height of the null point from the divertor plate (Xp), as presented in Fig. 1. The control variables are calculated in real time, using formulas statistically derived from the predicted equilibrium database of about 3000 cases by regression analysis [9, 12]. These formulas use 6-17 sets of predictor variables calculated from magnetic probes, the plasma current and the D coil current. The power supplies of the PF coils are controlled by a plasma feedback computer consisting of three 32-bit reduced instruction set computers (RISCs) MC88100 [14], and by a direct digital controller (DDC) with a 32-bit microcomputer (MC68030) for each power supply of PF coil, as presented in Fig. 2. The pipeline computer system organized by the RISCs executes the following tasks: (1) collecting data of magnetic probes, PF coil currents and reference values of control variables; (2) calculating control variables using statistically derived formulas; (3) calculating command values to PF coil power supplies following programmed control algorithms; (4) sending com-
Table 1 Inductance matrix of PF coils in JT-60U (units of henries)
F Va H D
F
Va
H
D
1.24 x 10 2
- 1 . 0 3 x 10 -2 3.37 x 10 -2
- 9 . 5 2 x 10 -5 6.21 x 10 -5 1.91 x 10 -3
8.92 -9.74 -2.43 2.10
a The connection o f the V coil is in the highly elongated mode.
× x x x
10 -4 10 -4 10 -4 10 -3
380
R. Yoshino / Fusion Engineering and Design 24 (1994) 375-388
Plosmo Feedbock Computer ,r-. . . . . . . . . . . . . . 7 II
- - - - -
1;' i.!
v~"
,
i /hlp~p"q
DDC
,
•
q
i
MATRIX I I~"EaI,÷BP"~ 'it . . . . . . . . . . G A 1N
9.~
~-"~!'"L~J__ ~,"_F-_O~C~'~'__!i'I,.,g,, V,L2 C, IF . . . . . . . . . . .
~
~v
co~
~
Lv
I
(eGch10ms }
~AR~,,AzJf,,X~, ~ pre- calculation . . . . . . . . . .
io
~
I
L ~ - IRogowski,BN,T probes IF J
Fig. 2. Control system of the plasma current and configuration in JT-60U. A plasma feedback computer controls the plasma current and the plasma configuration, and receives reference values from a supervisory computer each 10 ms. It calculates Ip, ARp, AZp and Xp from raw data of a Rogowski coil and BT and By magnetic probes, then calculates commands and sends them to the DDC of each PF coil power supply. In the figure, voltage control is performed in the OH(F) coil power supply, and current control is performed in the other power supplies. This control diagram is also used for the case presented in Fig. 3.
mand values to DDCs. The control of AZp is performed with a cycle time of 250 ps and a delay time of 500 gs, and the other control variables are executed with a cycle time of 500 ps and a delay time of 1 ms. A faster control loop is executed for the control of AZp. The vertical position is unstable, because of the plasma elongation of 1.31.8. The control algorithms of each PF coil (such as current control, voltage control, proportional control to the plasma current, and feedback control) can be changed individually according to a preset program. Here, non-interacting control is one of the control algorithms executed in a discharge. Furthermore, the set of the matrix gain can be changed five times during a discharge, according to a preset-time schedule. DDCs of V, H and D coil power supplies have local gains of Kv, Kn and KD. These local gains are 0.05(Tdecay/"Ccl ). Here, "('decayis the current decay time constant (=L/R) of each PF coil (in seconds), and Zc~ is the cycle time of each D D C (e.g. 500 gs). In contrast, the OH coil power supply is controlled by the voltage command from the plasma feedback computer.
3.2. Matrix gain Two 2 x 2 matrix gains are installed in a central feedback control computer for the configuration control in JT-60U. Here, a 4 x 4 matrix gain cannot be used, owing to the limit on the computation speed. Therefore, two sets of non-interacting controls, i.e. (lp and A R p ) and ( A Z p and X p ) , have been selected. In particular, a high mutual inductance between the O H coil and V coil (see Table 1) causes a strong interaction between the controls of Ip and a R p , so that their non-interacting control must be adopted. However, independent control with a diagonal matrix is enough for the control of (AZp and Xp), owing to the relatively low mutual inductance between the H and D coils. Differential control can be performed for the following four cases: Iv control by the F coil: •Rp control by the V coil; AZp control by the H coil; and Xp control by the D coil. Here the differential time of ARp control is selected to be 6 ms and that of AZp control is 6 - 1 0 ms. Differential times of other control variables are set to be zero. The time delay of the control system itself is
R. Yoshino / Fusion Engineering and Design 24 (1994) 375-388
about 2 ms and the shielding time of the vacuum vessel to the dipole magnetic field is 8.5 ms. The matrix A for the non-interacting control of (Ip and ARp) can be obtained from a circuit equation of Ip and Shafranov equation, with assumptions of tip = 0, /Tp = 0, ~ = 0 and A,~p = 0. Here, ~IpARp is used instead of ARp in a real system. Then, the relationship between the control variables of (lp, #olpARp) and the PF coil currents of (IoH and Iv) can be presented as
?,o.) = ~2 F -M"°" ~MpoH
-M"l-'r L. t3MpvI ] R, J
xr
]:AF
L#oIp~ARp]
",
]
L#oIp,$ARvJ
(20)
where Ao
:
In(8Rj'] \Up )
--]- t i p + - -
.°=['o :v}['o,+,
Here, r/v is the resistance of the V coil. The derivation of Eq. (22) is presented in Appendix B. Then, GFB can be written as 1 Gva =
0
l~-3 2
aL f0= 1 -R,>
Yv--
RG
(23)
Noninteracting Gain Matrix
Vv ~ R j
7 -1
1
A comparison of independent control and noninteracting control has been executed in a divertor plasma with the highly elongated mode of the V coil connection, as presented in Fig. 3. The reference ARp was swung by + 2 cm at 10 Hz. The matrix gain for Ip and ARp is changed from a non-interacting gain to a diagonal matrix gain at t = 8.8 s. The matrix gain GFB for the non-interacting control from 8.3 to 8.8 s is selected with the
Rj OVv
\ap/
1
~/v 1 + K v
fl = A o - 1 + A 0 - - - -
g = ln(8Rj~ + 4
0
3.3. Non-interacting control of lp and ARp
t3R
L
381
Diagonal Gain Matrix
2.1
Ao 7
I
T~
ref
2.0
1 OMpv eL.
27rRj
0R
Here, the shift of the plasma current center from the plasma geometrical center is assumed to be given by
1.9 2
# iJllllllllllllllllllll,
lllll,l,,llllJJrlllltll
ARrer p ~- +-2cm,JOHz
,i
-3
,:~ , ! ~ ,'~
<3
The derivation of Eq. (20) is presented in Appendix A. The matrix gain G of non-interacting control can be obtained according to Eq. (12). In JT-60U, feedback control gains are installed in two types of computer, which are a plasma feedback computer and DDCs, as presented in Fig. 2. Then, the matrix gain GFBinstalled in the plasma feedback computer for the control of (Ip and/~olpARp) c a n be presented with the gain (Kv) for a minor feedback loop in the DDC of a V coil power supply (see Fig. 2) as
-8
8.3
' " ' .... ' .... ' .... , .... , .... , .... ,,,,',,8,,37,m.... 8.8
TIME
9.3
(sec)
Fig. 3. Swing of the horizontal plasma position ARp. A reference command of ARn, denoted by a broken line, is swung by _+2 cm at 10 Hz. The matrix gain is changed from the non-interacting gain to a diagonal matrix gain at 8.8 s. Large fluctuations of 144 kA are observed in the plasma current under the independent control method (t = 8.8-9.3 s), but this is suppressed drastically to 8 kA by the non-interacting control method (t = 8.3-8.8 s).
382
R. Yoshino / Fusion Engineering and Design 24 (1994) 375 388
characteristic control frequencies of 6 Hz for Ip control and 30 Hz for ARp control, such that
10 ,04 Mv J
+8.0 x lO 3 × [ kt°~/p
- 1.o x 104]
]
Lgolp6ARp]
(24)
However, the matrix gain for independent control from 8.8 to 9.3 s is a diagonal matrix gain, such that 6Iv
J
0.0 Lpolp6ARpJ
- 1.0 x 104
(25)
An OH coil power supply is controlled by the voltage command, and a V coil power supply is controlled by the current command. The units of the parameters are as follows: Von in volts, Iv and I v in amperes, and Rp in meters. In contrast, the gain matrix of AZp and Xp is a constant diagonal matrix. ARp is overswung at 10 Hz, but this frequency is selected to demonstrate the effect of the non-interacting control. A large Iv fluctuation of 144 kA, which occurs inevitably in independent control from t = 8.8 to 9.3 s, is drastically suppressed to about 8 kA (5% of 144kA) by non-interacting control from t = 8.3 to 8.8 s. This result shows clearly the usefulness of non-interacting control. The frequency analysis comparing the non-interacting control method and the independent control method was performed for shape control in JT-60, as presented in Fig. 6 of Ref. [13]. According to this frequency analysis, at low swing frequencies of the control variables (e.g. 1 Hz), non-interacting control has no benefit and independent control is sufficient. At higher frequencies of about 50 Hz or more, both control methods cannot be used, owing to the delay effect of eddy currents. However, at medium frequency of 1020 Hz, the interaction between the control variables (inevitably caused by independent control) can be reduced drastically by non-interacting control. The same performance of the non-interacting control should be observed in JT-60U, so ARp is swung at 10 Hz.
4. Application to ITER-CDA
In ITER-CDA, two control loops were designed for plasma shape control [6]. A slow control loop is performed by 14 PF superconductor coils with the characteristic frequency up to about 0.3 Hz to control the plasma current and the plasma shape. A fast control loop is performed by in-vessel coils of normal conductors, with a characteristic control frequency of 5 10 Hz, which controls the vertical plasma position, stabilizing the positional instability resulting from the highly elongated plasma (~: ~ 2.0), The selected control variables in the slow control loop are Ip, R .... R~,,, Zp, R,, and Z,, as presented in Fig. 4, in which the locations of the 14 PF superconductor coils with u p - d o w n symmetry are also presented. The plasma current lp must be controlled at the required value according to the operational scenario. Rout and Ri. are the horizontal locations of the outside and inside outermost plasma surfaces at the same vertical height with the plasma current axis. Each of Ro., and R~,, determines the clearance between the plasma outermost surface and the first wall, which should be a reasonable value to achieve the divertot performance and to decrease the erosion of the first wall caused by the charge exchange.
E su 4U 5U
Dzu
2U 1U IL 2L
[] 7L
5L 4L
(Rx, Zx)
[~ 6L
[~5L Fig. 4. Control variables of the plasma configuration in tTERCDA, and locations of PF coils.
R. Yoshino / Fusion Engineering and Design 24 (1994) 375-388 R o u t should be controlled to obtain the good coupling of the lower hybrid wave with the plasma. Ri, should be controlled to obtain a clearance of about 10 cm or more, in order to avoid the plasma touching the inner first wall owing to an inward horizontal shift caused by a minor disruption. Control of Zp is necessary to equalize the heat load on to two divertor plates, settled at the top and bottom of the first wall. The low frequency component of the in-vessel coil current Iiu, which will be used for fast control of Zp, should be reduced to achieve the control margin of Zp. Thus, the reduction of the low frequency mode in /in is also the control objective of the slow control loop of Zp. Rn and Z, are the horizontal and vertical locations of the null point, which should be controlled to guide the separatrix lines onto the divertor plate. ITER-CDA has a double-null divertor configuration, and the locations of the two null points should be controlled. Swing of the separatrix lines on the divertor plate is required to decrease the heat flux effectively. It takes too much time to obtain several thousand equilibrium values for a database for proFurthermore,
Table 2 M a t r i x A in I T E R - C D A Ip
383
ducing a linear relationship as presented by Eq. (16). Therefore, matrix A was investigated according to Eq. (15), by moving each control variable while keeping the other variables in an equilibrium state. Here, the merit of using Eq. (15) is that the matrix A can be obtained without knowledge of the theoretical formulations of the control variables (such as force balance equations in the horizontal direction and the vertical direction). The reference equilibrium configuration is presented in Fig. 5(g). Then, a 14 × 6 matrix A has been obtained, as presented in Table 2 for ITER-CDA with 22 MA, R p / a p = 6.0/2.0 m, and bp/l i = 0.6/0.65. In the case of Zp, the plasma was moved rigidly in a vertical direction. The units of matrix A are kiloampere teslas. Each magnetic field produced by each vector of matrix A is presented in Fig. 5. The voltages of the coil power supplies RG are then estimated by (MA)2ztf, as presented in Table 3, by selecting each control speed at 400 kA s(22 MA per 75 s) for Ip, 2 cm s -1 for Rout, Rin, Zp and Zn, and 3 cm per 1.25 s (swing of 3 cm per 0.2Hz) for R o. As presented in Table 3, the matrix A is also useful for estimating the coil voltages required for configuration control.
( u n i t s o f k i l o a m p e r e teslas) Rout
Rin
Rn
Zn
Zp
PF1U PF2U PF3U PF4U PF5U PF6U PF7U
1.453 1.453 1.533 1.559 7.860 6.480 5.700
× × × × × × ×
10 ° 10° 10° 10 ° 10 -1 10 - 2 10 -3
-7.159 -7.159 3.477 5.193 -1.287 1.664 -1.039
× x × × × × ×
101 101 102 102 102 102 102
2.475 2.475 -4.604 2.178 5.673 -1.266 9.119
× × × × × x ×
101 101 101 101 101 102 101
2.800 2.800 -2.010 -9.200 1.412 4.400 -1.040
× × × × x × ×
101 101 102 10° 102 10 ° 101
-4.590 -4.590 - 1.530 -7.776 -1.485 1.451 -7.929
x × × × × × ×
10° 10 ° 10 ° 101 102 102 101
-2.300 -2.300 -5.530 -6.220 -1.190 -3.360 3.360
× × × × × × ×
101 10 I 10 l 10 l 101 10 l 10 l
PF1L PF2L PF3L PF4L PF5L PF6L PF7L
1.453 1.453 1.533 1.559 7.860 6.480 5.700
× × × × × × ×
10° 10 ° 10° 10° 10 -1 10 -2 10 3
-7.159 -7.159 3.477 5.193 -1.287 1.664 -1.039
× × × × × × ×
101 101 102 102 102 102 102
2.475 2.475 -4.604 2.178 5.673 -1.266 9.119
× × × × × × ×
101 101 101 101 101 102 101
2.800 2.800 -2.010 -9.200 1.412 4.400 -1.040
× × × × × × x
101 101 102 10° 102 10 ° 101
-4.590 -4.590 -1.530 -7.776 -1.485 1.451 -7.929
× × × × × × x
10° 10° 10° 101 102 102 101
2.300 2.300 5.530 6.220 1.190 3.360 -3.360
× × × × × × x
101 l0 t 10 ~ 10 ~ 10 j 101 101
C o n t r o l l e d p l a s m a is Ip = 22 M A , is in k i l o a m p e r e s .
Rp/ap = 6.0/2.0 m , a n d flp/li = 0.6/0.65. U n i t s o f Rout, Rin ,
Zp a n d
Z , a r e c e n t i m e t e r s , a n d Ip
R. Yoshino / Fusion Engineering and Design 24 (/994) 375.-388
384
[]
(b)
0
E3
i , i , 1 ,~1,
1 ,.,J.,~.
t i I I [ i r i I i
I0
E~
(c)
[]
(d
[]
(g)
5
v
0
N
-5
[] ,
E~
,
,
-I0
5
10 R
(m)
l
Fig. 5. Magnetic field produced by each current vector presented in matrix A: (a) Rm; (b) Rout; (c) Rn; (d) Z,,; (e) Zv; (1) lp; (g) A reference plasma configuration used to investigate each current vector in matrix A. Controlled plasma is 1o = 22 Ma, Rr/a r, = 6.0,,' 2.0 m, and ~p/lj = 0.6/0.65.
R. Yoshino / Fusion Engineering and Design 24 (1994) 375-388 Table 3 Required coil voltage RG for I T E R - C D A (units of kilovolts) 10
Rout
Rin
Rn
Z.
Zp
PF1U PF2U PF3U PF4U PF5U PF6U PF7U
1.48 1.40 1.02 0.36 0.46 2.63 1.59
0.00 -0.04 -0.19 -0.03 0.43 -3.13 1.23
-0.66 -0.34 0.53 0.07 - 1.37 3.70 -0.93
0.07 -0.12 -0.72 -0.12 1.44 -0.18 -0.06
0.08 0.06 -0.02 -0.12 -1.35 3.87 -0.57
-0.3 0.82 -1.3 -0.4 0.57 -3.2 0.79
PFIL PF2L PF3L PF4L PF5L PF6L PF7L
1.48 1.40 1.02 0.36 0.46 2.63 1.59
0.00 -0.04 -0.19 -0.03 0.43 -3.13 1.23
-0.66 -0.34 0.53 0.07 -1.37 3.70 -0.93
0.07 -0.12 -0.72 -0.12 1.44 -0.18 -0.06
0.08 0.06 -0.02 -0.12 -1.35 3.87 -0.57
-0.17 0.12 0.06 -0.004 0.24 -0.33 0.048
Equilibrium plasma and units are the same for Table 2.
5. Conclusions The deviation of the matrix gain of non-interacting control for the plasma configuration is presented in this paper. The relationship between small perturbations of the poloidal coil currents (6/) and small displacements of control variables from an equilibrium state (fiX), which can be presented as 61 = A f X , is of essential importance to determine the matrix gain of non-interacting control. The matrix A can be obtained from some theoretical formulas for some plasma parameters, from an equilibrium database by using regression analysis, or from small perturbations of an equilibrium plasma configuration, as presented in this paper. The non-interacting control technique has been applied to the control of the plasma configuration with single-null divertors in JT-60U. Comparisons of the independent control method and the noninteracting control method have been experimentally performed in the control of the plasma current Ip and the vertical positions of a plasma's geometrical center from the center of a vacuum vessel (ARp), swinging a reference of ARp by +2 cm at 10 Hz. When independent control is adopted, a large mutual inductance between the OH and V coils inevitably causes strong interac-
385
tion between the controls of Ip and ARp. However, the fluctuation of Ip is drastically suppressed by the non-interacting control method by about 5% of that in the independent control method. The multivariable non-interacting control method can be used generally, and can be applied to any tokamak machines and tokamak fusion reactors. As an example, the matrix A for an ITER-CDA plasma has been investigated, and the PF coil currents and voltages required for control of the plasma configuration have been calculated for selected control speed of the control variables, using the non-interacting control algorithm.
Acknowledgments Tha author wishes to thank the Japan Atomic Energy Research Institute, who have contributed to the JT-60U project, and he especially would like to express his gratitude to Mrs. M. Matsukawa and M. Takahasi, and Drs. M. Sugihara, K. Shinya, L. Zaharov, T. Kimura and Y. Shimomura for their fruitful discussions. He would like to express his gratitude to Dr. H. Ninomiya for his continuous support.
Appendix A: Derivation of Eq. (20) A 1. Relationship between Rp and P F coil currents
The Shafranov equation is 2
\
-"}- tip -+ap
-1- 2 1 r R j B v = 0
(A1)
When Ion and I v are the coil currents of the OH and V coils, the equilibrium vertical magnetic field can be presented as Bv = VOHIo, + Vvlv
(A2)
Here, vi (i = OH, V) can be presented as 1 dMpi vi - 27zRj d R j
(A3)
In JT-60U, VoH is much smaller than Vv. However, to achieve the generality of the equation, the vertical magnetic field produced by the OH coil is
386
R. Yoshino / Fusion Engineering and Design 24 (1994) 375- 388
not neglected. Here, Rj is the horizontal position of the plasma current center, ap is the averaged plasma minor radius. In JT-60U, control of the plasma horizontal position is performed for the geometrical center of the plasma column Rp. The relationship between Rp and Rj can be approximated by
tip + ~
Rj = Rp -}-
- 1
(A4)
From Eq. (A4), with assumptions of rip = 0,/~p 0 and Ii. = 0, the following relationship can be obtained: ~--
ap2{B li 2Rp\ '~p+ 2 - 1 ) ] / ~ p
/~j__[1
Then, with the definition
1
(15,
offo as
(A6)
The derivative of the Shafranov Eq. (A1) with Eq. (A6) can lead to
Af0dgp A0dRp A0( R, dt Ip
dG.
dlv5 (A7)
Here, f~ and Ao are defined as Rj 8Vv5 j] --- A0 - 1 + Ao( -RJ 8Vo. vo. OR,
R--7
(All)
Here, Lp is the plasma inductance, such that Lp=/~oRj(ln 8Rj a - ~ + ~l,.- 2)
(112)
Furthermore, from Eq. (A3), we can obtain 2~RjBz =
Ao
(')Mpi
8g-~l,=-Itolp~ -
Z
/ = OH,V
(113)
The derivative of Eq, (A12), with assumptions of rip = 0 and /] = 0, leads to \
ap
L dip
J d-t-
(A14)
dRp
(A15, In Eq. (A15), the relationship between Ip, Rp and the PF coil currents is presented. Here, g is defined as
ln(8Rj~ +/,. \ap] 2 -1
(A9)
Lp
dRp
gfo][
- #oAo
0
(A16)
A modified Shafranov Eq. (AI0) and a modified circuit equation of the plasma current (Eq. (A15)) can be combined as
- olo d7
-- 47zRj VOH d-~- -1- VV ~ - / / =
Ao 2
A3. Derivation of matrix A
2
d/v']
d/,.)
(A8)
li - 3 - -
d/oH
(
p--~- + g 0 g 0 # 0 I p ~ - -]-~plp-t-i=O~H, V M p i - d - T j = 0
g=
From Eqs. (AI), (A7) and (A9), the relationship between Rp and the OH and V coil currents can be presented as flf0/2oi p dRp
( dI, ?~MpidRj ) i=o~H.v M p i ~ - + ?Rj dt li =-0
Then, Eq. (AI 1) can be modified using Eqs, (A13) and (A14) as
l~j = foRp
+ tip +
+
dt
Eq. (A5) becomes
ap
dL v
dLp=#o(lnSRj+~_ 1~dRJ
ap2(flp+ li l )
Ao - In -8Rj -
dip
G¥
dlv
I . dRy,
(A10)
A2. Relationship between Ip, Rp and PF coil currents From the circuit equation of the plasma current, we have
__ M p o H
2 3MpoH
-
-
mpv
OMpv
d/o.
d,v +[-;'q
57
(A17)
R. Yoshino / Fusion Engineering and Design 24 (1994) 375-388 The control speed is much faster than the timescale of the plasma current decay time (Lp/?lp), so the resistivity drop of the plasma current can be ignored to investigate the feedback control gain. Then, the relationship between the PF coil currents and control variables, i.e. as in Eq. (20), can be obtained from Eq. (A17) as
387
Here, the mutual inductance matrix M is defined as
[Lo. Mo.v] M=
Monv
Lv ]
(B2)
and the voltages of the OH and V coils are controlled as
."on -~- =/2 63MpoH dlv
L ~7-
L
c~R
OMpvl
l / / 2 ~ R ~ j L- p ° A °
fofll
RjJ
[-l
dt dRp
r --MpoH
--Mpv 1-1 r 0Mpv / 2 ~ ]
Lp
/
Rj 1
L-#oA o
II
MpOH /~0---~'1 OMov
MA +
M~v
V
=
RG[
The circuit ec uations of the OH and V coil currents can be ~resented as
dlv
+
?
[M'°" #0-TY-f0/ 1 OMpv l M.v ;; ~:oj
d~ X
dt dRp
~oI.-~-
:rE°H--q°nl°nl
L e,,-rtv/v J
l |
(B4)
The matrix gain G can be obtained from Eq. (B4) for selected characteristic control frequencies offlp and fRp for Iv and Rp as
MA+
B1. Matrix gain G in the general form
~-
~o~p
b,0Ip(Rp - Rp)d
RG=
1 ~3Mpon
at dRp
1 ~Mvoufo] M"o.;o~-I
Appendix B: Derivation of Eq. (22)
[
i l
~ ~ f0 ~
fief [ P r~f O
Then, Eq. (A20) is the same as Eq. (5).
Lon Monv 1 Monv Lv ]
(B3)
gfO 1
/of, /
J = LPoIp6RpJ
-,iV
1
The state equation of the plasma current and the horizontal plasma position can then be obtained from Eqs. (A20) and (B1)-(B3) as
(AI9) the relationship between the displacements of the control variables (Mp and #0IprRp) from an equilibrium state and those of the PF coil currents O/OH and 6Iv) can be presented as
"l/OH
[ref p __];ef
RGL~Ip (R [ff - RDe~)J
(A18)
By defining the matrix A as
A--|2 OMpon L aR
t/vJLg21 g22_J [ r e f /ref 1 xI --P --P I kklolp(Rpef RIoOj V
d/~ X
6Iv
L Ev - qvlv ] =
(B1)
x
-
10Mvv
I
}
27ZflP 0 0 2~zfRp
(B5)
Thus, Eq. (B5) is the same as Eq. (12). B2. Matrix gain GFBinstalled in JT-6OU' s plasma feedback computer In JT-60U, a matrix gain feedback control is installed in two types of computer, which are a plasma feedback computer and a DDC for each thyristor bank of PF coil power supplies, as pre-
R. Yoshino / Fusion Engineering and Design 24 (1994) 375 388
388
sented in Fig. 2. E~%m and -~vr are calculated in the plasma feedback computer as
Here, GFB is defined as GvB--
A/r~yfJ =
lJkg[, ~
;
g=V~"Jk~,oldR;"~-R~) (B6)
H e r e , gVB (i,j = 1, 2) is the f e e d b a c k g a i n i n s t a l l e d in the p l a s m a f e e d b a c k c o m p u t e r . T h e c o m m a n d v o l t a g e E~'~ is sent d i r e c t l y to t h e p o w e r s u p p l y o f the O H coil as (see Fig. 2)
Eb~ '
=
rloHIoa + AEb'~
(BY)
The voltage command E~°m to the thyristor bank of the V coil is calculated in the DDC as E~,° ' ' = r]v{I~°m -+- K v ( I ~ ° " - Iv)}
(B8)
Here, Kv is the local feedback gain in the DDC (see Fig. 2) and "v'refis the reference coil current sent from the plasma feedback computer to the V coil DDC, such that
I~°m = Iv + A/'ref ~-v
(B9)
From Eqs. (B8) and (B9), E~¢°m becomes E~°~ = q v l v + r/v( 1 + Kv)AIF r Then Eq. (B6) can be presented as
[
(B10)
EbO~' - qonlou 1 E~/°m -- r/V/v ]
Lqv(1 + K v ) A I F r J = ~r/;
.
0 1F r/v( 1 + Kv)jLgVB g~;Sj
[
× Lmlp(RF' - np) =[~
0][1 qv 0 [
×
]
0 Ilion 1 +Kv
Ipf--IP °'-
l
r )j
°l[
l JLgl;" g~gJ (ml)
",,
(B12)
1jLg~,B g2~2BJ
T h e n , E q . (22) c a n be o b t a i n e d f r o m E q s . (B3), ( B l l ) a n d (B12) as
RG=
0 t/v
0
l+KvJ
v,3
(B13)
References [1] F.C. Schfiller and A.A.M. Oomens, Plasma engineering aspects of large tokamak operation, Fus. Eng. Des. 22 (1993) 35 55. [2] F. Schneider, Proc. l lth Symp, in Fusion Technology, Oxford, Vol. 2, 1978, pp. 1027 1032. [3] D. Ciscato, L. de Kock and P. Noll, Proc. llth Symp. in Fusion Technology, Oxford, Vol. 2, 1978, pp. 1033 1039. [4] J.M. Ane et al., Proc. 15th Symp. on Fusion Technology, Oxford, Vol. 1, 1988, pp. 657 661. [5] R.E. Bell et al., Proc. 13th Symp. in Fusion Engineering, VoL 1, 1990, pp. 467 470. [6] F. Hofmann et al., Nucl. Fus. 32 (1992) 897 902. [7] V.S. Mukhovatov and V.D. Schafranov, Nuc. Fus. 11 (197t) 605. [8] R.D. Wooley el al., Fus. Technol. 8 (1985) 1807 1812. [9] N. Hosogane et al., Nucl. Fus. 26 (1986) 657. M. Matsukawa et al., Plasma Phys. Control. Fus. 34 (1992) 907 921. [10] S. Kinoshita et al., General Atomics Rep. GA-A 19584, 1990. [l I] P.J. McCarthy et al., Proc. 19th EPS Conf. on Controlled Fusion and Plasma Physics, lnnsbruck, Vol. 1, 1992, pp. 459 462. [ 12] M. Matsukawa et at., Detecting method of plasma position by regression analysis, JAERI Rep., Japan Atomic Energy Research Institute, JAERI-M 92-073, 1992, pp. 16 19. [13] R.Yoshino et al., Control of divertor configuration in JT-60, Proc. 13th Symp. on Fusion Technology, Vol. I, 1984, pp. 1141 1146. [14] T. Kimura et al., VME multiprocessor system ['or control al the JT-60 upgrade, tEEE Trans. Nucl. Sci. 36 (1989) 1554 1558. [ 15] tt. Nimomiya et al., JT-60 upgrade device for confinement and steady state studies, Plasma Dev. Oper. I (1990) 43 65. [16] Y. Shimomura et al., ITER Poloidal Field System, ITER Doc. Ser. 27, IAEA, 1990, pp. 100 .114.