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Design of compact tokamak reactor with HTSC plasma stabilizing coils
Fusion Engineering and Design 51 – 52 (2000) 1077 – 1086 www.elsevier.com/locate/fusengdes
Design of compact tokamak reactor with HTSC plasma stabilizing coils T. Uchimoto, T. Yamada *, K. Miya Nuclear Engineering Research Laboratory, Graduate School of Engineering, The Uni6ersity of Tokyo, 22 -2 Shirakata-Shirane, Tokai, Naka Ibaraki, Japan 319 -1106
Abstract This paper presents a design of tokamak plasma with use of the high temperature superconducting coils as plasma stabilizer. Having regard to the current situation of the ITER project, two cases of designs were investigated: a smaller machine which has the same mission of the ITER/RCO (Q= 10), and an ignition machine which has the same major radius as ITER/RCO (R= 6.1 m). The same data base and formulas as ITER are here used and any innovative technology other than the HTSC stabilizing coils is not assumed. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Tokamak reactor; HTSC coils; Plasma stabilization
1. Introduction The authors have proposed the new method to stabilize plasmas position passively by applying high temperature superconducting (HTSC) coils [1]. HTSC coils are expected to stabilize the plasmas since HTSC coils prevent the change of magnetic flux induced by vertical motion of plasma. HTSC coils are robust against the thermal disturbance, so that they can be arranged in the vicinity of plasmas and effectively stabilize them. Advantages of the stabilizing system include the following compared with conventional ones: * Corresponding author. Tel.: +81-29-2878422; fax: + 8129-2878488. E-mail address: [email protected] (T. Yamada).
1. It requires no power supply unlike feedback control systems; and 2. Its stabilizing effect on plasmas is free of decay which is inevitable in that of eddy current induced in structures. Considerable plasma stabilization owing to HTSC coils provides the wider range of plasma elongation which is usually restricted to moderate values because of the vertical instability. In other words, HTSC coils enable compact reactors with highly elongated plasmas. We have displayed the feasibility of the HTSC coils for plasma stabilization and its potential to enable highly elongated plasma which leads to compact tokamak reactors [1]. In addition, we showed that HTSC coils mitigate VDEs followed by major disruptions as well as minor disruptions in the configuration of ITER/EDA [2].
0920-3796/00/$ - see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 0 - 3 7 9 6 ( 0 0 ) 0 0 4 3 9 - 7
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T. Uchimoto et al. / Fusion Engineering and Design 51–52 (2000) 1077–1086
The ITER project is now pursuing low-cost approaches reducing the technical objectives of the original design shown in the ITER Final Design Report [3]. Rather than trying to achieve ignition, the new objective is to demonstrate a net energy gain (Q) of 10 – 100. This is ITER Reduced Cost Option (ITER/RCO) and the cost will be cut down by half. In this study, having regard to the current situation of the ITER project, we introduce HTSC coils to ITER/RCO like tokamak reactors and show the enhancement of reactor potentiality owing to the resulting high elongation plasma. For the purpose, two cases are investigated: a smaller machine which has the same mission of the ITER/RCO (Q = 10) — we named HTSC tokamak-A, and an ignition machine
Fig. 1. Decision of design point in I –A space for HTSC tokamak-A (Q= 10).
which has the same major radius as ITER/RCO (R= 6.1 m) — HTSC tokamak-B. The reminder of the article is organized as follows. The plasma response model in the presence of HTSC coils — which is employed for the plasma stability analysis of HTSC tokamaks — is described in Section 2. Results and verifications of the design of HTSC tokamaks are contained in Section 3, and a summary and conclusions are given in Section 4.
2. Numerical formulation of plasma stability analysis
2.1. Linear plasma response model In this study, the behaviors of stabilized plasmas is exclusively of interest since stabilizing effect of HTSC coils on plasmas is investigated. Therefore, it is required to compute the plasma response accurately and efficiently in the case where change of plasma parameters is small. In the sense, the linear plasma response model, which assumes that plasma is supposed to be in equilibrium at each time due to the massless approximation, meets the requirement of this study. Here, we adopt the CREATE-L model [4] which is one of the linear plasma response model and embed HTSC shielding current analysis into it. The CREATE-L model is illustrated as follows. In axisymmetric condition, plasma equilibrium can be described by the Grad–Shafranov equation: Lc = Jpf + Ji = r where Lc = −
Fig. 2. Decision of operation point in POPCON plot for HTSC tokamak-A (Q =10).
dp f df + + Ji, dc mr dc
( 1 (c ( 1 (c − , (r m0r (r (z m0r (z
(1)
(2)
m0 is the magnetic permeability in the vacuum, c is the poloidal flux per radian, Ji is the external current density, Jpf is the toroidal component of the plasma current density, p is the pressure and f/r is the toroidal component of the flux density. Here, we assume the plasma current density profile to be a function of four parameters
T. Uchimoto et al. / Fusion Engineering and Design 51–52 (2000) 1077–1086
Fig. 3. FEM mesh used for the simulations with the code SCAPE and initial equilibrium result for HTSC tokamak-A.
(l, b0, am, an ) that can be associated with the physical quantities Ip, bp, li and q0: Jpf =l
n
b0r R +(1 −b0) 0 (1 − c( am )an, r R0
(3)
with c( =(c−cb )/(cb −ca ), where ca is the flux per radian at the plasma magnetic axis, cb the flux per radian at the plasma boundary [5]. Then, the perturbed plasma equilibrium can be expressed as Ldc =dJpf +dJi =
(Jpf (J (J dc + pf dq + pf ds (c (q (s + % dIk Jkf (r,z),
(4)
where q=(am, an, b0), s = (ca, cb, l), Ik is the current flowing in k-th coil and Jkf is the distribution function of the coil current. Ignoring nonaxisymmetirc contributions, Ohm’s law becomes
Fig. 4. Poloidal cross section of HTSC tokamak-A.
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Fig. 5. Time evolution of plasma parameters in the presence of HTSC plasma stabilizing coils (HTSC tokamak-A).
hJ=E +Eext = −
1 (c + Eext, r (t
(5)
where E is the electric field, h the electric resistivity and Eext the impressed field. Applying the same linearization procedure, the perturbed circuit equations for i-th coil is
2p
& Vcond
n
Jifdc: dV + % RikdIk =dui,
(6)
k=1
where Rik is the resistance matrix and ui is the applied voltage. Applying the Galerkin method to Eqs. 4 and 6,
we can obtain a linear matrix equation for perturbation of a set of currents in conductive shells. , L*dI: + RdI = du − L*ds; s
(7)
where L* is modified inductance matrix and dI is the first variation of a set of currents. Here, I and s can be regarded as the state variable of the system. Using the set of the state variable, expressed in the output equation which describes the plasma parameters of interest y such as positions of reference points at the separatrix and the plasma current centroid; y= AdI + du + Fds
(8)
T. Uchimoto et al. / Fusion Engineering and Design 51–52 (2000) 1077–1086
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Fig. 8. Decision of operation point in POPCON plot for HTSC tokamak-B (Q = ).
2.2. HTSC shielding current analysis Shielding currents in the HTSCs were estimated based on the flux flow and creep model [6] which properly describes the quickly changing shielding currents in superconductors like this case. The constitutive relations between the current density, J, and the electric filed, E, based on the model are as follows: Fig. 6. Time evolution of gap length between plasma surface and first wall (HTSC tokamak-A).
1. The creep region (05 J5Jc )
E(J)=2rc Jc sinh
Uo J U J exp − o , ku Jc ku Jc (9)
2. The flow region (Jc 5 J) E(J)=Ec + rf Jc (J/Jc − 1),
(10)
where Ec is the critical electric field, rc, the creep resistivity, u the temperature, Uo the pinning potential, k the Boltzmann constant, rf the flow resistivity and Jc the critical current density. Shielding currents in superconductors can be estimated by applying this constitutive relation to the conventional eddy current analysis [7]. Therefore, in order to introduce the shielding current analysis to the system equations, Eq. (7) is changed as follows: Fig. 7. Decision of design point in I –A space for HTSC tokamak-B (Q = ).
dF/dt + R(I)I= V,
(11)
T. Uchimoto et al. / Fusion Engineering and Design 51–52 (2000) 1077–1086
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where, Ri ·Ii =
&
E(Ji ) dl.
(12)
c
Because of this change, Eq. (11) become nonlinear and the Newton – Raphson method was applied to solve it.
3. Design of HTSC tokamaks
3.1. HTSC tokamak-A (Q = 10) Here, HTSC tokamak of Q =10 is pursued. Assumptions adopted here are to use the same data base and formula as ITER in principle and not to use any innovative technology other than HTSC stabilizing coils. In order to obtain an optimum design point of a HTSC tokamak, major design parameters are decided with 0-dimensional plasma analysis employing the I – A – Btf – k analysis methodology, which was introduced in the
ITER Conceptual Design Activity [8]. This methodology is to find the allowable design windows in the four-dimensional I–A–Btf –k space — where I is the plasma current, A is the aspect ratio, Btf is the peak field at the TF coils, and k is the elongation of plasma — considering the two constraints: (1) the minimum safety factor at the plasma edge, (2) radial build between plasma and inner leg of TF coil which includes inboard scrapeoff layer, first wall, inboard blanket and vacuum vessel. We determined apriori the basic parameters, k= 2.1, Btf = 12.5T, qc \ 3.0, tshield = 1.05 m. The elongation was chosen to the maximum value in the range that plasma positional stability is secured with HTSC coils after some iteration of plasma stability analysis was made. As 4-D surfaces cannot be visualized, we consider 2-D I–A plot of plasma current versus aspect ratio at constant values of Btf and k. The I–A plot of HTSC tokamak-A is shown in Fig. 1 with parameters of major radius. The hatched region in the figure
Fig. 9. FEM mesh used for the simulations with the code SCAPE and initial equilibrium result (HTSC tokamak-B).
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Fig. 10. Time evolution of plasma parameters in the presence of HTSC plasma stabilizing coils (HTSC tokamak-B).
corresponds to the possible design window bounded by the radial build constraint. We determine Ip = 16(MA) for A = 3.3 and R = 5.0(m) as the design point. This is very small compared with the size of ITER/RCO. Plasma OPeration CONtour (POPCON) plot [9] provides an indication of plasma operation point which satisfies Q = 10. To generate the contours of the auxiliary power required for plasma energy balance, ITER-89P [10] confinement scaling with a multiplier, H =2.0 was employed. The resulting contours of auxiliary power in the HTSC tokamak are shown in Fig. 2 together with beta limit based on Troyon scaling [11]. If we take a
fusion power of 650 MW, contour line of 650 MW gives n= 1.7× 1020/m3, T= 10.5 keV as the cross point with the curve of Q= 10. The elongation of HTSC tokamak is set to higher value, so that it is highly required to investigate whether or not the plasma equilibrium condition is satisfied and its positional stability is secured. For this purpose, the plasma response code with HTSC current analysis, SCAPE (Sperconducting Current Analysis code with perturbed Plasma Equilibrium model) was used to obtain the plasma equilibrium and to conduct the stability analysis. The FEM mesh for estimation of the matrices in Eq. (7) and equilibrium results are
T. Uchimoto et al. / Fusion Engineering and Design 51–52 (2000) 1077–1086
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Fig. 11. Time evolution of gap length between plasma surface and first wall (HTSC tokamak-B). Table 1 Comparison of major parameters between HTSC tokamaks and ITER/RCO Parameter
HTSC (Q = 10)
Major radius 5.0 (m) Minor radius 1.51 (m) Elongation 2.1 Plasma 16 current (MA) Fusion 650 power (MW) Energy gain 10 Pulse length 500 (s)
HTSC (Q = )
RCO (Japan)
6.0
6.1
1.88
1.8
2.1 20
650 500
1.7 13–14
500–600
10–20 300
shown in Fig. 3. Primary parameters determined in the preceding section are Ip = 16(MA), R= 5.0(m), A= 3.3 and k= 2.1. Double null system of magnetic separatrix is adopted. Stability analyses were conducted for the equilibrium obtained above in order to confirm that HTSC coils suppress the plasma positional instability. The conductive structures were modeled as an array of discrete conducting loops; the first wall, the back plates and vacuum vessel were approximated as axisymmetric rings of 50, 50 and 60, respectively to construct an equivalent electrical circuit. Electrical resistances are 5.66, 0.50 and 2.32 (mV) for the first wall, the back plate and the vacuum vessel, respectively. These values are evaluated on the data base given in the ITER design. In order to improve the plasma positional stability, four pairs of HTSC coils are installed at the back plate as shown in Fig. 4, after the location and cross-section of the HTSC coils are optimized with respect to their stabilizing effect. They are placed in the form of axisymmetric toroidal coils. Two coils of each pair are symmetric in position with regard to the vertical center and connected never to induce current during plasma start-up. Silver-sheathed Bi-2223 superconducting tape was considered as the material of high Tc superconducting coils [12]. The physical parameters of Bi-2223 tape in the computation are assumed to be that of 100 m length tape which have been achieved in the current technology. Time evolution of plasma parameters are evaluated with SCAPE code and are shown in Fig. 5. As for the disturbance of the plasma, we considered instantaneous bp drop of 0.15 and li drop of 0.18 recovering in 5 sec. The maximum deviations of plasma centroid are − 0.21 and −0.05 m in radial and vertical directions, respectively, and the centroid moves back to the original point, followed by the recovery of bp and li. The motion of the plasma separatrix in the presence of HTSC coils is shown in Fig. 6. The plasma at reference point P4 moves toward the first wall by 7 cm, but do not touch the wall because the initial gap length is 20 cm. These results indicate the plasma is stabilized by HTSC coils sufficiently, which supports for the enough plasma stabilization owing to HTSC coils and feasibility of the design of HTSC tokamak-A.
T. Uchimoto et al. / Fusion Engineering and Design 51–52 (2000) 1077–1086
3.2. HTSC tokamak-B (Q = ) We designed HTSC tokamak-B in the same manner as HTSC tokamak-A, determining apriori the basic parameters, k =2.1, Btf (T), qc \ 3.0, tshield =1.05 m. The elongation was chosen to the maximum value in the range that plasma positional stability is secured with HTSC coils. The Ip –A space of HTSC tokamak (k =2.1, Q= ) is shown in Fig. 7. The hatched region in the figure corresponds to the possible design window. We determine Ip =20(MA) for A = 3.2 and R= 6.0(m) as the design point. Plasma parameters for HTSC tokamak-B was determined so as to achieve Q = , considering a POPCON diagram shown in Fig. 8 together with beta limit. If we take a fusion power of 650 MW, contour line of 650 MW gives n =1.0× 1020/ m3, T =14 keV as the cross point with the curve of Paux =0. The elongation of HTSC tokamak-B is also set to higher value, so that plasma stability analyses were conducted to confirm the stabilization owing to HTSC coils. Fig. 9 displays the FEM mesh and equilibrium results for HTSC tokamak-B. Primary parameters determined in the above are Ip =20(MA), R = 6.0(m), A = 3.2 and k=2.1 The conductive structures were modeled as an array of discrete conducting loops; the first wall, the back plates and vacuum vessel were approximated as axisymmetric rings of 50, 50 and 60, respectively. Electrical resistances are 5.66, 0.50 and 2.32 (mV) for the first wall, the back plate and the vacuum vessel, respectively. Four pairs of HTSC coils are installed at the back plate in the same manner as HTSC tokamak-A, after their location and crosssection are optimized. Time evolution of plasma parameters are evaluated with SCAPE code and are shown in Fig. 10. The disturbance of the plasma is instantaneous bp drop of 0.2 and li drop of 0.08 recovering in 5 sec. It can be recognized that all the plasma parameters recover from the deviations after the disturbances of bp and li vanish. The motion of the plasma separatrix in the presence of HTSC coils is shown in Fig. 11. The plasma at reference point P1 moves toward the first wall by 7 cm, but do not touch the wall because the initial gap length is 20 cm. In the
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configuration of HTSC tokamak-B, the considerable plasma stabilization due to HTSC coils is observed as is the case with HTSC tokamak-A, which imply that the ignition machine is feasible in the same size as ITER/RCO applying HTSC coils.
4. Concluding remarks The design of HTSC tokamak was carried out under the same condition as the ITER/RCO activity, except that HTSC coils are applied. The major parameters of HTSC tokamak A and B are listed in Table 1 together with those of ITER/ RCO. Conclusions of these designs are summarized in the following. 1. Plasma stabilization owing to HTSC coils allows the high elongation of Section 2.1. This stabilization method offers an advantage that it does not require plasma sensors and power supply compared with the conventional active control system. We confirmed stable behavior of plasmas in the presence of HTSC coil via numerical analysis in the both designs of HTSC tokamak A and B. 2. Applying HTSC coils, major radius of plasma can be reduced to 5.0 m, keeping the same mission of ITER/RCO, the net energy gain of 10. 3. Applying HTSC coils, ignition can be attained with a plasma the same as major radius of the current ITER/RCO. To confirm the advantage of the proposed design option for ITER further, we should check if HTSC coils provide robust plasma stability — which is shown for one operating point in this article — for the whole plasma operating regime as well, in future.
References [1] T. Uchimoto, K. Miya. Application of high temperature superconductors to enhance nuclear fusion reactors, Fus. Technol. 36 (1999), in press. [2] T. Yamada, T. Uchimoto, K. Miya, Y. Nakamura. Application of HTSC coils for mitigation of VDE during a major disruption. Proceedings of 20th Symposium on
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[3] [4]
[5] [6]
[7]
T. Uchimoto et al. / Fusion Engineering and Design 51–52 (2000) 1077–1086 Fusion Technology, Marseille, France, 1998, pp. 599 – 602. Technical basis for the ITER final design report, cost review and safety analysis (FDR), IAEA, Vienna (1998) R. Albanese, F. Villone, The linearized CREATE-L plasma response model for the control of current, position and shae in tokamaks, Nucl. Fus. 38 (5) (1998) 723 – 738. J. Luxon, B. Brown. Magnetic analysis of non-circular cross-section tokamaks, Nucl. Fus. 22 (1982) 813 – 821. K. Yamafuji, Y. Mawatari, Electromagnetic properties of high Tc superconductors: relaxation of magnetization, Cryogenics 32 (1992) 569–577. Y. Yoshida, M. Uesaka, K. Miya, Magnetic field and force analysis of high Tc superconductor with flux flow and creep, IEEE Trans. Magn. 30 (1994) 3503 – 3506.
[8] L.J. Perkins et al. ITER parametric analysis and operational performance, ITER Documentation series No. 22, IAEA, Vienna, 1991. [9] W.A. Houlberg, S.E. Attenberger, L.M. Hively, Contour analysis of fusion reactor plasma performance, Nucl. Fus. 22 (1982) 935 – 945. [10] Uckan and ITER Physics Group: ITER physics design guidelines: 1989, ITER Documentation series No. 10, IAEA, Vienna, 1990. [11] F. Troyon, R. Gruber, H. Saurenmann, S. Semenzato, S. Succi, MHD-Limits to plasma confinement, Plasma Phys. Control. Fus. 26 (1984) 209 – 215. [12] S. Kobayashi, T. Kaneko, T. Kato, I. Fujikami, K. Sato, A novel scaling of magnetic field dependencies of critical currents for Ag-sheathed Bi-2223 superconducting tape, Physica C 258 (1996) 336 – 340.
. .
Design of compact tokamak reactor with HTSC plasma stabilizing coils T. Uchimoto, T. Yamada *, K. Miya Nuclear Engineering Research Laboratory, Graduate School of Engineering, The Uni6ersity of Tokyo, 22 -2 Shirakata-Shirane, Tokai, Naka Ibaraki, Japan 319 -1106
Abstract This paper presents a design of tokamak plasma with use of the high temperature superconducting coils as plasma stabilizer. Having regard to the current situation of the ITER project, two cases of designs were investigated: a smaller machine which has the same mission of the ITER/RCO (Q= 10), and an ignition machine which has the same major radius as ITER/RCO (R= 6.1 m). The same data base and formulas as ITER are here used and any innovative technology other than the HTSC stabilizing coils is not assumed. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Tokamak reactor; HTSC coils; Plasma stabilization
1. Introduction The authors have proposed the new method to stabilize plasmas position passively by applying high temperature superconducting (HTSC) coils [1]. HTSC coils are expected to stabilize the plasmas since HTSC coils prevent the change of magnetic flux induced by vertical motion of plasma. HTSC coils are robust against the thermal disturbance, so that they can be arranged in the vicinity of plasmas and effectively stabilize them. Advantages of the stabilizing system include the following compared with conventional ones: * Corresponding author. Tel.: +81-29-2878422; fax: + 8129-2878488. E-mail address: [email protected] (T. Yamada).
1. It requires no power supply unlike feedback control systems; and 2. Its stabilizing effect on plasmas is free of decay which is inevitable in that of eddy current induced in structures. Considerable plasma stabilization owing to HTSC coils provides the wider range of plasma elongation which is usually restricted to moderate values because of the vertical instability. In other words, HTSC coils enable compact reactors with highly elongated plasmas. We have displayed the feasibility of the HTSC coils for plasma stabilization and its potential to enable highly elongated plasma which leads to compact tokamak reactors [1]. In addition, we showed that HTSC coils mitigate VDEs followed by major disruptions as well as minor disruptions in the configuration of ITER/EDA [2].
0920-3796/00/$ - see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 0 - 3 7 9 6 ( 0 0 ) 0 0 4 3 9 - 7
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T. Uchimoto et al. / Fusion Engineering and Design 51–52 (2000) 1077–1086
The ITER project is now pursuing low-cost approaches reducing the technical objectives of the original design shown in the ITER Final Design Report [3]. Rather than trying to achieve ignition, the new objective is to demonstrate a net energy gain (Q) of 10 – 100. This is ITER Reduced Cost Option (ITER/RCO) and the cost will be cut down by half. In this study, having regard to the current situation of the ITER project, we introduce HTSC coils to ITER/RCO like tokamak reactors and show the enhancement of reactor potentiality owing to the resulting high elongation plasma. For the purpose, two cases are investigated: a smaller machine which has the same mission of the ITER/RCO (Q = 10) — we named HTSC tokamak-A, and an ignition machine
Fig. 1. Decision of design point in I –A space for HTSC tokamak-A (Q= 10).
which has the same major radius as ITER/RCO (R= 6.1 m) — HTSC tokamak-B. The reminder of the article is organized as follows. The plasma response model in the presence of HTSC coils — which is employed for the plasma stability analysis of HTSC tokamaks — is described in Section 2. Results and verifications of the design of HTSC tokamaks are contained in Section 3, and a summary and conclusions are given in Section 4.
2. Numerical formulation of plasma stability analysis
2.1. Linear plasma response model In this study, the behaviors of stabilized plasmas is exclusively of interest since stabilizing effect of HTSC coils on plasmas is investigated. Therefore, it is required to compute the plasma response accurately and efficiently in the case where change of plasma parameters is small. In the sense, the linear plasma response model, which assumes that plasma is supposed to be in equilibrium at each time due to the massless approximation, meets the requirement of this study. Here, we adopt the CREATE-L model [4] which is one of the linear plasma response model and embed HTSC shielding current analysis into it. The CREATE-L model is illustrated as follows. In axisymmetric condition, plasma equilibrium can be described by the Grad–Shafranov equation: Lc = Jpf + Ji = r where Lc = −
Fig. 2. Decision of operation point in POPCON plot for HTSC tokamak-A (Q =10).
dp f df + + Ji, dc mr dc
( 1 (c ( 1 (c − , (r m0r (r (z m0r (z
(1)
(2)
m0 is the magnetic permeability in the vacuum, c is the poloidal flux per radian, Ji is the external current density, Jpf is the toroidal component of the plasma current density, p is the pressure and f/r is the toroidal component of the flux density. Here, we assume the plasma current density profile to be a function of four parameters
T. Uchimoto et al. / Fusion Engineering and Design 51–52 (2000) 1077–1086
Fig. 3. FEM mesh used for the simulations with the code SCAPE and initial equilibrium result for HTSC tokamak-A.
(l, b0, am, an ) that can be associated with the physical quantities Ip, bp, li and q0: Jpf =l
n
b0r R +(1 −b0) 0 (1 − c( am )an, r R0
(3)
with c( =(c−cb )/(cb −ca ), where ca is the flux per radian at the plasma magnetic axis, cb the flux per radian at the plasma boundary [5]. Then, the perturbed plasma equilibrium can be expressed as Ldc =dJpf +dJi =
(Jpf (J (J dc + pf dq + pf ds (c (q (s + % dIk Jkf (r,z),
(4)
where q=(am, an, b0), s = (ca, cb, l), Ik is the current flowing in k-th coil and Jkf is the distribution function of the coil current. Ignoring nonaxisymmetirc contributions, Ohm’s law becomes
Fig. 4. Poloidal cross section of HTSC tokamak-A.
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T. Uchimoto et al. / Fusion Engineering and Design 51–52 (2000) 1077–1086
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Fig. 5. Time evolution of plasma parameters in the presence of HTSC plasma stabilizing coils (HTSC tokamak-A).
hJ=E +Eext = −
1 (c + Eext, r (t
(5)
where E is the electric field, h the electric resistivity and Eext the impressed field. Applying the same linearization procedure, the perturbed circuit equations for i-th coil is
2p
& Vcond
n
Jifdc: dV + % RikdIk =dui,
(6)
k=1
where Rik is the resistance matrix and ui is the applied voltage. Applying the Galerkin method to Eqs. 4 and 6,
we can obtain a linear matrix equation for perturbation of a set of currents in conductive shells. , L*dI: + RdI = du − L*ds; s
(7)
where L* is modified inductance matrix and dI is the first variation of a set of currents. Here, I and s can be regarded as the state variable of the system. Using the set of the state variable, expressed in the output equation which describes the plasma parameters of interest y such as positions of reference points at the separatrix and the plasma current centroid; y= AdI + du + Fds
(8)
T. Uchimoto et al. / Fusion Engineering and Design 51–52 (2000) 1077–1086
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Fig. 8. Decision of operation point in POPCON plot for HTSC tokamak-B (Q = ).
2.2. HTSC shielding current analysis Shielding currents in the HTSCs were estimated based on the flux flow and creep model [6] which properly describes the quickly changing shielding currents in superconductors like this case. The constitutive relations between the current density, J, and the electric filed, E, based on the model are as follows: Fig. 6. Time evolution of gap length between plasma surface and first wall (HTSC tokamak-A).
1. The creep region (05 J5Jc )
E(J)=2rc Jc sinh
Uo J U J exp − o , ku Jc ku Jc (9)
2. The flow region (Jc 5 J) E(J)=Ec + rf Jc (J/Jc − 1),
(10)
where Ec is the critical electric field, rc, the creep resistivity, u the temperature, Uo the pinning potential, k the Boltzmann constant, rf the flow resistivity and Jc the critical current density. Shielding currents in superconductors can be estimated by applying this constitutive relation to the conventional eddy current analysis [7]. Therefore, in order to introduce the shielding current analysis to the system equations, Eq. (7) is changed as follows: Fig. 7. Decision of design point in I –A space for HTSC tokamak-B (Q = ).
dF/dt + R(I)I= V,
(11)
T. Uchimoto et al. / Fusion Engineering and Design 51–52 (2000) 1077–1086
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where, Ri ·Ii =
&
E(Ji ) dl.
(12)
c
Because of this change, Eq. (11) become nonlinear and the Newton – Raphson method was applied to solve it.
3. Design of HTSC tokamaks
3.1. HTSC tokamak-A (Q = 10) Here, HTSC tokamak of Q =10 is pursued. Assumptions adopted here are to use the same data base and formula as ITER in principle and not to use any innovative technology other than HTSC stabilizing coils. In order to obtain an optimum design point of a HTSC tokamak, major design parameters are decided with 0-dimensional plasma analysis employing the I – A – Btf – k analysis methodology, which was introduced in the
ITER Conceptual Design Activity [8]. This methodology is to find the allowable design windows in the four-dimensional I–A–Btf –k space — where I is the plasma current, A is the aspect ratio, Btf is the peak field at the TF coils, and k is the elongation of plasma — considering the two constraints: (1) the minimum safety factor at the plasma edge, (2) radial build between plasma and inner leg of TF coil which includes inboard scrapeoff layer, first wall, inboard blanket and vacuum vessel. We determined apriori the basic parameters, k= 2.1, Btf = 12.5T, qc \ 3.0, tshield = 1.05 m. The elongation was chosen to the maximum value in the range that plasma positional stability is secured with HTSC coils after some iteration of plasma stability analysis was made. As 4-D surfaces cannot be visualized, we consider 2-D I–A plot of plasma current versus aspect ratio at constant values of Btf and k. The I–A plot of HTSC tokamak-A is shown in Fig. 1 with parameters of major radius. The hatched region in the figure
Fig. 9. FEM mesh used for the simulations with the code SCAPE and initial equilibrium result (HTSC tokamak-B).
T. Uchimoto et al. / Fusion Engineering and Design 51–52 (2000) 1077–1086
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Fig. 10. Time evolution of plasma parameters in the presence of HTSC plasma stabilizing coils (HTSC tokamak-B).
corresponds to the possible design window bounded by the radial build constraint. We determine Ip = 16(MA) for A = 3.3 and R = 5.0(m) as the design point. This is very small compared with the size of ITER/RCO. Plasma OPeration CONtour (POPCON) plot [9] provides an indication of plasma operation point which satisfies Q = 10. To generate the contours of the auxiliary power required for plasma energy balance, ITER-89P [10] confinement scaling with a multiplier, H =2.0 was employed. The resulting contours of auxiliary power in the HTSC tokamak are shown in Fig. 2 together with beta limit based on Troyon scaling [11]. If we take a
fusion power of 650 MW, contour line of 650 MW gives n= 1.7× 1020/m3, T= 10.5 keV as the cross point with the curve of Q= 10. The elongation of HTSC tokamak is set to higher value, so that it is highly required to investigate whether or not the plasma equilibrium condition is satisfied and its positional stability is secured. For this purpose, the plasma response code with HTSC current analysis, SCAPE (Sperconducting Current Analysis code with perturbed Plasma Equilibrium model) was used to obtain the plasma equilibrium and to conduct the stability analysis. The FEM mesh for estimation of the matrices in Eq. (7) and equilibrium results are
T. Uchimoto et al. / Fusion Engineering and Design 51–52 (2000) 1077–1086
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Fig. 11. Time evolution of gap length between plasma surface and first wall (HTSC tokamak-B). Table 1 Comparison of major parameters between HTSC tokamaks and ITER/RCO Parameter
HTSC (Q = 10)
Major radius 5.0 (m) Minor radius 1.51 (m) Elongation 2.1 Plasma 16 current (MA) Fusion 650 power (MW) Energy gain 10 Pulse length 500 (s)
HTSC (Q = )
RCO (Japan)
6.0
6.1
1.88
1.8
2.1 20
650 500
1.7 13–14
500–600
10–20 300
shown in Fig. 3. Primary parameters determined in the preceding section are Ip = 16(MA), R= 5.0(m), A= 3.3 and k= 2.1. Double null system of magnetic separatrix is adopted. Stability analyses were conducted for the equilibrium obtained above in order to confirm that HTSC coils suppress the plasma positional instability. The conductive structures were modeled as an array of discrete conducting loops; the first wall, the back plates and vacuum vessel were approximated as axisymmetric rings of 50, 50 and 60, respectively to construct an equivalent electrical circuit. Electrical resistances are 5.66, 0.50 and 2.32 (mV) for the first wall, the back plate and the vacuum vessel, respectively. These values are evaluated on the data base given in the ITER design. In order to improve the plasma positional stability, four pairs of HTSC coils are installed at the back plate as shown in Fig. 4, after the location and cross-section of the HTSC coils are optimized with respect to their stabilizing effect. They are placed in the form of axisymmetric toroidal coils. Two coils of each pair are symmetric in position with regard to the vertical center and connected never to induce current during plasma start-up. Silver-sheathed Bi-2223 superconducting tape was considered as the material of high Tc superconducting coils [12]. The physical parameters of Bi-2223 tape in the computation are assumed to be that of 100 m length tape which have been achieved in the current technology. Time evolution of plasma parameters are evaluated with SCAPE code and are shown in Fig. 5. As for the disturbance of the plasma, we considered instantaneous bp drop of 0.15 and li drop of 0.18 recovering in 5 sec. The maximum deviations of plasma centroid are − 0.21 and −0.05 m in radial and vertical directions, respectively, and the centroid moves back to the original point, followed by the recovery of bp and li. The motion of the plasma separatrix in the presence of HTSC coils is shown in Fig. 6. The plasma at reference point P4 moves toward the first wall by 7 cm, but do not touch the wall because the initial gap length is 20 cm. These results indicate the plasma is stabilized by HTSC coils sufficiently, which supports for the enough plasma stabilization owing to HTSC coils and feasibility of the design of HTSC tokamak-A.
T. Uchimoto et al. / Fusion Engineering and Design 51–52 (2000) 1077–1086
3.2. HTSC tokamak-B (Q = ) We designed HTSC tokamak-B in the same manner as HTSC tokamak-A, determining apriori the basic parameters, k =2.1, Btf (T), qc \ 3.0, tshield =1.05 m. The elongation was chosen to the maximum value in the range that plasma positional stability is secured with HTSC coils. The Ip –A space of HTSC tokamak (k =2.1, Q= ) is shown in Fig. 7. The hatched region in the figure corresponds to the possible design window. We determine Ip =20(MA) for A = 3.2 and R= 6.0(m) as the design point. Plasma parameters for HTSC tokamak-B was determined so as to achieve Q = , considering a POPCON diagram shown in Fig. 8 together with beta limit. If we take a fusion power of 650 MW, contour line of 650 MW gives n =1.0× 1020/ m3, T =14 keV as the cross point with the curve of Paux =0. The elongation of HTSC tokamak-B is also set to higher value, so that plasma stability analyses were conducted to confirm the stabilization owing to HTSC coils. Fig. 9 displays the FEM mesh and equilibrium results for HTSC tokamak-B. Primary parameters determined in the above are Ip =20(MA), R = 6.0(m), A = 3.2 and k=2.1 The conductive structures were modeled as an array of discrete conducting loops; the first wall, the back plates and vacuum vessel were approximated as axisymmetric rings of 50, 50 and 60, respectively. Electrical resistances are 5.66, 0.50 and 2.32 (mV) for the first wall, the back plate and the vacuum vessel, respectively. Four pairs of HTSC coils are installed at the back plate in the same manner as HTSC tokamak-A, after their location and crosssection are optimized. Time evolution of plasma parameters are evaluated with SCAPE code and are shown in Fig. 10. The disturbance of the plasma is instantaneous bp drop of 0.2 and li drop of 0.08 recovering in 5 sec. It can be recognized that all the plasma parameters recover from the deviations after the disturbances of bp and li vanish. The motion of the plasma separatrix in the presence of HTSC coils is shown in Fig. 11. The plasma at reference point P1 moves toward the first wall by 7 cm, but do not touch the wall because the initial gap length is 20 cm. In the
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configuration of HTSC tokamak-B, the considerable plasma stabilization due to HTSC coils is observed as is the case with HTSC tokamak-A, which imply that the ignition machine is feasible in the same size as ITER/RCO applying HTSC coils.
4. Concluding remarks The design of HTSC tokamak was carried out under the same condition as the ITER/RCO activity, except that HTSC coils are applied. The major parameters of HTSC tokamak A and B are listed in Table 1 together with those of ITER/ RCO. Conclusions of these designs are summarized in the following. 1. Plasma stabilization owing to HTSC coils allows the high elongation of Section 2.1. This stabilization method offers an advantage that it does not require plasma sensors and power supply compared with the conventional active control system. We confirmed stable behavior of plasmas in the presence of HTSC coil via numerical analysis in the both designs of HTSC tokamak A and B. 2. Applying HTSC coils, major radius of plasma can be reduced to 5.0 m, keeping the same mission of ITER/RCO, the net energy gain of 10. 3. Applying HTSC coils, ignition can be attained with a plasma the same as major radius of the current ITER/RCO. To confirm the advantage of the proposed design option for ITER further, we should check if HTSC coils provide robust plasma stability — which is shown for one operating point in this article — for the whole plasma operating regime as well, in future.
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[8] L.J. Perkins et al. ITER parametric analysis and operational performance, ITER Documentation series No. 22, IAEA, Vienna, 1991. [9] W.A. Houlberg, S.E. Attenberger, L.M. Hively, Contour analysis of fusion reactor plasma performance, Nucl. Fus. 22 (1982) 935 – 945. [10] Uckan and ITER Physics Group: ITER physics design guidelines: 1989, ITER Documentation series No. 10, IAEA, Vienna, 1990. [11] F. Troyon, R. Gruber, H. Saurenmann, S. Semenzato, S. Succi, MHD-Limits to plasma confinement, Plasma Phys. Control. Fus. 26 (1984) 209 – 215. [12] S. Kobayashi, T. Kaneko, T. Kato, I. Fujikami, K. Sato, A novel scaling of magnetic field dependencies of critical currents for Ag-sheathed Bi-2223 superconducting tape, Physica C 258 (1996) 336 – 340.
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