Toroidal Multipolar Expansion for Fast L-Mode Plasma Boundary Reconstruction in EAST

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1 Plasma Science and Technology, Vol.3, No.3, Jun. 2 Toroidal Multipolar Expansion for Fast L-Mode Plasma Boundary Reconstruction in EAST GUO Yong ), XIAO Bingjia ), LUO Zhengping ) Institute of Plasma Physics, Chinese Academy of Science, Hefei 233, China Abstract A new method for plasma boundary reconstruction, based on the toroidal multipolar expansion TME) scheme, is applied successfully in EAST. TME applies a limited number of toroidal multipolar moments based on toroidal coordinates to treat a two-dimensional problem of axisymmetric plasma equilibrium. The plasma boundary reconstructed by TME is consistent with the results by using EFIT. The method is sufficiently reliable and fast for real time shape control. Keywords: toroidal multipolar expansion, fast plasma boundary reconstruction, experiment advanced superconducting tokamak PACS: s, Fa Introduction An efficient and safe operation of tokamak discharge depends on fast and accurate identification of plasma position and shape. Unfortunately, the plasma shape could not be measured directly and is only evaluated by diagnostic data, such as the poloidal flux and magnetic field. Several approaches have been considered ]. Most methods consist in finding the plasma current distribution with the corresponding magnetic field in a good agreement with experimental data. The current distribution is modeled with a set of current filaments 3,4,6], smooth equilibrium flux function 8,9], or multipolar moment. Toroidal multipolar expansion TME), which adopts a multipolar moment model, has been used in the analysis of the MHD equilibrium for JET ], FTU 2] and TUMAN-3 3]. The formalism of multipolar moments associated with toroidal geometry leads to a simple scheme for plasma boundary reconstruction. The purpose of the present work is to make a detailed study of TME to further develop a code for fast plasma boundary reconstruction, which is not sensitive to measurement errors in determining the configuration from arbitrary shifts of the plasma position within the confinement region. In this paper, the TME technique is introduced in section 2. The benchmark by EFIT is shown in section 3. Section 4 gives a good sample for boundary reconstruction in EAST experiments, which is consistent with the real-time reconstructed shape in an actual shot. Finally, conclusions are drawn in section 5. 2 TME technique Unlike most codes defined in Cartesian co-ordinates or flux co-ordinates, TME describes plasma in the toroidal co-ordinates η,, φ), which is defined by the following relations with the Cartesian co-ordinates x, y, z), x a sinh η cosh η cos cos φ y a sinh η cosh η cos sin φ a sin z cosh η cos η < + < φ < ) where a a positive constant) is the pole of the system. As shown in Fig., the co-ordinate η defines a non-concentric tori of circular cross-section with major radii a/coth η and minor radii a/sinh η, while the coordinate defines the spheres passing through the pole R a. Fig. Poloidal projection of fully toroidal co-ordinates with the pole of the system a.78 m, η const lines) and const dashes), supported by the Major State Basic Research Development Program of China 973 program, No. 29GB3), National Natural Science Foundation of China No. 8359), and the Chinese Academy of Sciences with grant ID of KJCX3.SYW.N4

2 GUO Yong et al.: Toroidal Multipolar Expansion for Fast L-Mode Plasma Boundary Reconstruction in EAST We start from the Ampere s law given by B µ j. 2) In the case of axial symmetry / φ ), the magnetic field is independent of the toroidal angle φ and could be expressed Appendix A) as, B η cosh η cos )2 a 2 sinh η cosh η cos )2 B a 2 sinh η ) ψ ), 3) ψ η and the toroidal component of Eq. 2) is expressed as cosh η cos) 2 cosh η cos ψ ) a 3 η sinh η η + cosh η cos sinh η ψ ) ] µ j φ. 4) In the region where j φ, Eq. 2) could be expressed as cosh η cos η sinh η ) ψ + cosh η cos η sinh η ) ψ. 5) This is the so-called vacuum form of the Grad- Shafranov equation with a solution of the form ψ C + cosh η cos m a i n cosn) n +b i nsinn) ] sinh η P n /2 + cosh η cos k a e ncosn) + b e nsinn) ] sinhη Q n /2, n m and k ); 6) where Pn /2, Q n /2 are the Legendre functions of half integer 5]. a i n and b i n are symmetric and asymmetric internal multipolar moments of order n, respectively. a e n and b e n are symmetric and asymmetric external multipolar moments of order n, respectively. The coefficient vector C C, a i n, a e n, b i n, b e n] is constant in the region η η min, η max ], where there is no toroidal current Appendix B). In a least square sense, vector C is determined by fitting with the measured magnetic signals in this region through D C M. 7) Then, the poloidal flux distribution, described by Eq. 6), can be mapped. In Eq. 7), D is the response matrix determined only by the location of sensors, which is constant when the location of sensors and expansion orders are set, which could be pre-calculated before real-time boundary reconstruction. The vector M is given by the measured data at the corresponding sensors and is updated at every time-slice Appendix C). In most tokamaks, the magnetic probes are not placed along the direction of B η and B. The transform of variables should be carried out. For a magnetic probe placed at the point η, ) whose orientation is ϕ related to the horizontal, its measured value could be presented as below, BR B Z ] B mp B R cos ϕ + B Z sin ϕ, 8) cosh η cos sinh η sin a cosh η cos ) 2 a cosh η cos ) 2 sinh η sin cosh η cos a cosh η cos ) 2 a cosh η cos ) 2 ] Bη. 9) B After calculating the flux distribution, the plasma boundary can be searched by the definition of the last closed flux surface LCFS) contained entirely inside the vacuum vessel, which either touches the limiter or forms X-points) where the poloidal magnetic field is zero equivalently, ψ ). In EAST, the poloidal flux is monotonically decreasing from the center of the plasma towards the edge in this region. In TME, the plasma boundary is found in the usual way by finding the maximal flux value between various limiter points around the vessel as well as at separatrices if they exist inside the vessel. Usually, some active coils are placed in the vacuum vessel for special use. For EAST, the inside coils, IC and IC2, are connected in anti-series for the fast control of vertical instability Fig. ). These two coils strongly determine the flux distribution nearby. TME fits the field measurement after having removed the contribution from these coils, as in Eq. ). Finally, the analytic expression of the field produced by the coils is added back to the flux reconstruction, as shown in Eq. ) below, ψ ψ measurment ψ IC ψ IC2 B B measurment B IC B IC2, ) ψη, ) ψ fit + GR IC, Z IC, η, ) I IC +GR IC2, Z IC2, η, ) I IC2. ) 333

3 3 Benchmark calculation for L-mode plasma The EFIT code, with a grid size of.9 cm.86 cm for EAST, can be used to design the plasma configuration of the limiter, single null or double null shapes. The plasma current distribution 4] is defined by R J t J β c + β c ) R ] c x n ) m. 2) R c R Here, x ψ ψ M )/ψ B ψ M ), ψ M is the poloidal flux at the magnetic axis and ψ B is the boundary poloidal flux. β c.95, R c.75, m 2, while J and n are adjusted during the equilibrium reconstruction. In EAST, the measured magnetic data come from 35 flux loops and 38 magnetic probes. The locations of these sensors are shown in Fig. 2. TME uses theoretically calculated values of all magnetic data generated by EFIT as input to reconstruct the corresponding plasma boundary. Evaluation is made by comparing the EFIT designed boundary with the TME reconstructed boundary. In the calculation, the pole of the system a is chosen at R.78 m, close to the center of the vessel in EAST. The order of multipolar expansion is determined by the convergence property of the expansion base, fitting measurements number and position. By comparing the results of benchmark calculation with different Plasma Science and Technology, Vol.3, No.3, Jun. 2 orders, the order of m 3 for internal multipolar moment and k 7 for external multipolar moment are chosen in the following calculations. 3. Limiter configuration For a limiter configuration, the plasma is designed to intercept the inner limiter point at mid-plane with a boundary flux of.474 Wb/rad. In Fig. 3, the TMEreconstructed boundary solid line) is superimposed on the designed circular boundary dot line). The relative flux error at the EFIT designed boundary points is defined by δ ψ B ψ EFIT B ψ EFIT a ψ EFIT B, 3) with ψ B the TME-calculated flux at reference boundary points, ψ EFIT a and ψ EFIT B the reference fluxes at the axis and boundary, respectively. This error, shown in Fig. 4a), is below.25%. The distances between reference boundary points and corresponding reconstructed boundary points are smaller than mm, as shown in Fig. 4b). Table lists a comparison between EFIT designed values and TME calculated data. The discrepancies of the inner point and outer point at the mid-plane are less than mm, much less than the grid size in the EFIT calculation. The boundary flux calculated by TME is almost the same as the designed value. Fig.2 Location of a) flux loops circle) and b) magnetic probes square) in EAST Table. Comparison between TME-reconstructed boundary and reference boundary for a limiter configuration EFIT TME Difference c Psi bdry a Wb/rad) e-4 Inner point b m).358,.).358,.). Outer point b m) ,.) ,.) 7.e-4 a plasma boundary flux b location of inner points and outer points at the mid-plane c distance between point locations as calculated by EFIT and TME 334

4 GUO Yong et al.: Toroidal Multipolar Expansion for Fast L-Mode Plasma Boundary Reconstruction in EAST Fig.3 Reference boundary of limiter configuration designed by using EFIT, which intercepts the inside limiter point. The locations of plasma boundary points are denoted by the dots. The solid line denotes the TME-reconstructed boundary Fig.5 Reference boundary of single null configuration designed by using EFIT, whose X-point is located at.6635 m,.7692 m). The locations of plasma boundary points are denoted by the dots. The solid line denotes the TME-reconstructed boundary a) Relative flux error %) at the reference boundary points defined by Eq. 3), b) Distance cm) between the points of the reference boundary and the corresponding TMEreconstructed boundary Fig.4 Boundary points counted clockwise from the inner point at the mid-plane 3.2 Single null configuration The single null configuration, designed by EFIT, is of one X-point at R.6635 m and Z.7692 m with a flux value of.489 Wb/rad. The plasma boundary identified by TME is shown in Fig. 5. For the boundary points, the relative errors of the flux, shown in Fig. 6a) and defined by Eq. 3), are below 2.7%. The longest a) Relative flux error %) at the reference boundary points defined by Eq. 3), b) Distance cm) between the points of the reference boundary and the corresponding TMEreconstructed boundary Fig.6 Boundary points counted clockwise from the inner point at the mid-plane. distance between the designed boundary and reconstructed boundary, as shown in Fig. 6b), is.65 cm, while the mean distance is.24 cm. In Table 2, a comparison of important shape parameters is listed. The TME identified locations of X-point and inner point are close to the EFIT-designed values. The discrepancies are about the order of millimeters. 335

5 Table 2. Plasma Science and Technology, Vol.3, No.3, Jun. 2 Comparison between TME-reconstructed boundary and reference boundary for a single null configuration EFIT TME Difference Psi bdry a Wb/rad) e-4 X lower b m).6635,.7692).66,.773).4 Inner point c m).4449,.).4484,.).35 Outer point c m) 2.357,.) 2.355,.).55 a plasma boundary flux b location of X-point for single null configuration c locations of inner points and outer points at the mid-plane The discrepancy of the outer point at the mid-plane is slightly larger, but within the grid size set in EFIT. 3.3 Double null configuration The double null configuration, designed by EFIT, is with two null points at R.6239 m and Z +/.846 m with a flux value of.489 Wb/rad. The plasma boundary identified by TME is shown in Fig. 7. The relative flux errors for the boundary points are below 2%, shown in Fig. 8a). The mean distance between the reference boundary and reconstructed boundary, shown in Fig. 8b), is.46 cm. A comparison between the reference values and TME-calculated data is listed in Table 3. The locations of X-points and boundary points at mid-plane identified by TME are close to the designed values, within a discrepancy below 3 mm. Fig.7 Reference boundary of double null configuration designed by using EFIT, whose X-points are located at.6239 m,.846 m) and.6239 m,.846 m). The locations of the plasma boundary points are denoted by the dots. The solid line denotes the reconstructed boundary by TME a) Relative flux error %) at the reference boundary points defined by Eq. 3), b) Distance cm) between the points of the reference boundary and the corresponding TMEreconstructed boundary Fig.8 Boundary points counted clockwise from the inner point at the mid-plane Table 3. Comparison between TME-reconstructed boundary and reference boundary for a double null configuration EFIT TME Difference Psi bdry a Wb/rad) e-4 Psi upper b Wb/rad) e-4 X upper c m).6239,.846) ).24 Psi lower b Wb/rad) e-4 X lower c m).6239,.846).625,.84).25 Inner point d m).4342,.).4333,.) 9.e-4 Outer point d m) ,.) ,.).34 a plasma boundary flux b flux at upper and lower X-point for double null configuration c locations of upper and lower X-point for double null configuration d locations of inner points and outer points at the mid-plane 336

6 GUO Yong et al.: Toroidal Multipolar Expansion for Fast L-Mode Plasma Boundary Reconstruction in EAST 3.4 Results From the analysis, a good agreement is reached for all three reference configurations in the L-mode discharge. The location error of the plasma boundary is within.648 cm, while the grid size in EFIT is.9 cm.86 cm. The discrepancy is lower than the scale of inaccuracy of the numerical scheme underlying EFIT. The benchmark calculation proves that TME is reliable for plasma boundary reconstruction. 4 TME plasma boundary reconstruction for the L-mode discharge in EAST less than.5 cm. δ ψ B ψ rtefit B ψ rtefit a ψ rtefit B. 4) The related parameters for the double null configuration calculated by RT-EFIT and TME are listed in Table 4. The discrepancy of two boundaries is 2.59 cm at the inner point, and.24 cm at the outer point. TME basically identifies the X-point position with an accuracy of.5 cm. In the experiment, the magnetically measured data contain irremovable random noise. Sometimes, several magnetic data should be removed due to broken wires or a bad integrator. These factors would decline the accuracy of the reconstructed boundary. As a real-time equilibrium code, RT-EFIT is used to check the robustness of TME under real experimental conditions. As is shown in Fig. 9, for shot 4289 the plasma current starts to reach its flat-top at.4 s, is with a small oscillation from.4 s to 3.5 s, keeps flat to s, and then ramps down. The time-slice at 4.5 s is chosen for analysis. From the RT-EFIT results, at this time-slice, the plasma is in a double nulls configuration. The boundary flux is of.83 Wb/rad. Two X- points are located at R m, Z m and R m, Z m. The inner point and outer point at mid-plane are located at R.46 m, Z. and R m, Z.. Fig. Plasma boundary for shot 4289 at 4.5 s in EAST, when plasma reaches flattop, and is with two X-points. The plasma boundary identified by RT-EFIT is denoted by the dots, while the boundary reconstructed by TME is denoted by the solid line Fig.9 Waveform of the plasma current for shot Plasma current started to reach its flattop at.4 s, keeps smooth from 3.5 s to s, and then ramps down In the TME calculation, some useless signals have been removed as was done in the RT-EFIT calculation. In Fig., the TME boundary solid line) is superimposed on the RT-EFIT boundary dot) for the timeslice of 4.5 s. For most boundary segments, the relative flux error, shown in Fig. a) and defined by Eq. 4), is below 3%, and the discrepancy in boundaries reconstructed by the two methods, as shown in Fig. b), is a) Relative flux error %) at boundary points identified by RT-EFIT defined by Eq. 4), b) Distance cm) between the points of boundary designed by RT-EFIT and the TMEreconstructed boundary Fig. Boundary points counted clockwise from the inner point at the mid-plane 337

7 Table 4. Plasma Science and Technology, Vol.3, No.3, Jun. 2 Comparison between TME-reconstructed boundary and by RT-EFIT for shot 4289, at 4.5 s rt-efit TME Difference Psi bdry a Wb/rad) e-4 X upper b m).62579,.86699).6289,.8539).35 X lower b m).62478,.85689).6257,.8489).8 Inner point c m).46,.).3857,.).259 Outer point c m) 2.346,.) 2.322,.).24 a plasma boundary flux b locations of upper and lower X-point for double null configuration c locations of inner points and outer points at the mid-plane Fig.2 Temporal evolution of a) the major radius of the inner points and outer points at the mid-plane identified by RT- EFIT solid line) and TME dot), b) The boundary flux calculated by RT-EFIT solid line) and TME dot), c) Discrepancy in the location of upper X-point between RT-EFIT and TME when plasma is with an upper X-point, d) Discrepancy in the location of lower X-point between RT-EFIT and TEM when plasma is with a lower X-point TME can also follow the evolution of plasma shape. Fig. 2 shows the temporal evolution of a) the major radius of the inner point and outer point at mid-plane, b) the boundary flux, c) the discrepancy in the location of upper X-points identified by the two methods, and d) the discrepancy in the location of the lower X-points identified by the two methods. The discrepancies of the X-points are within.8 cm. The largest discrepancy of two boundaries is below 3. cm. Considering the grid size of 4.24 cm 7.27 cm in RT-EFIT, the discrepancies are close to the scale of inaccuracy of the numerical scheme underlying RT-EFIT. 5 Conclusions The new method using TME solves the Ampere s circuital equation by fitting measured results in the vacuum region. As shown in the benchmark calculation, the boundaries reconstructed by TME are consistent with the reference boundaries from the fixed boundary calculation for related configurations by using EFIT. The error in the boundary location is less than cm. The accuracy of the L-mode plasma boundary identification is proved. For the experimental data, the TMEreconstructed boundary matches the RT-EFIT boundary within a reasonable error of below 3 cm, and the discrepany in localizing X-points is below 2 cm. TME works well in the EAST experiment. On a Linux server with Intel R) Xeon TM) CPU 3.2 GHz, TME completes one plasma boundary reconstruction in 3 µs. The computational speed is fast enough for routine realtime shape control. TME provides a possible approach for EAST plasma boundary control. 338

8 GUO Yong et al.: Toroidal Multipolar Expansion for Fast L-Mode Plasma Boundary Reconstruction in EAST Appendix A B A where A is the magnetic vector potential. Poloidal flux: ψ B ds A dl R A φ, And R x 2 + y 2 a sinh η cosh η cos. For toroidal co-ordinates η,, φ), the Lame coefficients: x ) 2 ) 2 ) 2 y z h η + + η η η x ) 2 ) 2 ) 2 y z h + + ) 2 ) 2 ) 2 x y z h φ + + φ φ φ a cosh η cos, a cosh η cos, a sinh η cosh η cos R. Due to the axial symmetry, φ h η e η h e h φ e φ e coshη cos η e sinhη e φ )2 B A h η h h φ η φ a h η A η h A h φ A 2, sinhη η φ φ h η A η h A h φ A φ ) cosh η cos )2 ψ B η a 2, sinh η ) cosh η cos )2 ψ B a 2. sinh η η Appendix B The expression ] of the n-th order internal multipolar moments are evaluated as the integral of the current density J φ η, ) that flows inside the torus η. a i nη) µ R 3 b i n η) µ R 3 2 δ n n 2 /4 2 δ n n 2 /4 d d + η + η dη J φ η, ) sinh η Q m /2 cosh η ) cos n ) cosh η cos ) 5/2, dη J φ η, ) sinh η Q m /2 cosh η ) sin n ) cosh η cos ) 5/2. The external multipolar moments of n-th order ] are expressed as the integral of the current density J φ η, ) that flows outside the torus η. a e n η) µ R 3 b e n η) µ R 3 2 δ n n 2 /4 2 δ n n 2 /4 d d η η dη J φ η, ) sinh η P m /2 cosh η ) cos n ) cosh η cos ) 5/2, dη J φ η, ) sinh η P m /2 cosh η ) sin n ) cosh η cos ) 5/2. In the region η η min, η max ] located at vacuum region, there is no toroidal current. a i n µ R 3 µ R 3 2 δ n n 2 /4 2 δ n n 2 /4 a i n η max ) const d d + η + dη J φ η, ) sinh η Q m /2 cosh η ) cos n ) cosh η cos ) 5/2, η max dη J φ η, ) sinh η Q m /2 cosh η ) cos n ) cosh η cos ) 5/2. 339

9 a e n µ R 3 µ R 3 2 δ n n 2 /4 2 δ n n 2 /4 a e n η min ) const d d η η min Plasma Science and Technology, Vol.3, No.3, Jun. 2 dη J φ η, ) sinh η P m /2 cosh η ) cos n ) cosh η cos ) 5/2, dη J φ η, ) sinh η P m /2 cosh η ) cos n ) cosh η cos ) 5/2. In a similar way, b i n, b e n are constant in this region. Theorerically, η min is the minimum η line that does not contain PF coils, and η max is the maximum η line that is just outside the plasma region. Appendix C Coefficient vector: C C, a i,..., a i n,..., a e n, b i,..., b i n, b e,... b e n] T For the kth flux loop located at η, ), the element D k,i of matrix D corresponding to the ith element of C, is D k,, D k,2 sinh η P /2, cosh η cos D k,n+3 sinh η Q /2. cosh η cos For j n D k,j+2 sinh η P j /2 cos j), cosh η cos D k,n+j+3 sinh η Q j /2 cos j), cosh η cos D k,2n+j+3 sinh η P j /2 sin j), cosh η cos D k,3n+j+3 sinh η Q j /2 sin j), cosh η cos and M k, is the measured value of this flux loop, deducting the influence of IC and IC2. For the kth magnetic probe, located at η, ), according to Eqs. 4) and 5), the m-th element of matrix D is And B k,m η For j n 34 cosh η cos sinh η sin D k,m cos ϕ sin ϕ ] a cosh η cos ) 2 a cosh η cos ) 2 sinh η sin cosh η cos a cosh η cos ) 2 a cosh η cos ) 2 and B k,m Bη k,, are the m-th elements of B η and B, respectively. Then, Bη k,2 P /2 2 a 2 sinh η sin cosh η cos, Bη k,n+3 Q /2 2 a 2 sinh η sin cosh η cos, η P j /2 a 2 sinh η B k,j+2 Bη k,n+j+3 Q j /2 a 2 sinh η B k,m η B k,m. j sin j cosh η cos ) 3/2 2 sin cos j cosh ] η cos, j sin j cosh η cos ) 3/2 2 sin cos j cosh ] η cos,

10 GUO Yong et al.: Toroidal Multipolar Expansion for Fast L-Mode Plasma Boundary Reconstruction in EAST Bη k,2n+j+3 P j /2 a 2 j cos j cosh η cos ) 3/2 2 sin sin j cosh ] η cos, Bη k,3n+j+3 Q j /2 a 2 j cos j cosh η cos ) 3/2 2 sin sin j cosh ] η cos, and For j n B k,j+2 B k,n+j+3 B k,2n+j+3 B k,3n+j+3 B k,, B k,2 a 2 B k,n+3 a 2 cos j a 2 cos j a 2 sin j a 2 4 cosh η cos )3/2 P /2 + ] cosh η cos sinh η P 2 /2, 4 cosh η cos )3/2 Q /2 + ] cosh η cos sinh η Q 2 /2. cosh η cos ) 3/2 j 2 ) Pj /2 4 + ] cosh η cos sinh η P 2 j /2, cosh η cos ) 3/2 j 2 ) Q j /2 4 + ] cosh η cos sinh η Q 2 j /2, cosh η cos ) 3/2 j 2 ) Pj /2 4 + ] cosh η cos sinh η P 2 j /2, sin j cosh a 2 η cos ) 3/2 j 2 ) Q j /2 4 + ] cosh η cos sinh η Q 2 j /2, and M k, is the measured value of this magnetic probe that deducts the influence of IC and IC2. References Kurihara K. 2, Fus. Engineering and Design, 5 52: 49 2 Braams B J, Jilge W, Lackner K. 986, Nucl. Fusion, 26: Lister J B, Hofmann F, Morret M, et al. 997, Fusion Technol., 32: 32 4 Tsaun S, Jhang Hogun. 27, Fus. Engineering and Design, 82: 63 5 Windsor C G, Todd T N, Trotman D L, et al. 997, Fus. Technol., 32: 46 6 Amoskov V M, Belyakov V A, Bender S E, et al. 23, Plasma Physics Reports, 29: Lao L L, St John H, Stambaugh R D, et al. 985, Nucl. Fusion, 25: 6 8 Ferron J R, Walker M L, Lao L L, et al. 998, Nucl. Fusion, 38: 55 9 Gates D A, Ferron J R, Bell M, et al. 26, Nucl. Fusion, 46: 7 O Brien D P, Ellis J J, Lingertat J. 993, Nucl. Fusion, 33: 467 Alladio F, Crisanti F. 986, Nucl. Fusion, 26: 43 2 Sadeghi Y, Boncagni L, Calabro G, et al. 29, Performing real-time reconstruction of the magnetic flux of the FTU tokamak in an RTAI virtual machine using multi-polar current moments. Presented at the 4th international scientific conference on Physics and Control, September -4, 29, University of Catania, Sicily, Italy 3 Deshko G N, Kilovataya T G, Kuznetsov Yu K, et al. 983, Nucl. Fusion, 23: 39 4 Lao L L, John H St, Stambaugh R D. 985, Nucl. Fusion, 25, 42 5 Segura J, Gil A. 2, Comput. Phys. Commun., 24: 4 Manuscript received 3 March 2) Manuscript accepted 6 December 2) address of GUO Yong: yguo@ipp.ac.cn 34