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Feedback control of resistive wall modes by saddle coils in RFX-mod
Fusion Engineering and Design 82 (2007) 1064–1072
Feedback control of resistive wall modes by saddle coils in RFX-mod T. Bolzonella, M. Cavinato, E. Gaio, L. Grando, A. Luchetta ∗ , G. Manduchi, G. Marchiori, L. Marrelli, R. Paccagnella, A. Soppelsa, P. Zanca Consorzio RFX, Associazione EURATOM-ENEA sulla Fusione corso Stati Uniti 4, 35127 Padova, Italy Received 1 August 2006; received in revised form 4 May 2007; accepted 6 May 2007 Available online 29 June 2007
Abstract Active control of resistive wall modes (RWMs) is a key issue for high-beta operation of future tokamaks, including ITER. RFX has been recently modified to specifically address this issue by means of a new saddle coil system, which is the most powerful and flexible among all fusion devices, including 192 coils covering the whole torus. Unstable RWMs have been experimentally observed during the recent machine operation, showing poloidal and toroidal components m = 1, 2 < |n| < 7 and growth times of some tens of ms, consistent with theoretical predictions. RFX-mod addresses the control of RWMs by acting on the magnetic field configuration at the plasma boundary. The system aims not only at RWM stabilization, but also at interaction with tearing modes. By means of the system, successful experimental campaigns have been executed demonstrating the feasibility of RWM active control. The paper discusses on the system engineering and technological features and presents the results achieved. © 2007 Elsevier B.V. All rights reserved. Keywords: Resistive wall modes; MHD mode control; Feedback control; Digital control system
1. Introduction Magneto-hydrodynamic (MHD) instabilities arise in toroidal plasmas due to non-zero current and pressure profile gradients. The growth of external kink modes, referred to as resistive wall modes (RWM), limits the achievable  in tokamak advanced scenarios and ∗ Corresponding author. Tel.: +39 049 8295043; fax: +39 049 8700718. E-mail address: [email protected] (A. Luchetta).
0920-3796/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.fusengdes.2007.05.020
their stabilization is, thus, a key issue for the operation of ITER and future fusion reactors. These modes, unstable in the presence of a resistive wall, are stable when an infinitely conducting wall is placed close to the plasma surface. Since a passive ideal boundary, acting as a perfect conductor during the plasma pulse of a fusion device, is not feasible, the development of successful technology for active feedback control of MHD instabilities is mandatory. Efforts to address the problem are thus currently accomplished in fusion community [1–4].
T. Bolzonella et al. / Fusion Engineering and Design 82 (2007) 1064–1072
Plasma instabilities are usually described in terms of two-dimensional spatial Fourier decomposition along the poloidal and toroidal directions and the relevant harmonic components are indicated with the mode numbers m and n, respectively. RWMs have been predicted also for reverse field pinch (RFP) machines and, according to theory, growth of RWMs showing poloidal and toroidal components m = 1, 2 < |n| < 7 and growth times of some tens of ms have been observed experimentally in RFX-mod during the recent campaigns [5]. The RFP is a low q magnetic profile machine – with q(0) ≈ 0.5a/R decreasing monotonically to q(a) ≤ 0 – whose spectrum of unstable modes is dominated by those with m = 1 and n in a large range of values. In RFX (a = 0.459m, R = 2.0m) the modes with n ≤ −7, internally resonant tearing modes (TM), are responsible for the dynamo sustainment. The modes with −7 < n < 0 and n > 0 are, instead, non-resonant (internally or externally) RWMs. As a consequence, RFX-mod shows a very rich spectrum of MHD modes and is, thus, an excellent environment where to study the MHD mode phenomenology and the techniques for mode control. Fig. 1 shows by striped bars the amplitudes of m = 1 modes measured at the plasma edge in reference RFXmod pulses executed without mode control. RFX-mod addresses the control of MHD modes by acting on the magnetic configuration at the plasma boundary [6]. This corresponds to the concept of a discrete active shell as proposed for the first time in late 1980s [7]. To implement the concept, two major modifications were necessary on the RFX original load assembly: the installation of a dedicated coil system,
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to perform active control on non-axisymmetric radial magnetic field at the boundary, and the replacement of the previous shell, having long time constant for radial field penetration (∼450 ms), with a new one with lower time constant (∼60 ms), thus allowing the diffusion into the plasma region of external magnetic field on the timescale of the RFX-mod pulse (nominally 250 ms) [8]. In this paper, after a description of the components of the MHD mode control system, we present the successful control scheme implemented on the experiment along with the results achieved demonstrating the successful control of RWMs. The outstanding features of the system are highlighted in details with focus on the regimes that are now reachable due to the power of the control algorithms and the high robustness and flexibility of the system.
2. System overview The radial field coil system consists of a bidimensional mesh, including 192 equispaced saddle coils, mounted to cover the whole outer surface of the shell, as installation inside the shell was impossible because of mechanical constraints. The mesh can be thought of as made up by 4 toroidal and 48 poloidal arrays. With reference to the toroidal reference system used in RFX, the centre of the generic saddle coil is identified with the coordinates (r = 0.581 m, ϑi = iπ/2 rad, ϕj = jπ/24 rad) where 0 ≤ i ≤ 3 and 0 ≤ j ≤ 47. To the purpose of both measure and control, a pair of sensors is associated with each
Fig. 1. Spectra of radial field modes measured by the radial field probes. Striped and solid bars are computed as averages on measured mode amplitudes at the current flat-top of two sets of pulses executed without and with mode control, respectively.
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Table 1 Main parameters of saddle coils and power amplifiers
Voltage (V) Peak current (A) Turns Coil time constant L/R (s) Magneto motive force (kAt) Pulse-width modulation frequency (kHz)
Saddle coil
Power amplifier
650 400 60 5e−3/1 24 –
650 400 – – 10
radial field coil, including one pick-up coil, detecting both toroidal and poloidal components, and one saddle probe sensing the average radial field. The pick-up coils are located on the inner surface of the shell, aligned with the centre of the corresponding correction coil, whereas the saddle probes are mounted on the outer surface of the vacuum vessel on the projection of the correction coil onto the vacuum vessel itself. Each saddle coil is fed independently through its own power amplifier. Table 1 shows the power rating for both coils and power units. The system capability to generate field modes spans, in principle, in the range (m = 0, n = 0–23), (m = 1, n = −24 to 23, m = 2, n = 0–23), that is far beyond the
need for RWM control in RFX-mod, but allows also to study the TM control. The filter actions of both vacuum vessel and passive structures limit the magnetic field penetration. The vacuum vessel acts as a low-pass filter on the radial field measurements with cut-off frequency in the range of 1 kHz, whereas the shell along with the other passive structures has a dramatic shielding effect – with cutoff frequency of ∼16 Hz – on the radial field that the saddle coils can produced at the plasma edge. The relative position of shell, correction coils and field sensors represents the best compromise between the contrasting requirements of minimizing the shell proximity and maximizing the correction coil effectiveness [9]. The saddle coil, power amplifier and field sensor systems are described in details in [10–12], respectively. 3. Control system for MHD mode control Fig. 2 illustrates the architecture of the RFX-mod system for MHD mode control that is implemented as a general real time framework for feedback control (both hardware and software) integrating a high performance, distributed computing system with a
Fig. 2. Block diagram illustrating the architecture of the real time control system along with the I/O data flow.
T. Bolzonella et al. / Fusion Engineering and Design 82 (2007) 1064–1072
comprehensive analogue input/output section [13]. The system acquires the signals from both radial and ϑi,ϕj ϑi,ϕj and bt ) and produces the toroidal sensors (br ϑi,ϕj references ireq to drive the power amplifiers. Due to the large number of input/output channels, the system is split into four nodes exchanging data via a real time local area network (RT LAN). The framework, that can support whatever pre-elaboration and control algorithms conceivable in the space defined by the input/output section, implements a variety of alternative control algorithms [14]. In this paper we will discuss on the implementation of one specific control algorithm – the so-called virtual shell scheme – that has been extensively tested on the experiment and has produced very successful results. The active cancellation at the radius of the radial field sensors of the magnetic field, due to plasma instabilities and machine field errors, is referred to as virtual shell (VS), in analogy with the passive cancellation by an ideal superconducting shell. The active action could be implemented as the operation of 4 × 48 independent flux control loops, each one controlling the flux by one saddle coil on the feedback of the field measure from the corresponding radial field sensor. This realization, though simple and robust, is not suitable for toroidal devices, since it prevents from controlling the plasma equilibrium and position by external vertical field, as routinely done in most toroidal experiments, either tokamaks or RFP machines. In addition, this implementation does not allow to operate selectively on MHD modes, and cannot easily integrate control schemes relying on feedback signals originating from complex elaboration of magnetic field measurements. To override these limitations, the VS scheme is implemented in a more flexible way. At any control cycle,
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the radial and toroidal field processor nodes, shown in Fig. 2, compute in real time the two-dimensional spatial harmonic analyses on a complete sample of the corresponding magnetic field components, and distribute the results to the two controller nodes that process control loops and generate each one a half of the control references. The real time implementation of the twodimensional spatial Fourier analysis allows controlling the MHD modes selectively. By this scheme it is possible to permit the penetration of the external vertical field for plasma equilibrium and position control and, in addition, to study the effect of the control on single or multiple modes, for example the control of the sole RWMs, letting tearing modes evolves freely. To fulfil the time constraints of the mode control (10 ms growth time), the control cycle latency is kept less than 333 s, corresponding to 3 kHz of input sampling frequency. Fig. 3 shows in details the block diagram of the ϑi,ϕj VS scheme. The radial field br is measured by the 192 sensors and the real time 2D spatial fast Fourier transform (FFT) is computed to produce the Fourier components Brm,n . Feedback control is applied on a m,n subset of selected modes Bfeed , the selection consisting in a matrix preset in the pulse parameter-setting phase. On the selected modes, the inverse FFT is applied to ϑi,ϕj obtain the feedback control signals bfeed . The referm,n ence signals Bref for the modes to be controlled are preset directly in the Fourier transform domain and ϑi,ϕj transformed to space domain references bref through an inverse FFT. When a mode reference is preset to zero, the control system will operate to cancel the corresponding mode, whereas when it is preset to a complex number, the control system will act to produce the corresponding mode either static or rotating, depending on phase preset. A set of 192 independent
Fig. 3. Block diagram illustrating the implementation concept of the virtual shell scheme. The generic mode m, n can be either included in or excluded from control by pre-setting properly the selection matrix Sel.
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T. Bolzonella et al. / Fusion Engineering and Design 82 (2007) 1064–1072 ϑi,ϕj
PI regulators computes the field request breq on each radial sensor (ϑi , ϕj ) using as feedback the error signal ϑi,ϕj berr . The matrix M−1 S, taking into account the static mutual magnetic coupling of the saddle coils and sensors (M) and the radial sensor surfaces (S), transforms ϑi,ϕj the field requests into the current requests ireq that are sent, as current references, to the saddle coil power amplifiers implementing their own local current control loops. At parameter-setting level, the entire pulse can be subdivided into multiple segments, associated to different time spans during the pulse that can be preprogrammed independently. In this way, it is possible, for example, to program a first segment including control on a certain mode at the plasma rise, to let the mode free to evolve in a second segment and, finally, to re-apply control on the mode in a third segment. The algorithm can be summarized by the following equations, where the symbols have the meaning described above: Ki ireq = M −1 S Kp + (bref − bfeed ), s bfeed = FFT−1 (Sel · FFT(br ))
The VS, as described above, is applied on a toroidal surface at the radius of the radial field sensors. Alternative more complex schemes, computing the field distribution in the region external to the plasma, allow applying the VS at any radius in the no-plasma region. This requires the acquisition and processing of two magnetic field components, typically the radial and toroidal ones, to compute the real time spatial distribution of magnetic field [15]. By this technique, referred to as closer virtual shell and currently being experimented, the system can force the magnetic boundary at any radius between the plasma and the correction coils [16].
4. Advanced tests on MHD mode control Extended tests have been performed on the system by generating single modes to evaluate its capacity to produce pure spectra not affected by spurious modes. Fig. 4 shows, as an example, by transparent bars the measured normalized amplitudes of magnetic field modes when the system is used to generate only the m = 1 n = −7 mode. In the mode generation the mutual
Fig. 4. Generation of (m = 1 n = −7) mode. The figure displays the magnetic field spectra obtained without (transparent) and with decoupling (solid), respectively.
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Fig. 5. Generation of (m = 1 n = −7) mode. The figure displays the current spectra obtained without (transparent) and with decoupling (solid), respectively. When operating with decoupling m = 0, 2 n = 7 modes are present in the current to compensate the corresponding magnetic field modes.
coupling matrix M−1 S is chosen to be the Identity matrix, thus no coupling among coils is taken into account. In this case, due to toroidal geometry, the magnetic field spectrum achieved shows the presence of two spurious modes, the m = 0 n = 7 and m = 2 n = 7 ones, with noticeable amplitudes. To enhance the spectrum quality, a dynamic electromagnetic model has been developed and validated, and through this model the static matrix M has been obtained accounting for the mutual coupling among saddle coils and among saddle coils and sensors [17]. In solid bars the spectrum is displayed when the generation is performed using the decoupling block. As shown in figure, the decoupling implemented is very effective in suppressing spurious modes. Fig. 5 displays the current spectra in the same cases illustrated in the previous figure. When operating without decoupling the spectrum is clean, whereas with decoupling, modes arise in the current to compensate the corresponding spurious magnetic field modes. Analyses are currently under way to extend the model to the dynamic case to account for the field penetration into the load assembly with the aim of improving the system time response.
The capability of the system to control single modes has been checked by cancelling the pre-programmed perturbation m = 1 n = 0 and m = 0 n = 4 modes, produced with the field shaping and the toroidal windings, respectively. Fig. 6 shows the system feedback action
Fig. 6. Cancellation of disturbance m = 1 n = 0 and m = 0 n = 4 modes produced with the field shaping and toroidal windings, respectively.
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on both modes. As shown in figure both modes are cancelled in few tens of ms. The response time is acceptable in both cases, even though improvements are expected including in the model the passive structures.
5. RWM control Since the commissioning of the MHD mode control system, the VS control scheme has been used routinely in the machine experimental sessions, to control both RWMs and TMs, contributing significantly to the successful results obtained in the experimental campaigns [18]. In Fig. 1 the spectra of the radial magnetic field measured at the sensor radius and achieved as average on a set of pulses executed without and with mode control are compared. The figure illustrates clearly the system effectiveness in reducing the amplitudes of both RWMs and TMs. To characterize the RWM growth – depending on the reversal parameter F = bϕ (a)/bϕ (r) – and their control, extensive experimental campaigns have been carried out in 2006 at 600 kA and value of F ranging from −0.1 (shallow reversal) to −0.4 (deep reversal) [19]. In this experimental operation the VS scheme has allowed studying in the same pulse both the evolution and the
Fig. 7. Demonstration of feedback control on RWMs. Comparison among pulses programmed with the same parameters, but differing in RWM control.
control of the RWMs. Internal non-resonant RWMs turned out to be the most unstable modes when operating with shallow reversal, as anticipated by theory, whereas external non-resonant modes are dominant in deep reversal. Fig. 7 highlights a clear example of RWM mode dynamic and control in RFX-mod comparing three pulses programmed with the same parameters (the m = 1 n = −6 being the most unstable RWM), but differing in how RWMs are controlled. Pulse #17301 (dotted trace) is executed excluding RWMs m = 1 n = −6 to −3 from control. The m = 1 n = −6 unstable mode grows and when its amplitude reaches a few mT the pulse is early terminated. Pulse #17287 (solid trace) is exe-
Fig. 8. Power consumption associated to MHD mode control. From top to bottom, first panel shows the plasma current, second panel the power delivered by the power amplifiers for the MHD mode control, third panel shows the product VLoop × IPlasma , and fourth panel the m = 1 spectrum.
T. Bolzonella et al. / Fusion Engineering and Design 82 (2007) 1064–1072
cuted with full VS control (except, as usually, for the m = 1 n = 0 equilibrium field). The amplitude of the m = 1 n = −6 mode is kept at negligible values and the plasma current is well sustained up to 250 ms. Pulse #17304 is executed letting the m = 1 n = −3 to −6 free to grow until t = 150 ms and at this point in time control is applied on the modes reducing promptly the mode amplitude – in ∼10 ms – to very low values. These results are a clear demonstration of pure feedback stabilization of RWMs. To quantify the power needed in RFX-mod for mode control, Fig. 8 shows the power supplied by the power amplifiers to control the MHD modes by means of the VS scheme in pulse #19648 reaching 1 MA of plasma current. In the pulse the reference for the m = 1 n = 7, 8, 10, 11, 12 TM is not preset to cancel the modes, but to produce rotating modes with pre-programmed amplitudes and rotating frequencies The panels show, from above to below, the traces of the plasma current, the power supplied by the amplifiers, the power associated with the plasma Vloop × Iplasma , and the m = 1 spectrum, respectively. During the current flat-top the power needed to control the MHD mode is ∼1% of the plasma power. Active control has allowed a significant improvement of the plasma performance and the VS scheme plays a key role in the surprising extension of the pulse length achieved in the experiment. As shown in Fig. 9, a threefold increase in plasma dura-
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tion has been achieved and the pulse has been well controlled for over 300 ms (five shell times). This confirms similar results achieved with mode control in the EXTRAP T2R experiment at KTH in Stockholm where the same concepts have been tested in collaboration with RFX.
6. Conclusions RFX-mod represents a unique environment for advanced studies on MHD mode control and the flexibility and robustness of its control system have permitted a significant step ahead in the demonstration of the feasibility of the MHD mode active control. The control of MHD modes by imposing the magnetic boundary through active correction coils has proven very effective in RFX-mod to achieve significant stationary confinement improvements and full feedback stabilization of RWMs has been obtained in conditions where multiple unstable modes are present. The application of MHD control has produced remarkable and reproducible effects on the plasma. The robustness of the RWM feedback control qualifies the system as an important step for possible application in tokamaks and fusion reactors.
Acknowledgments This work was supported by the Euratom Communities under the contract of Association between EURATOM/ENEA. The views and opinions expressed herein do not necessarily reflect those of the European Commission.
References
Fig. 9. Plasma current and resistive loop voltage waveforms. Comparison between a typical pulse without active control and one with VS control.
[1] M. Okabayashi, J. Bialek, A. Bondeson, M.S. Chance, M.S. Chu, A.M. Garofalo, et al., Control of the resistive wall mode with internal coils in the DIII–D tokamak, Nucl. Fusion 45 (2005) 1715–1731. [2] H. Zohm, G. Gantenbein, A. Isayama, A. Keller, R.J. La Haye, M. Maraschek, et al., MHD limits to tokamak operation and their control, Plasma Phys. Control. Fusion 45 (2003) A163–A173. [3] Y. Liu, A. Bondeson, Y. Gribov, A. Polevoi, Stabilization of resistive wall modes in ITER by active feedback and toroidal rotation, Nucl. Fusion 44 (2004) 232–242.
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[4] P.R. Brunsell1, D. Yadikin, D. Gregoratto, R. Paccagnella, Y.Q. Liu, T. Bolzonella, et al., Active control of multiple resistive wall modes, Plasma Phys. Control. Fusion 47 (2005) B25–B36. [5] T. Bolzonella, L. Marrelli, R. Paccagnella, P. Zanca, G. Manduchi, Resistive wall mode growth and control in RFX-mod, in: Proceedings of the 33rd EPS Conference on Plasma Physics, June 19–23, 2006, Rome, Italy, 2006. [6] P. Sonato, R. Piovan, A. Luchetta, the RFX team, Control of non-axisymmetric magnetic fields for plasma enhanced performances: the RFX contribution, Fusion Eng. Des. 74 (2005) 97–107. [7] C.M. Bishop, An intelligent shell for the toroidal pinch, Plasma Phys. Control. Fusion 31 (1989) 1179–1189. [8] G. Marchiori, L. Grando, M. Cavinato, Characterization of RFX-mod conducting structures to optimise plasma start up and equilibrium control, in: Presented at 24th SOFT, September 11–15, Warsaw, Poland, 2006. [9] P. Sonato, G. Chitarin, P. Zaccaria, F. Gnesotto, S. Ortolani, A. Buffa, et al., Machine modification for active MHD control in RFX, Fusion Eng. Des. 66–68 (2003) 161–168. [10] G. Chitarin, S. Dal Bello, L. Grando, S. Peruzzo, The design of 192 saddle coils for RFX, Fusion Eng. Des. 66–68 (2003) 1055–1059. [11] V. Toigo, E. Gaio, N. Balbo, A. Tescari, The power supply system for the active control of MHD modes in RFX, Fusion Eng. Des. 66–68 (2003) 1143–1147. [12] N. Pomaro, F. Basso, Transducers and signal conditioners of the RFX new magnetic measurement system, Fusion Eng. Des. 74 (2005) 721–726.
[13] M. Cavinato, A. Luchetta, G. Manduchi, C. Taliercio, General-purpose framework for real time control in nuclear fusion experiments, IEEE Trans. Nucl. Sci. 53 (2006) 1002– 1008. [14] M. Cavinato, A. Luchetta, G. Manduchi, G. Marchiori, C. Taliercio, Distributed real time control in RFX-mod nuclear fusion experiment: commissioning and first results, IEEE Trans. Nucl. Sci. 53 (2006) 1015–1021. [15] P. Zanca, S. Martini, Reconstruction of the plasma surface in a RFP in the presence of non-axisymmetric perturbations, Plasma Phys. Control. Fusion 41 (1999) 1251–1275. [16] R. Paccagnella, P. Zanca, D. Terranova, MHD fluctuation in a Reversed Field Pinch with different magnetic boundaries produced using an active feedback coil system, in: Presented at 33rd EPS Conference on Plasma Physics, June 19–23, Rome, Italy, 2006. [17] G. Marchiori, A. Soppelsa, Development and validation of an electromagnetic model of the active control system of MHD modes in RFX-mod, in: Presented at 24th SOFT, September 11–15, Warsaw, Poland, 2006. [18] R. Paccagnella, S. Ortolani, P. Zanca, A. Alfier, T. Bolzonella, L. Marrelli, et al., Active feedback control of magnetic boundary for magnetic-hydro-dynamic stabilisation of a fusion plasma, Phys. Rev. Lett., 95, 075001 (2006). [19] S. Ortolani, the RFX team, Active MHD control experiments in RFX-mod, in: Proceedings of the 33rd EPS Conference on Plasma Physics, June 19–23, Rome, Italy, 2006.
Feedback control of resistive wall modes by saddle coils in RFX-mod T. Bolzonella, M. Cavinato, E. Gaio, L. Grando, A. Luchetta ∗ , G. Manduchi, G. Marchiori, L. Marrelli, R. Paccagnella, A. Soppelsa, P. Zanca Consorzio RFX, Associazione EURATOM-ENEA sulla Fusione corso Stati Uniti 4, 35127 Padova, Italy Received 1 August 2006; received in revised form 4 May 2007; accepted 6 May 2007 Available online 29 June 2007
Abstract Active control of resistive wall modes (RWMs) is a key issue for high-beta operation of future tokamaks, including ITER. RFX has been recently modified to specifically address this issue by means of a new saddle coil system, which is the most powerful and flexible among all fusion devices, including 192 coils covering the whole torus. Unstable RWMs have been experimentally observed during the recent machine operation, showing poloidal and toroidal components m = 1, 2 < |n| < 7 and growth times of some tens of ms, consistent with theoretical predictions. RFX-mod addresses the control of RWMs by acting on the magnetic field configuration at the plasma boundary. The system aims not only at RWM stabilization, but also at interaction with tearing modes. By means of the system, successful experimental campaigns have been executed demonstrating the feasibility of RWM active control. The paper discusses on the system engineering and technological features and presents the results achieved. © 2007 Elsevier B.V. All rights reserved. Keywords: Resistive wall modes; MHD mode control; Feedback control; Digital control system
1. Introduction Magneto-hydrodynamic (MHD) instabilities arise in toroidal plasmas due to non-zero current and pressure profile gradients. The growth of external kink modes, referred to as resistive wall modes (RWM), limits the achievable  in tokamak advanced scenarios and ∗ Corresponding author. Tel.: +39 049 8295043; fax: +39 049 8700718. E-mail address: [email protected] (A. Luchetta).
0920-3796/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.fusengdes.2007.05.020
their stabilization is, thus, a key issue for the operation of ITER and future fusion reactors. These modes, unstable in the presence of a resistive wall, are stable when an infinitely conducting wall is placed close to the plasma surface. Since a passive ideal boundary, acting as a perfect conductor during the plasma pulse of a fusion device, is not feasible, the development of successful technology for active feedback control of MHD instabilities is mandatory. Efforts to address the problem are thus currently accomplished in fusion community [1–4].
T. Bolzonella et al. / Fusion Engineering and Design 82 (2007) 1064–1072
Plasma instabilities are usually described in terms of two-dimensional spatial Fourier decomposition along the poloidal and toroidal directions and the relevant harmonic components are indicated with the mode numbers m and n, respectively. RWMs have been predicted also for reverse field pinch (RFP) machines and, according to theory, growth of RWMs showing poloidal and toroidal components m = 1, 2 < |n| < 7 and growth times of some tens of ms have been observed experimentally in RFX-mod during the recent campaigns [5]. The RFP is a low q magnetic profile machine – with q(0) ≈ 0.5a/R decreasing monotonically to q(a) ≤ 0 – whose spectrum of unstable modes is dominated by those with m = 1 and n in a large range of values. In RFX (a = 0.459m, R = 2.0m) the modes with n ≤ −7, internally resonant tearing modes (TM), are responsible for the dynamo sustainment. The modes with −7 < n < 0 and n > 0 are, instead, non-resonant (internally or externally) RWMs. As a consequence, RFX-mod shows a very rich spectrum of MHD modes and is, thus, an excellent environment where to study the MHD mode phenomenology and the techniques for mode control. Fig. 1 shows by striped bars the amplitudes of m = 1 modes measured at the plasma edge in reference RFXmod pulses executed without mode control. RFX-mod addresses the control of MHD modes by acting on the magnetic configuration at the plasma boundary [6]. This corresponds to the concept of a discrete active shell as proposed for the first time in late 1980s [7]. To implement the concept, two major modifications were necessary on the RFX original load assembly: the installation of a dedicated coil system,
1065
to perform active control on non-axisymmetric radial magnetic field at the boundary, and the replacement of the previous shell, having long time constant for radial field penetration (∼450 ms), with a new one with lower time constant (∼60 ms), thus allowing the diffusion into the plasma region of external magnetic field on the timescale of the RFX-mod pulse (nominally 250 ms) [8]. In this paper, after a description of the components of the MHD mode control system, we present the successful control scheme implemented on the experiment along with the results achieved demonstrating the successful control of RWMs. The outstanding features of the system are highlighted in details with focus on the regimes that are now reachable due to the power of the control algorithms and the high robustness and flexibility of the system.
2. System overview The radial field coil system consists of a bidimensional mesh, including 192 equispaced saddle coils, mounted to cover the whole outer surface of the shell, as installation inside the shell was impossible because of mechanical constraints. The mesh can be thought of as made up by 4 toroidal and 48 poloidal arrays. With reference to the toroidal reference system used in RFX, the centre of the generic saddle coil is identified with the coordinates (r = 0.581 m, ϑi = iπ/2 rad, ϕj = jπ/24 rad) where 0 ≤ i ≤ 3 and 0 ≤ j ≤ 47. To the purpose of both measure and control, a pair of sensors is associated with each
Fig. 1. Spectra of radial field modes measured by the radial field probes. Striped and solid bars are computed as averages on measured mode amplitudes at the current flat-top of two sets of pulses executed without and with mode control, respectively.
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Table 1 Main parameters of saddle coils and power amplifiers
Voltage (V) Peak current (A) Turns Coil time constant L/R (s) Magneto motive force (kAt) Pulse-width modulation frequency (kHz)
Saddle coil
Power amplifier
650 400 60 5e−3/1 24 –
650 400 – – 10
radial field coil, including one pick-up coil, detecting both toroidal and poloidal components, and one saddle probe sensing the average radial field. The pick-up coils are located on the inner surface of the shell, aligned with the centre of the corresponding correction coil, whereas the saddle probes are mounted on the outer surface of the vacuum vessel on the projection of the correction coil onto the vacuum vessel itself. Each saddle coil is fed independently through its own power amplifier. Table 1 shows the power rating for both coils and power units. The system capability to generate field modes spans, in principle, in the range (m = 0, n = 0–23), (m = 1, n = −24 to 23, m = 2, n = 0–23), that is far beyond the
need for RWM control in RFX-mod, but allows also to study the TM control. The filter actions of both vacuum vessel and passive structures limit the magnetic field penetration. The vacuum vessel acts as a low-pass filter on the radial field measurements with cut-off frequency in the range of 1 kHz, whereas the shell along with the other passive structures has a dramatic shielding effect – with cutoff frequency of ∼16 Hz – on the radial field that the saddle coils can produced at the plasma edge. The relative position of shell, correction coils and field sensors represents the best compromise between the contrasting requirements of minimizing the shell proximity and maximizing the correction coil effectiveness [9]. The saddle coil, power amplifier and field sensor systems are described in details in [10–12], respectively. 3. Control system for MHD mode control Fig. 2 illustrates the architecture of the RFX-mod system for MHD mode control that is implemented as a general real time framework for feedback control (both hardware and software) integrating a high performance, distributed computing system with a
Fig. 2. Block diagram illustrating the architecture of the real time control system along with the I/O data flow.
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comprehensive analogue input/output section [13]. The system acquires the signals from both radial and ϑi,ϕj ϑi,ϕj and bt ) and produces the toroidal sensors (br ϑi,ϕj references ireq to drive the power amplifiers. Due to the large number of input/output channels, the system is split into four nodes exchanging data via a real time local area network (RT LAN). The framework, that can support whatever pre-elaboration and control algorithms conceivable in the space defined by the input/output section, implements a variety of alternative control algorithms [14]. In this paper we will discuss on the implementation of one specific control algorithm – the so-called virtual shell scheme – that has been extensively tested on the experiment and has produced very successful results. The active cancellation at the radius of the radial field sensors of the magnetic field, due to plasma instabilities and machine field errors, is referred to as virtual shell (VS), in analogy with the passive cancellation by an ideal superconducting shell. The active action could be implemented as the operation of 4 × 48 independent flux control loops, each one controlling the flux by one saddle coil on the feedback of the field measure from the corresponding radial field sensor. This realization, though simple and robust, is not suitable for toroidal devices, since it prevents from controlling the plasma equilibrium and position by external vertical field, as routinely done in most toroidal experiments, either tokamaks or RFP machines. In addition, this implementation does not allow to operate selectively on MHD modes, and cannot easily integrate control schemes relying on feedback signals originating from complex elaboration of magnetic field measurements. To override these limitations, the VS scheme is implemented in a more flexible way. At any control cycle,
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the radial and toroidal field processor nodes, shown in Fig. 2, compute in real time the two-dimensional spatial harmonic analyses on a complete sample of the corresponding magnetic field components, and distribute the results to the two controller nodes that process control loops and generate each one a half of the control references. The real time implementation of the twodimensional spatial Fourier analysis allows controlling the MHD modes selectively. By this scheme it is possible to permit the penetration of the external vertical field for plasma equilibrium and position control and, in addition, to study the effect of the control on single or multiple modes, for example the control of the sole RWMs, letting tearing modes evolves freely. To fulfil the time constraints of the mode control (10 ms growth time), the control cycle latency is kept less than 333 s, corresponding to 3 kHz of input sampling frequency. Fig. 3 shows in details the block diagram of the ϑi,ϕj VS scheme. The radial field br is measured by the 192 sensors and the real time 2D spatial fast Fourier transform (FFT) is computed to produce the Fourier components Brm,n . Feedback control is applied on a m,n subset of selected modes Bfeed , the selection consisting in a matrix preset in the pulse parameter-setting phase. On the selected modes, the inverse FFT is applied to ϑi,ϕj obtain the feedback control signals bfeed . The referm,n ence signals Bref for the modes to be controlled are preset directly in the Fourier transform domain and ϑi,ϕj transformed to space domain references bref through an inverse FFT. When a mode reference is preset to zero, the control system will operate to cancel the corresponding mode, whereas when it is preset to a complex number, the control system will act to produce the corresponding mode either static or rotating, depending on phase preset. A set of 192 independent
Fig. 3. Block diagram illustrating the implementation concept of the virtual shell scheme. The generic mode m, n can be either included in or excluded from control by pre-setting properly the selection matrix Sel.
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T. Bolzonella et al. / Fusion Engineering and Design 82 (2007) 1064–1072 ϑi,ϕj
PI regulators computes the field request breq on each radial sensor (ϑi , ϕj ) using as feedback the error signal ϑi,ϕj berr . The matrix M−1 S, taking into account the static mutual magnetic coupling of the saddle coils and sensors (M) and the radial sensor surfaces (S), transforms ϑi,ϕj the field requests into the current requests ireq that are sent, as current references, to the saddle coil power amplifiers implementing their own local current control loops. At parameter-setting level, the entire pulse can be subdivided into multiple segments, associated to different time spans during the pulse that can be preprogrammed independently. In this way, it is possible, for example, to program a first segment including control on a certain mode at the plasma rise, to let the mode free to evolve in a second segment and, finally, to re-apply control on the mode in a third segment. The algorithm can be summarized by the following equations, where the symbols have the meaning described above: Ki ireq = M −1 S Kp + (bref − bfeed ), s bfeed = FFT−1 (Sel · FFT(br ))
The VS, as described above, is applied on a toroidal surface at the radius of the radial field sensors. Alternative more complex schemes, computing the field distribution in the region external to the plasma, allow applying the VS at any radius in the no-plasma region. This requires the acquisition and processing of two magnetic field components, typically the radial and toroidal ones, to compute the real time spatial distribution of magnetic field [15]. By this technique, referred to as closer virtual shell and currently being experimented, the system can force the magnetic boundary at any radius between the plasma and the correction coils [16].
4. Advanced tests on MHD mode control Extended tests have been performed on the system by generating single modes to evaluate its capacity to produce pure spectra not affected by spurious modes. Fig. 4 shows, as an example, by transparent bars the measured normalized amplitudes of magnetic field modes when the system is used to generate only the m = 1 n = −7 mode. In the mode generation the mutual
Fig. 4. Generation of (m = 1 n = −7) mode. The figure displays the magnetic field spectra obtained without (transparent) and with decoupling (solid), respectively.
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Fig. 5. Generation of (m = 1 n = −7) mode. The figure displays the current spectra obtained without (transparent) and with decoupling (solid), respectively. When operating with decoupling m = 0, 2 n = 7 modes are present in the current to compensate the corresponding magnetic field modes.
coupling matrix M−1 S is chosen to be the Identity matrix, thus no coupling among coils is taken into account. In this case, due to toroidal geometry, the magnetic field spectrum achieved shows the presence of two spurious modes, the m = 0 n = 7 and m = 2 n = 7 ones, with noticeable amplitudes. To enhance the spectrum quality, a dynamic electromagnetic model has been developed and validated, and through this model the static matrix M has been obtained accounting for the mutual coupling among saddle coils and among saddle coils and sensors [17]. In solid bars the spectrum is displayed when the generation is performed using the decoupling block. As shown in figure, the decoupling implemented is very effective in suppressing spurious modes. Fig. 5 displays the current spectra in the same cases illustrated in the previous figure. When operating without decoupling the spectrum is clean, whereas with decoupling, modes arise in the current to compensate the corresponding spurious magnetic field modes. Analyses are currently under way to extend the model to the dynamic case to account for the field penetration into the load assembly with the aim of improving the system time response.
The capability of the system to control single modes has been checked by cancelling the pre-programmed perturbation m = 1 n = 0 and m = 0 n = 4 modes, produced with the field shaping and the toroidal windings, respectively. Fig. 6 shows the system feedback action
Fig. 6. Cancellation of disturbance m = 1 n = 0 and m = 0 n = 4 modes produced with the field shaping and toroidal windings, respectively.
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on both modes. As shown in figure both modes are cancelled in few tens of ms. The response time is acceptable in both cases, even though improvements are expected including in the model the passive structures.
5. RWM control Since the commissioning of the MHD mode control system, the VS control scheme has been used routinely in the machine experimental sessions, to control both RWMs and TMs, contributing significantly to the successful results obtained in the experimental campaigns [18]. In Fig. 1 the spectra of the radial magnetic field measured at the sensor radius and achieved as average on a set of pulses executed without and with mode control are compared. The figure illustrates clearly the system effectiveness in reducing the amplitudes of both RWMs and TMs. To characterize the RWM growth – depending on the reversal parameter F = bϕ (a)/bϕ (r) – and their control, extensive experimental campaigns have been carried out in 2006 at 600 kA and value of F ranging from −0.1 (shallow reversal) to −0.4 (deep reversal) [19]. In this experimental operation the VS scheme has allowed studying in the same pulse both the evolution and the
Fig. 7. Demonstration of feedback control on RWMs. Comparison among pulses programmed with the same parameters, but differing in RWM control.
control of the RWMs. Internal non-resonant RWMs turned out to be the most unstable modes when operating with shallow reversal, as anticipated by theory, whereas external non-resonant modes are dominant in deep reversal. Fig. 7 highlights a clear example of RWM mode dynamic and control in RFX-mod comparing three pulses programmed with the same parameters (the m = 1 n = −6 being the most unstable RWM), but differing in how RWMs are controlled. Pulse #17301 (dotted trace) is executed excluding RWMs m = 1 n = −6 to −3 from control. The m = 1 n = −6 unstable mode grows and when its amplitude reaches a few mT the pulse is early terminated. Pulse #17287 (solid trace) is exe-
Fig. 8. Power consumption associated to MHD mode control. From top to bottom, first panel shows the plasma current, second panel the power delivered by the power amplifiers for the MHD mode control, third panel shows the product VLoop × IPlasma , and fourth panel the m = 1 spectrum.
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cuted with full VS control (except, as usually, for the m = 1 n = 0 equilibrium field). The amplitude of the m = 1 n = −6 mode is kept at negligible values and the plasma current is well sustained up to 250 ms. Pulse #17304 is executed letting the m = 1 n = −3 to −6 free to grow until t = 150 ms and at this point in time control is applied on the modes reducing promptly the mode amplitude – in ∼10 ms – to very low values. These results are a clear demonstration of pure feedback stabilization of RWMs. To quantify the power needed in RFX-mod for mode control, Fig. 8 shows the power supplied by the power amplifiers to control the MHD modes by means of the VS scheme in pulse #19648 reaching 1 MA of plasma current. In the pulse the reference for the m = 1 n = 7, 8, 10, 11, 12 TM is not preset to cancel the modes, but to produce rotating modes with pre-programmed amplitudes and rotating frequencies The panels show, from above to below, the traces of the plasma current, the power supplied by the amplifiers, the power associated with the plasma Vloop × Iplasma , and the m = 1 spectrum, respectively. During the current flat-top the power needed to control the MHD mode is ∼1% of the plasma power. Active control has allowed a significant improvement of the plasma performance and the VS scheme plays a key role in the surprising extension of the pulse length achieved in the experiment. As shown in Fig. 9, a threefold increase in plasma dura-
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tion has been achieved and the pulse has been well controlled for over 300 ms (five shell times). This confirms similar results achieved with mode control in the EXTRAP T2R experiment at KTH in Stockholm where the same concepts have been tested in collaboration with RFX.
6. Conclusions RFX-mod represents a unique environment for advanced studies on MHD mode control and the flexibility and robustness of its control system have permitted a significant step ahead in the demonstration of the feasibility of the MHD mode active control. The control of MHD modes by imposing the magnetic boundary through active correction coils has proven very effective in RFX-mod to achieve significant stationary confinement improvements and full feedback stabilization of RWMs has been obtained in conditions where multiple unstable modes are present. The application of MHD control has produced remarkable and reproducible effects on the plasma. The robustness of the RWM feedback control qualifies the system as an important step for possible application in tokamaks and fusion reactors.
Acknowledgments This work was supported by the Euratom Communities under the contract of Association between EURATOM/ENEA. The views and opinions expressed herein do not necessarily reflect those of the European Commission.
References
Fig. 9. Plasma current and resistive loop voltage waveforms. Comparison between a typical pulse without active control and one with VS control.
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