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Linearized models of the plasma response in the new RFX load assembly
Fusion Engineering and Design 74 (2005) 573–578
Linearized models of the plasma response in the new RFX load assembly P. Bettini a,∗ , M. Cavinato b , G. Marchiori b , F. Villone c a DIEGM, Universit` a di Udine, Via delle Scienze 208, I-33100 Udine, Italy Consorzio RFX, Associazione EURATOM-ENEA sulla Fusione, Corso Stati Uniti 4, I-35127 Padova, Italy Ass. EURATOM/ENEA/CREATE, DAEIMI, Universit`a di Cassino, Via Di Biasio 43, I-03043 Cassino (FR), Italy b
c
Available online 20 October 2005
Abstract The new RFX load assembly characterized by a thin copper shell requires using an active control system for the plasma equilibrium. A new version of the CREATE-L RFP plasma response model was developed as a tool for the control system design and to crosscheck results with another linearized model, derived by a different technique. The RFP version of the MAXFEA equilibrium code adapted to the new load assembly was used to produce benchmark data for validation tests. The results of comparisons performed imposing either the active coil currents or voltages are presented in the paper. © 2005 Elsevier B.V. All rights reserved.
1. Introduction At present, the design of control systems in fusion devices is generally based on linearized models of the plasma response at different equilibria. The CREATEL model, derived by linearizing the MHD equilibrium equation and Ohm’s law in the active and passive conductors and in the plasma, has been extended to reversed field pinch (RFP) plasmas and already successfully applied to the old RFX magnetic configuration [1,2]. The new RFX load assembly includes now a new 3 mm thick copper shell, close fitting to the vacuum vessel, which will allow to perform active control ∗ Corresponding author. Tel.: +39 0432 558291; fax: +39 0432 558292. E-mail address: [email protected] (P. Bettini).
0920-3796/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.fusengdes.2005.06.170
experiments by means of a set of 192 saddle coils surrounding the torus [3]. The much shorter time constant of the shell requires also an active control of the plasma equilibrium. One of the aims of this paper is to apply the CREATE-L plasma response model also to study in details the effects of the structural modifications of RFX machine on the open loop plasma response. Since experimental data are not yet available, a two dimensional FEM code (MAXFEA), solving the ideal MHD free boundary problem in axisymmetric geometry, has been used to provide a set of equilibrium reference data, also in terms of coil currents and virtual measures from pick-up and flux loop probes [4]. As a first step in the validation procedure, in particular to assess the accuracy of the electromagnetic model of the passive structures implemented in the codes, a crosscheck has been carried out on a shot without plasma. As a sec-
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P. Bettini et al. / Fusion Engineering and Design 74 (2005) 573–578
ond step, complete comparisons have been performed in the case of pulses with plasma both imposing the active coil currents and the voltages as inputs.
2. The new RFX magnetic poloidal field system The poloidal magnetic field system of RFX (Fig. 1) includes the field shaping winding (F), the magnetizing winding (M), the vacuum vessel (V) and the thin copper stabilizing shell (S). An integrated system of magnetic field pick-up and flux probes, installed both on the outer and inner surface of vessel and shell, will provide signals to drive the field shaping coil power supply. The thin copper shell provides the plasma equilibrium and limits the growth of fast instabilities on the short time scale by means of its induced currents. As these currents diffuse (time constant 50 ms), the plasma equilibrium shall be controlled by means of the F winding and MHD modes shall be controlled by means of the 192 saddle coils. The linearized CREATE-L plasma response model, used to predict the plasma current, shape and position response to voltages applied to the control coils and to
known βp and li variations, is derived by linearizing the MHD equilibrium equation and Ohm’s law in the active and passive conductors and in the plasma. The plasma is assumed to be in permanent MHD equilibrium and to be described by a small number of global parameters s(t). The profiles of the poloidal flux current density f(ψ) and the kinetic pressure p(ψ) are given as known quantities, adopting the expressions described in [2]. The dynamic evolution of currents flowing in conductors is described in the linearized form: L∗
d(␦I) d(␦s) + R ␦I + LS = B1 ␦U dt dt
(1)
where U is the active coil voltage vector, B1 is a matrix with appropriate zeros and ones to apply voltages to the active coils, LS = ∂Ψ /∂s, and L* = ∂Ψ /∂I is the modified inductance matrix. For control purposes, it may be useful to recast (1) in standard state-space form, as: d(␦I) d(␦s) = A ␦I + B ␦U + ES dt dt with obvious definitions.
Fig. 1. Poloidal magnetic field system: probes (vloop, Bp), vessel, shell, M (1/20) and F (1/8) coils.
(2)
P. Bettini et al. / Fusion Engineering and Design 74 (2005) 573–578
575
A number of outputs has been defined in the form: ␦y = CY ␦I + FS ␦s
(3)
for magnetic fields or fluxes variations produced at given RFX positions: • Poloidal flux density probes Integrated System of Internal Sensors (ISIS) array: Eight pick-up probes located inside the vessel; INT array: Eight pick-up probes located outside the vessel, on the inner surface of the shell; EXT array: Eight pick-up probes located on the outer surface of the shell; • Flux-loop probes: Eight axisymmetric loops located on the outer surface of the vessel. CREATE-L model is computed using a mesh made of 26,707 second-order triangular elements with 53,500 nodes. The mesh used by MAXFEA consists of 26,780 first-order triangular mesh with 13,402 nodes.
3. Numerical validation on shots without plasma
Fig. 3. Time evolution of the m = 1 poloidal mode of the shell toroidal current density (—: CREATE-L, - - -: reference 50 ms exponential).
In Fig. 2 we present a comparison of the current density distribution in the new shell, along the poloidal angle, predicted by the two models at two time instants. The agreement between the two models is good both at t = 0.14 s, corresponding to the peak induced current, where a clear m = 1 mode appears, and at t = 0.64
A shot without plasma, programmed in the previous experimental phase of RFX (#14,209) to apply a vertical field step of about 60 mT, was chosen to assess the electromagnetic modelling of the passive structures.
Fig. 2. Comparison of the current density distribution in the shell elements (o: MAXFEA, ×: CREATE-L).
Fig. 4. Time evolution of reference and reconstructed signals (o: CREATE-L, —: MAXFEA). Top: vessel current, Middle: poloidal field at ISIS pick-up coil #2. Bottom: poloidal flux at flux-loop probe #5.
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P. Bettini et al. / Fusion Engineering and Design 74 (2005) 573–578
s in a phase where a more complex mode structure is observed. In Fig. 3 the time evolution of the m = 1 poloidal mode of the shell toroidal current density predicted by the CREATE-L model is presented along with an exponential fit with a time constant T = 56 ms. The time evolutions of vessel current, poloidal induction flux and field calculated by MAXFEA in selected positions are compared with the corresponding quantities computed by the model in Fig. 4.
4. Numerical validation on shots with plasma In order to perform comparisons in the case of pulses with plasma the system matrices of the linearized model have been obtained considering an equilibrium consistent with the integral parameters measured in a typical RFX pulse (Ip = 720 kA, li = 1.3, fa = 1.41 Tm and fb = −0.14 Tm, α = 11.32, where li is the normalized internal inductance, fa and fb are the poloidal current density flux function at the axis and the boundary, respectively, α is a parameter related to the equilibrium plasma current profile). Since the plasma is open loop stable some tests have been carried out imposing the evolutions of the active coil (M, F) currents so as to evaluate the accuracy of the coupling of plasma and active coils with the passive conducting structures. This approach allows to neglect all the effects related to uncertainty of the electrical circuit parameters, such as the additional inductances or resistances which are series connected to each couple of F coils, or other stray parameters. Fig. 5 presents the time evolution of the poloidal field inside the vessel (ISIS array), the poloidal flux on its outer surface and the poloidal field “sensed” by some pick-up coils of the INT and EXT arrays as well. In view of using the model as a tool for the design of plasma current and equilibrium control system, validation tests imposing voltage waveforms have then been started. However, since no mechanism accounting for the plasma resistive flux consumption is present in the model, it was preferred to maintain the plasma current as an input term. It should be now reminded that another linearized model of the plasma response in the new RFX load assembly was derived by applying a different perturbation technique in MAXFEA and it was already used for the design of a first equilibrium controller [5]. Our aim is to make available a second model
Fig. 5. Time evolutions of reference and reconstructed signals (o: CREATE-L, —: MAXFEA). From top to bottom: poloidal field at ISIS pick-up coil #2, poloidal flux at flux-loop probe #5, poloidal field at INT pick-up coil #6, poloidal field at EXT pick-up coil #3.
to crosscheck results before the final implementation of the controller. The electrical topology of the RFX poloidal field circuit (Fig. 6) is rather complex and it required solving some numerical problems in the nonlinear simulations, due to the high value of the ground resistances, which increases the numerical stiffness of the system. To compare the two models while assuring the plasma equilibrium, it was also decided to use voltage waveforms computed in non-linear simulations with the already available plasma equilibrium controller. Results in terms of poloidal field at different positions are satisfactory as shown in Fig. 7, where two examples from ISIS and EXT arrays are given. A good agreement was also consistently observed between the outer coil currents, which most contribute to the equilibrium field configuration, as it can be seen in the same figure. Further tests are now under way to improve the results for the inner coils for which the calculation of the coupling terms is more critical, for instance due to
P. Bettini et al. / Fusion Engineering and Design 74 (2005) 573–578
577
Fig. 6. Poloidal field circuit.
the narrow shape and the proximity to the magnetizing winding.
5. Conclusions A version of the CREATE-L model of RFP plasma response was developed for the new RFX load assembly. Validation tests were carried out benchmarking the model outputs with the corresponding data provided by a non-linear FEM MHD equilibrium code both in the case of shots without and with plasma. The comparisons made either imposing the active coil currents or voltages showed a satisfactory agreement between the time evolutions of the poloidal magnetic field and flux variations. Preliminary results are encouraging and further tests with imposed voltages are foreseen in view of the design of the plasma current and equilibrium control system.
References Fig. 7. Time evolutions of reference and reconstructed signals (o: CREATE-L, —: MAXFEA). From top to bottom: ISIS pick-up coil #2, EXT array pick-up coil #7, field shaping F6 and F8 currents.
[1] R. Albanese, F. Villone, The linearized CREATE-L plasma response model for the control of current, position and shape in tokamaks, Nucl. Fusion 38 (5) (1998) 723–738.
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P. Bettini et al. / Fusion Engineering and Design 74 (2005) 573–578
[2] R. Albanese, P. Bettini, M. Guarnieri, G. Marchiori, F. Villone, Linearized models for RFX configurations, Fusion Eng. Des. 56/57 (2002) 733–738. [3] P. Sonato, G. Chitarin, P. Zaccaria, F. Gnesatto, S. Ortolani, A. Buffa, et al., Machine modifications for active MHD control in RFX, Fusion Eng. Des. 66/68 (2003) 161–168.
[4] P. Bettini, M. Cavinato, G. Marchiori, 2D non-linear model of Reversed Field Pinch plasma evolution, Nucl. Fusion 43 (2003) 119–129. [5] M. Cavinato, G. Marchiori, Design of the new RFX equilibrium active control system, presented at the 23rd SOFT, September 2004.
Linearized models of the plasma response in the new RFX load assembly P. Bettini a,∗ , M. Cavinato b , G. Marchiori b , F. Villone c a DIEGM, Universit` a di Udine, Via delle Scienze 208, I-33100 Udine, Italy Consorzio RFX, Associazione EURATOM-ENEA sulla Fusione, Corso Stati Uniti 4, I-35127 Padova, Italy Ass. EURATOM/ENEA/CREATE, DAEIMI, Universit`a di Cassino, Via Di Biasio 43, I-03043 Cassino (FR), Italy b
c
Available online 20 October 2005
Abstract The new RFX load assembly characterized by a thin copper shell requires using an active control system for the plasma equilibrium. A new version of the CREATE-L RFP plasma response model was developed as a tool for the control system design and to crosscheck results with another linearized model, derived by a different technique. The RFP version of the MAXFEA equilibrium code adapted to the new load assembly was used to produce benchmark data for validation tests. The results of comparisons performed imposing either the active coil currents or voltages are presented in the paper. © 2005 Elsevier B.V. All rights reserved.
1. Introduction At present, the design of control systems in fusion devices is generally based on linearized models of the plasma response at different equilibria. The CREATEL model, derived by linearizing the MHD equilibrium equation and Ohm’s law in the active and passive conductors and in the plasma, has been extended to reversed field pinch (RFP) plasmas and already successfully applied to the old RFX magnetic configuration [1,2]. The new RFX load assembly includes now a new 3 mm thick copper shell, close fitting to the vacuum vessel, which will allow to perform active control ∗ Corresponding author. Tel.: +39 0432 558291; fax: +39 0432 558292. E-mail address: [email protected] (P. Bettini).
0920-3796/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.fusengdes.2005.06.170
experiments by means of a set of 192 saddle coils surrounding the torus [3]. The much shorter time constant of the shell requires also an active control of the plasma equilibrium. One of the aims of this paper is to apply the CREATE-L plasma response model also to study in details the effects of the structural modifications of RFX machine on the open loop plasma response. Since experimental data are not yet available, a two dimensional FEM code (MAXFEA), solving the ideal MHD free boundary problem in axisymmetric geometry, has been used to provide a set of equilibrium reference data, also in terms of coil currents and virtual measures from pick-up and flux loop probes [4]. As a first step in the validation procedure, in particular to assess the accuracy of the electromagnetic model of the passive structures implemented in the codes, a crosscheck has been carried out on a shot without plasma. As a sec-
574
P. Bettini et al. / Fusion Engineering and Design 74 (2005) 573–578
ond step, complete comparisons have been performed in the case of pulses with plasma both imposing the active coil currents and the voltages as inputs.
2. The new RFX magnetic poloidal field system The poloidal magnetic field system of RFX (Fig. 1) includes the field shaping winding (F), the magnetizing winding (M), the vacuum vessel (V) and the thin copper stabilizing shell (S). An integrated system of magnetic field pick-up and flux probes, installed both on the outer and inner surface of vessel and shell, will provide signals to drive the field shaping coil power supply. The thin copper shell provides the plasma equilibrium and limits the growth of fast instabilities on the short time scale by means of its induced currents. As these currents diffuse (time constant 50 ms), the plasma equilibrium shall be controlled by means of the F winding and MHD modes shall be controlled by means of the 192 saddle coils. The linearized CREATE-L plasma response model, used to predict the plasma current, shape and position response to voltages applied to the control coils and to
known βp and li variations, is derived by linearizing the MHD equilibrium equation and Ohm’s law in the active and passive conductors and in the plasma. The plasma is assumed to be in permanent MHD equilibrium and to be described by a small number of global parameters s(t). The profiles of the poloidal flux current density f(ψ) and the kinetic pressure p(ψ) are given as known quantities, adopting the expressions described in [2]. The dynamic evolution of currents flowing in conductors is described in the linearized form: L∗
d(␦I) d(␦s) + R ␦I + LS = B1 ␦U dt dt
(1)
where U is the active coil voltage vector, B1 is a matrix with appropriate zeros and ones to apply voltages to the active coils, LS = ∂Ψ /∂s, and L* = ∂Ψ /∂I is the modified inductance matrix. For control purposes, it may be useful to recast (1) in standard state-space form, as: d(␦I) d(␦s) = A ␦I + B ␦U + ES dt dt with obvious definitions.
Fig. 1. Poloidal magnetic field system: probes (vloop, Bp), vessel, shell, M (1/20) and F (1/8) coils.
(2)
P. Bettini et al. / Fusion Engineering and Design 74 (2005) 573–578
575
A number of outputs has been defined in the form: ␦y = CY ␦I + FS ␦s
(3)
for magnetic fields or fluxes variations produced at given RFX positions: • Poloidal flux density probes Integrated System of Internal Sensors (ISIS) array: Eight pick-up probes located inside the vessel; INT array: Eight pick-up probes located outside the vessel, on the inner surface of the shell; EXT array: Eight pick-up probes located on the outer surface of the shell; • Flux-loop probes: Eight axisymmetric loops located on the outer surface of the vessel. CREATE-L model is computed using a mesh made of 26,707 second-order triangular elements with 53,500 nodes. The mesh used by MAXFEA consists of 26,780 first-order triangular mesh with 13,402 nodes.
3. Numerical validation on shots without plasma
Fig. 3. Time evolution of the m = 1 poloidal mode of the shell toroidal current density (—: CREATE-L, - - -: reference 50 ms exponential).
In Fig. 2 we present a comparison of the current density distribution in the new shell, along the poloidal angle, predicted by the two models at two time instants. The agreement between the two models is good both at t = 0.14 s, corresponding to the peak induced current, where a clear m = 1 mode appears, and at t = 0.64
A shot without plasma, programmed in the previous experimental phase of RFX (#14,209) to apply a vertical field step of about 60 mT, was chosen to assess the electromagnetic modelling of the passive structures.
Fig. 2. Comparison of the current density distribution in the shell elements (o: MAXFEA, ×: CREATE-L).
Fig. 4. Time evolution of reference and reconstructed signals (o: CREATE-L, —: MAXFEA). Top: vessel current, Middle: poloidal field at ISIS pick-up coil #2. Bottom: poloidal flux at flux-loop probe #5.
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P. Bettini et al. / Fusion Engineering and Design 74 (2005) 573–578
s in a phase where a more complex mode structure is observed. In Fig. 3 the time evolution of the m = 1 poloidal mode of the shell toroidal current density predicted by the CREATE-L model is presented along with an exponential fit with a time constant T = 56 ms. The time evolutions of vessel current, poloidal induction flux and field calculated by MAXFEA in selected positions are compared with the corresponding quantities computed by the model in Fig. 4.
4. Numerical validation on shots with plasma In order to perform comparisons in the case of pulses with plasma the system matrices of the linearized model have been obtained considering an equilibrium consistent with the integral parameters measured in a typical RFX pulse (Ip = 720 kA, li = 1.3, fa = 1.41 Tm and fb = −0.14 Tm, α = 11.32, where li is the normalized internal inductance, fa and fb are the poloidal current density flux function at the axis and the boundary, respectively, α is a parameter related to the equilibrium plasma current profile). Since the plasma is open loop stable some tests have been carried out imposing the evolutions of the active coil (M, F) currents so as to evaluate the accuracy of the coupling of plasma and active coils with the passive conducting structures. This approach allows to neglect all the effects related to uncertainty of the electrical circuit parameters, such as the additional inductances or resistances which are series connected to each couple of F coils, or other stray parameters. Fig. 5 presents the time evolution of the poloidal field inside the vessel (ISIS array), the poloidal flux on its outer surface and the poloidal field “sensed” by some pick-up coils of the INT and EXT arrays as well. In view of using the model as a tool for the design of plasma current and equilibrium control system, validation tests imposing voltage waveforms have then been started. However, since no mechanism accounting for the plasma resistive flux consumption is present in the model, it was preferred to maintain the plasma current as an input term. It should be now reminded that another linearized model of the plasma response in the new RFX load assembly was derived by applying a different perturbation technique in MAXFEA and it was already used for the design of a first equilibrium controller [5]. Our aim is to make available a second model
Fig. 5. Time evolutions of reference and reconstructed signals (o: CREATE-L, —: MAXFEA). From top to bottom: poloidal field at ISIS pick-up coil #2, poloidal flux at flux-loop probe #5, poloidal field at INT pick-up coil #6, poloidal field at EXT pick-up coil #3.
to crosscheck results before the final implementation of the controller. The electrical topology of the RFX poloidal field circuit (Fig. 6) is rather complex and it required solving some numerical problems in the nonlinear simulations, due to the high value of the ground resistances, which increases the numerical stiffness of the system. To compare the two models while assuring the plasma equilibrium, it was also decided to use voltage waveforms computed in non-linear simulations with the already available plasma equilibrium controller. Results in terms of poloidal field at different positions are satisfactory as shown in Fig. 7, where two examples from ISIS and EXT arrays are given. A good agreement was also consistently observed between the outer coil currents, which most contribute to the equilibrium field configuration, as it can be seen in the same figure. Further tests are now under way to improve the results for the inner coils for which the calculation of the coupling terms is more critical, for instance due to
P. Bettini et al. / Fusion Engineering and Design 74 (2005) 573–578
577
Fig. 6. Poloidal field circuit.
the narrow shape and the proximity to the magnetizing winding.
5. Conclusions A version of the CREATE-L model of RFP plasma response was developed for the new RFX load assembly. Validation tests were carried out benchmarking the model outputs with the corresponding data provided by a non-linear FEM MHD equilibrium code both in the case of shots without and with plasma. The comparisons made either imposing the active coil currents or voltages showed a satisfactory agreement between the time evolutions of the poloidal magnetic field and flux variations. Preliminary results are encouraging and further tests with imposed voltages are foreseen in view of the design of the plasma current and equilibrium control system.
References Fig. 7. Time evolutions of reference and reconstructed signals (o: CREATE-L, —: MAXFEA). From top to bottom: ISIS pick-up coil #2, EXT array pick-up coil #7, field shaping F6 and F8 currents.
[1] R. Albanese, F. Villone, The linearized CREATE-L plasma response model for the control of current, position and shape in tokamaks, Nucl. Fusion 38 (5) (1998) 723–738.
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P. Bettini et al. / Fusion Engineering and Design 74 (2005) 573–578
[2] R. Albanese, P. Bettini, M. Guarnieri, G. Marchiori, F. Villone, Linearized models for RFX configurations, Fusion Eng. Des. 56/57 (2002) 733–738. [3] P. Sonato, G. Chitarin, P. Zaccaria, F. Gnesatto, S. Ortolani, A. Buffa, et al., Machine modifications for active MHD control in RFX, Fusion Eng. Des. 66/68 (2003) 161–168.
[4] P. Bettini, M. Cavinato, G. Marchiori, 2D non-linear model of Reversed Field Pinch plasma evolution, Nucl. Fusion 43 (2003) 119–129. [5] M. Cavinato, G. Marchiori, Design of the new RFX equilibrium active control system, presented at the 23rd SOFT, September 2004.