- Email: [email protected]
Linearized models for a new magnetic control in MAST
Fusion Engineering and Design 88 (2013) 1091–1096
Contents lists available at ScienceDirect
Fusion Engineering and Design journal homepage: www.elsevier.com/locate/fusengdes
Linearized models for a new magnetic control in MAST G. Artaserse a,∗ , F. Maviglia b , R. Albanese b , G.J. McArdle c , L. Pangione c a
Associazione Euratom-ENEA sulla Fusione, Via Enrico Fermi 45, I-00044 Frascati (RM), Italy Associazione Euratom-ENEA-CREATE sulla Fusione, Via Claudio 21, I-80125 Napoli, Italy c EURATOM/CCFE Fusion Association, Culham Science Centre, Abingdon, Oxon, OX14 3DB, UK b
h i g h l i g h t s
We applied linearized models for a new magnetic control on MAST tokamak. A suite of procedures, conceived to be machine independent, have been used. We carried out model-based simulations, taking into account eddy currents effects. Comparison with the EFIT flux maps and the experimental magnetic signals are shown. A current driven model for the dynamic simulations of the experimental data have been performed.
a r t i c l e
i n f o
Article history: Available online 4 February 2013 Keywords: MAST Linearized model Tokamak Magnetic control Equilibrium State space model
a b s t r a c t The aim of this work is to provide reliable linearized models for the design and assessment of a new magnetic control system for MAST (Mega Ampère Spherical Tokamak) using rtEFIT, which can easily be exported to MAST Upgrade. Linearized models for magnetic control have been obtained using the 2D axisymmetric finite element code CREATE L. MAST linearized models include equivalent 2D axisymmetric schematization of poloidal field (PF) coils, vacuum vessel, and other conducting structures. A plasmaless and a double null configuration have been chosen as benchmark cases for the comparison with experimental data and EFIT reconstructions. Good agreement has been found with the EFIT flux map and the experimental signals coming from magnetic probes with only few mismatches probably due to broken sensors. A suite of procedures (equipped with a user friendly interface to be run even remotely) to provide linearized models for magnetic control is now available on the MAST linux machines. A new current driven model has been used to obtain a state space model having the PF coil currents as inputs. Dynamic simulations of experimental data have been carried out using linearized models, including modelling of the effects of the passive structures, showing a fair agreement. The modelling activity has been useful also to reproduce accurately the interaction between plasma current and radial position control loops. © 2013 Euratom-ENEA Association sulla Fusione. Published by Elsevier B.V. All rights reserved.
1. Introduction MAST is a spherical tokamak (ST), which presents a compact “cored apple” shape and a lower aspect ratio, up down symmetric usually operating in a double null divertor (DND) magnetic configuration. Plasmas with elongated cross section are vertically unstable hence subjected to vertical displacement events (VDEs) which affecting the tokamaks operation [1,2]. MAST is equipped with extensive and advanced diagnostics, with a digital control system which includes density feedback control and a novel optical system for plasma radial position control. Real time equilibrium reconstruction, based on rtEFIT [3], has been deployed in the
∗ Corresponding author. Tel.: +39 06 9400 5906; fax: +39 06 9400 5735. E-mail address: [email protected] (G. Artaserse).
control system. Reliable linearized models are necessary for the design and assessment of a new magnetic control system for MAST using rtEFIT, which can easily be exported to MAST Upgrade. In Section 2 of this paper the MAST machine modelling activity is introduced. The porting of the XSCTools to MAST is treated in Section 3. Benchmark cases are analyzed in Section 4 with dynamic simulations including eddy currents, comparing the model predictions with flux map reconstructions and experimental magnetic signals. Summary and conclusions are given in Section 5. 2. MAST modelling MAST has a central solenoid (P1), which provides the magnetic flux used to control the plasma current, and a set of up down symmetric PF coil sets (as shown in Fig. 1) connected by six independent circuits. The P2 coil can be used to achieve the desired DND
0920-3796/$ – see front matter © 2013 Euratom-ENEA Association sulla Fusione. Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fusengdes.2012.12.033
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G. Artaserse et al. / Fusion Engineering and Design 88 (2013) 1091–1096
finite element method (FEM) code CREATE L [4]. The electromagnetic model used to obtain the linearized model includes an equivalent 2D axisymmetric modelling of PF circuits, vacuum vessel, and other conducting structures. Both active and passive coils are schematized with a dedicated set of circuit equations coupled to electromagnetic and magneto hydrodynamic (MHD) equations, in order to take into account the plasma presence. As reported in [4], starting from a nonlinear form of the problem, the dynamics of active and passive conductors is determined by circuit equations (where is the poloidal flux per radian linked to each circuit and R is the resistance matrix associated to each circuit):
d /dt + R i = u
(1)
T
[ , y] = (i, w)
linearizing Eq. (1) and assuming that the perturbed quantities are iı = i0 + i, uı = u0 + u, wı = w0 + w, yı = y0 + y (where i0 , u0 , w0 and y0 are the reference values) we obtain
L∗ di/dt + R i = u − L∗E dw/dt
(2)
y = Ci + Dw
Fig. 1. MAST upper part of poloidal field coils system.
configuration and compensate the stray field from the solenoid. The flux from the startup coil P3 is used to obtain the poloidal field null and then the breakdown. A vertical field is applied from the P4 and P5 coils, for the radial position control. Shape and elongation depend both on the plasma internal profile and how the total vertical field current is balanced between P4 and P5. The P6 coils are connected in up-down anti-series and used exclusively for vertical position feedback control. Metallic structures sources of passive currents in MAST (see Fig. 2) are the vessel, the mechanical supports and the PF coil cases, which are continuous. The modelling activity carried out for this study uses a linearized dynamic model of MAST provided by 2D axisymmetric
where L∗ = ∂ /∂i, L∗E = ∂ /∂w, C = ∂y/∂i, D = ∂y/∂w. The linearized model (2) describes the electromagnetic behaviour of plasma surrounded by conducting structures and is valid in the neighbourhood of an equilibrium point. The state variables i are the coils (active and passive currents) and plasma current Ip ; the inputs u are the applied voltages (or currents), whereas poloidal beta ˇP and internal inductance i play the role of disturbances (non-controllable inputs w). The outputs y are field and flux values and some basic plasma parameters of interest (plasma current moments, X point position, gaps, strike points, triangularity, elongation, etc.). Assuming: x = i, A = −(L∗ )
−1
R, B = (L∗ )
−1
, E = −(L∗ )
−1 ∗ LE
we obtain the state space form of (2) [4]:
dx/dt = A x + B u + E dw/dt y = Cx + Dw
(3)
L* is an inductance matrix modified by the presence of the plasma linearization carried out using any MHD equilibrium code using incremental ratios (e.g., /i) or Jacobian matrix (available with Newton’s method). 3. Porting of XSCTools on MAST
Fig. 2. MAST upper part of the conducting structures.
CREATE L code is part of a suite of procedures, the XSCTools (eXtreme Shape Controller Tools) [5] written in MatLab with a Graphical User Interface (GUI), to design and validate plasma shape controllers. They have been designed to be machine independent (to be used and ‘easily’ ported on any tokamaks) by using ASCII files containing a standard description of the tokamak. Custom versions of the XSCTools for MAST both in Windows and Linux environments to be run also remotely have been developed and installed. To extend the XSCTools on MAST we consider: (i) a first order FEM mesh with 51,016 triangles and 25,585 nodes (built using the pdetool of MatLab). Note that we designed a less detailed first wall especially in the proximity of the central rod to avoid unnecessary computational efforts; (ii) circuit schematization of poloidal field coil connections and passive structures; and (iii) a subset of reliable experimental magnetic signals (providing magnetic field and flux) fit in the least square sense to obtain the plasma current density profile parameters for a given experimental configuration.
G. Artaserse et al. / Fusion Engineering and Design 88 (2013) 1091–1096
1093
Fig. 4. Equilibrium configuration calculated by the XSCTools for MAST pulse #24542 @0.27 s.
close to the central column and in the low field side region. As for the flux measurements, the reconstruction error is less than 1%.
Fig. 3. Comparison between the reconstructed and the experimental measurements: (a) pick up coils in the low field side region and (b) flux loops. The arrows show mismatches probably due to broken sensors.
4.1.2. Plasma equilibrium case #[email protected] s The benchmark pulse is a DND configuration, with upper dominant X point. Fig. 4 shows the equilibrium provided by the XSCTools, and in particular the reconstructed boundary (outermost closed magnetic surface) at the time instant of interest. A good agreement has been found with the EFIT flux map (see Fig. 5) interpolated into the XSCTools mesh. Still a good agreement with the field and fluxes signals with the same few mismatches of plasmaless case. The reconstruction error for a large amount of pick up coils is <8% and for the fluxes measurements still the error less than 1%.
4. Benchmark cases 4.2. Dynamic simulations To provide reliable linearized models for a new magnetic control in MAST we firstly tried to asses if the XSCTools were properly implemented. As validating procedure a plasmaless and a DND configuration have been chosen for the comparison with experimental data and EFIT reconstructions. Once established the reliability of the reconstructed equilibrium point we linearized and carried out the dynamic simulations of experimental data including modelling of the effects of the passive structures. 4.1. Magnetic reconstruction 4.1.1. Plasmaless case #[email protected] s Fig. 3 shows the comparison between the reconstructed and the experimental measurements for the whole set of signals provided by pick up coils and flux loops, not only those of the subset chosen for the least square fit. Good agreement has been found with the experimental signals coming from magnetic probes with only few mismatches probably due to broken sensors. The reconstruction error for many of them is less than 10% for the pick-up coils, both
Once obtained a reliable equilibrium reconstruction we could proceed to linearize the model in the neighbourhood of the equilibrium point. The open loop dynamic simulations have been carried out using two different models: one with eddy and the other without eddy currents to assess the benefits of inserting the eddy currents into the model. To simulate the experimental data using the model without eddy currents we simply used the output matrix (C) of the linearized model in (3), while for the simulation using the model with eddy currents we used a current driven model [6] to obtain a state space model having the PF coils currents as inputs in addition to the disturbance terms. The need to use a current driven model comes from that the voltage measurements in MAST are too noisy to be used in the dynamic simulations. In terms of formulation rewriting the circuit equation (2) in a simple way, we obtain ˙ L x˙ + R x = S u − LE w
(4)
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G. Artaserse et al. / Fusion Engineering and Design 88 (2013) 1091–1096
obtaining a new state space model [6], where pe is the new state variable: ˜ e + B ˜ p˙ e = Ap ˜ e + D ˜ y = Cp
(7)
−1 −1 −1 ˜ ˜ = −Re L−1 ˜ ˜ assuming A e , B = [Re Le Lea Re Le LEe ], C = Ce Le , D = −1 −1 T [Ca − Ce Le Lea D − Ce Le LEe ], = [xa w] . Note that contain the currents in the poloidal field circuits xa and the disturbances w as the new inputs of the system. The current driven model (7) has a lower order than the original one (pe has the same dimension of xe ) because it totally ignores the electric dynamics on the poloidal coils and assumes that the value of the currents can be arbitrarily imposed. Moreover the growth rate associated to the unstable mode of the system (7) of the matrix ˜ = −Re L−1 A e is bigger than that one of system (4) of the matrix A = −1 −(L∗ ) R, due the reduced order of the new system A > A˜ .
4.2.1. Plasmaless case #1 To understand the contribution of using the eddy currents in the CREATE L linearized model, we have chosen a calibration pulse. In shot #24936 the P2U, P3U, P4U, P5U currents are individually calibrated. Dynamic simulations have been carried out using y = C x for the case without eddy currents. To insert the eddy currents effects on the dynamic predictions we use a simplified formulation of (7), since in this case the absence of plasma the disturbances term w disappear:
Fig. 5. Comparison of the flux maps provided by XSCTools and EFIT for MAST pulse #24542 @0.27 s.
where following, we divide the contribution of the states x in passive currents xe and active currents xa , x = [xe xa ]T . In this way we get:
Le
Lea
Lae
La
x˙ e x˙ a
+
Re
0
0
Ra
xe xa
=
0 Sa
u−
LEe
˙ w
(5)
LEa
then we define a new state vector pe associated to the passive currents, pe = Le xe + Lea xa + LEe w which yields the following new system
˙ p˙ e = Le x˙ e + Lea x˙ a + LEe w −1 −1 xe = L−1 e pe − Le Lea xa − Le LEe w
(6)
˜ 0 pe + B ˜ 0 xa p˙ e = A ˜ 0 pe + D ˜ 0 xa y=C
(8)
−1 −1 ˜ 0 = −Re L−1 ˜ ˜ ˜ where A e , B0 = [Re Le Lea ], C0 = Ce Le , D0 = [Ca − Ce L−1 L ]. ea e Fig. 6 shows the linearized model predictions of some of the magnetic signals versus the experimental one, in both cases with and without taking into the modelling the effects of parasitic currents in the metallic structures facing the plasma. We choose a simulation time windows of 40 ms when only P2U was acting. We discovered a fair agreement using the model with eddy currents, so the decision for the further open loop analysis to take into account them into the model. Even the transition (see Fig. 6 on the left) from ramp up to flat top of P2U currents is well reproduced by the linearized model.
Fig. 6. MAST open loop simulation of plasmaless case #24936 with (grey line, red in the web version) and without (upward pointing triangles grey line, green in the web version) eddy currents: comparison with experimental (dark line, blue in the web version) signals coming from a pick-up coil (left) and a flux loop (right).
G. Artaserse et al. / Fusion Engineering and Design 88 (2013) 1091–1096
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Fig. 7. MAST open loop simulation dry run #24007 with (grey line, red in the web version) eddy currents: comparison with experimental (dark line, blue in the web version) signals coming from a pick-up coil (left) and a flux loop (right).
Fig. 8. MAST open loop simulation of plasma run case #12758 with (grey line, red in the web version) eddy currents: comparison with various experimental (dark line, blue in the web version) signals: (a) pick-up coils, (b) PF coil cases, (c) plasma current, and (d) plasma radial position.
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G. Artaserse et al. / Fusion Engineering and Design 88 (2013) 1091–1096
4.2.2. Plasmaless case #2 We also analyzed the standard MAST dry run #24007 already used for magneto-static calibrations in Section 4.1. This time we carried out dynamic simulations using system (8). We also predict the currents flowing in the PF circuits, which are measured (P2, P3, P4 and P5) and here considered as passive conductors. Fig. 7 shows a good agreement between linearized model predictions and experimental data. Some discrepancies, especially for the vertical magnetic signals (see Fig. 7 on the right) in the low field side region, due to some passive structures model inaccuracies (e.g. equivalent geometry, material properties, 3D effects), are under investigation. 4.2.3. Plasma run case, DND #12758 The chosen equilibrium reconstruction is 0.15 s, when fast phenomena do not occur and before the observed plasma current oscillation due to deliberately driven radial position oscillation without compensation so as to measure the coupling with Ip . We carried out dynamic simulations using system (7) taking into accounts the eddy currents effects and disturbances. We simulate the magnetic signals, plasma current, eddy currents flowing in the PF coils cases and plasma column radial position. Fig. 8 shows a fair agreement between the simulations and the experimental data, reproducing accurately the interaction between plasma current and radial position (see Fig. 8c and d) control loops. 5. Summary and conclusions Linearized models for a new magnetic control on MAST have been obtained using a 2D axisymmetric modelling of PF circuits, vacuum vessel, and other conducting structures.
The suite of procedures XSCTools, conceived to be machine independent, have been applied to MAST analysis and installed on Linux environment so as to be run both on-site and remotely. These set of procedures can be easily ported on MAST Upgrade. The benchmark cases show good agreement of the model-based simulations with the EFIT flux maps and the experimental magnetic signals. Only few mismatches have been noticed: some are probably due to broken, misaligned or not calibrated measurement; others, currently under investigations are likely due to inaccurate modelling of the geometry or the material properties of some conducting structures. The authors applied a current driven model to obtain a state space model for the dynamic simulations of the experimental data. The linearized models including the effects of the passive structures show fair agreement with the experimental data. The modelling activity has been useful also to reproduce accurately the interaction between plasma current and radial position control loops. Better results could be achieved providing a more detailed schematization of the passive structures. References [1] R. Albanese, M. Mattei, F. Villone, Nuclear Fusion 44 (2004). [2] A. Portone, R. Albanese, R. Fresa, M. Mattei, G. Rubinacci, F. Villone, Fusion Engineering and Design 74 (2005). [3] J.R. Ferron, M.L. Walker, L.L. Lao, H.E. St. John, D.A. Humphreys, J.A. Leuer, Nuclear Fusion 38 (1998). [4] R. Albanese, F. Villone, Nuclear Fusion 38 (1998). [5] G. De Tommasi, R. Albanese, G. Ambrosino, M. Ariola, M. Mattei, A. Pironti, et al., IEEE Transactions on Plasma Science 35 (3) (2007). [6] M. Ariola, A. Pironti, Advanced in Fusion Control, 1st ed., Springer, 2008, pp. 73–75.
Contents lists available at ScienceDirect
Fusion Engineering and Design journal homepage: www.elsevier.com/locate/fusengdes
Linearized models for a new magnetic control in MAST G. Artaserse a,∗ , F. Maviglia b , R. Albanese b , G.J. McArdle c , L. Pangione c a
Associazione Euratom-ENEA sulla Fusione, Via Enrico Fermi 45, I-00044 Frascati (RM), Italy Associazione Euratom-ENEA-CREATE sulla Fusione, Via Claudio 21, I-80125 Napoli, Italy c EURATOM/CCFE Fusion Association, Culham Science Centre, Abingdon, Oxon, OX14 3DB, UK b
h i g h l i g h t s
We applied linearized models for a new magnetic control on MAST tokamak. A suite of procedures, conceived to be machine independent, have been used. We carried out model-based simulations, taking into account eddy currents effects. Comparison with the EFIT flux maps and the experimental magnetic signals are shown. A current driven model for the dynamic simulations of the experimental data have been performed.
a r t i c l e
i n f o
Article history: Available online 4 February 2013 Keywords: MAST Linearized model Tokamak Magnetic control Equilibrium State space model
a b s t r a c t The aim of this work is to provide reliable linearized models for the design and assessment of a new magnetic control system for MAST (Mega Ampère Spherical Tokamak) using rtEFIT, which can easily be exported to MAST Upgrade. Linearized models for magnetic control have been obtained using the 2D axisymmetric finite element code CREATE L. MAST linearized models include equivalent 2D axisymmetric schematization of poloidal field (PF) coils, vacuum vessel, and other conducting structures. A plasmaless and a double null configuration have been chosen as benchmark cases for the comparison with experimental data and EFIT reconstructions. Good agreement has been found with the EFIT flux map and the experimental signals coming from magnetic probes with only few mismatches probably due to broken sensors. A suite of procedures (equipped with a user friendly interface to be run even remotely) to provide linearized models for magnetic control is now available on the MAST linux machines. A new current driven model has been used to obtain a state space model having the PF coil currents as inputs. Dynamic simulations of experimental data have been carried out using linearized models, including modelling of the effects of the passive structures, showing a fair agreement. The modelling activity has been useful also to reproduce accurately the interaction between plasma current and radial position control loops. © 2013 Euratom-ENEA Association sulla Fusione. Published by Elsevier B.V. All rights reserved.
1. Introduction MAST is a spherical tokamak (ST), which presents a compact “cored apple” shape and a lower aspect ratio, up down symmetric usually operating in a double null divertor (DND) magnetic configuration. Plasmas with elongated cross section are vertically unstable hence subjected to vertical displacement events (VDEs) which affecting the tokamaks operation [1,2]. MAST is equipped with extensive and advanced diagnostics, with a digital control system which includes density feedback control and a novel optical system for plasma radial position control. Real time equilibrium reconstruction, based on rtEFIT [3], has been deployed in the
∗ Corresponding author. Tel.: +39 06 9400 5906; fax: +39 06 9400 5735. E-mail address: [email protected] (G. Artaserse).
control system. Reliable linearized models are necessary for the design and assessment of a new magnetic control system for MAST using rtEFIT, which can easily be exported to MAST Upgrade. In Section 2 of this paper the MAST machine modelling activity is introduced. The porting of the XSCTools to MAST is treated in Section 3. Benchmark cases are analyzed in Section 4 with dynamic simulations including eddy currents, comparing the model predictions with flux map reconstructions and experimental magnetic signals. Summary and conclusions are given in Section 5. 2. MAST modelling MAST has a central solenoid (P1), which provides the magnetic flux used to control the plasma current, and a set of up down symmetric PF coil sets (as shown in Fig. 1) connected by six independent circuits. The P2 coil can be used to achieve the desired DND
0920-3796/$ – see front matter © 2013 Euratom-ENEA Association sulla Fusione. Published by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.fusengdes.2012.12.033
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G. Artaserse et al. / Fusion Engineering and Design 88 (2013) 1091–1096
finite element method (FEM) code CREATE L [4]. The electromagnetic model used to obtain the linearized model includes an equivalent 2D axisymmetric modelling of PF circuits, vacuum vessel, and other conducting structures. Both active and passive coils are schematized with a dedicated set of circuit equations coupled to electromagnetic and magneto hydrodynamic (MHD) equations, in order to take into account the plasma presence. As reported in [4], starting from a nonlinear form of the problem, the dynamics of active and passive conductors is determined by circuit equations (where is the poloidal flux per radian linked to each circuit and R is the resistance matrix associated to each circuit):
d /dt + R i = u
(1)
T
[ , y] = (i, w)
linearizing Eq. (1) and assuming that the perturbed quantities are iı = i0 + i, uı = u0 + u, wı = w0 + w, yı = y0 + y (where i0 , u0 , w0 and y0 are the reference values) we obtain
L∗ di/dt + R i = u − L∗E dw/dt
(2)
y = Ci + Dw
Fig. 1. MAST upper part of poloidal field coils system.
configuration and compensate the stray field from the solenoid. The flux from the startup coil P3 is used to obtain the poloidal field null and then the breakdown. A vertical field is applied from the P4 and P5 coils, for the radial position control. Shape and elongation depend both on the plasma internal profile and how the total vertical field current is balanced between P4 and P5. The P6 coils are connected in up-down anti-series and used exclusively for vertical position feedback control. Metallic structures sources of passive currents in MAST (see Fig. 2) are the vessel, the mechanical supports and the PF coil cases, which are continuous. The modelling activity carried out for this study uses a linearized dynamic model of MAST provided by 2D axisymmetric
where L∗ = ∂ /∂i, L∗E = ∂ /∂w, C = ∂y/∂i, D = ∂y/∂w. The linearized model (2) describes the electromagnetic behaviour of plasma surrounded by conducting structures and is valid in the neighbourhood of an equilibrium point. The state variables i are the coils (active and passive currents) and plasma current Ip ; the inputs u are the applied voltages (or currents), whereas poloidal beta ˇP and internal inductance i play the role of disturbances (non-controllable inputs w). The outputs y are field and flux values and some basic plasma parameters of interest (plasma current moments, X point position, gaps, strike points, triangularity, elongation, etc.). Assuming: x = i, A = −(L∗ )
−1
R, B = (L∗ )
−1
, E = −(L∗ )
−1 ∗ LE
we obtain the state space form of (2) [4]:
dx/dt = A x + B u + E dw/dt y = Cx + Dw
(3)
L* is an inductance matrix modified by the presence of the plasma linearization carried out using any MHD equilibrium code using incremental ratios (e.g., /i) or Jacobian matrix (available with Newton’s method). 3. Porting of XSCTools on MAST
Fig. 2. MAST upper part of the conducting structures.
CREATE L code is part of a suite of procedures, the XSCTools (eXtreme Shape Controller Tools) [5] written in MatLab with a Graphical User Interface (GUI), to design and validate plasma shape controllers. They have been designed to be machine independent (to be used and ‘easily’ ported on any tokamaks) by using ASCII files containing a standard description of the tokamak. Custom versions of the XSCTools for MAST both in Windows and Linux environments to be run also remotely have been developed and installed. To extend the XSCTools on MAST we consider: (i) a first order FEM mesh with 51,016 triangles and 25,585 nodes (built using the pdetool of MatLab). Note that we designed a less detailed first wall especially in the proximity of the central rod to avoid unnecessary computational efforts; (ii) circuit schematization of poloidal field coil connections and passive structures; and (iii) a subset of reliable experimental magnetic signals (providing magnetic field and flux) fit in the least square sense to obtain the plasma current density profile parameters for a given experimental configuration.
G. Artaserse et al. / Fusion Engineering and Design 88 (2013) 1091–1096
1093
Fig. 4. Equilibrium configuration calculated by the XSCTools for MAST pulse #24542 @0.27 s.
close to the central column and in the low field side region. As for the flux measurements, the reconstruction error is less than 1%.
Fig. 3. Comparison between the reconstructed and the experimental measurements: (a) pick up coils in the low field side region and (b) flux loops. The arrows show mismatches probably due to broken sensors.
4.1.2. Plasma equilibrium case #[email protected] s The benchmark pulse is a DND configuration, with upper dominant X point. Fig. 4 shows the equilibrium provided by the XSCTools, and in particular the reconstructed boundary (outermost closed magnetic surface) at the time instant of interest. A good agreement has been found with the EFIT flux map (see Fig. 5) interpolated into the XSCTools mesh. Still a good agreement with the field and fluxes signals with the same few mismatches of plasmaless case. The reconstruction error for a large amount of pick up coils is <8% and for the fluxes measurements still the error less than 1%.
4. Benchmark cases 4.2. Dynamic simulations To provide reliable linearized models for a new magnetic control in MAST we firstly tried to asses if the XSCTools were properly implemented. As validating procedure a plasmaless and a DND configuration have been chosen for the comparison with experimental data and EFIT reconstructions. Once established the reliability of the reconstructed equilibrium point we linearized and carried out the dynamic simulations of experimental data including modelling of the effects of the passive structures. 4.1. Magnetic reconstruction 4.1.1. Plasmaless case #[email protected] s Fig. 3 shows the comparison between the reconstructed and the experimental measurements for the whole set of signals provided by pick up coils and flux loops, not only those of the subset chosen for the least square fit. Good agreement has been found with the experimental signals coming from magnetic probes with only few mismatches probably due to broken sensors. The reconstruction error for many of them is less than 10% for the pick-up coils, both
Once obtained a reliable equilibrium reconstruction we could proceed to linearize the model in the neighbourhood of the equilibrium point. The open loop dynamic simulations have been carried out using two different models: one with eddy and the other without eddy currents to assess the benefits of inserting the eddy currents into the model. To simulate the experimental data using the model without eddy currents we simply used the output matrix (C) of the linearized model in (3), while for the simulation using the model with eddy currents we used a current driven model [6] to obtain a state space model having the PF coils currents as inputs in addition to the disturbance terms. The need to use a current driven model comes from that the voltage measurements in MAST are too noisy to be used in the dynamic simulations. In terms of formulation rewriting the circuit equation (2) in a simple way, we obtain ˙ L x˙ + R x = S u − LE w
(4)
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G. Artaserse et al. / Fusion Engineering and Design 88 (2013) 1091–1096
obtaining a new state space model [6], where pe is the new state variable: ˜ e + B ˜ p˙ e = Ap ˜ e + D ˜ y = Cp
(7)
−1 −1 −1 ˜ ˜ = −Re L−1 ˜ ˜ assuming A e , B = [Re Le Lea Re Le LEe ], C = Ce Le , D = −1 −1 T [Ca − Ce Le Lea D − Ce Le LEe ], = [xa w] . Note that contain the currents in the poloidal field circuits xa and the disturbances w as the new inputs of the system. The current driven model (7) has a lower order than the original one (pe has the same dimension of xe ) because it totally ignores the electric dynamics on the poloidal coils and assumes that the value of the currents can be arbitrarily imposed. Moreover the growth rate associated to the unstable mode of the system (7) of the matrix ˜ = −Re L−1 A e is bigger than that one of system (4) of the matrix A = −1 −(L∗ ) R, due the reduced order of the new system A > A˜ .
4.2.1. Plasmaless case #1 To understand the contribution of using the eddy currents in the CREATE L linearized model, we have chosen a calibration pulse. In shot #24936 the P2U, P3U, P4U, P5U currents are individually calibrated. Dynamic simulations have been carried out using y = C x for the case without eddy currents. To insert the eddy currents effects on the dynamic predictions we use a simplified formulation of (7), since in this case the absence of plasma the disturbances term w disappear:
Fig. 5. Comparison of the flux maps provided by XSCTools and EFIT for MAST pulse #24542 @0.27 s.
where following, we divide the contribution of the states x in passive currents xe and active currents xa , x = [xe xa ]T . In this way we get:
Le
Lea
Lae
La
x˙ e x˙ a
+
Re
0
0
Ra
xe xa
=
0 Sa
u−
LEe
˙ w
(5)
LEa
then we define a new state vector pe associated to the passive currents, pe = Le xe + Lea xa + LEe w which yields the following new system
˙ p˙ e = Le x˙ e + Lea x˙ a + LEe w −1 −1 xe = L−1 e pe − Le Lea xa − Le LEe w
(6)
˜ 0 pe + B ˜ 0 xa p˙ e = A ˜ 0 pe + D ˜ 0 xa y=C
(8)
−1 −1 ˜ 0 = −Re L−1 ˜ ˜ ˜ where A e , B0 = [Re Le Lea ], C0 = Ce Le , D0 = [Ca − Ce L−1 L ]. ea e Fig. 6 shows the linearized model predictions of some of the magnetic signals versus the experimental one, in both cases with and without taking into the modelling the effects of parasitic currents in the metallic structures facing the plasma. We choose a simulation time windows of 40 ms when only P2U was acting. We discovered a fair agreement using the model with eddy currents, so the decision for the further open loop analysis to take into account them into the model. Even the transition (see Fig. 6 on the left) from ramp up to flat top of P2U currents is well reproduced by the linearized model.
Fig. 6. MAST open loop simulation of plasmaless case #24936 with (grey line, red in the web version) and without (upward pointing triangles grey line, green in the web version) eddy currents: comparison with experimental (dark line, blue in the web version) signals coming from a pick-up coil (left) and a flux loop (right).
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Fig. 7. MAST open loop simulation dry run #24007 with (grey line, red in the web version) eddy currents: comparison with experimental (dark line, blue in the web version) signals coming from a pick-up coil (left) and a flux loop (right).
Fig. 8. MAST open loop simulation of plasma run case #12758 with (grey line, red in the web version) eddy currents: comparison with various experimental (dark line, blue in the web version) signals: (a) pick-up coils, (b) PF coil cases, (c) plasma current, and (d) plasma radial position.
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G. Artaserse et al. / Fusion Engineering and Design 88 (2013) 1091–1096
4.2.2. Plasmaless case #2 We also analyzed the standard MAST dry run #24007 already used for magneto-static calibrations in Section 4.1. This time we carried out dynamic simulations using system (8). We also predict the currents flowing in the PF circuits, which are measured (P2, P3, P4 and P5) and here considered as passive conductors. Fig. 7 shows a good agreement between linearized model predictions and experimental data. Some discrepancies, especially for the vertical magnetic signals (see Fig. 7 on the right) in the low field side region, due to some passive structures model inaccuracies (e.g. equivalent geometry, material properties, 3D effects), are under investigation. 4.2.3. Plasma run case, DND #12758 The chosen equilibrium reconstruction is 0.15 s, when fast phenomena do not occur and before the observed plasma current oscillation due to deliberately driven radial position oscillation without compensation so as to measure the coupling with Ip . We carried out dynamic simulations using system (7) taking into accounts the eddy currents effects and disturbances. We simulate the magnetic signals, plasma current, eddy currents flowing in the PF coils cases and plasma column radial position. Fig. 8 shows a fair agreement between the simulations and the experimental data, reproducing accurately the interaction between plasma current and radial position (see Fig. 8c and d) control loops. 5. Summary and conclusions Linearized models for a new magnetic control on MAST have been obtained using a 2D axisymmetric modelling of PF circuits, vacuum vessel, and other conducting structures.
The suite of procedures XSCTools, conceived to be machine independent, have been applied to MAST analysis and installed on Linux environment so as to be run both on-site and remotely. These set of procedures can be easily ported on MAST Upgrade. The benchmark cases show good agreement of the model-based simulations with the EFIT flux maps and the experimental magnetic signals. Only few mismatches have been noticed: some are probably due to broken, misaligned or not calibrated measurement; others, currently under investigations are likely due to inaccurate modelling of the geometry or the material properties of some conducting structures. The authors applied a current driven model to obtain a state space model for the dynamic simulations of the experimental data. The linearized models including the effects of the passive structures show fair agreement with the experimental data. The modelling activity has been useful also to reproduce accurately the interaction between plasma current and radial position control loops. Better results could be achieved providing a more detailed schematization of the passive structures. References [1] R. Albanese, M. Mattei, F. Villone, Nuclear Fusion 44 (2004). [2] A. Portone, R. Albanese, R. Fresa, M. Mattei, G. Rubinacci, F. Villone, Fusion Engineering and Design 74 (2005). [3] J.R. Ferron, M.L. Walker, L.L. Lao, H.E. St. John, D.A. Humphreys, J.A. Leuer, Nuclear Fusion 38 (1998). [4] R. Albanese, F. Villone, Nuclear Fusion 38 (1998). [5] G. De Tommasi, R. Albanese, G. Ambrosino, M. Ariola, M. Mattei, A. Pironti, et al., IEEE Transactions on Plasma Science 35 (3) (2007). [6] M. Ariola, A. Pironti, Advanced in Fusion Control, 1st ed., Springer, 2008, pp. 73–75.