Lidia Piron Consorzio RFX, Euratom-ENEA Association, and University of Padova, Italy
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Transcript of Lidia Piron Consorzio RFX, Euratom-ENEA Association, and University of Padova, Italy
9 October 2008, European Doctorate in Fusion Science and Engineering
Lidia Piron
Consorzio RFX, Euratom-ENEA Association, andUniversity of Padova, Italy
Experimental characterization and numerical modeling of the active control
of resistive MHD modes in RFX-mod
9 October 2008, European Doctorate in Fusion Science and Engineering
RFX-mod and its MHD active control system
Feedback control of Tearing Modes (TMs)
Avoid wall locking of TMs
Error fields control
Future plans: integration of a control system model and a plasma model
Outline
9 October 2008, European Doctorate in Fusion Science and Engineering
a=0.459m, R0=2m
RFX-mod, Padova, Italy
The largest reversed field pinch in the world
1.5MA plasma current up to now at low magnetic field
B(a)< 0.1T
(target 2MA)
Multi-mode feedback control of MHD modes and Error Fields
9 October 2008, European Doctorate in Fusion Science and Engineering
The edge radial magnetic field br is controlled by saddle coils independently fed
Assembly of saddle coils on vacuum vessel (mid’ 2004)
RFX-mod active coils system
4(pol) x 48(tor) =192
saddle coils (Cycle latency <400 s)
Sensors for br, bT and is associated to each saddle coil
Stabilizing shell : tshell~50ms
9 October 2008, European Doctorate in Fusion Science and Engineering
The dynamo mechanism
A large portion of BT and BP is generated by currents flowing in the plasma, through a dynamo mechanism
€
E
€
η J+ =?
9 October 2008, European Doctorate in Fusion Science and Engineering
The dynamo mechanism
BT and BP are generated by currents flowing in the plasma, through a dynamo mechanism
€
E
€
η J+ =
Edinamo is produced by resistive MHD modes, identified as Tearing Modes (TMs) or dynamo modes
[H. Ji et S. C. Prager, Magnetohydrodynamics 38 (2002) 191 ]
Edynamo
€
˜ v × ˜ B
9 October 2008, European Doctorate in Fusion Science and Engineering
q(r)
r(m)
TMs spectrum
m=1, n=-7
m=0, n=1,2,3,4, …
m=1, n=-8m=1, n=-9
€
q(r) =rBT
RBp
Safety Factor:
9 October 2008, European Doctorate in Fusion Science and Engineering
TMs effects
Core: confinement degradation Edge: Non axi-symmetric deformations of the Last Closed Flux Surface, (r)
m=1, n=-7m=1, n=-8m=1, n=-9
r/a
b r (m
T)
(r)
9 October 2008, European Doctorate in Fusion Science and Engineering
TMs effects
Controlling the edge radial field br:
- (r) is reduced and, as conseguence,
the plasma wall interaction
- transition to a QSH regime
[P. Martin et al., PPCF 49 (2007) A177 ]
(r) produces plasma-wall interaction,
which is affected by the phase-locking
and wall-locking phenomena
9 October 2008, European Doctorate in Fusion Science and Engineering
Inputs: br
Outputs: coil current
reference
To control br
Plasma
Digital Controller
Power Amplifier
An algorithm for Feedback control: Virtual Shell
[C. M. Bishop, Plasma Phys. Control. Fusion 31 (1989) 1179 ]
Sensors
bext
9 October 2008, European Doctorate in Fusion Science and Engineering
The problem of sidebands in VS Discrete grid of coils (MxN, M=48, N=4) sideband harmonics
The VS scheme is applied to the raw measurements, which contain the high m-n sidebands
The sidebands are computed from the coils currents using a cylindrical vacuum model, and are real-time subtracted from the measurements
€
br,cm +l M ,n +k N (rs)
l ,k{ }∈Ζ2 −0
∑
9 October 2008, European Doctorate in Fusion Science and Engineering
The problem of sidebands in VS Discrete grid of coils (MxN, M=48, N=4) sideband harmonics
The VS scheme is applied to the measurements, which contain the high m-n sidebands
The sidebands are computed from the coils currents using a cylindrical vacuum model, and are real-time subtracted from the measurements
The Clean Mode Control algorithm [P.Zanca et al, Nucl. Fusion 47 (2007) ]
0
€
brm,n (rs) = br,DFT
m,n − br,cm +l M ,n +k N (rs)
l,k{ }∈Ζ2 −0
∑
9 October 2008, European Doctorate in Fusion Science and Engineering
Sidebands correction
brbr, DFT m,n bf, DFT m,n
Irefm,n
Icoilm,n
The Clean Mode Control Scheme
Plasma
Digital Controller PowerSupplies
DFT Magnetic analysis
br, DFT m,n
brm,n
9 October 2008, European Doctorate in Fusion Science and Engineering
Effects of active control on TMs
Reduction of m=1 deformation of the Last Closed Flux Surface,
Clean Mode Control
Virtual Shell
No MHD active control
time (s)
Ip (M
A)
Ip (MA)
(m
)
The highest current for a RFP up to now
Significant increase of pulse length
Virtual ShellClean Mode Control
No control
9 October 2008, European Doctorate in Fusion Science and Engineering
Effects of active control on TMs
Spontaneous transition to a quasi-single-helicity state (QSH)
The magnetic chaos is reduced, i.e. the confinement is improved
time (s)
b T / B
r (a)
(%)
Radius (mm)
Te (e
V)
m=1, n=-7 < m=1, n=-8 to -16 >
9 October 2008, European Doctorate in Fusion Science and Engineering
Active control on TMs: effects on the mode-rotation
Effects not only in the amplitudes but also in the phase of the modes
TMs start to rotate in CMC
Distribution of the median of the rotation frequency for the m=1, n=-7 TM for two ensembles of VS and CMC discharges
Phase dynamics of m=1, n=-7
Phas
e (ra
d)
time (s)
Freq (Hz)
Phas
e (ra
d)
Virtual Shell
Clean Mode Control
9 October 2008, European Doctorate in Fusion Science and Engineering
Control of the direction of rotation: Complex Gain
The reference value for the applied field of the CMC algorithm for each mode is
The direction of the mode rotation is determined by the sign of the phase m,n on the dominant m=1,n=-7 mode
€
bm,ncoil (t) = −KP ,m,n exp(iϕ m,n )br,m,n (t)
Complex proportional gain
Phas
e (ra
d)
time (ms)
b-7 1,
r (a) (
mT)
9 October 2008, European Doctorate in Fusion Science and Engineering
Experiments with Complex Gain: Multiple Modes
Complex gains of opposite sign have been set on TMs with n=-8 to n=-16
The direction of the phase rotation follows the alternate pattern
Freq
(Hz)
n
9 October 2008, European Doctorate in Fusion Science and Engineering
In the next campaign: Sweeping control Mitigation of the wall-locking phenomena appling rotation perturbations with a sweeping frequency as suggested by results obtained in the DIII-D experiment
[F. Volpe, 34º EPS conference, 2007 ]
Rotating fields unlock the mode and sustain rotation at up to 60 Hz
Proposed by P. Piovesan et F. Volpe in RFX-mod
Phase and amplitude of 2/1 NTM in DIII-D experiment
9 October 2008, European Doctorate in Fusion Science and Engineering
Error Fields
Control system acts not only on TMs but also on Error Fields
time (s)
pha
se (r
ad)
m=1, n=-7
m=1, n=-6
time (s)
pha
se (r
ad)
phas
e (ra
d)
phas
e (ra
d)
r(a)T
r(a)T
9 October 2008, European Doctorate in Fusion Science and Engineering
Effects of the shell axisimmetries on TMS locking
Clear effect of the gap of the shell on TMs locking
Non uniformities of the
passive structures must be
taken into account to
improve the control of TMs
Poloidal gap
Toroidal position
p.d.
f.
9 October 2008, European Doctorate in Fusion Science and Engineering
Electromagnetic Model
Dynamic ElectroMagnetic (EM) model
[G. Marchiori, Fusion Eng. Des. 82 (2007) 1015]
- State space rapresentation
- Accurate description of the passive structures and of the
mutual inductance between sensor and active coils
Inputs: 48 x 4 voltages applied to the saddle coils
Outputs: 48 x 4 magnetic fluxes measured by the sensor coils
9 October 2008, European Doctorate in Fusion Science and Engineering
Comparison EM model - experiment
SimulationMeasure
9 October 2008, European Doctorate in Fusion Science and Engineering
Torque balance ModelThe evolution of the amplitudes and phases of several TMs can be explained by a torque balance model in cylindrical geometry
[P. Zanca et al., Nucl. Fusion 47 (2007) 1425 ]
The model is based on
- Newcomb equation solution for TMs radial profiles
- the thin shell dispersion relation to describe the effects of
homogeneous passive structures
- simplified one pole transfer function of the power supply and
saddle coils dynamics
9 October 2008, European Doctorate in Fusion Science and Engineering
The TM phase evolution is ruled by a balance between
Torque balance Model
€
δTEMm,n + δTV
m,n = 0
A threshold of the proportional gain is found after which the
modes begin to rotate
Viscous torque:due to fluid motion
EM torque:due to feedback coilsand image currents
9 October 2008, European Doctorate in Fusion Science and Engineering
Qualitative comparison between the simulation and the experimental trend
Measure
(
rad/
s)
kp
m=1, n=-7
Simulation
Kp/2400
m=1, n=-8
(
rad/
s)
Torque balance Model
9 October 2008, European Doctorate in Fusion Science and Engineering
With this integration we aim to:
- Optimize the control system
- Test new algorithms before the real-time implementation
- Improve the system performances both in terms of plasma wall interaction and reduction of the core stochasticity as a secondary effect
Next step
9 October 2008, European Doctorate in Fusion Science and Engineering
R0=2, a=0.459
r(m)
plasmavessel
rvi=0.475 rve=0.505
τv=3ms
copper shell rwi=0.5125, τw=0.1s, δ
w=3mm
coils grid
c=0.58
9 October 2008, European Doctorate in Fusion Science and Engineering
A net of MxN saddle coils covering a torus can produce radial fields with helicities up to m=M/2 and |n| up to N/2 together with an infinite number of sideband harmonics
Nyquist’s sampling theorem states that the DFT harmonics (i.e. the Fourier coefficients of a discrete periodic sequence) correspond to Fourier harmonics only if the spectrum is contained within the Nyquist frequency. If this condition is violated the aliasing phenomenon occurs: i.e. Fourier harmonics with high toroidal number appear in the DFT spectrum at a lower toroidal number
Es: if the system of 48x4 RFX-mod coils is generating an m=1, n=-7 radial field, all the sideband harmonics, e.g. the m=1,n=-55, m=1,n=41, etc., will all be aliased into the m=1,n=-7 DFT coefficient
Sideband harmonics- Nyquist theorem
9 October 2008, European Doctorate in Fusion Science and Engineering
PID feedback law
The field harmonic produced by the saddle coils is given by a PID feedback law
€
bm,ncoil (t) = −KPbm,n
r (t) − K I dt0
t
∫ bm,nr (t) − KD
d
dtℑ (bm,n
r (t), fcut )
Cleaned harmonic
The derivative action is performed on a one pole low filtered version of the cleaned harmonic, with a cut-off frequency of 300Hz
9 October 2008, European Doctorate in Fusion Science and Engineering
The deformation of the LCFS tends to show a secondary local maximum of similar amplitude.
The maximum of the deformation jumps between these two locations.
Experiments with Complex Gain: Multiple Modes
Consequently, complex gains were set only on low amplitude high n (n<-12) TMs, while different proportional and proportional-derivative gains were set for dominant modes,in order to vary the modes’ phase locking.
9 October 2008, European Doctorate in Fusion Science and Engineering
Feedback equations
Feedback acquired signal wiffnm
r rratrb ≤≤);,(,
Delays introduced by the digital acquisition, feedback operations and coils power supply modelled as one-pole filter law, with a cut-off frequency of 80Hz, corresponding to a delay 2ms
The CMC feedback law (PID):
€
IDFTm,n (t) =
KP + (iω / (1+ iωτ d ))KD + (1 / iω)K I
1+ iωΔtbr
m,n (rf , t)
€
τ d gives the cut-off of the low-pass filter applied to the derivative gain (5ms)
9 October 2008, European Doctorate in Fusion Science and Engineering
Feedback equations
Feedback acquired signal:
wiffnm
rnm
rnm
rd rratrbwwdt
d≤≤=+ );,(,,,τ
Power supply equation modelled as
msIIIdt
dc
nmREF
nmc
nmcc 5.0;,,, ==+ ττ
9 October 2008, European Doctorate in Fusion Science and Engineering
The (MN) DFT harmonics must not be confused with the Fourier modes
Sideband effect
( ) ( )θθ nmi
nm
nmr
r erbrb +
Ζ∈∑=,
, )(,,
Fourier modes are defined by the analytical series
DFT harmonics are affected by aliasing:
The shape factor is due to the finite extent of the sensors
€
br,DFTm,n = br
p,q (rs) f (p,q)l,k{ }εZ
∑
€
f ( p,q) =sin(q(Δφ / 2))
q(Δφ / 2)
sin(p(Δφ / 2))
p(Δφ / 2)€
p = m + lM
q = n + kN
9 October 2008, European Doctorate in Fusion Science and Engineering
( ) ( ) ( ) Ζ∈⋅++ℑ= ∫ −++ ++kldIeNknMlmrtrb
t
t
nmDFT
tANknMlmcr
NknMlm
,,)(,,,
0
, ,,, ξξξ
( ) ( ) ;,,, ,20
2
000
nmcm
cm Anmf
R
rn
R
rnI
R
rnKnmr ⎟⎟
⎠
⎞⎜⎜⎝
⎛′⎟⎟
⎠
⎞⎜⎜⎝
⎛′=ℑ μ
;11
00
2
0
2
0
2
,
⎟⎟⎠
⎞⎜⎜⎝
⎛′⎟⎟
⎠
⎞⎜⎜⎝
⎛′
⎟⎟⎠
⎞⎜⎜⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛+
=
Rbn
IRbn
KRbn
Rbn
mA
mmb
nm
τ
Vacuum formulas for sidebands
The radial field produced by the coil in the absence of plasma is:
It is obtained using the standard cylindrical vacuum solution for the m.f.in terms of the modified Bessel function and adopting the thin shelldispersion relation
€
τ b = μ0bδbσ b
9 October 2008, European Doctorate in Fusion Science and Engineering
Vacuum solution Newcomb’s solution
( )222/1 εnm +In the Newcomb’s equation, the plasma terms scale as
Are small for sidebands, due to to their large poloidal and toroidal mode numbers (M=48, N=4)
Vacuum formulas for sidebands
9 October 2008, European Doctorate in Fusion Science and Engineering
Shell region
wiww
nmr
nmr r
r
b
t
b δστσ 0
,,
0 ; =∂∂
=∂∂
More general than the thin-shell relation since the variation of the radial field across the shell is taken into account.
wwir
wir
nmr
wir
nmr
w r
b
t
bδ
δσ+
∂∂
=∂∂ ,,
0
A diffusion equation valid in the limit δw<<rw is adopted according to C.G.Gimblett, Nucl. Fusion 26 (1986) 617