d. moreau3rd itpa meeting on steady state operation and energetic particles, st petersburg (russia)...

D. Moreau 3rd ITPA Meeting on Steady State Operation and Energetic Particles, St Petersburg (Russia) July 2003

ALGORITHMS FOR REAL-TIME PROFILE CONTROL AND

STEADY STATE ADVANCED TOKAMAK OPERATION

D. Moreau1, 2, F. Crisanti3, L. Laborde2, X. Litaudon2, D. Mazon2,

P. De Vries4, R. Felton5, E. Joffrin2, M. Lennholm2, A. Murari6,

V. Pericoli-Ridolfini3, M. Riva3, T. Tala7, L. Zabeo2, K.D. Zastrow5

and contributors to the EFDA-JET workprogramme.

1EFDA-JET Close Support Unit, Culham Science Centre, Abingdon, OX14 3DB, U. K.2Euratom-CEA Association, CEA-Cadarache, 13108, St Paul lez Durance, France

3Euratom-ENEA Association, C.R. Frascati, 00044 Frascati, Italy4Euratom-FOM Association, TEC Cluster, 3430 BE Nieuwegein, The Netherlands

5Euratom-UKAEA Association, Culham Science Centre, Abingdon, U. K.6Euratom-ENEA Association, Consorzio RFX, 4-35127 Padova, Italy

7Euratom-Tekes Association, VTT Processes, FIN-02044 VTT, Finland


OUTLINE

1. Quick summary of early experiments on real-time control of the ITB temperature gradient

2. Technique for controlling the current and pressure profiles in high performance tokamak plasmas with ITB's : a technique which offers the potentiality of retaining the distributed character of the plasma parameter profiles.

3. First experiments using the simplest, lumped-parameter, version of this technique for the current profile :

3.1. Control of the q-profile with one actuator : LHCD

3.2. Control of the q-profile with three actuators : LHCD, NBI, ICRH


D. Mazon et al, PPCF 44 (2002) 1087

53570 BT=2.6T Ip=2.2MA

Hold T* = 0.025

(above threshold)

0.025

The normalised Te gradientat the ITB location

is controlled during 3.4sby the ICRH power

in a regime with Ti =Te

Saturation of ICRH at 6.5sloss of control

Single-loop ITB control with ICRH


T* = 0.025

1.1 x 1016 n/s Disruption avoidance

ITB controlled for > 7s

ITB starts with qmin=3 at 4.5s

LH is lost at 6.5 s (radiation problem)

The q profile evolves

At 8s qmin=2 NBI and ICRH

decreases due to the control and disruption is avoided

Double-loop ITB control with ICRH+NBI (1)


53690 BT=3.4T Ip=2MA

ICRH T*

NBI neutron rate


(T*)réf=0.025

(Rnt)réf=0.9

Control

ITB controlled during 7.5s

Vloop ≈ 0

Better stability with control at slightly lower performance

#53521 reaches beta limit (collapses)

Double-loop ITB control with ICRH+NBI (2)


53697 / 53521 BT=3.4T Ip=1.8MA

ICRH T*

NBI neutron rate

LH holds the q-profile frozen


q-profile relaxation with/without LHCD

Inner ITB linked with negative shear region

53570 BT=2.6T Ip=2.2MA

LHCD off

MHD activity at 47.4 s

53697 BT=3.4T Ip=1.8MA

LHCD on



Challenges of advanced profile control

Previous experiments were based on scalar measurements

characterising the profiles (T*max) and/or other global parameters (li)

2. Multiple time-scale system + loop interaction

Energy confinement time Resistive time

Nonlinear interaction between p(r) and j(r)

HOWEVER

1. ITB = pressure and current (+ rotation ...) profiles

Multiple-input multiple-output distributed parameter system (MIMO + DPS)

Need more information on the space-time structure of the system

Identify a high-order operator model around the target steady state

and try model-based DPS control using SVD techniques

D. Moreau et al., EFDA-JET preprint PR(03) 23, to be published in Nucl. Fus.


Approximate Model andSingular Value Decomposition

t 1

0 0(x, t) dt' dx' (x,x', t t') (x', t') Y K P

K = Linear response function (Y = [current, pressure] ; P = heating/CD power)

Laplace transform :

1

0( , ) ' ( , ', ) ( ', )x s dx x x s x sY K P

Kernel singular value expansion in terms of orthonormal right and left singular functions + System reduction through Truncated SVD (best least square approximation) :

1

( , ', ) ( , ) ( ) ( ', )i i ii

x x s x s s x s

K W V


Set of input trial function basis

( , ) ( ) ( )1

Mx s x s residuali ii

PP CInput power deposition profiles :

With 3 power actuators :

( ) 0 0

( ) 0 ( ) 0

0 0 ( )

u xix v xi i

w xi

C

where ui(x), vi(x) and wi(x) correspond to LHCD, NBI and ICRH. The space spanned by these basis functions should more or less contain the accessible power deposition profiles.

If only the powers of the heating systems are actuated (e.g. no LH phasing)

then M =1 :

( , ) ( ) ( )x s x s residualk k

V C VInput singular vectors :

( , ) ( ) ( )x s x s residual PP CInput power deposition profiles :

and C(x) can be integrated into the operator K


Set of output trial function basis

( , ) ( ) ( )1

Nx s x s residualj jj

QY DOutput profiles :

and

Output singular functions : ( , ) ( ) ( )1

Nx s x s residualjk kjj

W D

With 2 profiles (current, pressure) :

( ) 0( )

0 ( )

a xjxj b xj

D

Identification of the operator KGalerkin’s method : residuals spatially orthogonal to each basis function

Q(s) = KGalerkin(s) . P(s)

1

0( , ) ' ( , ', ) ( ', )x s dx x x s x sY K P


Real time reconstruction of the safety factor profile (1)

The q-profile reconstruction uses the real-time data from the magnetic measurements and from the interfero-polarimetry, and a parameterization of the magnetic flux surface geometry

q(x)ddc

0c

1xc

2x2c

3x3c

4x4

a1xa

2x2a

3x3

D. Mazon et al, PPCF 45 (2003) L47L. Zabeo et al, PPCF 44 (2002) 2483


Trial function basis for q(x) or (x)=1/q(x)

If the real-time equilibrium reconstruction uses a particular set of trial functions, then one should take the same set for the controller design.

Otherwise, the family of basis functions must be chosen as to reproduce as closely as possible the family of profiles assumed in the "measurements".

EXAMPLE (with the parameterization used in JET and c0 ≈ 0) :

q(x)dd1c

1'xc

2' x2c

3' x3

a0' a

1'xa

2' x2 6-parameter family

1. A set of N = 6 basis functions bi(x) can be obtained through differentiation of the rational fraction with respect to the coefficients

2. Alternatively, one can choose N ≤ 6 cubic splines for bi(x)

q(x)or (x) Qi1

N .bi(x)with :approximated by


Choice of the trial function basis for q(x)

L. Laborde et al.

Galerkin residuals : qexp-qbasis

3 splines

4 splines

5 splines

6 splines

Differentiation of 6 rational fractions

q ref

q exp

q basis

qref : #58474 at t=51.8 s


Choice of the trial function basis for q(x)

3 splines 4 splines 5 splines 6 splines 6 differentiations

L. Laborde et al.

Galerkin residual mean squares : 1

0

2basisexp dx ] q-[q


What does the controller minimize ?

( , ) ( ) ( )1

Nx s x s residualj jj

QY DOutput profiles :

Setpoint profiles : ( ) ( )int , int1

Nx x residualjsetpo j setpoj

QY D

GOAL = minimize [Y(s=0) – Ysetpoint] . [Y(s=0) – Ysetpoint]

Define scalar product to minimize a least square quadratic form :


Identification of the first singular valuesand singular functions of K for the TSVD

EXPERIMENTAL

Q(s) = KGalerkin(s) . P(s)

ˆ K (s)KGalerkin

(s) (s)V(s)(s)(s) WK ˆˆ

W(s). ˆ W (s)

B = + . (Cholesky decomposition)

The best approximation for k, Wk and Vk in the Galerkin sense in the chosen trial function basis bi(x) is then obtained by performing the SVD of a matrix related by KGalerkin through :

ˆ K (s)


Pseudo-modal control scheme

SVD provides decoupled open loop relation betweenmodal inputs (s) =V+P and modal outputs (s) = W+BQ

Truncated diagonal system (≈ 2 or 3 modes) : (s) = (s) . (s)

STEADY STATE DECOUPLINGUse steady state SVD (s=0) to design a Proportional-Integral controller

(s) = G(s).(s) = gc [1+1/(i.s)] . -1. (s)


Proposed proportional-integral controller :

P(s) = gc [1+1/(i.s)] . [V0 0-1 W0

+ B] . Q = Gc . Q

• Closed loop transfer function ensures steady state convergence to the least square integral difference with no offset, i. e.

P(s=0) = Poptimal

• Choose gc and i to ensure closed-loop stability [i.e. Im (poles sk) < 0]

Closed-loop transfer function (PI control)

To minimize the difference between the steady state profiles and the reference ones in the least square sense :

Min (Q Qref ) (Q Qref)

Q = KGalerkin . P Solution : Poptimal = [V0 0-1 W0

+ B] . Qref


Initial experiments with the lumped-parameter version of the algorithm with 1 actuator

q-profile control with LHCD power

The accessible targets are restricted to a one-parameter family of profiles

With 5 q-setpoints : no problem if the q-profile tends to "rotate" when varying the power.

With only the internal inductance some features of the q-profile shape could be missed (e.g. reverse or weak shear in the plasma core).

Applying an SVD technique with 5 q-setpoints may not allow to reach any one of the setpoints exactly, but could minimize the error on the profile shape.


Lumped-parameter version of the algorithm with 1 actuator

q-profile control with LHCD powerat 5 radii


5-point q-profile control with LHCD power steady state

D. Mazon et al, PPCF 45 (2003) L47


Initial experiments with the lumped-parameter version of the algorithm with 3 actuatorsSVD for 5-point q-profile control


Initial experiments with the lumped-parameter version of the algorithm with 3 actuators

2-mode TSVD for 5-point q-profile control

KT = 1W1.V1+ + 2W2.V2

+

D. Moreau et al., EFDA-JET preprint PR(03) 23 F. Crisanti et al, EPS Conf. (2003)


Initial experiments with the lumped-parameter version of the algorithm with 3 actuators

2-mode TSVD for 5-point q-profile control

D. Moreau et al., EFDA-JET preprint PR(03) 23 F. Crisanti et al, EPS Conf. (2003)


Conclusions and perspectives (1)

A fairly successful control of the safety factor profile was obtained with the lumped-parameter version of TSVD algorithm.

This provides an interesting starting basis for a future experimental programme at JET, aiming at the sustainement and control of ITB's (q(r) + T* criterion) in fully non-inductive plasmas and with a large fraction of bootstrap current.

The potential extrapolability of the technique to strongly coupled distributed-parameter systems with a larger number of controlled parameters (density, pressure, current profiles ...) and actuators (with more flexibility in the deposition profiles), is an attractive feature.


Conclusions and perspectives (2)

If the nonlinearities of the system or the difference between various time scales were to be too important, the TSVD technique can be extended to model-predictive control, at the cost of a larger computational power, and by performing the identification of the full frequency dependence of the linearized response of the system to modulations of the input parameters, rather than simply of its steady state response.

It provides an interesting tool for an integrated plasma control in view of steady state advanced tokamak operation, including the control of the plasma shape and current, as in conventional tokamak operation, but also of the primary flux and of the safety factor, temperature and density profiles (and fusion power in a burning plasma), etc...

d. moreau3rd itpa meeting on steady state operation and energetic particles, st petersburg (russia)...

Documents