1. D. Moreau3rd 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. Moreau 1, 2, F. Crisanti 3, L. Laborde 2, X. Litaudon 2, D. Mazon 2, P. De Vries 4, R. Felton 5, E. Joffrin 2, M. Lennholm 2, A. Murari 6, V. Pericoli-Ridolfini 3, M. Riva 3, T. Tala 7, L. Zabeo 2, K.D. Zastrow 5 and contributors to the EFDA-JET workprogramme. 1 EFDA-JET Close Support Unit, Culham Science Centre, Abingdon, OX14 3DB, U. K. 2 Euratom-CEA Association, CEA-Cadarache, 13108, St Paul lez Durance, France 3 Euratom-ENEA Association, C.R. Frascati, 00044 Frascati, Italy 4 Euratom-FOM Association, TEC Cluster, 3430 BE Nieuwegein, The Netherlands 5 Euratom-UKAEA Association, Culham Science Centre, Abingdon, U. K. 6 Euratom-ENEA Association, Consorzio RFX, 4-35127 Padova, Italy 7 Euratom-Tekes Association, VTT Processes, FIN-02044 VTT, Finland

2. D. Moreau3rd ITPA Meeting on Steady State Operation and Energetic Particles, St Petersburg (Russia) July 2003 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

3. D. Moreau3rd ITPA Meeting on Steady State Operation and Energetic Particles, St Petersburg (Russia) July 2003 D. Mazon et al, PPCF 44 (2002) 1087 53570 B T =2.6T Ip=2.2MA Hold  T * = 0.025 (above threshold) 0.025 The normalised T e gradient at the ITB location is controlled during 3.4s by the ICRH power in a regime with T i =T e Saturation of ICRH at 6.5s loss of control Single-loop ITB control with ICRH

4. D. Moreau3rd ITPA Meeting on Steady State Operation and Energetic Particles, St Petersburg (Russia) July 2003  T * = 0.025 1.1 x 10 16 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) D. Mazon et al, PPCF 44 (2002) 1087 53690 B T =3.4T Ip=2MA ICRH  T * NBI neutron rate

5. D. Moreau3rd ITPA Meeting on Steady State Operation and Energetic Particles, St Petersburg (Russia) July 2003 (  T *) réf =0.025 (R nt ) 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) D. Mazon et al, PPCF 44 (2002) 1087 53697 / 53521 B T =3.4T Ip=1.8MA ICRH  T * NBI neutron rate LH holds the q-profile frozen

6. D. Moreau3rd ITPA Meeting on Steady State Operation and Energetic Particles, St Petersburg (Russia) July 2003 q-profile relaxation with/without LHCD Inner ITB linked with negative shear region 53570 B T =2.6T Ip=2.2MA LHCD off MHD activity at 47.4 s 53697 B T =3.4T Ip=1.8MA LHCD on D. Mazon et al, PPCF 44 (2002) 1087

7. D. Moreau3rd ITPA Meeting on Steady State Operation and Energetic Particles, St Petersburg (Russia) July 2003 Challenges of advanced profile control Previous experiments were based on scalar measurements characterising the profiles (  T *max) and/or other global parameters (l i ) 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.

8. D. Moreau3rd ITPA Meeting on Steady State Operation and Energetic Particles, St Petersburg (Russia) July 2003 Approximate Model and Singular Value Decomposition K = Linear response function ( Y = [current, pressure] ; P = heating/CD power) Laplace transform : Kernel singular value expansion in terms of orthonormal right and left singular functions + System reduction through Truncated SVD (best least square approximation) :

9. D. Moreau3rd ITPA Meeting on Steady State Operation and Energetic Particles, St Petersburg (Russia) July 2003 Set of input trial function basis Input power deposition profiles : With 3 power actuators : where u i (x), v i (x) and w i (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 : Input singular vectors : Input power deposition profiles : and C (x) can be integrated into the operator K

10. D. Moreau3rd ITPA Meeting on Steady State Operation and Energetic Particles, St Petersburg (Russia) July 2003 Set of output trial function basis Output profiles : and Output singular functions : With 2 profiles (current, pressure) : Identification of the operator K Galerkin’s method : residuals spatially orthogonal to each basis function Q(s) = K Galerkin (s). P(s)

11. D. Moreau3rd ITPA Meeting on Steady State Operation and Energetic Particles, St Petersburg (Russia) July 2003 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 D. Mazon et al, PPCF 45 (2003) L47L. Zabeo et al, PPCF 44 (2002) 2483

12. D. Moreau3rd ITPA Meeting on Steady State Operation and Energetic Particles, St Petersburg (Russia) July 2003 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 c 0 ≈ 0) : 6-parameter family 1. A set of N = 6 basis functions b i (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 b i (x) with : approximated by

13. D. Moreau3rd ITPA Meeting on Steady State Operation and Energetic Particles, St Petersburg (Russia) July 2003 Choice of the trial function basis for q(x) L. Laborde et al. Galerkin residuals : q exp -q basis 3 splines 4 splines 5 splines 6 splines Differentiation of 6 rational fractions q ref q exp q basis q ref : #58474 at t=51.8 s

14. D. Moreau3rd ITPA Meeting on Steady State Operation and Energetic Particles, St Petersburg (Russia) July 2003 Choice of the trial function basis for q(x) 3 splines4 splines5 splines6 splines6 differentiations L. Laborde et al. Galerkin residual mean squares :

15. D. Moreau3rd ITPA Meeting on Steady State Operation and Energetic Particles, St Petersburg (Russia) July 2003 What does the controller minimize ? Output profiles : Setpoint profiles : GOAL = minimize [ Y (s=0) – Y setpoint ]. [ Y (s=0) – Y setpoint ] Define scalar product to minimize a least square quadratic form :

16. D. Moreau3rd ITPA Meeting on Steady State Operation and Energetic Particles, St Petersburg (Russia) July 2003 Identification of the first singular values and singular functions of K for the TSVD EXPERIMENTAL Q(s) = K Galerkin (s). P(s) B =  +.  (Cholesky decomposition) The best approximation for  k, W k and V k in the Galerkin sense in the chosen trial function basis b i (x) is then obtained by performing the SVD of a matrix related by K Galerkin through :

17. D. Moreau3rd ITPA Meeting on Steady State Operation and Energetic Particles, St Petersburg (Russia) July 2003 Pseudo-modal control scheme SVD provides decoupled open loop relation between modal inputs  (s) =V + P and modal outputs  (s) = W + BQ Truncated diagonal system (≈ 2 or 3 modes) :  (s) =  (s).  (s) STEADY STATE DECOUPLING Use steady state SVD (s=0) to design a Proportional-Integral controller  (s) = G(s).  (s) = g c [1+1/(  i.s)].   -1.  (s)

18. D. Moreau3rd ITPA Meeting on Steady State Operation and Energetic Particles, St Petersburg (Russia) July 2003 Proposed proportional-integral controller : P(s) = g c [1+1/(  i.s)]. [V 0  0 -1 W 0 + B].  Q = G c.  Q Closed loop transfer function ensures steady state convergence to the least square integral difference with no offset, i. e. P(s=0) = P optimal Choose g c and  i to ensure closed-loop stability [i.e. Im (poles s k ) < 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 : Q = K Galerkin. P Solution : P optimal = [V 0  0 -1 W 0 + B]. Q ref

19. D. Moreau3rd ITPA Meeting on Steady State Operation and Energetic Particles, St Petersburg (Russia) July 2003 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.

20. D. Moreau3rd ITPA Meeting on Steady State Operation and Energetic Particles, St Petersburg (Russia) July 2003 Lumped-parameter version of the algorithm with 1 actuator q-profile control with LHCD power at 5 radii

21. D. Moreau3rd ITPA Meeting on Steady State Operation and Energetic Particles, St Petersburg (Russia) July 2003 5-point q-profile control with LHCD power steady state D. Mazon et al, PPCF 45 (2003) L47

22. D. Moreau3rd ITPA Meeting on Steady State Operation and Energetic Particles, St Petersburg (Russia) July 2003 Initial experiments with the lumped-parameter version of the algorithm with 3 actuators SVD for 5-point q-profile control

23. D. Moreau3rd ITPA Meeting on Steady State Operation and Energetic Particles, St Petersburg (Russia) July 2003 Initial experiments with the lumped-parameter version of the algorithm with 3 actuators 2-mode TSVD for 5-point q-profile control K T =  1 W 1.V 1 + +  2 W 2.V 2 + D. Moreau et al., EFDA-JET preprint PR(03) 23 F. Crisanti et al, EPS Conf. (2003)

24. D. Moreau3rd ITPA Meeting on Steady State Operation and Energetic Particles, St Petersburg (Russia) July 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)

25. D. Moreau3rd ITPA Meeting on Steady State Operation and Energetic Particles, St Petersburg (Russia) July 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.

26. D. Moreau3rd ITPA Meeting on Steady State Operation and Energetic Particles, St Petersburg (Russia) July 2003 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...