ALGORITHMS FOR REAL-TIME PROFILE CONTROL AND STEADY STATE ADVANCED TOKAMAK OPERATIOND. 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. Zastrow5and 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, France3Euratom-ENEA Association, C.R. Frascati, 00044 Frascati, Italy4Euratom-FOM Association, TEC Cluster, 3430 BE Nieuwegein, The Netherlands5Euratom-UKAEA Association, Culham Science Centre, Abingdon, U. K.6Euratom-ENEA Association, Consorzio RFX, 4-35127 Padova, Italy7Euratom-Tekes Association, VTT Processes, FIN-02044 VTT, Finland
OUTLINE • Quick summary of early experiments on real-time control of the ITB temperature gradient • 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. • 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
53570 BT=2.6T Ip=2.2MA Hold rT* = 0.025 (above threshold) The normalised Te gradient at the ITB location is controlled during 3.4s by the ICRH power in a regime with Ti =Te Saturation of ICRH at 6.5s loss of control 0.025 Single-loop ITB control with ICRH D. Mazon et al, PPCF 44 (2002) 1087
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 1.1 x 1016 n/s rT* = 0.025 Double-loop ITB control with ICRH+NBI (1) 53690 BT=3.4T Ip=2MA ICRH rT* NBI neutron rate D. Mazon et al, PPCF 44 (2002) 1087
(Rnt)réf=0.9 ITB controlled during 7.5s Vloop ≈ 0 Better stability with control at slightly lower performance #53521 reaches beta limit (collapses) (T*)réf=0.025 Control Double-loop ITB control with ICRH+NBI (2) 53697 / 53521 BT=3.4T Ip=1.8MA ICRH rT* NBI neutron rate LH holds the q-profile frozen D. Mazon et al, PPCF 44 (2002) 1087
53570 BT=2.6T Ip=2.2MA 53697 BT=3.4T Ip=1.8MA LHCD off MHD activity at 47.4 s LHCD on q-profile relaxation with/without LHCD Inner ITB linked with negative shear region D. Mazon et al, PPCF 44 (2002) 1087
Challenges of advanced profile control Previous experiments were based on scalar measurements characterising the profiles (rT*max) and/or other global parameters (li) • Need more information on the space-time structure of the system HOWEVER 1. ITB = pressure and current (+ rotation ...) profiles Multiple-input multiple-output distributed parameter system (MIMO + DPS) 2. Multiple time-scale system + loop interaction Energy confinement time Resistive time Nonlinear interaction between p(r) and j(r) 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 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 throughTruncated SVD (best least square approximation) :
Input power deposition profiles : Input power deposition profiles : Input singular vectors : Set of input trial function basis With 3 power actuators : 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 : and C(x) can be integrated into the operator K
Output profiles : and Output singular functions : Set of output trial function basis With 2 profiles (current, pressure) : Identification of the operator K Galerkin’s method : residuals spatially orthogonal to each basis function Q(s) = KGalerkin(s) . P(s)
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 L. Zabeo et al, PPCF 44 (2002) 2483 D. Mazon et al, PPCF 45 (2003) L47
6-parameter family Trial function basis for q(x) or i(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) : with : approximated by 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 exp q basis q ref 3 splines 4 splines 5 splines 6 splines Differentiation of 6 rational fractions Choice of the trial function basis for q(x) qref : #58474 at t=51.8 s Galerkin residuals : qexp-qbasis L. Laborde et al.
3 splines 4 splines 5 splines 6 splines 6 differentiations Choice of the trial function basis for q(x) Galerkin residual mean squares : L. Laborde et al.
What does the controller minimize ? Output profiles : Setpoint profiles : GOAL = minimize[Y(s=0) –Ysetpoint]. [Y(s=0) –Ysetpoint] Define scalar product to minimize a least square quadratic form :
EXPERIMENTAL Q(s) = KGalerkin(s) . P(s) The best approximationfor sk, 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 : B = D+ . D (Cholesky decomposition) Identification of the first singular valuesand singular functions of K for the TSVD
Pseudo-modal control scheme • SVD provides decoupled open loop relation between • modal inputs a(s) =V+P and modal outputs b(s) = W+BQ • Truncated diagonal system (≈ 2 or 3 modes) :b(s) = S(s) . a(s) STEADY STATE DECOUPLING Use steady state SVD (s=0) to design a Proportional-Integral controller (s) = G(s).db(s) = gc [1+1/(ti.s)] . S-1. db(s)
Q = KGalerkin . P Solution : Poptimal = [V0 S0-1 W0+ B] . Qref Proposed proportional-integral controller : P(s) =gc [1+1/(ti.s)]. [V0 S0-1 W0+ B] . dQ = Gc . dQ • 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 ti 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 :
Initial experiments with the lumped-parameter version of the algorithm with 1 actuatorq-profile control with LHCD power The accessible targets are restricted to aone-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 actuatorq-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
F. Crisanti et al, EPS Conf. (2003) D. Moreau et al., EFDA-JET preprint PR(03) 23 Initial experiments with the lumped-parameter version of the algorithm with 3 actuators2-mode TSVD for 5-point q-profile control KT = 1W1.V1+ + 2W2.V2+
F. Crisanti et al, EPS Conf. (2003) D. Moreau et al., EFDA-JET preprint PR(03) 23 Initial experiments with the lumped-parameter version of the algorithm with 3 actuators2-mode TSVD for 5-point q-profile control
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...