# Presentation: 802059

## 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, Italy4Euratom-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

- 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

## Single-loop ITB control with ICRH

**Single-loop ITB control with ICRH**

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

**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

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

**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

## 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 (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 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) :

## 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

## 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) = **K**Galerkin(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

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

***L. 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) :

- 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)

- with :

- approximated by

## Choice of the trial function basis for q(x)

***L. Laborde et al.***

- Galerkin residuals : qexp-qbasis

-     qref :   #58474 at t=51.8 s

## Choice of the trial function basis for q(x)

***L. Laborde et al.***

- Galerkin residual mean squares :

## 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** :

## Identification of the first singular valuesand singular functions of K for the TSVD

## 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) = gc [1+1/(i.s)] . -1. (s)

## Closed-loop transfer function (PI control)

**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 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 powerat 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 actuatorsSVD 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

**K**T = 1**W**1.**V**1+ + 2**W**2.**V**2+

## Initial experiments with the lumped-parameter version of the algorithm with 3 actuators2-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...**