ITPA-IOS Joint Experiment # 6 - bpsi.nucleng.kyoto-u.ac.jpbpsi.nucleng.kyoto-u.ac.jp/itpa2011/protect/Day2_PM/08-4-DMoreau... · D. Moreau, ITPA-IOS Meeting, Kyoto, October 18-21,

D. Moreau, ITPA-IOS Meeting, Kyoto, October 18-21, 2011 LEHIGH U N I V E R S I T Y

First Integrated Magnetic and Kinetic Control

for AT Scenarios on DIII-D

D. Moreau CEA-DSM-IRFM, Cadarache, 13108 St-Paul-lez-Durance Cedex, France

E. Schuster, J. Barton, D. Boyer, W. Shi, W. Wehner

Lehigh University, Bethlehem, PA 18015, USA

M. Walker, J. Ferron, D. Humphreys, A. Hyatt, B. Johnson, B. Penaflor, F. Turco, and many DIII-D collaborators

General Atomics, San Diego, CA 92186, USA

1

ITPA-IOS Joint Experiment # 6.1


Outline

•  ARTAEMIS model structure and integrated control

•  Control of the internal poloidal flux profile, ψ(x)

•  Integrated control of ψ(x) and βN

•  Lehigh University controller and ι(x) control

•  Proposal for 2012 and beyond

2


The ARTAEMIS (grey-box) model-based approach

D. Moreau et al., Nucl. Fusion 48 (2008) 106001

3

ARTAEMIS is a set of algorithms that use singular perturbation methods : (i)  a semi-empirical system identification method (ii) a model-based, 2-time-scale, control algorithm for magneto-kinetic plasma state

What could a minimal state space model look like ? Are there natural state variables and input variables ? How are they coupled ?

!

"#(x, t)" t

= L#,# x{ } $ #(x, t) + L#,% x{ } $V&(x,t)T(x,t)'

( )

*

+ , + L#, n x{ } $ n(x, t)

!

+ L", P(x) P(t)+Vext (t)

!

"#n(x, t)# t

= Ln,$ x{ } % $(x, t) + Ln,& x{ } %V'(x,t)T(x,t)(

) *

+

, - + Ln, n x{ } % n(x, t) +Ln,P (x) P(t)

!

"## t

V$(x,t)T(x,t)%

& '

(

) * = L+,, x{ } - ,(x, t) + L+,+ x{ } -

V$(x,t)T(x,t)%

& '

(

) * + L+, n x{ } - n(x, t) +L+,P (x) P(t)

etc …

Generic structure of linearized flux-averaged plasma transport equations :


The ARTAEMIS controller design and parameters for combined ψ(x) and βN control

4

•  Singular perturbation analysis è Near-optimal control = Optimal control up to O(ε2)"

0

!

" X+(t)Q X(t) dt + 0

!

" u+(t) R u(t) dtThe dynamics minimizes"

x1

x2! ! x( )"! target x( )#$ %&

2dx

using a given set of SVD modes (controller order)"

minimizes"

•  The slow proportional + integral feedback tracks a steady state that"

•  The fast proportional feedback loop maintains the kinetic variables, e. g. βN, on a trajectory which is consistent with the slow magnetic state evolution, ψ(x, t)."

given weight matrices, Q and R, with X = controlled variables and u = actuators

x1

x2! ! x( )"! target x( )#$ %&

2dx + ! "N ""N ,t arget

#$ %&2

to control simultaneously ψ(x) and βN"

minimizes"


Control of the internal poloidal flux profile : ψ(x) = Ψ(x) – Ψboundary

5

x1

x2! ! x( )"! target x( )#$ %&

2dxThe controller minimizes"

with actuator constraints and optimal gain matrices that depend on controller parameters : •  4 actuators = NB-Co, NB-Bal, ECCD (5 gyros), Vsurf (NB 210R unavailable on 09/13) •  R-matrix : actuator weight fixed by considering actuator headroom (MW & Volts ?) •  Q-matrix : same weight on 9 different controlled radii (x = 0.1, …, 0.9) •  Controller order = 2 (proportional + integral control) •  Weight on integral control in the Q-matrix = 4, 10, 25, respectively, on the 3 examples below :

2.5 3 3.5 4 4.5 5 5.5 60

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Time

Qua

drat

ic c

ost f

unct

ion

#146410 : Cost function (EcoilGp= 5 IntWeight=4)

Psi profile control

33L max15L max

EC minn = 1 mode

#146410 : IntWeight = 4

2.5 3 3.5 4 4.5 5 5.50

0.005

0.01

0.015

0.02

0.025#146407 : Cost function (EcoilGp= 15 IntWeight=10)

Time

Qua

drat

ic c

ost f

unct

ion

Psi profile control

33L max15L max

n = 1 mode

#146407 : IntWeight = 10

2 2.5 3 3.5 4 4.5 5 5.5 60

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

Time

Qua

drat

ic co

st fu

nctio

n

#146416 : Cost function (EcoilGp= 7.5 IntWeight=25)

NBCO min

Psi profile control

n = 2 mode

n = 2 mode15L max

#146416 : IntWeight = 25


Control of the poloidal flux profile (x = 0.1, 0.2, … 0.9)

6

2.5 3 3.5 4 4.5 5 5.5 60

0.5

1

1.5

Time

Psi (

x =

0.1

0.9

)

#146410 : Psi1 Psi9 (EcoilGp= 5 IntWeight=4)

Psi profile control

n = 1 modeEC min

33L max15L max

#146410 : IntWeight = 4

2.5 3 3.5 4 4.5 5 5.50

0.5

1

1.5#146407 : Psi1 Psi9

Time

Psi (

x =

0.1

0.9

)

Psi profile control

33L max15L max

n = 1 mode

#146407 : IntWeight = 10

2 2.5 3 3.5 4 4.5 5 5.5 60

0.5

1

1.5

Time

Psi (

x =

0.1

0.9

)

#146416 Psi1 Psi9 (EcoilGp= 7.5 IntWeight=25)

n = 2 mode

Psi profile control

NBCO minn = 2 mode

15L max

#146416 : IntWeight = 25

Control : 3.5 s - 6 s Control : 3.5 s - 5 s Control : 2.5 s - 6 s

x = 0.1

x = 0.2

x = 0.3

x = 0.4

x = 0.5

x = 0.6

x = 0.7

x = 0.8

x = 0.9


Control of the poloidal flux profile ψ(x) @ t = 2.5 s, 4 s, 6 s

7

#146410 : IntWeight = 4 #146416 : IntWeight = 25

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

Normalized radius

Polo

idal

flux

(Wb)

#146410 : Psi(x) @ t=2.5 s, 4 s, 6 s

t = 2.5 st = 4 st = 6 sPsi(x) targets

Slow

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

Normalized radius

Polo

idal

flux

(Wb)

#146416 : Psi(x) @ t= 2.5 s, 4 s and 6 s

t = 2.5 st= 4 s (overshoot)t = 6 sPsi(x) targets

Fast (overshoot)


with actuator constraints and optimal gain matrices that depend on controller parameters : •  5 actuators = NB-Co, NB-Bal, NB-Cnt, ECCD (6 gyros), Vsurf •  R-matrix : actuator weights fixed by considering actuator headroom (MW & Volts ?) •  Q-matrix : same weight on 9 controlled radii for ψ(x), x=0.1, 0.2, … 0.9 •  Weight on βN control : λ = 0.3 •  Controller order = 2 or 3 (proportional + integral control) •  Weight on integral control in the Q-matrix = 25 and 10, respectively, in the 2 examples below :

2.5 3 3.5 40

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Time

Qua

drat

ic c

ost f

unct

ion

#146453 Cost function (betaN weight = 0.31, control order = 3, IntWeight = 10)

Psi partbetaN partTotal

n = 1mode

EC min21R min

Actuator saturation

Simultaneous control of the ψ(x) profile and βN 5 actuators : NB-co, NB-bal, NB-cnt, ECCD, Vsurf

8

x1

x2! ! x( )"! target x( )#$ %&

2dx +! "N ""N ,target

#$ %&2The controller minimizes"

#146420 Control order = 2 IntWeight = 25

#146453 Control order = 3 IntWeight = 10

2.5 3 3.5 40

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1#146420 : Cost function (betaN weight = 0.31, control order = 2, IntWeight = 25)

Time

Qua

drat

ic c

ost f

unct

ion

Psi partbetaN part

n = 1 moden = 2 mode

MHD


1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.50

0.5

1

1.5#146455 : Psi1 Psi9 (betaN weight = 0.31, control order = 3, IntWeight = 10)

TimePs

i (x

= 0.

1 0

.9)

MaximumPsi(x) overshoot

Psi profile + betaN control with 5 actuators

n = 2 mode

x = 0.1 x = 0.2 x = 0.3

x = 0.4

x = 0.5

x = 0.6

x = 0.7

x = 0.8

x = 0.9

radii

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

Normalized radius

Psi(x

)

#146455 : Psi(x) targets and achieved profiles

t = 0.5 st = 4.25 s (overshoot)t = 6 sPsi targets (#140076 @ 3.4 s)

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.51.4

1.5

1.6

1.7

1.8

1.9

2

2.1

2.2#146455 : betaN

Time

beta

N

betaN targetbetaN

Psi profile + betaN control

n = 2 mode

Simultaneous control of the ψ(x) profile and βN

(shot # 146455 : control starting @ t = 1.5 s)

9

βN control

ψ-profile control

ψ(x= 0.1, 0.2, …0.9) & targets (lines)

Time



(shot # 146455 : control starting @ t = 1.5 s)

10

5 actuators (no saturation)

Vsurf actuator control & Ip

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Time

PNB

(MW

), PE

CC

D (M

W),

Vsur

f (V)

#146455 : All Actuators for combined control of Psi(x) and betaN

NBCONBBALNBCNTECCDVsurf (x10)


MaximumPsi(x) overshoot

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.50

0.2

0.4

0.6

0.8

1

Time

Vsur

f (V)

/ Ip

(MA)

#146455 : Vsurf actuator

Req. VsurfDel. VsurfIp


Cost function minimization

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.50

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01#146455 : Cost function (betaN weight = 0.31, control order = 3, IntWeight = 10)

Time

Qua

drat

ic c

ost f

unct

ion

Psi partbetaN partTotal

Psi(x) overshoot

n = 2 mode

Psi profile + betaN control with 5 actuators


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

1.2

1.4

Normalized radius

Psi (

Wb)

#146463 : Psi(x) targets and achieved profiles

t = 0.5 st = 2.4 s (overshoot)t = 5 sPsi targets (#136212 @ 2 s)

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60

0.5

1

1.5

2

2.5

3

Time

beta

N, I

p (M

A), n

=1 m

ode,

n=2

mod

e#146463 : betaN (control weight = 0.31, control order = 3, IntWeight = 10)

betaN targetbetaNIpn=1 mode (x 0.04)n=2 mode (x 0.04)

Psi profile + betaN control (5 actuators)

MHD

Ip


shot # 146463 : control starting @ t = 1 s (ramp-up)

11

βN control

ψ-profile control

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60

0.5

1

1.5#146463 : Psi1 Psi9 (betaN weight = 0.31, control order = 3, IntWeight = 10)

TimePs

i (x

= 0.

1 0

.9)

n=1 / n=2 modes


x = 0.1 x = 0.2 x = 0.3 x = 0.4

x = 0.5 x = 0.6 x = 0.7

x = 0.8

x = 0.9

radii

ψ(x= 0.1, 0.2, …0.9) & targets (lines)

Time



shot # 146463 : control starting @ t = 1 s (ramp-up)

12

5 actuators (MHD è NB-Bal saturation)

Vsurf actuator control & Ip

Cost function minimization

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60

1

2

3

4

5

6#146463 : All actuators for combined control of Psi(x) and betaN

Time

PNB

(MW

), PE

CC

D (M

W),

Vsur

f (V)

NBCONBBALNBCNTECCDVsurf (x10)


0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60

0.2

0.4

0.6

0.8

1

1.2

#146463 : Vsurf actuator

Time

Vsur

f (V)

, Ip

(MA)

Req. VsurfDel. VsurfIp


Vsurf controlFeedfwd : 0.4 VProp gain : 7.5

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Time

Qua

drat

ic c

ost f

unct

ion

& n=

1 m

ode

#146463 : Cost function (betaN weight = 0.31, control order = 3, IntWeight = 10)

Psi partbetaN partn=1 mode (inverted)


Psi(x) overshoot

D. Moreau, ITPA-IOS Meeting, Kyoto, October 18-21, 2011 LEHIGH U N I V E R S I T Y 13

•  Lehigh University Approach to Feedback Control Design:

Data-driven Model-based Current Profile Control

–  Both static and dynamic control-oriented plasma response models are embedded in the control synthesis.

–  A cost functional is defined to quantify the control objectives: •  Tracking error minimization •  Disturbance rejection •  Control power minimization

–  Stabilizing controllers, which are robust against model uncertainties, are synthesized by minimizing different norms (H∞ and H2 norms) of the cost functional subject to the control-oriented model.

–  These controllers do not need PID-like empirical tuning.

–  The controllers are augmented with model-based anti-windup

compensators to overcome detriments effects due to actuator saturation.


Data-driven Model-based Control of ι(x) =1/q(x)

–  In the DIII-D PCS different magnetic profiles (ψ(ρ), ι(ρ), q(ρ) or θ(ρ)=∂ψ/∂ρ) can now be obtained in real time from a complete equilibrium reconstruction using data from the MSE diagnostic.

–  Figure on the left illustrates both simulated and experimental evolutions for the rotational transform ι(ρ) at normalized radii ρ=0.2, 0.4, 0.5, 0.6, 0.8 for shot146419.

–  Tracking of the ι(ρ) target profile was achieved by regulating: •  Total plasma current •  Co-injection and balanced beam powers

(counter-injection beam not available) •  Total ECH/ECCD power

–  Artificial input disturbances were introduced at t=3.5 sec.


Data-driven Model-based Control of ι(x) =1/q(x)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Normalized

Time:2.538 Sec.

TargetMeasuredControl point

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Normalized

Time:3.498 Sec.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Normalized

Time:4.018 Sec.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normalized

Time:5.998 Sec.


Tight regulation is achieved before t=3.5 sec.

A significant tracking error in the inner part of the profile is noted after the artificial disturbance is introduced at t=3.5 sec.

The controller reacts by decreasing the tracking error at the inner point and slightly increasing the error at the outer points with the ultimate goal of improving the overall profile tracking.

Ip regulation was poor due to setup problem.

More control authority is expected with stronger Ip regulation and counter injection beam.


•  Control of ψ(x), ι(x) = 1/q(x) and combined control of ψ(x) and βN have been achieved for the first time on DIII-D using either 4 or 5 actuators :

Co-NBI, (Cnt-NBI), Bal-NBI, ECCD, Vsurf

•  New PCS with profile control algorithm was qualified and worked perfectly

•  ψ(x) and βN control was switched on during current ramp-up.

•  Combined feedback control of ψ(x) and βN was successful up to [1s-6s]

Proposal for ITPA-IOS 2012 :

Integrated magnetic and kinetic plasma control

•  More experiments on DIII-D with different target current profiles and βN.

•  Add control of the rotation and/or Ti profiles (real-time CER)

•  Start profile control experiments on other devices (Tore Supra, TCV …?)

Summary and proposal for 2012 …

16

ITPA-IOS Joint Experiment # 6 - bpsi.nucleng.kyoto-u.ac.jpbpsi.nucleng.kyoto-u.ac.jp/itpa2011/protect/Day2_PM/08-4-DMoreau... · D. Moreau, ITPA-IOS Meeting, Kyoto, October 18-21,

Documents