Plasma thermal characterization in nuclear fusion context
L Attar1, TP Tran1, L Perez2, E Moulay3, R Nouailletas4, S Brémond4,
L Autrique1
1
LARIS-ISTIA, University of Angers, France
LTN-IUT, University of Nantes, France
3
XLIM, University of Poitiers, France
4
IRFM, CEA-Cadarache, France
2
Corresponding author: [email protected]
Abstract. In this communication, estimation of space and time varying thermal diffusivity and
heating source in a nonlinear parabolic partial differential equation (PDE) describing heat
transport in tokamak plasma is considered. An iterative regularization method based on a
Conjugate Gradient Method (CGM) is implemented to deal with model and measurement
errors. The estimation strategy is investigated in the infinite dimensional framework. Real data
from the Tore Supra (TS) tokamak are provided to evaluate the performance of the proposed
strategy. A demonstration of a software dedicated to quasi on line parametric identification is
proposed.
1. Introduction
The huge amount of energy produced by thermonuclear fusion motivates scientists to investigate this
reaction which can be considered as a new source of sustainable energy resource without highly
radioactive waste and without major risks. One of the most developed modern devices to achieve this
goal is the tokamak [1]. In such reactor, heat transport is commonly modelled using partial differential
equations (PDE) systems [2]. Our field of interest is the estimation of space and time dependent
thermal diffusivity and heat source in tokamak plasma.
The proposed strategy is based on inverse heat conduction problem resolution. It is well known that
inverse heat conduction problem are usually ill-posed since stability cannot be obtained (even a small
measurement noise induces large variation of the identified parameter). However, accurate solutions
can be determined using regularization methods. In such a way, an iterative approach based on the
Conjugate Gradient Method (CGM, [3]) is relevant for minimizing the effect of random perturbations
in measurement as well as for dealing with model uncertainties. The particularity of our approach lies
in the estimation of both the diffusion coefficient (thermal diffusivity) and the source term without a
priori assumption by using real data coming from a TS tokamak pulse. A preliminary study has been
investigated in [4] in order to identify thermal diffusivity (depending on space and time) in a thermal
system described by PDE.
In the following, resolution of an inverse thermal problem in 1D geometry is investigated. In this
context it is usual to wait the acquisition of all measurements before starting the identification
procedure. However, when the inversion is realized for control or diagnosis purposes, an online
identification can be an interesting mean to reduce the calculation time.
The article is organized as follows. In order to identify both parameters considering experimental
observations, output error minimization is performed. Numerical results are predicted from the direct
problem resolution issued from the mathematical model. The modelling of heat transfers in plasma is
presented in section 2. Minimization method is based on an iterative descent scheme in order to solve
the inverse problem formulated in section 3. In this section, iterative regularization method (CGM) is
detailed in order to present (for each algorithm iteration) how the descent direction is computed
(numerical resolution of an adjoint problem) and how to determine the descent depth (numerical
resolution of a sensitivity problem). In section 4, real data (considering an experimental campaign on
TS tokamak) are shown and several strategies are discussed in order to develop quasi online
identification. These strategies are issued from previous authors works ([5] [6]). Several concluding
remarks and outlooks are addressed in the last section.
2. Direct problem
From the experimental point of view, this method is devoted to the identification of electron thermal
diffusivity and power density in tokamak plasma. The modelling of phenomena in such fusion reactor
is quite complex and has been widely studied from early works [7] [8]. In the specific context of
robust stabilization of the current profile in tokamak plasmas, control-oriented model is proposed in
[9]. Several recent approaches for control of the magnetic flux are presented in [10] [11] [12] [13] but
the question remains about the control of the coupled system “magnetic flux and thermal state”. In
such a context, it is crucial to identify several key parameters in the diffusion-convection equation
[14].
Let us consider the following notations in order to introduce the direct problem. The state of the
system is the plasma temperature denoted by Te (electron temperature) and in plasma physics it is
(
)
convenient to use the electronvolt 1 ev ≈ 1.610−19 J as a unit of temperature by using Boltzmann
constant kb ≈ 1.3810
−23
−1
J.K . Then for typical magnetic confinement fusion plasma corresponding to
15000 × 1.6 × 10 −19
≈ 174 × 10 6 K. Assuming toroidal axisymmetric
−23
1.38 × 10
assumptions and averaging over the so-called magnetic surface, the geometrical domain can be
reduced to a 1D geometry across the magnetic surfaces, where physical dynamics occur only along the
minor plasma radius a in m. Dimensionless space variable for such configuration is denoted by
15 keV, temperature is
x ∈ [ 0,1] and time variable is t ∈ 0, t f  in seconds ( t f is the final time for the considered
experimentation). Let us denote by ne ( x, t ) in m-3 the electron density. The following equation issued
from the energy conservation principle is proposed in [14]:
∂T 
3 ∂ ( neTe )
1 ∂
− 2  xneα e  = g
2 ∂t
xa ∂x 
∂x 
(1)
where α ( x, t ) is the thermal diffusivity in m2 .s-1 and g ( x, t ) is a global heat source in W.m−3
which can be considered as an input for the thermal system. Heating source is absorbed by the plasma
from an external heating system, usually Ohmic, radiofrequency waves or neutral beams injection. The
function g ( x, t ) is positive for heating and negative for cooling. Let us consider the following term:
θ ( x, t ) = ne ( x , t ) kbTe ( x, t ) which can be considered as a thermal energy density and can be expressed
in J.m-3 . Equation (1) can be reformulated as:
∂ne 
3 ∂θ
1 ∂  ∂θ 
1 ∂
− 2  xα
 = g − 2  xα Te
2 ∂t xa ∂x 
∂x 
xa ∂x 
∂x 
(2)
∂n 
1 ∂
xα Te e  and α ( x, t ) have to be identified and for

2
xa ∂x 
∂x 
the purposes of notation, the first parameter is globally denoted g ( x, t ) . Then, the parabolic PDE
system describing the thermal phenomena in tokamak plasma is formulated as follows:
In the following, g ∗ ( x, t ) = g ( x, t ) −

∀ ( x, t ) ∈ [ 0,1] × 0, t f 


∀x ∈ [ 0,1]




∀t ∈ 0, t f 

3 ∂θ
1 ∂  ∂θ 
− 2
xα
=g
∂x 
2 ∂t xa ∂x 
θ ( x,0 ) = θ 0 ( x )
θ (1, t ) = ne (1, t ) Te (1, t )
and
(3)
∂θ ( x, t )
∂x
=0
x =0
where θ 0 ( x ) is the initial distribution. On the boundary x = 1 , electron density ne (1, t ) and
electron temperature Te (1, t ) are assumed to be known. Except for academic situations, direct problem
is usually solved according to numerical methods such as finite element method [15] [16]. In the
following, this numerical method is implemented considering Comsol® solver.
3. Inverse problem
3.1. Inverse ill-posed problem formulation
Inverse problems formulation aims to identify the causes of a phenomenon (physical, biological…)
from the observation of its effects. If one or several input parameters p ∈ P are unknown, then an
inverse problem can be solved considering real measurements. For our specific application,
observations are provided by the TS tokamak. In this context, inverse problem can be formulated as
follows: find the unknown parameters α ( x, t ) and g ( x, t ) such that the simulated thermal energy
density provided by the resolution of direct problem (modelling thermal phenomena in the tokamak
plasma) is closed to the measurement θˆ ( x, t ) provided by sensors. It is usual to investigate such
inversion as a minimization problem where a quadratic cost-function has to be minimized: find p∗
such that:
p ∗ = arg min J (θ , p )
p∈P
1

2
ˆ ( x, t ) dxdt 
θ
θ
= arg min 
x
,
t
;
p
−
(
)
p∈P
 2 [0,1]×∫∫0,t f 





(
)
where θ ( x , t ; p ) is the solution of the direct problem ( 3) obtained with parameter p = {α , g } .
Let us consider in the absence of a priori knowledge that α ( x , t ) and g ( x, t ) are assumed to be
continuous piecewise linear functions dependent on space and time:
Nt Nx
Nt Nx
j =1 i =1
j =1 i =1
α ( x, t ) = ∑∑α ij si ( x ) s j ( t ) and g ( x, t ) = ∑∑gij si ( x ) s j ( t )
(4)
where Nt and Nx are the number of time and space discretization steps for the parametrization,
si ( x ) and s j ( t ) are basis hat functions. The unknown thermal diffusivity matrix is denoted by
α = α 
and the unknown flux is denoted by G =  gij  Nx×Nt . Then, N = Nx × Nt unknown
coefficients have to be identified for each matrix. Let us consider P = [α , G]Nx×2 Nt the matrix of the
ij
Nx×Nt
unknown coefficients and a discrete formulation of the minimization problem: find P ∗ such that:
P ∗ = arg min J (θ , P )
P∈ℝ Nx×2 Nt
 1 t f Ns

2
= arg min  ∫ ∑ θ ( xi , t ; P ) − θˆi ( t ) dt 
P∈ℝ Nx×2 Nt  2 0 i =1


(
)
( 3)
where θ ( x , t ; P ) is the solution of the direct problem
obtained with parameter
P = [α , G]Nx×2 Nt , θˆi ( t ) is the measurement provided by sensor i located at xi and N s the number of
sensors. The conjugate gradient method (see for instance [17] [18]) is implemented in order to identify
the unknown parameters P = [α , G]Nx×2 Nt . For this algorithm iterative resolution of three well-posed
problems is required:
• the direct problem to calculate the cost-function J (θ , P k ) and estimate the quality of the estimate
P k at iteration k;
• the sensitivity problem to calculate the descent depth in the descent direction;
• the adjoint problem to determine the gradient of the cost-function and thus to define the next
descent direction (see for instance [19] [20]).
3.2. Adjoint problem
The adjoint problem is formulated in order to determine at iteration

 ∂J 
∂J   ∂J 
 the gradient of the cost function J defined by:
∇J k = k =  
, 


g
∂
  ∂αij 
∂P


ij

 Nx×Nt 
Nx× Nt

(
)
J θ , Pk =
((
) ( ))
∆t Ns NT
∑ ∑ θ xi , t j ; P k − θˆi t j
2 i =1 j =1
k
:
2
(5)
where NT is the number of observations obtained for each sensor (NT is related to the time sample).
There is no relation between spatial location of sensors and spatial discretization of unknown
parameters then Ns and Nx can be different. (NT and Nt can also be different). The cost-function
gradient is issued from the following continuous Lagrangian formulation where ψ is the Lagrange
multiplier:
tf 1
 3 ∂θ 1 ∂  ∂θ  2 
l (θ , P,ψ ) = J (θ , P ) + ∫ ∫  a 2
−
 xα
 − a g ψ dxdt
∂t x ∂x 
∂x 

0 0 2
(6)
Lagrangian variation is:
δ l (θ ; P;ψ ) =
∂l
∂l
∂l
δθ + δ P +
δψ
∂θ
∂P
∂ψ
(7)
Lagrangian multiplier ψ is chosen such that:
∂l
∂l
δψ = 0 and
δθ = 0
∂ψ
∂θ
(8)
Moreover since θ is solution of the direct problem (3), then:
δ l (θ , P,ψ ) =
∂l
δP
∂P
(9)
Considering (6) and (7), we have:
tf 1
∫ ∫E ( x, t ) δθ dxdt +
δl =
00
tf 1
tf 1
1 ∂ 
∂θ
− ∫ ∫
 xδα
∂x
0 0  x ∂x 
Ns
(
 3 2 ∂δθ
∫ ∫  2 a ∂t
0 0

ψ dxdt −

tf 1
1 ∂ 

0 0
∫ ∫  x ∂x  xα
∂δθ
∂x

 ψ dxdt

(10)
 2 
 + a δ g ψ dxdt


)
where E ( x, t ) = ∑ θ ( xi , t ; P ) − θˆi ( t ) δ Dxi ( x ) and δ Dxi is the Dirac distribution related to sensor i.
i =1
After several integrations by parts and considering the sensitivity problem (in the next section), the
following PDE system has to be satisfied by the Lagrangian multiplier [20]:

∂ψ 1 ∂ 
∂ψ 
∀ ( x, t ) ∈ [ 0,1] ×  0, t f 
+
 xα
=E
∂t x ∂x 
∂x 


ψ ( x, t f ) = 0
∀x ∈ [ 0,1]

∂ψ (.)

=0
ψ (1, t ) = 0 and
∀t ∈  0, t f 
∂x x = 0

(11)
Let us remark that the condition ψ ( x, t f ) = 0 is a final time condition. If ψ is solution of the
adjoint problem (11) then equations (9) are satisfied and cost function gradients can be computed as
follows:
∂J t 1  ∂ψ ( x, t ) ∂θ ( x, t ) 
= ∫∫ 
 si ( x ) s j ( t ) dxdt and
∂α ij 0 0 
∂x
∂x 
f
3.3.
∂J t 1
= ∫∫ψ ( x, t ) si ( x ) s j ( t ) dxdt
∂g ij 0 0
f
(12)
3.4. Sensitivity problem
Sensitivity problem are introduced in order to analyse how the variation of parameters δα ( x , t ) and
δ g ( x , t ) acts on the variation of the system state δθ ( x , t ) . In the studied case, the following PDE
system (13 ) is considered in order to obtain δθ ( x , t ) induced by δα ( x , t ) :

3a 2 ∂δθ 1 ∂  ∂δθ  1 ∂ 
∂θ 
∀ ( x, t ) ∈ [ 0,1] × 0, t f  2 ∂t − x ∂x  xα ∂x  = x ∂x  xδα ∂x 





δθ ( x,0) = 0
∀x ∈ [ 0,1]

∂δθ
∀t ∈ 0, t f 
δθ (1, t ) = 0 and
=0


∂x x=0

(13)
PDE system (14 ) is considered in order to obtain δθ ( x , t ) induced by a small heating source
variation δ g ( x , t ) :

3a 2 ∂δθ 1 ∂  ∂δθ 
2


∀
,
∈
0,1
×
0,
x
t
t
(
)
[
]

 f  2 ∂t − x ∂x  α ∂x  = a δ g



δθ ( x,0) = 0
∀x ∈ [ 0,1]

∂δθ
∀t ∈ 0, t f 
=0
δθ (1, t ) = 0 and


∂x x=0

(14)
The iterative scheme for the descent algorithm is given by:
Pk +1 = Pk − γ k +1dk +1
(15)
k +1
where γ
is the descent depth in the direction d k +1 at iteration k+1.
Nx NT
γ
k +1
=
)
(
∑ ∑ θ ( xi , t j ; P k ) − θˆi ( t j ) δθdk +1 ( xi , t j ; P k )
i =1 j =1
Nx NT
(
)
∑ ∑ δθdk +1 xi , t j ; P k 
i =1 j =1
2
(16)
3.5. CGM algorithm
In order to obtain a stable solution P = [α ]Nx× Nt , [G ]Nx× Nt  , three well-posed problems (direct
problem (3), adjoint problem (11), sensitivity problems (13) and (14)) have to be solved at each
iteration k of the minimization algorithm. For usual descent methods, at each iteration k a new value of
the unknown parameter P k +1 is obtained as follows:
α k +1 = α k + γ δα k and G k +1 = G k + γ δ G k
(17)
where the corrections
J (θ , P
k +1
) < J (θ , P ) .
γ δα k
and
γ δ Gk
at each iteration are chosen such that
k
The CGM algorithm can be presented as follows:
• Step 1: k = 0 , initialization of P 0 ;
• Step 2:
k
considering direct problem (3) and measurements
o estimation of the cost function J P
( )
θˆi ( t ) , i = 1,⋯, Ns ;
( )
o if J P k ≤ J stop then P k can be considered as a correct estimation of P and the algorithm is
stopped ;
o else go to step 3;
∂J k
∂J k
and
for i = 1,⋯ , Nx and j = 1,⋯ , NT
∂α ijk
∂g ijk
considering numerical solution of adjoint problem (11) and equation (12). Then, descent direction
at iteration k is defined as:
d k +1 =  d ijk +1 
= − ∇ J k + β k d k −1
(18)
Nx × 2 NT
• Step 3: evaluation of the cost function gradients
where β k =
M =
∇J
∇J
k 2
k −1 2
(except for β 0 = 0 ) where
.
is the Frobenius matrix norm
Nt Nx
∑∑M
2
ij
;
j =1 i =1
• Step 4: evaluation of the descent depth γ k +1 ∈ ℝ related to the descent direction d k +1 considering
equation (16) ;
• Step 5: estimation of the new parameter P k +1 = P k + γ k +1d k +1
• Step 6: k + 1 → k and go to step 2.
The most important stages are the gradient calculation (Step 3) and the descent depth estimation
(Step 4). Both numerical steps are obtained considering PDE systems well posed in Hadamard sense.
4. CGM Implementation and numerical results
The aim of our approach is to find the parameters α ( x, t ) and g ( x, t ) which minimize the output
error. Thus, the main goal is that the simulated state θ ( x , t ) according to these identified parameters
is closed to the real measurements θˆ obtained during a TS tokamak pulse
4.1. Offline identification
The previous algorithm is implemented considering experimental measurements performed during the
Tore Supra tokamak pulse 47673. Normalized cost function defined as
iteration in Figure 1.
( )
J (θ , P )
J θ , Pk
0
is drawn versus
10
10
0
-1
10
-2
10
-3
0
50
100
150
200
250
iteration
Figure 1. Cost function evolution versus iteration k
The stop criterion J stop is difficult to establish without a prior assumptions. For measurements
disturbed by a Gaussian noise (mean equal to zero and standard deviation denoted by σ ) admissible
threshold of minimization can be defined as J stop = NT × Ns × σ 2 . Iteration number acts as a
regularization parameter, and minimization algorithm has to be stopped before taking into account the
noise. If the identification is continued then parameters distributions α ( x, t ) and g ( x, t ) are identified
in order to describe the noise. Since noise is meaningless, disturbed distributions obtained for a large
number of iterations have to be considered with extreme caution and in the studied configuration,
results are retained at iteration 40.
thermal diffusivity
4
3
2
1
0
15
1
10
0.8
0.6
5
0.4
0.2
t
0
0
x
Figure 2. Estimated thermal diffusivity α ( x, t ) for pulse TS#47673
x 10
5
5
x 10
4
heating source
heating source
4
3
2
1
0
3
2
1
0
0
15
0
1
10
0.2
5
0.8
0.4
0.6
5
0.6
10
0.4
0.8
0.2
0
t
0
15
x
1
x
t
Figure 3. Estimated heating source g ( x, t ) for pulse TS#47673
Comparison between the estimated thermal energy density θ ( x, t ) and measurements are shown in
Figure 4. Identified distributions of α ( x, t ) and g ( x, t ) estimated by CGM are in adequacy with the
measured thermal energy density. The proposed method for simultaneous identification of plasma
thermal diffusivity and heating source using the iterative regularization method leads to a correct
identification.
4
2
x 10
15000
1.8
1.6
θ in J.m
-3
1.4
10000
1.2
1
0.8
5000
0.6
0.4
0.2
0
0
2
4
6
8
t in s
10
12
14
16
0
0
0.2
0.4
0.6
0.8
1
x
Figure 4. Real (continuous line -) and estimated (stars *) thermal energy density θ ( x, t ) with
CGM for pulse TS#47673
4.2. Quasi on line identification
Previous results highlight the efficiency of the CGM for identification purpose. However, results are
obtained offline (once the experimentation is stopped). This major drawback avoids elaboration
(during the process) of control laws based on a better knowledge of thermal state. To reduce the global
computation time and to determine thermal diffusivity or heating source in the plasma during the
experimentation, CGM is implemented considering a time interval Ti = τ i− ,τ i+  ⊂ T =  0, t f  which
slides on the total time horizon with a step ∆ti > 0 to identify the values of unknown parameters
{α ( x, t ) ; g ( x, t )} .
Ti
Ti
Once the values of the parameters are accurately estimated on the interval Ti , the identification
window Ti +1 = τ i− + ∆ti− ,τ i+ + ∆ti+  = τ i−+1 ,τ i++1  moves on the horizon time considering initialization
α Tk =0 ( x, t ) = αT ( x,τ i+ ) and gTk =0 ( x, t ) = gT ( x,τ i+ ) . Initial thermal state of the direct problem
i +1
i
(
corresponds to θ x,τ
i +1
−
i +1
i
) . Several strategies based on this approach are proposed below.
• Strategy A : constant time window size with constant overlap. For this first strategy, the time
interval of the window Ti = τ i− ,τ i+  ⊂ T used to identify α Ti ( x, t ) ; g Ti ( x, t ) is fixed such as
{
}
τ − τ is constant (for example τ − τ = 1 s ). This first strategy is based on a constant offset of
Ti with ∆t < τ i+ − τ i− to ensure identification ranges overlap. For example an offset value
∆t = 0.25 s (25% of the overall time of identification on Ti ) can be considered.
• Strategy B: constant time window size with adaptive overlap. For this second method, let us
consider the time interval Ti = τ i− ,τ i+  ⊂ T such as τ i+ − τ i− = 1 s . Identification on this interval is
+
i
−
i
+
i
−
i
performed during a CPU time equivalent to ti . When the identification is satisfactory then
immediately a new interval is considered τ i++1 = τ i+ + ti or, if τ i++1 − τ i+ > 1 then τ i++1 = τ i+ + 0.5 and
τ i−+1 = τ i++1 − 1 . Moreover identification process is launched only if the new measurements are not in
{
}
adequacy with the temperatures predicted using the previous identification of α Ti ( x, t ) ; g Ti ( x, t ) .
If measurements are not with temperature calculated with the direct problem, a new time interval is
considered equal to Ti +1 = τ i− + ti ,τ i+ + ti  ⊂ T with ti is computational time required for the
resolution of the direct problem.
• Strategy C: adaptive time window size and adaptive overlap. Window size is determined quasi on
line in order to take into account the arrival of new measurements.
Previous strategies can also be improved considering an adaptation of the sequential initialization
of CGM. Let us consider that the next identification window is Ti +1 = τ i−+1 ,τ i++1  . Both previous
methods have been implemented considering an initialization based on the last identified values on the
previous time interval. In the following, initialization is based on the predicted behavior of unknown
parameters on the next time interval Ti+1 . This prediction is based on the time derivative of
{α ( x, t ) ; g ( x, t )} calculated on the previous time interval T . The aim of such initialization is to try
Ti
Ti
i
to be more adequate with the previous behavior of identified parameters.
In preliminary studies, it has been shown that time required for identification is divided by 40 by
using this quasi online adaptation. A software demonstration will be proposed during the demonstrator
session in order to illustrate the quasi online identification method implemented for a real scenario
obtained on Tore-Supra fusion reactor. Effect of window size, overlap, initialization improvement will
be highlighted.
5. Concluding remarks
An inverse problem of space and time dependant parameters estimation has been investigated in the
infinite dimension framework. Mathematical model of heat transfer in tokamak plasma is based on 1D
axisymmetric parabolic partial differential equation. Numerical results are provided from the finite
element method implemented in COMSOL software interfaced with Matlab. In order to estimate the
space and time thermal diffusivity and heating source, the conjugate gradient method is proposed for
output error minimization. Such inverse heat conduction problems are ill-posed and the minimization
method acts as an iterative regularization method where iteration number can be considered as a
regularization parameter. Without loss of generalities, unknowns parameters are expressed as
continuous piecewise linear functions. For a real scenario (performed on the Tore Supra tokamak),
results show the efficiency of the chosen methodology to handle parameter and input estimation for
heat transport in tokamak plasmas. Moreover an online implementation of the CGM for identification
purposes is discussed. Experimental results of quasi online parametric identification will be presented
by the means of a numerical software developed in the LARIS institute.
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