- Email: [email protected]
Modelling and experimental validation of RFX-mod Tokamak shaped discharges
Fusion Engineering and Design xxx (xxxx) xxx–xxx
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Fusion Engineering and Design journal homepage: www.elsevier.com/locate/fusengdes
Modelling and experimental validation of RFX-mod Tokamak shaped discharges ⁎
Domenico Abatea, , Giuseppe Marchioria, Fabio Villoneb a b
Consorzio RFX, Corso Stati Uniti 4, 35127 Padova, Italy Consorzio CREATE, DIETI, Università di Napoli Federico II, Via Claudio 21, Napoli, Italy
A R T I C LE I N FO
A B S T R A C T
Keywords: Linearized plasma response model Rogowski coil Total plasma current sensitivity Iterative procedure Plasma equilibrium
A general procedure for computing plasma equilibria through the CREATE-L code has been applied to RFX-mod shaped tokamak discharges in order to produce very accurate models with respect to experimental data. It involves the solution of a constrained non-linear minimization problem to estimate the CREATE-L free parameters by using an iterative scheme trying to minimize the discrepancy between the poloidal magnetic field experimentally measured by pick-up coils and the one simulated by the model. The procedure has been applied considering eleven experimental shots spanning the whole range of poloidal beta achieved in the RFX-mod tokamak. A preliminary sensitivity analysis showed a non-negligible dependence of static equilibria on variations of the total plasma current with respect to the measurement provided by Rogowski coils. Thus, the total plasma current has been set as an additional degree of freedom in the minimization problem assuming values between the Rogowski measurement and the value of the discrete line integral of the poloidal magnetic field measured by the pick-up coils. In all cases under analysis, the iterative procedure showed that the most accurate equilibrium is obtained with a plasma current higher than the Rogowski measurement but lower than the pickup coils line integral.
1. Introduction Shaped Tokamak discharges with an insertable polarized electrode have been executed in RFX-mod to achieve H-mode regime [1]. Equilibrium magnetic configurations with different plasma shapes have been experimentally produced and analysed by means of the linearized plasma response model CREATE-L [2]. So far, plasma linearized models were produced using experimental values for the input equilibrium currents (i.e. active coil and plasma total current) [3], while the plasma current density parameters have been chosen on the basis of the available measurements on poloidal beta and internal inductance. This procedure leads to a noticeable inaccuracy in the fitting of magnetic field measurements, which was higher, with respect to standard tokamak operations, in all the experimental plasmas related to the Hmode campaign – these experiments are characterized by enhanced plasma shape conditions to explore its role in the L-H transition. In this paper, we describe a general iterative procedure [4] for computing plasma equilibria, and the related linearized plasma response models, through the CREATE-L code with a high level of accuracy with respect to experimental data. In addition, the procedure has been tested with the PEGASOS plasma equilibrium code reproducing
⁎
the same tokamak experimental shots and additionally two Reversed Field Pinch (RFP) shots. In all cases under analysis, the iterative procedure shows that the most accurate equilibrium is obtained with a plasma current higher than the Rogowski measurement but lower than the pick-up coils line integral. The paper is organized as follows. Section 2 describes the formulation of the iterative procedure and the methodology adopted for its application to RFX-mod experimental data. Section 3 reports the results and their impact on RFX-mod experiment. Finally, Section 4 draws the conclusions. 2. Problem formulation The iterative procedure involves the solution of a constrained nonlinear minimization problem to estimate the CREATE-L free parameters of the current density by using an iterative scheme trying to minimize the discrepancy between the experimental poloidal magnetic fields and the one simulated by the model [4]. The iterative procedure proposed is a simplified version of the one presented in [3] and it is in principle valid for any equilibrium code, since it is related to the estimation of the best values of the free parameters of the code to describe the reference
Corresponding author. E-mail address: [email protected] (D. Abate).
https://doi.org/10.1016/j.fusengdes.2018.11.055 Received 12 September 2018; Received in revised form 14 November 2018; Accepted 29 November 2018 0920-3796/ © 2018 Elsevier B.V. All rights reserved.
Please cite this article as: Abate, D., Fusion Engineering and Design, https://doi.org/10.1016/j.fusengdes.2018.11.055
Fusion Engineering and Design xxx (xxxx) xxx–xxx
D. Abate et al.
the total plasma current has been set as an additional degree of freedom in the minimization problem leading to have W = [αM , αN , β0, Ip] with 4 different constraints on their possible values (Table 2). The total plasma current and the parameter related to the poloidal beta (β0 ) have been allowed to assume variations up to 10% of their experimental values; for the total plasma current this correspond to the value obtained by computing the discrete line integral of the magnetic poloidal field measured by the pick-up coils as we will see in next section. The boundaries on the other two dof have been chosen considering the typical physical conditions for RFX-mod tokamak plasmas: the parameter related to the on axis safety factor (αN ) is assumed to vary around 1, accordingly to the standard inductive tokamak scenario, while the parameter related to the internal inductance (αM ) covers a plasma current density profile from peaked to broader shape. The search of the solution is carried out using the fmincon function of MATLAB with the default interior point algorithm. The algorithm satisfies constraints at all iterations and solves a sequence of approximate minimization problems. The approximate problem is a sequence of equality constrained problems which are easier to solve than the original one; the method is then based on the method of Lagrange multipliers and the Karush-Kuhn-Tucker (KKT) conditions. By default, the algorithm first attempts to take a direct step, i.e. Newton step, for the solution of the KKT equations via a linear approximation. If it cannot, it attempts a conjugate gradient CG step. Since the Hessian is unknown, the algorithm computes a quasi-Newton approximation to the Hessian of the Lagrangian at each iteration. Details about the interior point algorithm can be found in the MATLAB optimization toolbox user's guide. Before starting the iterative procedure a preliminary phase is needed in order to get a starting model in which the free parameters are fixed to the typical values of RFX-mod standard tokamak operations [3] (i.e. αM = 0.7, αN = 1.001, β0 = 0.1) while the active coil currents, including the internal-external saddle coil circuit which provides the vertical stability of the elongated plasmas, are the experimental ones. The equilibrium poloidal magnetic flux values of this starting model will be used as initial guess for launching the CREATE-L code in the iterative procedure. In order to start the iterative procedure, we followed the so called “multi-start” approach by defining a certain number of initial points, Ni, corresponding to Ni values for each degree of freedom, from where the search for a minimum begins; thus, the vector W is converted in a matrix whose dimension is Ni×Ndof. In this way, it is possible to locate a minimum by approaching it from many different directions. The number of initial points is the factor that mainly determines the computational time of the procedure. For this reason different preliminary runs, with an increasing number of initial points, have been performed; the minimum computational time obtained for 10 initial points was approximately 2.7 h on a PC equipped with two 6-core Intel Xeon CPU X5660 @ 2.8 GHz processors. Nevertheless, because of the highly nonlinear structure of the problem, there is no explicit law governing the computational time. One of the most demanding case has been obtained with 80 initial points and required above 40 h of computation; nevertheless a similar result in terms of accuracy (i.e. F value) was reached with only 10 initial points and a much shorter computational time. Since a detailed convergence analysis was not the focus of this work and we are looking for local minima indicating an optimal value for the set of four parameters in Eq. (2), ten starting points are usually a good compromise to find an accurate solution in a reasonable computational time. The iterative procedure has been applied to all the experimental shots reported in Table 1; in order to analyse the results obtained with the procedure we define a relative percentage variation factor, which will be used as a relative percentage error for many physical quantities compared to their experimental measurements (e.g. poloidal magnetic field, total plasma current):
Table 1 Shot numbers under analysis, equilibrium time instants, poloidal beta and plasma regimes. Shot
teq [s]
βp
Plasma regime
36922 39036 39039 39040 39068 39084 39122 39123 39135 39136
0.1 0.5 0.5 0.5 0.4 0.71 0.85 0.85 0.5 0.5
0.1 0.5 0.5 0.5 0.5 0.75 0.8 1 0.7 0.65
Low-β Intermediate-β Intermediate-β Intermediate-β Intermediate-β Increased-β H-mode H-mode Increased-β Increased-β
experimental data as accurately as possible. Once a plasma current density profile is established by a mathematical parametrization of it, the problem is to determine the best values of the profile parameters. Eleven experimental shots have been identified and considered in this study: all of them are Upper Single Null tokamak configurations spanning the whole range of poloidal beta achieved in the RFX-mod tokamak (low-β, intermediate-β, biased induced H-mode regime) as reported in Table 1. In our case, by following the parameterization proposed in [6], the three parameters are (αM , αN , β0) related to the physical quantities (li , q0 , βp) . This leads to search the solution of a constrained non-linear minimization problem, in which we want to minimize a parameter that relates the values of the experimental poloidal field measured by the 8 magnetic pick-up coils located on the inner surface of the stabilizing shell with the computed values given by the model. It has been chosen to minimize the difference between the measured and the computed values of the poloidal field normalized with respect to the experimental values, as defined in Eq. (1): Nsensors
F=
∑ i=1
(Bθi experimental − Bθi simulated )2 |Bθi experimental|
(1)
where Nsensors = 8 is the number of magnetic pick-up coils able to measure the poloidal component Bθ of the magnetic field, i.e. the tangential component. The flux loops measurements have not been included, in order to avoid a further degree of freedom related to the relative weight of magnetic fields and fluxes in the error functional, or to avoid a multi-objective optimization. A posteriori we can say that the quality of the fit is adequate also using the magnetic field measurements alone. The objective function F is defined for each set of computed and experimental values of poloidal magnetic field related to a given set of degrees of freedom (dof). Therefore the problem can be stated as a nonlinear constrained minimization problem of finding a vector of dof, W = [αM , αN , β0], that is a local minimum to the scalar function F subject to constraints on the allowable values ofW :
⎧ minW F ⎨ ⎩L≤W≤U
(2)
where L and U are the lower and upper boundary values of W defined in Table 2. A preliminary sensitivity analysis showed a non-negligible dependence of static equilibria on variations of the total plasma current with respect to the measurement provided by Rogowski coils (IRog) [5]. Thus, Table 2 Lower and upper boundaries values for each degree of freedom. Bounds
αM
αN
β0
Ip
L U
0.5 2
0.9 1.1
βp-0.1βp βp+0.1βp
IRog IRog+ΔIRog
2
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Table 3 Iterative procedure results in terms of objective function F for all the experimental shots under analysis. Shot
F starting case
F iterative results
εF [%]
36922 39036 39039 39040 39068 39084 39122 39123 39135 39136
1.5148 2.7048 6.44457 0.3981 1.4717 1.2914 0.6725 3.3476 1.3329 1.3963
0.1224 0.4905 1.2414 0.1963 0.1529 0.4451 0.1810 0.7169 0.0775 0.3226
−91.92 −81.87 −80.74 −50.69 −89.61 −65.53 −73.09 −78.58 −94.19 −76.90
εV =
V − V* ∙100 V*
Table 4 Relative percentage error comparisons between preliminary results and the one obtained from the iterative procedure for Bθ measurements in a reference shot. Shot no. 39068 Bθ measurements θ [°]
ε preliminary [%]
ε procedure [%]
27 72 117 162 207 252 297 342
9.62 3.27 28.78 2.44 −3.97 −7.79 2.92 17.26
1.35 0.20 −12.54 −2.05 −1.85 −3.17 −0.37 −1.47
(3)
where V is the quantity considered (e.g. Bθ, F, Ip) and V* is the reference quantity used as comparison (e.g. the experimental value, the reference case). The εV parameter leads us to show the improvements obtained in terms of lower values of F by the iterative procedure with respect to the starting model obtained from the preliminary phase; in addition, the relative percentage error εV will be used also to compare some physical quantities computed by the CREATE-L model with respect to experimental values, such as the poloidal magnetic field as we will see in Section 3. 3. Results All the eleven equilibria computed by CREATE-L code with the iterative procedure are characterized by a lower value of F with respect to the one that is obtained from the preliminary phase (i.e. without the iterative procedure). In fact, as it can be seen from Table 3, the relative percentage reduction of the objective function F goes from 50% up to the 95%. This inevitably leads to more accurate results also in terms of poloidal magnetic fields computed by the CREATE-L model with respect to the experimental measurements. The associated relative percentage error for each sensor measurement is around the 5% for all the shots under analysis, as it can be seen from Fig. 1 where three typical cases are considered each one representing a different plasma regime (36922 low-β, 39068 intermediate-β, 39122 biased induced H-mode regime). Some exceptions could involve the values related to the pickup coils located in proximity of the X-point (θ = 117°) where the poloidal magnetic field goes to zero. As it can be seen from Table 4 and Fig. 1, the relative percentage
Fig. 2. Iterative procedure results in terms of plasma boundary and poloidal magnetic flux topology: CREATE-L with iterative procedure (Black), CREATE-L without the iterative procedure (Red) and plasma boundary reconstruction from magnetic measurements (Blue) and first wall (Green) (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article).
error for shot no. 39068 is about 15% but in any case far below 30% which is the value of the preliminary phase. It is important to stress that lower values of F lead also to a better match between CREATE-L computed plasma boundary and the one reconstructed from magnetic diagnostic measurements [7] as it can be seen from Fig. 2; the CREATE-L plasma boundary (Black line) obtained with the iterative procedure matches very well the reconstructed one (Blue dashed line) and improves the starting CREATE-L model (Red line) obtained in the preliminary phase. The improvement in terms of plasma boundary is also reflected in terms of poloidal magnetic flux topology, since the iterative procedure allows in finding the X-point magnetic configuration (Black line) and not a limiter one as for the starting model (Red line). A comparison between the Rogowski coil measurement and the total plasma current value obtained by performing the discrete line integral of the poloidal magnetic fields measured by pick-up coils revealed a relative percentage difference of this line integral (with respect to the Rogowski value) in the order of 10% as reported in fifth column of Table 5 (see εIp(Bθ)). Since the effect of the induced current in the vessel is already taken into account by subtracting its contribution in the measurements, and in any case it is negligible for tokamak discharges far away from the ramp-up phase, this discrepancy has to be considered as a kind of systematic "experimental uncertainty" which is present for
Fig. 1. Relative percentage error with respect to the measured poloidal magnetic fields for three typical experimental shots. 3
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which is consistent with the other low-current experimental shots previously analyzed in tokamak configuration. These results highlight a better accuracy of the Rogowski coil for high-current RFP operations with respect to the low-current RFP and tokamak ones. This suggests the possibility of an uncertainty on the calibration of the Rogowski coil which is evident at low-current operation. In fact, the Rogowski coil was usually calibrated considering only RFP magnetic configuration with a simple subtraction of a constant related to the poloidal voltage loop integral, i.e. the toroidal magnetic flux. This simple correction provided good results in high current RFP discharges but not in the low-current ones, irrespective of being in RFP or tokamak configuration.
Table 5 Comparison between total plasma current values. Shot
IRog [kA]
ICREATE-L [kA]
εIp,CREATE-L [%]
εIp(Bθ)
36922 39036 39039 39040 39068 39084 39122 39123 39135 39136
56.11 59.37 61.84 64.84 60.91 63.63 60.94 56.89 49.51 51.01
58.24 62.47 65.94 66.11 64.03 65.83 62.42 57.57 52.16 53.86
3.80 5.22 6.64 1.96 5.12 3.46 2.43 1.21 5.35 5.58
9.90 9.55 8.92 9.30 9.07 8.49 9.08 10.22 9.89 9.19
4. Conclusions all the duration of the plasma discharge in all the eleven tokamak shots analysed. The results of the iterative procedure revealed a clear dependence of the plasma equilibrium on variations of the total plasma current with respect to the Rogowski measurement. In fact, as reported in Table 5, the most accurate equilibrium is always found by the iterative procedure with an increased value of total plasma current (ICREATE-L) with respect to the Rogowski measurement (IRog). Increased values of total plasma current with respect to the measured values from 1% to 7% (εIp, CREATE-L) are necessary to fit the experimental data in terms of poloidal magnetic fields and plasma boundary reconstruction. It can be noticed also that this increased value of plasma current is always lower than the level of experimental uncertainty (i.e. 10%). The same results have been obtained by applying the iterative procedure with PEGASOS plasma equilibrium code which solves the same equations of CREATE-L but with a different computational approach (i.e. Cell Method [8]). In addition, the iterative procedure has been applied with PEGASOS code in reproducing a high-current RFP plasma equilibrium (i.e. shot no. 29283@ 0.1 s, Ip = 2 MA) and a lowcurrent one (i.e. shot no. [email protected] s, Ip = 169kA), both in axisymmetric approximation. For both cases, the only free parameter was the total plasma current since the plasma current density profile has been set from experimental estimation by means of the parametrization proposed in [9]. Thus, for the high-current RFP case, the equilibrium found by the procedure has a percentage increment of total plasma current (εIp, PEGASOS) of 0.45%, which is very small compared to the 10% of tokamak equilibria. On the other hand, the low-current RFP case has a percentage increment of total plasma current of about 9%,
An iterative procedure for the production of accurate plasma equilibria, and the related linearized plasma response models, has been developed and tested successfully on several experimental RFX-mod shaped tokamak discharges. The procedure allowed obtaining very accurate models in terms of both computed poloidal magnetic fields with respect to the pick-up coil measurements, and plasma boundary which matches very well the one reconstructed from magnetic measurements. The procedure also revealed a sensitivity of plasma equilibria on total plasma current; a comparison between tokamak and RFP results suggests the possibility of an uncertainty on the calibration of the Rogowski coil in low plasma current operations. The iterative procedure here proposed will be used for the analysis of existing tokamak equilibria data with different plasma shape and regimes in view of starting the operations of the new RFX-mod2 experiment; in addition, the procedure is general and can be applied in principle to other devices with minor modifications. References [1] [2] [3] [4] [5] [6] [7] [8]
M. Spolaore, et al., Nucl. Fusion (2017). R. Albanese, F. Villone, Nucl. Fusion 38 (5) (1998) 723. G. Marchiori, et al., FED 108 (2016) 81–91. J. Luxon, B. Brown, Nucl. Fusion 22 (6) (1982) 813. D. Abate, et al., 43rd EPS Conference, 4-8 July, (2016). J. Blum, Numerical Simulation and Optimal Control in Plasma Physics, (1989). O. Kudlacek, et al., Phys. Plasmas 22 (10) (2015) 102503. E. Tonti, The Mathematical Structure of Classical and Relativistic Physics, Springer, 2013. [9] P. Bettini, et al., Nucl. Fusion 43 (2) (2003) 119.
4
Contents lists available at ScienceDirect
Fusion Engineering and Design journal homepage: www.elsevier.com/locate/fusengdes
Modelling and experimental validation of RFX-mod Tokamak shaped discharges ⁎
Domenico Abatea, , Giuseppe Marchioria, Fabio Villoneb a b
Consorzio RFX, Corso Stati Uniti 4, 35127 Padova, Italy Consorzio CREATE, DIETI, Università di Napoli Federico II, Via Claudio 21, Napoli, Italy
A R T I C LE I N FO
A B S T R A C T
Keywords: Linearized plasma response model Rogowski coil Total plasma current sensitivity Iterative procedure Plasma equilibrium
A general procedure for computing plasma equilibria through the CREATE-L code has been applied to RFX-mod shaped tokamak discharges in order to produce very accurate models with respect to experimental data. It involves the solution of a constrained non-linear minimization problem to estimate the CREATE-L free parameters by using an iterative scheme trying to minimize the discrepancy between the poloidal magnetic field experimentally measured by pick-up coils and the one simulated by the model. The procedure has been applied considering eleven experimental shots spanning the whole range of poloidal beta achieved in the RFX-mod tokamak. A preliminary sensitivity analysis showed a non-negligible dependence of static equilibria on variations of the total plasma current with respect to the measurement provided by Rogowski coils. Thus, the total plasma current has been set as an additional degree of freedom in the minimization problem assuming values between the Rogowski measurement and the value of the discrete line integral of the poloidal magnetic field measured by the pick-up coils. In all cases under analysis, the iterative procedure showed that the most accurate equilibrium is obtained with a plasma current higher than the Rogowski measurement but lower than the pickup coils line integral.
1. Introduction Shaped Tokamak discharges with an insertable polarized electrode have been executed in RFX-mod to achieve H-mode regime [1]. Equilibrium magnetic configurations with different plasma shapes have been experimentally produced and analysed by means of the linearized plasma response model CREATE-L [2]. So far, plasma linearized models were produced using experimental values for the input equilibrium currents (i.e. active coil and plasma total current) [3], while the plasma current density parameters have been chosen on the basis of the available measurements on poloidal beta and internal inductance. This procedure leads to a noticeable inaccuracy in the fitting of magnetic field measurements, which was higher, with respect to standard tokamak operations, in all the experimental plasmas related to the Hmode campaign – these experiments are characterized by enhanced plasma shape conditions to explore its role in the L-H transition. In this paper, we describe a general iterative procedure [4] for computing plasma equilibria, and the related linearized plasma response models, through the CREATE-L code with a high level of accuracy with respect to experimental data. In addition, the procedure has been tested with the PEGASOS plasma equilibrium code reproducing
⁎
the same tokamak experimental shots and additionally two Reversed Field Pinch (RFP) shots. In all cases under analysis, the iterative procedure shows that the most accurate equilibrium is obtained with a plasma current higher than the Rogowski measurement but lower than the pick-up coils line integral. The paper is organized as follows. Section 2 describes the formulation of the iterative procedure and the methodology adopted for its application to RFX-mod experimental data. Section 3 reports the results and their impact on RFX-mod experiment. Finally, Section 4 draws the conclusions. 2. Problem formulation The iterative procedure involves the solution of a constrained nonlinear minimization problem to estimate the CREATE-L free parameters of the current density by using an iterative scheme trying to minimize the discrepancy between the experimental poloidal magnetic fields and the one simulated by the model [4]. The iterative procedure proposed is a simplified version of the one presented in [3] and it is in principle valid for any equilibrium code, since it is related to the estimation of the best values of the free parameters of the code to describe the reference
Corresponding author. E-mail address: [email protected] (D. Abate).
https://doi.org/10.1016/j.fusengdes.2018.11.055 Received 12 September 2018; Received in revised form 14 November 2018; Accepted 29 November 2018 0920-3796/ © 2018 Elsevier B.V. All rights reserved.
Please cite this article as: Abate, D., Fusion Engineering and Design, https://doi.org/10.1016/j.fusengdes.2018.11.055
Fusion Engineering and Design xxx (xxxx) xxx–xxx
D. Abate et al.
the total plasma current has been set as an additional degree of freedom in the minimization problem leading to have W = [αM , αN , β0, Ip] with 4 different constraints on their possible values (Table 2). The total plasma current and the parameter related to the poloidal beta (β0 ) have been allowed to assume variations up to 10% of their experimental values; for the total plasma current this correspond to the value obtained by computing the discrete line integral of the magnetic poloidal field measured by the pick-up coils as we will see in next section. The boundaries on the other two dof have been chosen considering the typical physical conditions for RFX-mod tokamak plasmas: the parameter related to the on axis safety factor (αN ) is assumed to vary around 1, accordingly to the standard inductive tokamak scenario, while the parameter related to the internal inductance (αM ) covers a plasma current density profile from peaked to broader shape. The search of the solution is carried out using the fmincon function of MATLAB with the default interior point algorithm. The algorithm satisfies constraints at all iterations and solves a sequence of approximate minimization problems. The approximate problem is a sequence of equality constrained problems which are easier to solve than the original one; the method is then based on the method of Lagrange multipliers and the Karush-Kuhn-Tucker (KKT) conditions. By default, the algorithm first attempts to take a direct step, i.e. Newton step, for the solution of the KKT equations via a linear approximation. If it cannot, it attempts a conjugate gradient CG step. Since the Hessian is unknown, the algorithm computes a quasi-Newton approximation to the Hessian of the Lagrangian at each iteration. Details about the interior point algorithm can be found in the MATLAB optimization toolbox user's guide. Before starting the iterative procedure a preliminary phase is needed in order to get a starting model in which the free parameters are fixed to the typical values of RFX-mod standard tokamak operations [3] (i.e. αM = 0.7, αN = 1.001, β0 = 0.1) while the active coil currents, including the internal-external saddle coil circuit which provides the vertical stability of the elongated plasmas, are the experimental ones. The equilibrium poloidal magnetic flux values of this starting model will be used as initial guess for launching the CREATE-L code in the iterative procedure. In order to start the iterative procedure, we followed the so called “multi-start” approach by defining a certain number of initial points, Ni, corresponding to Ni values for each degree of freedom, from where the search for a minimum begins; thus, the vector W is converted in a matrix whose dimension is Ni×Ndof. In this way, it is possible to locate a minimum by approaching it from many different directions. The number of initial points is the factor that mainly determines the computational time of the procedure. For this reason different preliminary runs, with an increasing number of initial points, have been performed; the minimum computational time obtained for 10 initial points was approximately 2.7 h on a PC equipped with two 6-core Intel Xeon CPU X5660 @ 2.8 GHz processors. Nevertheless, because of the highly nonlinear structure of the problem, there is no explicit law governing the computational time. One of the most demanding case has been obtained with 80 initial points and required above 40 h of computation; nevertheless a similar result in terms of accuracy (i.e. F value) was reached with only 10 initial points and a much shorter computational time. Since a detailed convergence analysis was not the focus of this work and we are looking for local minima indicating an optimal value for the set of four parameters in Eq. (2), ten starting points are usually a good compromise to find an accurate solution in a reasonable computational time. The iterative procedure has been applied to all the experimental shots reported in Table 1; in order to analyse the results obtained with the procedure we define a relative percentage variation factor, which will be used as a relative percentage error for many physical quantities compared to their experimental measurements (e.g. poloidal magnetic field, total plasma current):
Table 1 Shot numbers under analysis, equilibrium time instants, poloidal beta and plasma regimes. Shot
teq [s]
βp
Plasma regime
36922 39036 39039 39040 39068 39084 39122 39123 39135 39136
0.1 0.5 0.5 0.5 0.4 0.71 0.85 0.85 0.5 0.5
0.1 0.5 0.5 0.5 0.5 0.75 0.8 1 0.7 0.65
Low-β Intermediate-β Intermediate-β Intermediate-β Intermediate-β Increased-β H-mode H-mode Increased-β Increased-β
experimental data as accurately as possible. Once a plasma current density profile is established by a mathematical parametrization of it, the problem is to determine the best values of the profile parameters. Eleven experimental shots have been identified and considered in this study: all of them are Upper Single Null tokamak configurations spanning the whole range of poloidal beta achieved in the RFX-mod tokamak (low-β, intermediate-β, biased induced H-mode regime) as reported in Table 1. In our case, by following the parameterization proposed in [6], the three parameters are (αM , αN , β0) related to the physical quantities (li , q0 , βp) . This leads to search the solution of a constrained non-linear minimization problem, in which we want to minimize a parameter that relates the values of the experimental poloidal field measured by the 8 magnetic pick-up coils located on the inner surface of the stabilizing shell with the computed values given by the model. It has been chosen to minimize the difference between the measured and the computed values of the poloidal field normalized with respect to the experimental values, as defined in Eq. (1): Nsensors
F=
∑ i=1
(Bθi experimental − Bθi simulated )2 |Bθi experimental|
(1)
where Nsensors = 8 is the number of magnetic pick-up coils able to measure the poloidal component Bθ of the magnetic field, i.e. the tangential component. The flux loops measurements have not been included, in order to avoid a further degree of freedom related to the relative weight of magnetic fields and fluxes in the error functional, or to avoid a multi-objective optimization. A posteriori we can say that the quality of the fit is adequate also using the magnetic field measurements alone. The objective function F is defined for each set of computed and experimental values of poloidal magnetic field related to a given set of degrees of freedom (dof). Therefore the problem can be stated as a nonlinear constrained minimization problem of finding a vector of dof, W = [αM , αN , β0], that is a local minimum to the scalar function F subject to constraints on the allowable values ofW :
⎧ minW F ⎨ ⎩L≤W≤U
(2)
where L and U are the lower and upper boundary values of W defined in Table 2. A preliminary sensitivity analysis showed a non-negligible dependence of static equilibria on variations of the total plasma current with respect to the measurement provided by Rogowski coils (IRog) [5]. Thus, Table 2 Lower and upper boundaries values for each degree of freedom. Bounds
αM
αN
β0
Ip
L U
0.5 2
0.9 1.1
βp-0.1βp βp+0.1βp
IRog IRog+ΔIRog
2
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Table 3 Iterative procedure results in terms of objective function F for all the experimental shots under analysis. Shot
F starting case
F iterative results
εF [%]
36922 39036 39039 39040 39068 39084 39122 39123 39135 39136
1.5148 2.7048 6.44457 0.3981 1.4717 1.2914 0.6725 3.3476 1.3329 1.3963
0.1224 0.4905 1.2414 0.1963 0.1529 0.4451 0.1810 0.7169 0.0775 0.3226
−91.92 −81.87 −80.74 −50.69 −89.61 −65.53 −73.09 −78.58 −94.19 −76.90
εV =
V − V* ∙100 V*
Table 4 Relative percentage error comparisons between preliminary results and the one obtained from the iterative procedure for Bθ measurements in a reference shot. Shot no. 39068 Bθ measurements θ [°]
ε preliminary [%]
ε procedure [%]
27 72 117 162 207 252 297 342
9.62 3.27 28.78 2.44 −3.97 −7.79 2.92 17.26
1.35 0.20 −12.54 −2.05 −1.85 −3.17 −0.37 −1.47
(3)
where V is the quantity considered (e.g. Bθ, F, Ip) and V* is the reference quantity used as comparison (e.g. the experimental value, the reference case). The εV parameter leads us to show the improvements obtained in terms of lower values of F by the iterative procedure with respect to the starting model obtained from the preliminary phase; in addition, the relative percentage error εV will be used also to compare some physical quantities computed by the CREATE-L model with respect to experimental values, such as the poloidal magnetic field as we will see in Section 3. 3. Results All the eleven equilibria computed by CREATE-L code with the iterative procedure are characterized by a lower value of F with respect to the one that is obtained from the preliminary phase (i.e. without the iterative procedure). In fact, as it can be seen from Table 3, the relative percentage reduction of the objective function F goes from 50% up to the 95%. This inevitably leads to more accurate results also in terms of poloidal magnetic fields computed by the CREATE-L model with respect to the experimental measurements. The associated relative percentage error for each sensor measurement is around the 5% for all the shots under analysis, as it can be seen from Fig. 1 where three typical cases are considered each one representing a different plasma regime (36922 low-β, 39068 intermediate-β, 39122 biased induced H-mode regime). Some exceptions could involve the values related to the pickup coils located in proximity of the X-point (θ = 117°) where the poloidal magnetic field goes to zero. As it can be seen from Table 4 and Fig. 1, the relative percentage
Fig. 2. Iterative procedure results in terms of plasma boundary and poloidal magnetic flux topology: CREATE-L with iterative procedure (Black), CREATE-L without the iterative procedure (Red) and plasma boundary reconstruction from magnetic measurements (Blue) and first wall (Green) (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article).
error for shot no. 39068 is about 15% but in any case far below 30% which is the value of the preliminary phase. It is important to stress that lower values of F lead also to a better match between CREATE-L computed plasma boundary and the one reconstructed from magnetic diagnostic measurements [7] as it can be seen from Fig. 2; the CREATE-L plasma boundary (Black line) obtained with the iterative procedure matches very well the reconstructed one (Blue dashed line) and improves the starting CREATE-L model (Red line) obtained in the preliminary phase. The improvement in terms of plasma boundary is also reflected in terms of poloidal magnetic flux topology, since the iterative procedure allows in finding the X-point magnetic configuration (Black line) and not a limiter one as for the starting model (Red line). A comparison between the Rogowski coil measurement and the total plasma current value obtained by performing the discrete line integral of the poloidal magnetic fields measured by pick-up coils revealed a relative percentage difference of this line integral (with respect to the Rogowski value) in the order of 10% as reported in fifth column of Table 5 (see εIp(Bθ)). Since the effect of the induced current in the vessel is already taken into account by subtracting its contribution in the measurements, and in any case it is negligible for tokamak discharges far away from the ramp-up phase, this discrepancy has to be considered as a kind of systematic "experimental uncertainty" which is present for
Fig. 1. Relative percentage error with respect to the measured poloidal magnetic fields for three typical experimental shots. 3
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which is consistent with the other low-current experimental shots previously analyzed in tokamak configuration. These results highlight a better accuracy of the Rogowski coil for high-current RFP operations with respect to the low-current RFP and tokamak ones. This suggests the possibility of an uncertainty on the calibration of the Rogowski coil which is evident at low-current operation. In fact, the Rogowski coil was usually calibrated considering only RFP magnetic configuration with a simple subtraction of a constant related to the poloidal voltage loop integral, i.e. the toroidal magnetic flux. This simple correction provided good results in high current RFP discharges but not in the low-current ones, irrespective of being in RFP or tokamak configuration.
Table 5 Comparison between total plasma current values. Shot
IRog [kA]
ICREATE-L [kA]
εIp,CREATE-L [%]
εIp(Bθ)
36922 39036 39039 39040 39068 39084 39122 39123 39135 39136
56.11 59.37 61.84 64.84 60.91 63.63 60.94 56.89 49.51 51.01
58.24 62.47 65.94 66.11 64.03 65.83 62.42 57.57 52.16 53.86
3.80 5.22 6.64 1.96 5.12 3.46 2.43 1.21 5.35 5.58
9.90 9.55 8.92 9.30 9.07 8.49 9.08 10.22 9.89 9.19
4. Conclusions all the duration of the plasma discharge in all the eleven tokamak shots analysed. The results of the iterative procedure revealed a clear dependence of the plasma equilibrium on variations of the total plasma current with respect to the Rogowski measurement. In fact, as reported in Table 5, the most accurate equilibrium is always found by the iterative procedure with an increased value of total plasma current (ICREATE-L) with respect to the Rogowski measurement (IRog). Increased values of total plasma current with respect to the measured values from 1% to 7% (εIp, CREATE-L) are necessary to fit the experimental data in terms of poloidal magnetic fields and plasma boundary reconstruction. It can be noticed also that this increased value of plasma current is always lower than the level of experimental uncertainty (i.e. 10%). The same results have been obtained by applying the iterative procedure with PEGASOS plasma equilibrium code which solves the same equations of CREATE-L but with a different computational approach (i.e. Cell Method [8]). In addition, the iterative procedure has been applied with PEGASOS code in reproducing a high-current RFP plasma equilibrium (i.e. shot no. 29283@ 0.1 s, Ip = 2 MA) and a lowcurrent one (i.e. shot no. [email protected] s, Ip = 169kA), both in axisymmetric approximation. For both cases, the only free parameter was the total plasma current since the plasma current density profile has been set from experimental estimation by means of the parametrization proposed in [9]. Thus, for the high-current RFP case, the equilibrium found by the procedure has a percentage increment of total plasma current (εIp, PEGASOS) of 0.45%, which is very small compared to the 10% of tokamak equilibria. On the other hand, the low-current RFP case has a percentage increment of total plasma current of about 9%,
An iterative procedure for the production of accurate plasma equilibria, and the related linearized plasma response models, has been developed and tested successfully on several experimental RFX-mod shaped tokamak discharges. The procedure allowed obtaining very accurate models in terms of both computed poloidal magnetic fields with respect to the pick-up coil measurements, and plasma boundary which matches very well the one reconstructed from magnetic measurements. The procedure also revealed a sensitivity of plasma equilibria on total plasma current; a comparison between tokamak and RFP results suggests the possibility of an uncertainty on the calibration of the Rogowski coil in low plasma current operations. The iterative procedure here proposed will be used for the analysis of existing tokamak equilibria data with different plasma shape and regimes in view of starting the operations of the new RFX-mod2 experiment; in addition, the procedure is general and can be applied in principle to other devices with minor modifications. References [1] [2] [3] [4] [5] [6] [7] [8]
M. Spolaore, et al., Nucl. Fusion (2017). R. Albanese, F. Villone, Nucl. Fusion 38 (5) (1998) 723. G. Marchiori, et al., FED 108 (2016) 81–91. J. Luxon, B. Brown, Nucl. Fusion 22 (6) (1982) 813. D. Abate, et al., 43rd EPS Conference, 4-8 July, (2016). J. Blum, Numerical Simulation and Optimal Control in Plasma Physics, (1989). O. Kudlacek, et al., Phys. Plasmas 22 (10) (2015) 102503. E. Tonti, The Mathematical Structure of Classical and Relativistic Physics, Springer, 2013. [9] P. Bettini, et al., Nucl. Fusion 43 (2) (2003) 119.
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