Open AccessArticle

Modeling Approaches for Accounting Radiation-Induced Effect in HVDC-GIS Design for Nuclear Fusion Applications

Francesco Lucchini

^1,2

Alessandro Frescura

^2,3

Kenji Urazaki Junior

¹,

Nicolò Marconato

^1,2,*

and

Paolo Bettini

^1,2,3

Department of Industrial Engineering, University of Padova, Via Gradenigo 6/a, 35131 Padova, Italy

Consorzio RFX (CNR, ENEA, INFN, Università di Padova, Acciaierie Venete SpA), Corso Stati Uniti 4, 35127 Padova, Italy

Centro Ricerche Fusione (CRF), University of Padova, Corso Stati Uniti 4, 35127 Padova, Italy

Author to whom correspondence should be addressed.

Appl. Sci. 2024, 14(24), 11666; https://doi.org/10.3390/app142411666

Submission received: 9 November 2024 / Revised: 6 December 2024 / Accepted: 12 December 2024 / Published: 13 December 2024

(This article belongs to the Special Issue Novel Approaches and Challenges in Nuclear Fusion Engineering)

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Abstract

This paper examines the modeling approaches used to analyze the electric field distribution in high-voltage direct-current gas-insulated systems (HVDC-GISs) used for the acceleration grid power supply (AGPS) of neutral beam injectors (NBIs). A key challenge in this context is the degradation of dielectric performance due to radiation-induced conductivity (RIC), a phenomenon specific to the harsh radioactive environments near fusion reactors. Traditional models for gas conductivity in HVDC-GISs often rely on constant or nonlinear conductivity formulations, which are based on experimental data but fail to capture the effects of external ionizing radiation that triggers RIC. To address this limitation, a more advanced approach, the drift–diffusion recombination (DDR) model, is used, as it more accurately represents gas ionization and the influence of radiation fields. However, this increased accuracy comes at the cost of higher computational complexity. This paper compares the different modeling strategies, discussing their strengths and weaknesses, with a focus on the capabilities in evaluating the charge accumulation and the RIC phenomenon.

Keywords:

HVDC; gas-insulated systems; transmission line; AGPS; neutral beam injector (NBI); radiation-induced conductivity (RIC)

1. Introduction

High-voltage direct-current (HVDC) equipment plays a fundamental role in the development and operation of neutral beam injectors (NBIs) installed in experimental thermonuclear fusion devices [1].

The NBI, together with ion cyclotron resonant heating and electron cyclotron resonant heating, is one of the heating and current drive systems currently adopted in experimental magnetic confinement fusion (MCF) devices. The negative NBI (NNBI) accelerates negative ions through a series of acceleration grids polarized at different voltage levels. The DC voltage polarizing each acceleration stage is provided by the acceleration grid power supply (AGPS). HVDC gas-insulated systems (GISs) are required to keep the size of the AGPS small and to ensure the dielectric strength of the system. Nowadays, the largest and most powerful HVDC-GIS components for the AGPS are installed in the Megavolt ITER Injector & Concept Advancement facility hosted at Consorzio RFX laboratories in Padua. HVDC-GIS components such as the transmission line (TL), high-voltage bushing, DC filter [2], and diode rectifiers [3] are required to ensure the −1 MV insulation, corresponding to five acceleration stages (−200 kV each) of the NNBI grids [4,5]. HVDC-GISs are required for the deployment of the AGPS for NNBIs to be installed in future experimental MCFs such as the Divertor Tokamak Test (DTT) facility, under construction in Frascati, Italy [6], and the China Fusion Engineering Test Reactor (CFETR) [7], aimed at bridging the gap between ITER and fusion demonstration power plants. The NNBI of DTT accelerates ions up to an energy of 510 keV [8,9], while that of CFETR reaches 1 MeV [10].

Differently from the commercial HVDC gas-insulated lines (GILs), the TL of the AGPS is multi-conductor, indeed all the voltage levels of the ion acceleration grids are transmitted within the same line. The design of the TL requires particular attention to what concerns both the conductors’ mutual positioning and the cross-section diameter selection. Indeed, conflicting demands must be met. From one side, reducing the overall volume of the TL is advantageous to reduce the amount of insulating gas. However, the selected geometry must satisfy the insulation requirements to avoid partial discharges and breakdowns.

Usually, commercial HVDC-GILs are placed underground [11]; however, this is not true for HVDC-GISs of NNBI AGPS. Due to the harsh radioactive environment in which HVDC-GISs will be forced to operate when placed in the proximity of MCF devices, the dielectric performances of the insulating gas can degrade quickly given radiation-induced conductivity (RIC) effects [12].

Advanced numerical tools tailored for specific applications are required to address these issues and sustain the engineering and operational phases. In the literature, different approaches are proposed to analyze the electric field distribution within HVDC-GISs, each characterized by pros and cons. This paper aims to compare the different modeling approaches for HVDC-GISs used in fusion technology and their capability to capture fundamental phenomena affecting the dielectric performances of the systems.

This paper is organized as follows. Section 2 briefly recalls the surface charge accumulation phenomenon within HVDC-GIS equipment. In Section 3, the RIC phenomenon is described. The electric field modeling approaches are described in Section 4. The conduction mechanism within the gas is explained in Section 5; in particular, an equivalent gas conductivity model accounting for the RIC phenomenon is introduced in Section 6. Numerical results are shown in Section 7. A discussion about the outcomes of the research is given in Section 8. Finally, Section 9 summarizes the main outcomes of this work.

2. Surface Charge Accumulation

In this section, the mechanisms of surface charge accumulation are briefly recalled. The electro-quasistatic (EQS) formulation is suited for modeling the capacitive-to-resistive transition of the electric field in HVDC systems [13,14].

It is known from the literature that the charge accumulating along the interface between dielectric materials with different properties, i.e., the solid insulators and the dielectric gas, is responsible for the degradation of insulation performances of HVDC-GIS components and represents one of their major fault causes [15,16]. The surface charge not only distorts the electric field distribution within the apparatus but also may trigger the surface flashover of the insulator [17,18]. The charge deposited on the solid insulator surface is subdivided into hetero-charge and homo-charge [19], where the former has the opposite polarity of the applied DC voltage, while the latter has the same polarity of the closest electrode. Experimental results show that hetero-charge is mainly due to conduction within the dielectric gas. Homo-charge, instead, is mainly driven by the nonlinear conductivity of the solid insulator and interface layer.

The surface charge density

ϱ_{S}

is due to the jump of the normal components of the electric displacement fields in the solid insulator, labeled with suffix I, and the dielectric gas, with suffix G:

ϱ_{S} = n \cdot (D_{I} - D_{G}),

(1)

where

n

is the unit normal vector pointing in the outward direction of the solid insulator domain and

D = ε E

, where

ε = ε_{0} ε_{r}

is the electric permittivity, with

ε_{0}

being the vacuum permittivity and

ε_{r}

being the relative permittivity. The dynamics of surface charge accumulation are recognized to be driven by three conduction mechanisms [20]:

Electric conduction within the gas.
Electric conduction in solid insulators.
Electric conduction of the interface layer.

Considering a portion of the interface layer of thickness

δ

as illustrated in Figure 1, the dynamic of surface charge accumulation is governed by the following ordinary differential equation (ODE):

\frac{d ϱ_{S}}{d t} = n \cdot (J_{I} - J_{G}) - \nabla_{S} \cdot (σ_{S} E_{t}),

(2)

where

J_{I}

is the current density in the solid domain,

J_{G}

is the current density in the gas,

σ_{S}

[S] is the interface layer electric conductivity,

E_{t}

the tangential electric field strength, and

\nabla_{S}

represents the surface divergence. It is worth mentioning that, generally,

σ_{S}

is a nonlinear function of electric field itself in the form [21]:

σ_{S} (E_{t}) = σ_{S 0} exp (α_{S} E_{t}),

(3)

where

1.2 < α_{S} < 1.4

[mm/kV] is the usual range of the nonlinearity exponent.

The charge accumulation can be reduced by an accurate design of the solid insulator shape [22] or by using innovative insulating materials, for example, field grading materials [23]. From the point of view of the authors, this last point requires intense industrial participation to reduce construction costs and make this technology available for future HVDC-GISs.

3. Effect of External Ionization Field

External ionization sources, such as radioactive elements in the ground and cosmic radiation, consistently influence the dielectric performance of HVDC gas-insulated equipment [14].

In the realm of nuclear fusion applications, HVDC-GISs are forced to operate in a harsh environment due to high ration levels, mainly

γ

-rays and neutrons coming from the MCF device. In the early stages of the development of NNBI of ITER, the issue of radiation drove the re-design of the SF

_{6}

-insulated high-voltage bushing (HVB), connecting the last part of the TL to the ion acceleration grids. Indeed, the estimated absorbed dose rate in SF

_{6}

of about 0.1 Gy/s results in unacceptable levels of ionization current

I_{s a t}

flowing in the gas [24]. This is the RIC phenomenon. In the case of ITER NNBI, the SF

_{6}

-insulated HVB is substituted with a SF

_{6}

to vacuum HVB rated for the −1 MV voltage level [5,25]. An experimental formula for estimating

I_{s a t}

for a given volume of gas v, with a pressure P [MPa], and subjected to an absorbed dose rate

ϕ

I_{s a t} = f_{g a s} \times v \times P \times ϕ

(4)

where

f_{g a s}

is a gas-dependent constant [24]. As illustrated in Figure 2, showing the Tokamak hall of DTT, the HVB connecting the TL to the ion source is subjected to the irradiation level of the machine [9].

Equation (4) can in principle be used to estimate the saturation current; however, the trend expressed by such an equation is based on experimental results with controlled geometries and cannot reproduce the effective electrode configuration of the component. To obtain information about the actual current density within the insulating gas under a high-ionization field for general geometries, more accurate numerical models are required. The way the ionization field affects the dielectric gas conductivity

σ_{G}

is a central topic for modeling HVDC-GISs for nuclear fusion equipment.

4. Electric Field Modeling

Modeling the electric field transition of HVDC systems from the initial (capacitive) to the final (resistive) distribution requires the solution of the EQS limit of Maxwell’s equations, which, in terms of the scalar electric potential

φ

, reads as follows:

\nabla \cdot [ε \nabla (\frac{\partial φ}{\partial t})] + \nabla \cdot (σ \nabla φ) = 0 .

(5)

The solution of Equation (5) requires the specification of

ε

and

σ

for each domain (gas or solid). The solid insulator, usually realized with cast epoxy resin or Alumina-filled epoxy resin [26], is characterized by nonlinear conductivity function on temperature T and electric field strength E:

σ_{I} (T, E) = σ_{I, 0} exp (α_{I} E + β_{I} T),

(6)

where

α_{I}, β_{I}

are two parameters. The functional dependence, even for FGMs, is usually extracted from experimental campaigns [27]. The conductivity of gas, in the simpler case, is usually kept constant; however, it may depend on different parameters, such as the temperature, electric field, pressure, and relative humidity percentage, i.e.,

σ_{G} = σ_{G} (T, E, P, % R H)

. This functional dependence is difficult to argue a priori for general geometries, insulating gases, and operational conditions, even if some attempts are made to extrapolate it from experimental results. For example, in [28], an analytical expression for

σ_{G}

of pure SF

_{6}

has been obtained by fitting experimental data, resulting in the following functional form:

σ_{G} (E, P, T) = σ_{G, 0} [α + β {(γ + \frac{E}{E_{c_{1}}})}^{ζ}] {(τ + ϵ \frac{E}{E_{c_{2}}})}^{- ι} exp (ξ P) exp (ν T)

(7)

where

σ_{G, 0}

is a base-level conductivity and

α

β

γ

ζ

τ

ϵ

ι

ξ

, and

ν

are parameters to be determined by fitting experimental data. In this expression, while the exponential dependence on gas pressure and temperature is quite trivial, the dependence on the electric field norm exhibits two different trends, embodied by the two field-dependent terms:

E_{c_{1}}

E_{c 2}

. The former captures the gas conductivity behavior below a first threshold electric field norm value defined by

E_{c_{1}}

, while the latter term describes the effect on these quantities at higher electric field regions. The electric field magnitude at which this second trend becomes important in the description of the gas is embodied by

E_{c_{2}}

. It is important to notice that the choice of both

E_{c_{1}}

and

E_{c_{2}}

is not trivial and affects the extrapolation accuracy and quality of all the other parameters in (7).

Simplified Approach

A simplified approach for the electric field modeling, when the displacement current is negligible, has been proposed by E. Volpov at the onset of numerical investigations into HVDC-GISs [29]. The surface charge dynamics, as described by Equation (2), are updated with an iterative approach. At a prescribed time step in the absence of free volumetric charge density, the unknown scalar electric potential

φ

is computed within each domain (gas and solid) by solving the Laplace equation: [29]:

∆ φ = 0 .

(8)

The surface charge is calculated by the continuity Equation (2), where the current in the gas and insulator are calculated as

J = σ E = - σ \nabla φ,

(9)

where

σ

is a specified electric conductivity. Therefore, the surface charge density

ρ_{S}

at the next time step is obtained. This surface charge density distribution affects the electric field map, and the procedure is repeated iteratively. It is worth mentioning that this method corresponds to the EQS when neglecting the displacement current since obtained as a sequence of stationary states. Moreover, the method is well suited for this application, since can take into account also of the free volumetric charge density

ρ_{G}

in the gas by solving the Poisson equation,

∆ φ = - \frac{ρ_{G}}{ε_{G}},

(10)

as described in the next paragraph.

5. Modeling Charge Transport in Dielectric Gas

The HVDC-GIS components are always subjected to irradiation fields coming from the surrounding environment. This radiation may be due to natural environmental radiation or may occur more severely in the vicinity of MCFs, as described in Section 3. In both cases, the irradiation field ionizes the dielectric gas with the consequent production of ion pairs. As described in Section 3, this fact cannot be neglected when dealing with nuclear fusion devices and should be taken carefully into account for accurate modeling due to the highly radioactive environment they are forced to operate.

Accurately modeling the ionization effect in the presence of high radiation levels, without using experimentally based formulas, is of great interest for the development of HVDC-GISs for nuclear fusion applications [1,21].

In this section, a drift–diffusion reaction (DDR) model is described to couple the radiation field with the production of free ions in the gas. The procedure for the latter case is described in [30]. The DDR model deals with positive and negative ion species, neglecting the presence of free electrons. This assumption is certainly valid for common dielectric gases used in HVDC-GIS apparatus, such as sulfur hexafluoride (SF

_{6}

). Indeed, SF

_{6}

is characterized by a high electron affinity (or electronegativity). Ideally, due to this property, all the free electrons in the gas are captured. Even in the case of a discharge, SF

_{6}

represents a good electric-arc extinction medium, precisely because of its high electron affinity, making it so widespread and difficult to substitute in HVDC applications despite its high environmental impact.

Positive and negative ions are described by their number densities [m

^{- 3}

n^{+}

and

n^{-}

. The source of ion pairs due to the external ionizing radiation field is described by the parameter S [m

^{- 3}

^{- 1}

]. Two mechanisms of migration occur in the gas. The first one is due to the electric force acting on the ions with mobility coefficients

μ^{+}

and

μ^{-}

^{2}

/(V·s)]. The second mechanism is the diffusion of ions with diffusion coefficients (

D^{+}

D^{-}

) that are defined, e.g., through the Einstein relation [31]:

D^{\pm} = μ^{\pm} \frac{k_{B} T}{e},

(11)

where

k_{B}

is the Boltzmann constant and e is the elementary charge. Recombination of ion pairs also occurs and is described by the coefficient R [m

^{3}

/s]. It is worth mentioning that the transport parameters of the gas are generally a function of the electric field, pressure, and temperature [32,33].

By combining these mechanisms, the following partial differential equations (PDEs) can be written for the local time variation of the ion densities:

\begin{matrix} \frac{\partial n^{+}}{\partial t} = S - R n^{+} n^{-} - \nabla \cdot (n^{+} μ^{+} E_{G}) + D^{+} ∆ n^{+}, \end{matrix}

(12)

\begin{matrix} \frac{\partial n^{-}}{\partial t} = S - R n^{+} n^{-} + \nabla \cdot (n^{-} μ^{-} E_{G}) + D^{-} ∆ n^{-} . \end{matrix}

(13)

where the electric field in the gas

E_{G}

is evaluated in a self-consistent way through the solution of the Gauss’s law:

\nabla \cdot E_{G} = \frac{e (n^{+} - n^{-})}{ε_{G}} .

(14)

The boundary and initial conditions for the solution of Equations (12) and (13) are described in [34], as follows:

A Dirichlet condition for $n^{+}$ is imposed in the boundaries, where $n \cdot E_{G} < 0$ , i.e., $n^{+}$ = 0, while a natural Neumann condition is imposed to $n^{-}$ therein, i.e., $- n \cdot D^{-} \nabla n^{-}$ = 0.
A Dirichlet condition for $n^{-}$ is imposed in the boundaries, where $n \cdot E_{G} > 0$ , i.e., $n^{-}$ = 0, while a natural Neumann condition is imposed to $n^{+}$ therein, i.e., $- n \cdot D^{+} \nabla n^{+}$ = 0.

Here,

n

denotes the unit vector normal to the gas domain. For what concerns the initial values of

n^{+} (t = 0)

and

n^{-} (t = 0)

, zero number densities can be assumed [35], or the equilibrium condition expressed by [31], as follows:

n^{+} (t = 0) = n^{-} (t = 0) = \sqrt{\frac{S}{R}} .

(15)

The equivalent current density flowing in the gas can be obtained by combining Equations (12) and (13), resulting in the following:

J_{G} = e [E_{G} (μ^{+} n^{+} + μ^{-} n^{-}) - \nabla (D^{+} n^{+} - D^{-} n^{-})] .

(16)

It is worth mentioning that the DDR model can be incorporated in the simplified approach described in Section 4, by solving the Poisson problem defined by Equation (10) in the gas, instead of (8).

It is important to note that in certain situations, such as studying the corona discharge phenomenon, the role of free electron collisions with gas molecules must be taken into account [36]. In these cases, a comprehensive set of electron–molecule collision reactions including ionization, excitation, attachment, and elastic collisions becomes essential.

6. An Equivalent Gas Conductivity for RIC Phenomenon

At the beginning of the EM analyses aimed at evaluating the insulation performances of SF

_{6}

for the TL of ITER NBI, an equivalent

σ_{G}

accounting for the RIC to be used in the EQS model, was defined from Equation (4) [37]:

σ_{G} = \frac{L}{A} \frac{I_{s a t}}{∆ V},

(17)

where L is the electrodes gap, A is the cross-section of TL, and

∆ V = 1

MV is the voltage. This expression is only valid for a simple electrode configuration; thus, a generalization is required. In this section, an attempt to obtain a relation linking gas conductivity to ion-pair generation rate S is carried out by noticing that if current density data are taken from a DDR simulation and plotted versus the corresponding electric field intensity; a recurrent pattern can be found to describe the link between these two quantities.

In Figure 3a, an example of this kind of plot is reported, taking current density and electric field values at each node of the tetrahedral mesh implemented for the DDR simulation. As can be observed, two leading trends emerge in the points’ distribution, described by two main data populations clustering around straight lines with different slopes. From a qualitative point of view, the distinctive feature of these two populations is ionic concentration in the region of the computational domain where data are collected: points around the lower slope line in each dataset belong to regions with lower concentration than points around the higher slope line. This can be confirmed by the fact that for very low values of S, the higher slope pattern tends to vanish, being completely absent if S is ideally set to zero. In Figure 3b, a comparison of the population for two different values of S is given. The red dataset corresponds to

S = 3 \times 10^{7}

^{- 3}

^{- 1}

, while the blue data are taken at

S = 10^{8}

^{- 3}

^{- 1}

. As can be observed by comparing the two datasets, the slope of both lines increases with S. In particular, the growth is linear in the case of the lower slope, while the higher slope shows a nonlinear behavior that tends to stabilize for high values of ion-pairs’ generation rate.

All the previous considerations motivate the idea to exploit this behavior of gas conductivity to obtain an alternative simulation approach by extrapolating an expression to be enforced in place of Equation (7), which has the non-trivial drawback of requiring to fit experimental data to a functional form defined by nine different parameters, which can be extremely difficult and finally lead to large confidence intervals on the identified parameters. Furthermore, the computational burden of such an alternative approach is still comparable to a standard EQS, since the only change in the implementation is the imposition of a gas conductivity linked to S, but constant inside the gas medium. This last aspect constitutes a fundamental advantage with respect to the DDR approach: indeed, despite its several interesting features and accuracy, the solution of two additional nonlinear drift–diffusion equations strongly affects its computational burden, possibly preventing a wide application in the field of nuclear fusion where the simulation of large-scale devices, e.g., the HVB inside the DTT Tokamak hall (see Figure 2), is often an unavoidable necessity [1].

To extract an explicit relation between conductivity and the generation rate, several DDR simulations were obtained for different values of S. The lower slope trend was considered to be representative of the average behavior of the gas, while the higher slope trend is considered as a deviation from the average, which is a reasonable assumption when S is not too high since regions of the computational domain where ions’ concentration is considerably higher than the average are kept to a small percentage of the total gas domain. From this procedure, a relation between

σ_{G}

and S can be obtained as follows:

σ_{G} = κ S + σ_{0}

(18)

where

σ_{0}

is the conductivity of the non-irradiated gas (set to

10^{- 19}

S/m), and a constant

κ

[Sm

^{2}

s] was found by this process to describe this linear dependence accurately. It is worth explaining that this behavior was found to be independent of the model geometry. Additionally, during the data analysis conducted to derive the given relation, a phenomenon was identified that helps explain the discrepancy between the results obtained from this relation and those from DDR simulations at high values of S. Indeed, as the generation rate grows, data points start to cluster not only around the line described by Equation (18), but the emergence of points populations following alternative regimes, i.e., clustering around straight lines with different slopes, can be observed. This is explained by the fact that ions are not uniformly spread inside the gas but concentrate in specific regions due to the effect of the electric field. Then, if the concentration of ions is low enough at low generation rates to allow an average description of gas conductivity via a single value, the presence of regions with a high ion concentration at high radiation levels, following a different J-E relation with respect to medium concentration regions, is not negligible anymore. Thus, the observed alternative conductivity regimes become important, preventing the application of a single value of

σ_{G}

to all the computational domains. Considering this aspect and the range of applicability of the proposed method, data obtained from the described simulations seem to suggest its validity for S, ranging from

10^{7}

10^{9}

^{- 3}

^{- 1}

. However, it is important to notice that this range could be strongly linked to the amount of gas taken into account during the simulation and to the specific geometry considered, since the presence of a larger volume available for ion motion can lower their tendency to cluster inside large high-concentration regions (where the proposed linear model is not a good approximation of gas conductivity due to the phenomena described above), thus keeping the majority of the computational domain to average concentration level and allowing an accurate application of the linear dependence of conductivity to S. Nevertheless, the existence of several regions, identified by their ionic concentration, where conductivity follows different specific trends, shows the need to obtain a relation for gas conductivity characterized by an explicit dependence on both ion-pair generation rate and electric field intensity (thus indirectly taking into account ions’ concentration levels), which is left to future work.

7. Numerical Results

In this section, the modeling methodologies presented in the previous sections are compared. In Section 7.1, the surface charge accumulation trend for a 2D axisymmetric HVDC-GIS test chamber is examined. In Section 7.2, the ionization effect upon the insulating gas of a multi-conductor TL for the AGPS is evaluated.

The models are implemented within the commercial Finite Element Software (FEM) COMSOL^® Multiphysics 6.2. In particular, the Electric Currents (ec) module is applied for the whole computational domain, with appropriate boundary conditions along the HV and LV electrodes. To add the DDR model, different possibilities are proposed in the literature. One is to use the Weak Form (w) as described in [31], or the Transport of Diluted Species (tds) module as pointed out in [30]. The latter approach is advantageous for the implementation of the boundary conditions to be applied to

n^{+}

and

n^{-}

since they are directly available within the Open Boundary options.

As described in [30], the tds is coupled with the ec for the electric field evaluation. In principle, this step requires a fictitious

σ_{G}

to be specified in the gas, usually set to be a very small constant value. The coupling between the tds model and the ec occurs at the gas–solid insulator interface where the charge accumulation phenomenon proceeds. Therein, a Boundary Current Source realizing

n \cdot (J_{I} - J_{G}),

(19)

with

J_{G}

given by Equation (16), is added to the model.

It is worth noting that, even though

σ_{G}

is set to a very small value, a post-processing analysis of the electric field/current characteristic of the gas clearly shows that this assumption is not fulfilled and the gas conductivity follows a nonlinear field-dependent pattern, supposing that temperature and gas pressure are kept constant, even when steady state is reached in a transient time-domain simulation.

7.1. Post-Insulator in SF $_{6}$

In this section, the models previously described are compared to establish the surface charge accumulation phenomenon along the gas–solid interface of a post-insulator embedded in a gas chamber filled with SF

_{6}

. The 2D axisymmetric geometry of the model, also considered in [31,38,39], is illustrated in Figure 4. The solid insulator of radius

r_{I} = 48

mm and height

h_{I} = 140

mm is fed from the top by a high-voltage (HV) electrode, while the chamber of radius

r_{O} = 180

mm and height

h_{O} = 284

mm is the low-voltage (LV) electrode. The SF

_{6}

filling the chamber has a pressure of

P = 0.5

MPa absolute, and the corresponding transport parameters to be used in the DDR model are listed in Table 1. The temperature is considered fixed at

T = 293.15

K in the subsequent simulations. The solid insulator is made of alumina-filled epoxy resin with

ε_{r, I} = 5

σ_{I} = 3.33 \cdot 10^{- 18}

S/m, and

σ_{S} = 0

The voltage applied to the HV electrode is increased from 0 V to 15 kV with a rise time of

10^{- 3}

s, while the LV electrode is referred to as ground potential (0 V). The simulation runs for

t = 8000

h, by using a backward differentiation formula (BDF) or maximum order 2. The trend of

ϱ_{S} (t)

, evaluated with the tds approach, is illustrated in Figure 5. For low external ionization, as is the case,

σ_{G}

is much less than

σ_{I}

, and a comparison with the EQS formulation with constant conductivity is made by using

σ_{G} = 1.23 \cdot 10^{- 17}

S/m (corresponding to

κ = 4.5 \cdot 10^{- 19}

). Using quadratic basis functions for

φ

and linear basis functions for

n^{+}, n^{-}

, the DDR model results in 65 k plus 30 k internal degrees of freedom (DoFs), while the EQS with constant conductivity has 50k DoFs, reflecting the increased computational burden. Moreover, the DDR model is nonlinear. Therefore, it requires, e.g., the Newton–Raphson algorithm for the solution. Conversely, the EQS with constant

σ

, described by Equation (5), is linear with respect to

φ

The steady-state trend of

ϱ_{S}

evaluated with the two models is reported in Figure 6. It can be seen that the two strategies are in good agreement.

Under a high ionization rate,

S = 3 \cdot 10^{9}

^{- 3}

^{- 1}

, the steady-state trend of

ϱ_{S}

is driven by the gas conductivity. As can be seen in Figure 7, the linear relation expressed by Equation (18), with

σ_{G} = 4.7 \cdot 10^{- 7}

S/m, is not able to replicate the trend obtained with the DDR model due to the limitations of this model for high S. As illustrated in Figure 8, if compared to the low-S regime, the

n^{-}

distribution drastically changes for a high S, reflecting the spread of the population in the J-E plane.

As can be seen in Figure 8, the density of negative ions in a high-ionization regime drastically differs from that in a low-ionization regime.

The charge density in the solid insulator after

t = 8000

h is compared for the low- and high-ionization rates. From the results shown in Figure 9, it can be seen that for the low-ionization case, the homo-charge is predominant, while hetero-charge dominates in the high-ionization regime.

7.2. Ionization Effect of TL

The RIC phenomenon of a multi-conductor TL is investigated in this section. The TL carrying three HV conductors polarized at −500 kV, −334 kV, and −167 kV, resembles the one conceptualized for the DTT NBI [1,8]. The three conductors are placed inside an enclosure of radius

r_{O} = 650

mm, referred to as ground potential (0 V).

As can be seen in Figure 2, the last part of the TL lying within the Tokamak hall, has a length L of

\approx 7.5

m and is subjected to the irradiation field coming from the machine. The absorbed dose rate

ϕ

, assuming SF

_{6}

as insulating gas pressurized at

P = 0.5

MPa absolute, is approximately 1.5 mGy/s. The relation between S and

ϕ

requires the knowledge of gas density

ρ

and the energy for the generation of one ion-pair

W_{i o n - p a i r}

[1]:

S = \frac{ϕ ρ}{W_{i o n - p a i r}} .

(20)

Using Equation (20), with

W_{i o n - p a i r} = 34

eV, the equivalent ion-pair generation rate is

S \approx 10^{15}

^{- 3}

^{- 1}

A 2D model of the cross-section of the TL, as illustrated in Figure 10, is built in COMSOL^®, and the electric current [A/m] at the grounded shell is measured in steady-state conditions as

i_{G} = \int_{Γ} J_{G} \cdot n d Γ,

(21)

where

n

is the outward unit normal vector. The results in natural environmental radiation, with the parameters listed in Table 1, and under the RIC phenomenon due to the radiation produced by the Tokamak, are listed in Table 2.

The estimated power losses in the gas under the RIC phenomenon are approximated as

P_{l o s s} = ∆ V \times i_{G} \times L

, with

∆ V = 500

kV, resulting in 4.8 kW. For a comparison,

P_{l o s s}

is also estimated based on the experimental law expressed by Equation (4) using

f_{g a s} = 2.06

, following [24], and

v \approx 8.3

^{3}

, resulting in

I_{s a t} = 12.8

mA, corresponding to

P_{l o s s} = 6.4

kW. This amount of heat can be considered tolerable for the TL operation, even considering the short operational pulse of the NBI in the DTT scenario [8]. Indeed, if the power losses due to the current

i_{G}

are too high, severe countermeasures affecting the final design of the components (e.g., addition of screens for the irradiation flux), must be conceived. It is worth mentioning that the RIC phenomenon and related power losses in the gas estimated from Equation (4) were sufficient to justify the substitution of SF

_{6}

insulation with vacuum insulation for the ITER NBI ion source [40]. Thus, the estimation of power losses under RIC conditions is of fundamental importance.

8. Discussion

The DDR model described in Section 5 is a viable tool to describe the behavior of an insulating gas under general conditions. Indeed, once the transport parameters of gas ions (e.g.,

μ^{+}

and

μ^{-}

) are obtained (e.g., in ad hoc experimental equipment [41]), the dynamic of the ions and conductivity can be determined. Directly specifying

σ_{G}

can be misleading especially in high-ionization conditions. The major differences between a standard EQS and the DDR approach are summarized in Table 3. This table can be used as a guideline to select the appropriate model for the specific case.

It is worth mentioning that the major issue in using EQS with specified gas conductivity is the inability to reproduce the high-ionization regime since the selection of a unique value for

σ_{G}

is not trivial; thus, results are highly affected by this choice, as shown in Figure 7. Moreover, if ions are injected from the electrodes or defects [31], the conductivity has local variations which cannot be captured with simple conductivity models. For these reasons, the DDR model is preferable if a detailed description of the gas in terms of ion dynamics is required. Under specific conditions, a detailed description of the phenomena in the gas is mandatory to gain knowledge of the reliability and performance of HVDC-GIS. As mentioned in Section 2, the surface charge is one of the major fault causes in these systems. As can be seen in Figure 7, the

ϱ_{S}

distribution is strongly affected by ion dynamics in the high-ionization regime, motivating the use of detailed gas models in such conditions to prevent early fault of the component.

The major obstacle in using the DDR model is the nonlinear time-dependent nature of the governing PDEs. The nonlinearity must be carefully addressed from a numerical viewpoint, e.g., by using the Newton–Raphson algorithm. This intrinsically increases the computational burden, solving the EQS with the DDR model for the gas results in two additional equations, i.e., Equations (12) and (13). A conductivity model

σ_{G} (S)

is preferable to simplify the numerical analysis as explained in Section 6; however, the simple relation expressed by Equation (17) does not apply to general geometries. Thus, the definition of a

σ_{G} (S)

from DDR results, extendable to general geometries, is a viable solution.

Application of Equation (18) seems to be a better approach to catch RIC effects, accurately describing dielectric gas behavior, provided that the model conditions are such that high-ionic-concentration regions are not dominant inside the computational domain. Furthermore, its numerical performances are comparable to the solution of a traditional EQS simulation, allowing an easier investigation to be performed of large-scale and complex devices, as usually is the case in the field of nuclear fusion, whose solution can often require taking into account several hundreds of millions of degrees of freedom.

9. Conclusions

The modeling approaches for solving the electric field distribution in HVDC-GISs for nuclear fusion applications were investigated in this paper. Formulations based on constant or nonlinear conductivity models for gas conductivity were extensively used. However, these approaches are usually based on experimental data and do not account for the influence of external ionization fields triggering the RIC phenomenon. A DDR model is preferable to model gas ionization correctly, at the cost of increasing the computational burden due to the nonlinear behavior of the governing equations for the ions in the gas. This issue can be overcome through the definition of functional dependence of the conductivity on ionization. An expression for the gas conductivity based on DDR results could be used to model the ionization effects in general electrode configurations, making a step forward with respect to the simple relation used in the past. Future research activities will be focused on the extension of the equivalent gas conductivity proposed in this paper to include the effect of electric field and match the results of DDR in the high-ionization regime.

Author Contributions

Conceptualization, F.L., A.F. and N.M.; methodology, F.L. and A.F.; formal analysis, F.L. and A.F.; writing—original draft preparation, F.L. and A.F.; writing—review and editing, F.L., A.F., K.U.J., N.M. and P.B. All authors have read and agreed to the published version of the manuscript.

Funding

Project financially supported by the BIRD 2023 Research Program of University of Padova (prot. BIRD235204).

Data Availability Statement

The original contributions presented in this study are included in this article; further inquiries can be directed to the corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

Abbreviations

The following abbreviations are used in this manuscript:

HVDC	High-voltage direct current
NBI	Neutral beam injector
MCF	Magnetic confinement fusion
AGPS	Acceleration grid power supply
GIS	Gas-insulated system
TL	Transmission line
DTT	Divertor Tokamak Test
CFETR	China Fusion Engineering Test Reactor
GIL	Gas-insulated line
RIC	Radiation-induced conductivity
EQS	Electro-quasistatic
HVB	High-voltage bushing
DDR	Drift–diffusion reaction

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Figure 1. Portion of solid–gas interface layer

Γ_{S}

of thickness

δ

. The normal and tangent vectors (

n

and

t

, respectively) and the direction of current densities across the layer are also drawn.

Figure 1. Portion of solid–gas interface layer

Γ_{S}

of thickness

δ

. The normal and tangent vectors (

n

and

t

, respectively) and the direction of current densities across the layer are also drawn.

Figure 2. Overview of DTT Tokamak hall with highlighted parts.

Figure 3. (a) Example of J-E population clustering. (b) Example of J-E populations for different values of S.

Figure 4. Geometrical model of the 2D axisymmetric HVDC-GIS chamber.

Figure 5. Trend of

ϱ_{S} (t)

along the gas–solid insulator interface. The arc length is measured from the bottom to the top of the interface.

Figure 5. Trend of

ϱ_{S} (t)

along the gas–solid insulator interface. The arc length is measured from the bottom to the top of the interface.

Figure 6. Surface charge density

ϱ_{S}

t = 8000

h along the gas–solid insulator interface obtained with the DDR and EQS with constant

σ_{G}

. Low-ionization regime. The arc length is measured from the bottom to the top of the interface.

Figure 6. Surface charge density

ϱ_{S}

t = 8000

h along the gas–solid insulator interface obtained with the DDR and EQS with constant

σ_{G}

. Low-ionization regime. The arc length is measured from the bottom to the top of the interface.

Figure 7. Surface charge density

ϱ_{S}

t = 8000

h along the gas–solid insulator interface obtained with the DDR and EQS with constant

σ_{G}

. High-ionization regime.

Figure 7. Surface charge density

ϱ_{S}

t = 8000

h along the gas–solid insulator interface obtained with the DDR and EQS with constant

σ_{G}

. High-ionization regime.

Figure 8. Negative ions’ distribution after

t = 8000

h. (a) Low-ionization regime. (b) High-ionization regime.

Figure 8. Negative ions’ distribution after

t = 8000

h. (a) Low-ionization regime. (b) High-ionization regime.

Figure 9. Charge density distribution in the solid insulator after

t = 8000

h. (a) Homo-charge in low-ionization regime. (b) Hetero-charge in high-ionization regime.

Figure 9. Charge density distribution in the solid insulator after

t = 8000

h. (a) Homo-charge in low-ionization regime. (b) Hetero-charge in high-ionization regime.

Figure 10. Cross -section of TL.

Table 1. Transport parameters for SF

_{6}

P = 0.5

MPa and

T = 293.15

Table 1. Transport parameters for SF

_{6}

P = 0.5

MPa and

T = 293.15

Parameter	Value
$μ^{+}$	4.8 $\cdot 10^{- 6}$ m $^{2}$ /(V·s)
$μ^{-}$	4.8 $\cdot 10^{- 6}$ m $^{2}$ /(V·s)
R	1.74 $\cdot 10^{- 13}$ m $^{3}$ /s
S	2.7 $\cdot 10^{7}$ m $^{- 3}$ s $^{- 1}$

Table 2. Current flowing out of the grounded enclosure of TL.

	Low Ionization	High Ionization
$\| i_{G} \|$ [A/m]	5.2 $\cdot 10^{- 10}$	1.3 $\cdot 10^{- 3}$

Table 3. Main differences between standard EQS and DDR models for the gas.

Approach	Advantages	Disadvantages
EQS with specified $σ_{G}$	• Simple implementation	• Does not account for ions’ distribution
	• Effective tool for low-ionization regime	• Ions’ injection cannot be considered
		• Not trivial selection of $σ_{G}$ in high-ionization regime. Need for a functional dependence $σ_{G} = σ_{G} (S)$
	• Account for ions’ distribution	• Intrinsically nonlinear model
DDR	• Effective tool for low- and high-ionization regimes	• Requires additional Equations (12) and (13) to be solved
	• Ions’ injection can be considered

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Lucchini, F.; Frescura, A.; Urazaki Junior, K.; Marconato, N.; Bettini, P. Modeling Approaches for Accounting Radiation-Induced Effect in HVDC-GIS Design for Nuclear Fusion Applications. Appl. Sci. 2024, 14, 11666. https://doi.org/10.3390/app142411666

AMA Style

Lucchini F, Frescura A, Urazaki Junior K, Marconato N, Bettini P. Modeling Approaches for Accounting Radiation-Induced Effect in HVDC-GIS Design for Nuclear Fusion Applications. Applied Sciences. 2024; 14(24):11666. https://doi.org/10.3390/app142411666

Chicago/Turabian Style

Lucchini, Francesco, Alessandro Frescura, Kenji Urazaki Junior, Nicolò Marconato, and Paolo Bettini. 2024. "Modeling Approaches for Accounting Radiation-Induced Effect in HVDC-GIS Design for Nuclear Fusion Applications" Applied Sciences 14, no. 24: 11666. https://doi.org/10.3390/app142411666

APA Style

Lucchini, F., Frescura, A., Urazaki Junior, K., Marconato, N., & Bettini, P. (2024). Modeling Approaches for Accounting Radiation-Induced Effect in HVDC-GIS Design for Nuclear Fusion Applications. Applied Sciences, 14(24), 11666. https://doi.org/10.3390/app142411666

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Modeling Approaches for Accounting Radiation-Induced Effect in HVDC-GIS Design for Nuclear Fusion Applications

Abstract

1. Introduction

2. Surface Charge Accumulation

3. Effect of External Ionization Field

4. Electric Field Modeling

Simplified Approach

5. Modeling Charge Transport in Dielectric Gas

6. An Equivalent Gas Conductivity for RIC Phenomenon

7. Numerical Results

7.1. Post-Insulator in SF $_{6}$

7.2. Ionization Effect of TL

8. Discussion

9. Conclusions

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

Abbreviations

References

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Modeling Approaches for Accounting Radiation-Induced Effect in HVDC-GIS Design for Nuclear Fusion Applications

Abstract

1. Introduction

2. Surface Charge Accumulation

3. Effect of External Ionization Field

4. Electric Field Modeling

Simplified Approach

5. Modeling Charge Transport in Dielectric Gas

6. An Equivalent Gas Conductivity for RIC Phenomenon

7. Numerical Results

7.1. Post-Insulator in SF ⁢ 6

7.2. Ionization Effect of TL

8. Discussion

9. Conclusions

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

Abbreviations

References

Share and Cite

Article Metrics

Article Access Statistics

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7.1. Post-Insulator in SF $_{6}$