DC active power filter based on model predictive control for DC bus overvoltage suppression of accelerator grid power supply

1.   Introduction

Steady-state high-performance discharges in fusion reactors, such as the International Thermonuclear Experimental Reactor (ITER), necessitate optimized density and pressure profiles to maintain plasma confinement [1]. One potential approach for controlling these profiles involves using an adaptable fueling system capable of varying the mass density and velocity of fuel particles [2]. Conventional fueling techniques, such as gas puffing, neutral beam injection, or pellet injection, may not be sufficient for achieving deep fueling in reactor-grade fusion plasma [3]. By contrast, high-velocity compact torus (CT) injection can access the core region of the tokamak plasma [4]. Hence, CT injection emerges as a promising strategy for deep fueling in forthcoming large-scale fusion reactors [5, 6].

CT, a high-density plasmoid, undergoes magnetic confinement under a helical magnetic field comprising both toroidal and poloidal components [7]. The apparatus employed for CT generation is commonly referred to as a CT injector utilizing a magnetized coaxial plasma gun. The CT generation process can be summarized as follows: (1) application of a bias magnetic field and injection of gas using fast-opening valves, (2) initiation of an electrical discharge between a pair of coaxial electrodes to form the CT, and (3) acceleration of the formed CT by another pair of coaxial electrodes. Magnetic reconnection significantly influences the CT formation process, while the Lorentz force plays a pivotal role in both plasma formation and acceleration. The CT structure exhibits remarkable resilience and can attain the velocities required for central fueling in future fusion-grade devices [8, 9].

Perkins et al and Parks proposed using CT injection for fueling fusion devices [10, 11]. The first non-disruptive CT injection experiment occurred in 1994 at the tokamak de varennes (TdeV) using a compact toroid fueler (CTF) [12]. Subsequently, in 1999, the Himeji Institute of Technology (HIT) developed a CT injector (HIT-CTI) to demonstrate CT fueling in high-confinement mode (H-mode) plasmas on the JFT-2M tokamak [13, 14]. Notably, tangential CT injection into the STOR-M tokamak can induce H-mode-like discharges [15]. Furthermore, CT technology can be applied to field-reversed configuration (FRC) and reversed field pinch (RFP) devices, such as the C-2 and Keda Torus eXperiment (KTX) devices [16–18]. The discharge parameters of the HL-2A device are very close to those of the KTX device. Therefore, there are no theoretical and technical difficulties in applying the CT fueling technology to the HL-2A device. In the initial design of ITER, there were no requirements for the tritium breeding ratio (TBR). With the changing goals of ITER, CT fueling technology has provided possibilities for achieving the goals of the TBR of the ITER.

The interaction between a CT and the magnetized tokamak plasma, which governs the CT trajectory and the fuel deposition process, remains largely uncharted [19]. Perkins et al suggested that comprehending the dynamics of the CT when it traverses the tokamak plasma necessitates quantifying several interrelated factors, including CT decay, tilting, field line reconnection, deceleration in the external field, and CT expansion/contraction [10]. In summary, they contend that the CT experiences deceleration due to the magnetic field gradient and rotation due to magnetic torque. Parks introduced the conducting sphere (CS) model to describe the CT’s motion within a tokamak [11]. This model posits the CT as a perfect CS with an embedded toroidal magnetic field (and hence a magnetic moment) that displaces the surrounding field lines during its transit through the tokamak [20]. According to this model, as the CT moves through the tokamak plasma, the CT’s magnetic pressure equates to the tokamak’s magnetic pressure at the CT’s surface, assuming the CT’s shape and structure remain unchanged. During its movement, the CT encounters the magnetic field gradient force and magnetohydrodynamic (MHD) wave drag force, both acting in opposition to the CT’s velocity. It is noteworthy that the CT eventually comes to a halt and, after a dwell time, is repelled from the tokamak. Bozhokin employed Parks’ CS model to simulate the CT trajectory when the initial velocity vector of the CT is oriented at an angle to the tokamak’s major radius direction [21]. Newcomb further refined the CS model [22], providing a systematic discussion of the MHD wave drag experienced by a CS traversing a magnetized plasma. Xiao et al summarized the content of the CS model, incorporating a selected MHD wave drag coefficient [19]. They derived equations delineating the CT’s motion within a tokamak and computed its trajectory within the ITER tokamak. Additionally, they were the first to examine the advantages of tangential CT injection and determine the optimal direction of the initial magnetic moment. It is worth mentioning that they also conducted numerical simulations of CT trajectories in the STOR-M tokamak discharges for both tangential and vertical injections using the CS model, and reported experimental results of CT tangential and vertical injection in the STOR-M [23]. However, the numerical calculation results and the trajectory of the model were not verified because few CT injection experiments have been conducted in large fusion devices.

Hwang et al [24] delved into an examination of how the shape and compressibility of a CT impact its propagation into a low-beta plasma using the CS model. In parallel, Suzuki et al introduced an alternative model, termed the non-slipping (NS) model, which incorporates considerations of magnetic pressure and the magnetic field force associated with the target magnetic field [25, 26]. The utilization of numerical simulations employing MHD equations has indicated that the CT’s behavior falls within the range of predictions generated by both the CS and NS models. Liu et al simulated the behavior of low-beta CT injected into a hot strongly magnetized plasma, and investigated the effect of the presence of shock waves in tokamak plasma for the plasma fueling by CT [27]. At present, the shock waves, generated by CT injecting into tokamak plasma, have not been obviously observed in variable devices. By considering the incompressible condition of tokamak plasma, we temporarily ignore the shock/wavefront mechanism by the CT fueling in tokamak plasmas. This conveys that treating the in-motion CT within the tokamak as a CS is a reasonable and valid approach. It is noteworthy that Olynyk and Morelli were pioneers in the adoption of the CS model developed by Xiao et al [19]. They extended this model’s capabilities, developing a novel six-degree-of-freedom model and effectively applying it to the design of CT injectors [20].

To conclude, it is imperative to underline that current research and dynamic analyses focused on CT penetration into the tokamak predominantly emphasize its behavior within the confines of the tokamak plasma. This emphasis leaves unaddressed the non-plasma region traversed by the CT post-injection. Nevertheless, as the CT enters the tokamak region from the injection point, it traverses the vacuum magnetic field region (${\text{Δ}} x$) devoid of plasma. Importantly, even within this vacuum magnetic field region, the CT is subject to the influence of the tokamak’s toroidal magnetic field. Consequently, the CT experiences magnetic field gradient forces and magnetic torques within this region. These forces induce a decrease in CT velocity, attributed to the magnetic field gradient force, alongside tilting and rotation of the CT, a consequence of the magnetic torque. Notably, within next-generation reactor-grade fusion devices like ITER, the vacuum magnetic field region extends over a length of approximately 211 cm [28]. This region includes the blanket portion of the fusion device, encompassing a length of 45 cm for ITER’s blanket [29] and 145 cm for CFETR’s blanket [30]. This expansion of the vacuum magnetic field region in conjunction with the increasing size of fusion devices results in a concomitant decrease in CT velocity, exerting significant impacts on the CT’s dynamic behavior. Moreover, the extended radial dimension of the vacuum magnetic field region elongates the CT’s traversal time, offering increased opportunities for fuel particle diffusion, which, in turn, can lead to the loss of particles and the decay of the magnetic field. The dispersion of fuel particles within the vacuum magnetic field region compromises the overall fueling efficiency of the CT since not all particles can reach their intended deposition regions within the tokamak. To address this challenge and optimize fueling efficiency, meticulous design and precise control of the CT trajectory are imperative. Achieving this objective entails minimizing the time spent in the vacuum region and extending the acceleration electrodes into the vacuum magnetic field region. This approach effectively mitigates particle diffusion and maximizes the count of fuel particles reaching their intended deposition regions. Such optimization can be realized through a range of strategies, including the fine-tuning of CT injection parameters such as injection angle, velocity, and magnetic moment direction. The systematic adjustment of these parameters empowers researchers to enhance the CT trajectory, augment fueling efficiency, curtail fuel particle loss, and optimize the effectiveness of CT-based fueling within fusion reactors.

In this study, we simulated the motion of CTs that were injected into the HL-2A and ITER devices. These were offered as representative examples to analyze the effects of the vacuum magnetic field region on the CT trajectory within the tokamak, including penetration parameters. Additionally, we meticulously evaluated injection strategies, encompassing both perpendicular and tangential CT injection into the tokamak. These assessments were based on a refined dynamic model tailored for the CT trajectory within the tokamak plasma. Section 2 discusses an expansion of the existing CS model to incorporate the vacuum magnetic field region. Section 3 discusses the computation of CT penetration parameters when it is perpendicularly injected into the tokamak, factoring in the presence of the vacuum magnetic field region. Furthermore, the determination of the optimal initial injection velocity of the CT to maximize the efficacy of CT injection is discussed. Section 4 focuses on the strategic aspects of tangential CT injection into a tokamak, providing insights into optimal injection angles and the preferred direction for the initial magnetic moment in CT injection for both HL-2A and ITER devices. Finally, section 5 presents a summary of the study.

2.   Dynamic model of CT injection into tokamak plasma

In this model, the CT is approximated as a CS with a fixed radius, denoted as ${r}_{\mathrm{C}\mathrm{T}}$. Figure 1 illustrates the CT injection into the tokamak, delineating the vacuum magnetic field region and the bulk plasma region. The interaction between the CT and the tokamak encompasses: (a) the exclusion of the external tokamak magnetic field from the CT, (b) CT rotation driven by the CT’s magnetic moment within the external magnetic field, and (c) MHD wave drag [19].

Figure  1.  Schematic of the CT injection into the tokamak plasma. The red portion represents the bulk plasma region, while the colorless region between the purple line (radius at which CT is injected) and the red region (plasma region) corresponds to the vacuum magnetic field region.

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2.1   Vacuum magnetic field region

When the CT is ejected from the injector outlet and subsequently enters the tokamak region, it initially traverses a segment of the vacuum magnetic field region (${\text{Δ}} x$ as depicted in figure 1) devoid of plasma. The vacuum magnetic field region is defined as the space between the plasma and the initial CT injection point. It is also assumed that the injection location is smaller than the radial position of the toroidal field coils. Consequently, the tokamak’s toroidal magnetic field, denoted as ${B}_{\mathrm{t}\mathrm{o}\mathrm{k}}$, varies inversely with the major radius position $R$:

where ${B}_{0}$ denotes the magnetic field at the tokamak major radius position ${R}_{0}$. In this plasma-free region, the drag arising from the excitation of MHD waves is disregarded, and the forces acting on the CT encompass solely (a) the exclusion of the external tokamak magnetic field from the CT, and (b) CT rotation induced by the CT’s magnetic moment within the external tokamak toroidal magnetic field (the small poloidal field was neglected in this study).

The toroidal magnetic field within the tokamak exhibits a non-uniform distribution characterized by a radial inward gradient. Consequently, a radial outward magnetic field force (${F}_{\nabla B}$) is exerted on the CT. The expression for ${F}_{\nabla B}$ is given by

Here, ${U}_{\nabla B}$ denotes the work done by the CT to exclude the tokamak magnetic field lines, which can be estimated as [19]:

$\dfrac{{B}_{\mathrm{t}\mathrm{o}\mathrm{k}}^{2}}{2{\mu }_{0}}$ denotes the energy density of the tokamak toroidal magnetic field. In this study, we analyzed the interaction between the CT’s magnetic moment and the tokamak’s toroidal magnetic field. The magnetic torque can be described as:

The magnetic potential energy of the CT can be expressed as follows:

where $\overrightarrow{\mu }$ denotes the magnetic moment of the CT, and its can be determined as follows [21]:

where the sign of $\pm$ denotes parallel and antiparallel, respectively. It is worth noting that, due to the characteristics generated by CT, there are only two types of magnetic moment vectors and velocity vectors: parallel and antiparallel [19]. The angle between $\overrightarrow{\mu }$ and ${\overrightarrow{B}}_{\mathrm{t}\mathrm{o}\mathrm{k}}$ is denoted by $\lambda$, as illustrated in figure 1. ${B}_{\mathrm{C}\mathrm{T}}$ denotes the average field in the CT, and the coefficient $k\ $($0\leqslant k\leqslant 1$) is a coupling factor depending on the current distribution within the CT.

2.2   Bulk plasma region

In addition to the forces detailed in section 2.1, the CT experiences a drag force. According to Newcomb’s model [22], the magnitude of the MHD wave drag experienced by a CT moving through a magnetized plasma can be described by the following equation:

Here, ${V}_{\mathrm{A}}$ represents the Alfvén velocity of the tokamak plasma. The negative sign implies that the direction of the MHD wave drag (${\overrightarrow{F}}_{\omega }$) opposes that of the CT velocity (${\overrightarrow{v}}_{\mathrm{C}\mathrm{T}}$). ${\rho }_{\mathrm{t}\mathrm{o}\mathrm{k}}$ denotes the mass density of the tokamak plasma, while I denotes a coefficient with the following expression:

where ${V}_{\mathrm{S}}$ denotes the sound velocity of the tokamak plasma. The coefficient depends on the parameters of the tokamak plasma.

2.3   Dynamic equations for CT motion

$\left(\dfrac{R}{{R}_{0}},\phi\right)$ is used to denote the position of the CT’s center of mass in a two-dimensional polar coordinate system and $\alpha$ is used to denote the angle between the magnetic moment vector and the $\overrightarrow{x}$-axis. Throughout the CT’s motion within the tokamak, we assumed that the size and shape of the CT remain constant. For convenience, we introduced a dimensionless time variable:

where

and dimensionless functions of $\tau$ are defined by

Here, these dimensionless functions describe the position, velocity, and rotation information of CT in polar coordinate. The equations governing the motion of the CT can be derived through the approach outlined in references [19, 21] by considering the interaction between the CT and the plasma and magnetic field within the tokamak. The equations can be written as a first order differential equation system:

where

and ${\dot{Y}}_{i}$ represents the derivatives with respect to the dimensionless time $\tau$. ${\rho }_{\mathrm{C}\mathrm{T}}$ denotes the CT mass density. We use a Lagrangian approach and numerically solve the differential equations. When the CT is within the regions of the vacuum magnetic field region without plasma, i.e., ${R}_{0}+ a < R\leqslant{R}_{0}+a+ {\text{Δ}} x$ and $R < {R}_{0}-a$, the coefficient ${Q}_{0}$ is 0.

Equation (20) describes the radial acceleration experienced by the CT during its motion within the tokamak. The term ${C}_{0}\dfrac{\mathrm{s}\mathrm{i}\mathrm{n}({Y}_{2}-{Y}_{3})}{{Y}_{1}^{2}}$ originates from the force acting on the CT magnetic dipole within the external magnetic field. As depicted in figure 1, the angle $\lambda$ between $\overrightarrow{\mu }$ and ${\overrightarrow{B}}_{\mathrm{t}\mathrm{o}\mathrm{k}}$ is

which determines whether the ${C}_{0}\dfrac{\mathrm{s}\mathrm{i}\mathrm{n}({Y}_{2}-{Y}_{3})}{{Y}_{1}^{2}}$ term accelerates or decelerates the CT. The magnetic dipole term in the CT’s motion equation can vary in sign and magnitude because of the oscillation of the CT’s magnetic moment direction around the direction of the external magnetic field. The specific value of this term depends on the angular amplitude of the oscillation. Notably, this term represents the force pulling the CT toward the inner wall, as in the Bozhokin model [21] and the Xiao model [19]. The term $-{Q}_{0}{Y}_{4}$ arises from the MHD wave drag, as given by equation (20). The term $\dfrac{{E}_{0}}{{Y}_{1}^{3}}$ in equation (20) arises from the tokamak toroidal magnetic field gradient force and is always positive, indicating an outward repelling force. Equation (21) describes the toroidal acceleration of the CT during its motion in a tokamak. The terms ${C}_{0}\dfrac{\mathrm{c}\mathrm{o}\mathrm{s}({Y}_{2}-{Y}_{3})}{{Y}_{1}^{3}}$ and $-{Q}_{0}{Y}_{5}$ in these equations arise from the magnetic dipole and the MHD wave drag, respectively.

Equation (22) describes the rotational motion of the CT under the torque $\overrightarrow{\mu }\times {\overrightarrow{B}}_{\mathrm{t}\mathrm{o}\mathrm{k}}$. The period of the rotational motion at ${R}_{0}$ can be calculated using the following equation:

where

is the moment of inertia of the CT.

In the model shown in figure 1 and based on the aforementioned assumptions, the initial conditions can be expressed as follows:

where ${\alpha }_{0}$ represents parallel injection, and ${\alpha }_{0}-{\text{π}}$ represents anti-parallel injection. ${V}_{\mathrm{C}\mathrm{T}0}$ denotes the initial injection velocity of the CT. For parallel (antiparallel) injection, the initial angle between $\overrightarrow{\mu }$ (magnetic moment direction) and ${\overrightarrow{B}}_{\mathrm{t}\mathrm{o}\mathrm{k}}$ (magnetic field direction) is smaller (larger) than $\dfrac{{\text{π}} }{2}$. Normally, the initial magnetic moment direction and the initial velocity direction of the CT are either the same or opposite. Parallel injection and antiparallel injection can be achieved by controlling the current direction in a solenoid coil in the formation region of the CT injector [17].

3.   Perpendicular injection

In this section, we present the results of numerical calculations for the trajectory and penetration parameters of CTs when perpendicularly injected into HL-2A and ITER devices. These simulations employ normalized parameters, facilitating the extrapolation of results to other tokamak devices, such as HL-3 and BEST tokamaks.

3.1   Initial parameters for numerical calculations

The CT is perpendicularly injected into the mid-plane of the HL-2A tokamak, as illustrated in figure 2. Notably, the injection point on the HL-2A tokamak is located on the vacuum vessel wall, as depicted in figure 2(b). Currently, the diameter of the CT injection flange on the vacuum vessel wall of the HL-2A device measures 10 cm. HL-2A is a medium-sized tokamak with a radial length of the vacuum magnetic field region ${ {\text{Δ}} x}_{1}=0.21\;\mathrm{m}$ [31]. Following parameter calculations and theoretical simulations, we have established that the mass and velocity of the CT plasma used for CT injection experiments in HL-2A are $50\;\mathrm{\mu }\mathrm{g}$ and $200\;\mathrm{k}\mathrm{m}/\mathrm{s}$, respectively. Table 1 provides the parameters for HL-2A [32, 33] and the proposed CT injector. The magnetic field selected for the CT model is 0.4 T, a value previously measured in CTIX [34] and widely adopted in numerical calculations [20]. In the HL-2A tokamak, as well as within the CT operating with deuterium fuel, the average mass per deuteron is $\bar{m}=3.34\times {10}^{-27}\;\mathrm{k}\mathrm{g}$.

Table  1.  HL-2A and CT parameters.

Parameters	Values
HL-2A major radius, ${R}_{0}$ (m)	1.65
HL-2A minor radius, $a$ (m)	0.40
HL-2A magnetic field, ${B}_{0}$ (T)	2.0
HL-2A plasma density, ${n}_{0}$ (${\mathrm{m}}^{-3}$)	$3.0\times {10}^{19}$
HL-2A averaged temperature, $\bar{T}$ (keV)	5
CT outer radius, ${r}_{\mathrm{C}\mathrm{T}}$ (m)	0.05
CT density, ${n}_{\mathrm{C}\mathrm{T}}$ (${\mathrm{m}}^{-3}$)	$2.98\times {10}^{22}$
CT initial velocity, ${V}_{\mathrm{C}\mathrm{T}0}$ (km/s)	200
CT magnetic field, ${B}_{\mathrm{C}\mathrm{T}}$ (T)	0.4

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Figure  2.  (a) Schematic (top view) of CT perpendicularly injected into the HL-2A and ITER devices. The blue lines and parameters represent the HL-2A device, while the pink lines and parameters represent the ITER device. (b) Schematic (side view) illustrating the perpendicular injection of CT into the HL-2A device.

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The perpendicular injection of the CT into the ITER device is also analyzed to assess the effects of the vacuum magnetic field region on CT penetration parameters. Figure 2(a) provides a schematic diagram (top view) of CT perpendicular injection into the ITER device. Notably, for CT injection into ITER, the exit of the CT injector is situated outside the TF coils. Figure 3 illustrates the calculated TF magnitude in HL-2A as an example. Assuming that the outer (inner) part of the TF coil is located at major radius ${R}_{\mathrm{T}\mathrm{F}-\mathrm{o}}$ (${R}_{\mathrm{T}\mathrm{F}-\mathrm{i}}$), the magnetic field generated by the TF coils in tokamaks can be described in two regions: a normal toroidal magnetic field region, ${R}_{\mathrm{T}\mathrm{F}-\mathrm{i}} < R < {R}_{\mathrm{T}\mathrm{F}-\mathrm{o}}$, where the magnitude of the toroidal magnetic field is inversely proportional to the major radius, and a fringe magnetic field region, ${R < R}_{\mathrm{T}\mathrm{F}-\mathrm{i}}$ or ${R > R}_{\mathrm{T}\mathrm{F}-\mathrm{o}}$, where the field gradient is steep, and the magnitude of the fringe magnetic field decreases significantly within a short distance from the TF coils. It should be mentioned that the magnetic field profile in the fringe region is closely approximate to the exact profile of the HL-2A device in figure 3. For CT injection into the HL-2A device, the CT injection position is located on the wall of the vacuum chamber. At this time, the fringe magnetic field region has almost no effect on the penetration parameters of CT, as our model only analyzes the CT motion after it being ejected from the outlet beyond this fringe region. For the next-generation reactor-grade fusion devices like ITER, if the CT injection window allows extending the CT injector electrode to the TF coil position, the fringe magnetic field still has no effect on the CT penetration parameters. Due to our study’s focus on the trajectory and penetration parameters following the CT injection into the tokamak, the impact of the magnetic field in the fringe region is considered negligible in the model. Therefore, the starting position of the CT is located on the inner side of the TF coil, and the exact profile of the magnetic field in the fringe region has no effect on the motion of the CT. The initial velocity of the CT, used in this model, corresponds to the CT’s velocity after it passing over the fringe magnetic field region. This assumption is made for calculating the CT trajectory in the ITER tokamak, where the length of the vacuum magnetic field region is ${ {\text{Δ}} x}_{2}=2.11\;\mathrm{m}$ [28, 35]. Due to the CT’s motion to the TF coil position through the fringe field region, the relationship between the CT’s magnetic moment direction ($\overrightarrow{\mu }$) and the toroidal magnetic field direction (${\overrightarrow{B}}_{\mathrm{t}\mathrm{o}\mathrm{k}}$) of the tokamak is unknown. For simplicity, we assume that initially, the CT’s magnetic moment and ITER’s toroidal magnetic field direction are perpendicular. Table 2 provides the parameters for ITER and the CT design parameters. For ITER’s parameters, we consider the example of plasma operation at full technical performance (15 MA/5.3 T) [36] in the model. The tokamak plasma has a density of ${n}_{0}=1\times {10}^{20}\;{\mathrm{m}}^{-3}$, and the average plasma temperature is $\bar{T}=10\;\mathrm{k}\mathrm{e}\mathrm{V}$ [37]. Concerning the CT parameters, we adopt the CT design parameters for ITER as given in reference [20]. The core plasma in ITER consists of 50% deuterium (D) and 50% tritium (T) [38]. The average mass per ion is $\bar{m}=4.19\times {10}^{-27}\;\mathrm{k}\mathrm{g}$.

Figure  3.  Relationship between the magnetic field of the HL-2A tokamak and the normalized major radius position.

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Table  2.  ITER and CT parameters.

Parameters	Values
ITER major radius, ${R}_{0}$ (m)	6.2
ITER minor radius, $a$ (m)	2.0
ITER magnetic field, ${B}_{0}$ (T)	5.3
ITER plasma density, ${n}_{0}$ (${\mathrm{m}}^{-3}$)	$1.0\times {10}^{20}$
ITER averaged temperature, $\bar{T}$ (keV)	10
CT outer radius, ${r}_{\mathrm{C}\mathrm{T}}$ (m)	0.1
CT density, ${n}_{\mathrm{C}\mathrm{T}}$ (${\mathrm{m}}^{-3}$)	$7.36\times {10}^{22}$
CT initial velocity, ${V}_{\mathrm{C}\mathrm{T}0}$ (km/s)	300
CT magnetic field, ${B}_{\mathrm{C}\mathrm{T}}$ (T)	0.4

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3.2   CT trajectory and penetration parameters

The Alfvén wave velocity (${V}_{\mathrm{A}}$) and sound wave velocity (${V}_{\mathrm{S}}$) in HL-2A at $R={R}_{0}$ are given as follows:

and

respectively. Figure 4 depicts the calculated trajectory of a CT injected into HL-2A at a velocity of ${V}_{\mathrm{C}\mathrm{T}0}=\mathrm{ }200\;\mathrm{k}\mathrm{m}/\mathrm{s}$ and an injection angle of ${\alpha }_{0}={\text{π}}$. The time scale employed for normalization is ${t}_{0}=\dfrac{{R}_{0}}{{V}_{\mathrm{A}}}=0.29$. In figure 4, the thick black lines depict the normalized outer wall $\left(\dfrac{{R}_{0}+a+ {\text{Δ}} x}{{R}_{0}} \right)$ of the HL-2A device, the thick green line signifies the normalized outer radius $\left(\dfrac{{R}_{0}+a}{{R}_{0}}\right)$ and inner radius $\left(\dfrac{{R}_{0}-a}{{R}_{0}} \right)$ of the tokamak plasma, and the pink dashed line represents the normalized position of the magnetic axis $\left( \dfrac{R}{{R}_{0}}=1 \right)$ of the tokamak plasma. The normalized period for the rotation of the CT’s magnetic moment is ${T}_{\mathrm{r}0}=8.49$. The magnetic torque coefficient used is $k=1$ and the MHD wave drag coefficient in equation (8) is $I=\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }0.06$.

Figure  4.  Trajectory of a CT injected into HL-2A at an injection angle of ${\text{π}}$ with respect to the tokamak major radius direction, presented in a normalized Cartesian coordinate system $\dfrac{y}{{R}_{0}}$ versus $\dfrac{x}{{R}_{0}}$, (a) illustrates the trajectory without considering the vacuum magnetic field region, while (b) shows the trajectory considering the vacuum magnetic field region. The solid red line represents the CT’s trajectory in HL-2A, and the blue arrows indicate the magnetic dipole orientations at normalized time intervals separated by ${\text{Δ}} \tau =2$.

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Comparing figures 4(a) and (b), we observe that when the vacuum magnetic field region is not considered, deeper penetration occurs in contrast to the case when it is considered. Table 3 compiles the CT penetration parameters for two cases: Case I with ${ {\text{Δ}} x}_{1}=0\;\mathrm{m}$ and Case II with ${ {\text{Δ}} x}_{1}=0.21\;\mathrm{m}$. Without accounting for the vacuum magnetic field region, the CT penetrates the tokamak until reaching a minimum major radial position of ${R}_{\mathrm{p}}=1.19\;\mathrm{m}$ at time ${t}_{\mathrm{p}}=6.88\;\mathrm{\mu }\mathrm{s}$. The relative radial penetration location, ${\rho }_{\mathrm{p}}=\dfrac{{R}_{\mathrm{p}}-{R}_{0}}{a}$, is −1.17, indicating full penetration of the plasma cross-section. The CT takes a dimensionless time of ${ {\text{Δ}} \tau }_{{R}_{0}+a}=3.66$ $({ {\text{Δ}} t}_{{R}_{0}+a}=1.06\;\mathrm{\mu }\mathrm{s})$ to penetrate through the vacuum magnetic field region $({ {\text{Δ}} x}_{1}=0.21\;\mathrm{m})$. Simultaneously, the velocity of the CT decreases by ${{ {\text{Δ}} V}_{\mathrm{C}\mathrm{T}}}_{{R}_{0}+a}=7.96\;\mathrm{k}\mathrm{m}/\mathrm{s}$, indicating a insignificant loss of kinetic energy of ${ {\text{Δ}} E}_{\mathrm{k}}=1.65\;\mathrm{J}$ within the vacuum magnetic field region. The CT’s trajectory is depicted by the red curve in figure 4(b) with the following penetration parameters: ${t}_{\mathrm{p}}=8.04\;\mathrm{\mu }\mathrm{s}$, ${R}_{\mathrm{p}}=1.22\;\mathrm{m}$ or ${\rho }_{\mathrm{p}}=-1.09$. To conclude, when the CT is perpendicularly injected into the HL-2A device at high velocity, considering the vacuum magnetic field region results in a smaller relative penetration depth compared to disregarding it. This reduction of ${\text{Δ}} {R}_{\mathrm{p}}=0.03\;\mathrm{m}$ indicates a 0.08 decrease in the relative penetration depth for the HL-2A tokamak when accounting for the vacuum magnetic field region compared to the scenario where this region is not considered.

Table  3.  Penetration parameters for the CT into the HL-2A.

Cases	${ {\text{Δ}} x}_{1}$ (m)	${ {\text{Δ}} t}_{{R}_{0}+a}$ ($\mathrm{\mu }\mathrm{s}$)	${{ {\text{Δ}} V}_{\mathrm{C}\mathrm{T}}}_{{R}_{0}+a}$ ($\mathrm{k}\mathrm{m}/\mathrm{s}$)	${ {\text{Δ}} E}_{\mathrm{k}}$ ($\mathrm{J}$)	${t}_{\mathrm{p}}$ ($\mathrm{\mu }\mathrm{s}$)	${R}_{\mathrm{p}}$ ($\mathrm{m}$)	${\rho }_{\mathrm{p}}$
Case I	0	0	0	0	6.88	1.19	−1.17
Case II	0.21	1.06	7.96	1.65	8.04	1.22	−1.09

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The trajectory of the CT injected into ITER at a velocity ${V}_{\mathrm{C}\mathrm{T}0}=300\;\mathrm{k}\mathrm{m}/\mathrm{s}$ with an injection angle ${\alpha }_{0}={\text{π}}$ is similar to the CT trajectory in the HL-2A device as shown in figure 4. In ITER, the Alfvén and sound wave velocities are ${V}_{\mathrm{A}}=7308\;\mathrm{k}\mathrm{m}/\mathrm{s}$ and ${V}_{\mathrm{S}}=618\;\mathrm{k}\mathrm{m}/\mathrm{s}$, respectively. The time-scale used for the normalization purpose is ${t}_{0}=\dfrac{{R}_{0}}{{V}_{\mathrm{A}}}=0.85$. The normalized period for the CT rotational motion is ${T}_{\mathrm{r}0}=6.33$. The magnetic torque coefficient $k$ equals 1, and the MHD wave drag coefficient in equation (8) is $I=\mathrm{ }0.06$. Table 4 summarizes the CT penetration parameters for the following two cases: Case I with ${ {\text{Δ}} x}_{2}=0\;\mathrm{m}$ and Case II with ${ {\text{Δ}} x}_{2}=2.11\;\mathrm{m}$. When the vacuum magnetic field region is not considered, deeper penetration and a shorter penetration time occur at the minimum major radial position. As anticipated, with an increase in the length of the vacuum magnetic field region, the CT’s consumption of kinetic energy during penetration of this region notably increases, and the relative penetration depth the CT reaches substantially decreases. The CT penetrates through the vacuum magnetic field region ${ {\text{Δ}} x}_{2}=2.11\;\mathrm{m}$ within a dimensionless time interval of ${ {\text{Δ}} \tau }_{{R}_{0}+a}=8.61\;({ {\text{Δ}} t}_{{R}_{0}+a}=7.32\;\mathrm{\mu }\mathrm{s})$. During this period, the CT’s velocity decreases from ${V}_{\mathrm{C}\mathrm{T}0}=300\;\mathrm{k}\mathrm{m}/\mathrm{s}$ to $274.99\;\mathrm{k}\mathrm{m}/\mathrm{s}$, and CT kinetic energy consumption is ${ {\text{Δ}} E}_{\mathrm{k}}=403.51\;\mathrm{J}$. If the minimum radial position point reached by the CT corresponds to the final deposition position of the CT fuel, the CT’s final deposition position is at the normalized minimum major radial position point $\dfrac{{R}_{\mathrm{p}}}{{R}_{0}}=0.77$ (${R}_{\mathrm{p}}=4.79\;\mathrm{m}$) outside the vacuum magnetic field region. By contrast, in the vacuum magnetic field region, the CT’s final deposition position is at the normalized minimum major radial position point $\dfrac{{R}_{\mathrm{p}}}{{R}_{0}}=0.82$ (${R}_{\mathrm{p}}=5.06\;\mathrm{m}$). Therefore, a radial difference of $\text{Δ}R_{\mathrm{p}}=27\mathrm{\ c}\mathrm{m}$ is observed in the final deposition position of the CT fuel, indicating that the relative penetration depth of the CT being perpendicularly injected into ITER is reduced by 0.13 for the ITER tokamak when the vacuum magnetic field region is considered compared to when that region is not considered.

Table  4.  Penetration parameters for the CT into the ITER.

Cases	${ {\text{Δ}} x}_{2}$ (m)	${ {\text{Δ}} t}_{{R}_{0}+a}$ ($\mathrm{\mu }\mathrm{s}$)	${{ {\text{Δ}} V}_{\mathrm{C}\mathrm{T}}}_{{R}_{0}+a}$ ($\mathrm{k}\mathrm{m}/\mathrm{s}$)	${ {\text{Δ}} E}_{\mathrm{k}}$ ($\mathrm{J}$)	${t}_{\mathrm{p}}$ ($\mathrm{\mu }\mathrm{s}$)	${R}_{\mathrm{p}}$ ($\mathrm{m}$)	${\rho }_{\mathrm{p}}$
Case I	0	0	0	0	18.81	4.79	−0.70
Case II	2.11	7.32	25.01	403.51	26.62	5.06	−0.57

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Parks [11], Perkins et al [10], and Bozhokin [21] have suggested that instant magnetic reconnection may occur between the CT and the tokamak when the CT’s magnetic moment aligns with the toroidal magnetic field of the tokamak. This alignment could potentially lead to the deposition of CT fuel particles into the tokamak. The trajectory of the CT in the HL-2A device is presented in figure 4. In the dynamic model for the CT trajectory within the tokamak discharge, the angle between the CT’s magnetic moment $\overrightarrow{\mu }$ and the tokamak’s toroidal magnetic field ${\overrightarrow{B}}_{\mathrm{t}\mathrm{o}\mathrm{k}}$ is denoted by $\lambda$, as expressed in equation (27). As illustrated in figure 5, the term $\lambda =0$ corresponds to the points at which the “dipole angle” (i.e., the angle between the CT magnetic moment $\overrightarrow{\mu }$ and the tokamak toroidal magnetic field ${\overrightarrow{B}}_{\mathrm{t}\mathrm{o}\mathrm{k}}$) reaches its minimum. Due to the non-zero reconnection time, the fuel will be “smeared” over a certain range, rather than being deposited right at the center. CT reconnection and fuel deposition are possible at any of these points. If the Sweet-Parker current sheet model [39] is employed, the timescale for magnetic reconnection can be estimated as ${\tau }_{\mathrm{r}\mathrm{e}\mathrm{c}}=\sqrt{{\tau }_{\mathrm{A}}{\tau }_{\mathrm{R}}}$, where ${\tau }_{\mathrm{A}}=\dfrac{{r}_{\mathrm{C}\mathrm{T}}}{{V}_{\mathrm{A}}}$ represents the Alfvén time scale and ${\tau }_{\mathrm{R}}=\dfrac{{\mathrm{\mu }}_{0}{r}_{\mathrm{C}\mathrm{T}}^{2}}{\mathrm{\eta }}$ represents the resistive diffusion timescale. Alternatively, if the Petschek model [40] is adopted, the timescale for magnetic reconnection is estimated as ${\tau }_{\mathrm{r}\mathrm{e}\mathrm{c}}\approx {\tau }_{\mathrm{A}}$. Notably, ${\tau }_{\mathrm{r}\mathrm{e}\mathrm{c}}$ can be calculated based on the Alfvén velocity ${V}_{\mathrm{A}}$ of the tokamak plasma, using tokamak parameters. Because the physical mechanism of magnetic reconnection is still elusive, we evaluate the magnetic reconnection based on parameters specific to the CT. For clarity, ${V}_{\mathrm{A}}$ is used to represent the Alfvén velocity of the tokamak plasma, and ${V}_{\mathrm{A}-\mathrm{C}\mathrm{T}}$ is used to represent the Alfvén velocity of the CT plasma.

Figure  5.  Time-dependent angle between the CT dipole moment $\overrightarrow{\mu }$ and the tokamak magnetic field ${\overrightarrow{B}}_{\mathrm{t}\mathrm{o}\mathrm{k}}$, considering the vacuum magnetic field region. The interval between $0\;\mathrm{\mu }\mathrm{s}$ and the moment marked by the blue dashed line corresponds to the duration during which the CT moves within the vacuum magnetic field region.

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The dipole angle depicted in figure 5(a) corresponds to the angle between the CT’s magnetic dipole direction (indicated by the blue arrows in figure 4(b)) and the tokamak’s toroidal magnetic field direction. Thus, during the CT’s motion in the HL-2A device, $\lambda =0$ could occur at six positions, with the first occurrence in the vacuum magnetic field region. The initial $\lambda =0$ position in the vacuum magnetic field region of HL-2A corresponds to $t=0.85\;\mathrm{\mu }\mathrm{s}$. If the parameters of HL-2A plasma are employed to calculate magnetic reconnection based on the previous calculation results, the Alfvén velocity of the HL-2A tokamak plasma is ${V}_{\mathrm{A}}=5630\;\mathrm{k}\mathrm{m}/\mathrm{s}$, which corresponds to an Alfvén time of ${\tau }_{\mathrm{A}}=\dfrac{{{r}}_{\mathrm{C}\mathrm{T}}}{{V}_{\mathrm{A}}}=0.01\;\mathrm{ }\mathrm{\mu }\mathrm{s}$. For the magnetic reconnection process between the CT’s magnetic field lines and the tokamak’s magnetic field lines of HL-2A, the estimated timescale for magnetic reconnection according to the Sweet-Parker current sheet model is ${\tau }_{\mathrm{r}\mathrm{e}\mathrm{c}}=\sqrt{{\tau }_{\mathrm{A}}{\tau }_{\mathrm{R}}}=2.75\;\mathrm{\mu }\mathrm{s}$, while the estimated timescale according to the Petschek model is ${\tau }_{\mathrm{r}\mathrm{e}\mathrm{c}}\approx {\tau }_{\mathrm{A}}=0.01\;\mathrm{\mu }\mathrm{s}$. If the parameters of the CT plasma are employed to calculate magnetic reconnection for the CT plasma utilized in the HL-2A injection, the Alfvén velocity is ${V}_{\mathrm{A}-\mathrm{C}\mathrm{T}}=35.72\;\mathrm{k}\mathrm{m}/\mathrm{s}$, corresponding to an Alfvén time of ${\tau }_{\mathrm{A}-\mathrm{C}\mathrm{T}}=\dfrac{{r}_{\mathrm{C}\mathrm{T}}}{{V}_{\mathrm{A}-\mathrm{C}\mathrm{T}}}=34.48\;\mathrm{\mu }\mathrm{s}$. The estimated timescale for the magnetic reconnection according to the Sweet-Parker current sheet model is ${\tau }_{\mathrm{r}\mathrm{e}\mathrm{c}}=\sqrt{{\tau }_{\mathrm{A}-\mathrm{C}\mathrm{T}}{\tau }_{\mathrm{R}}}=34.48\;\mathrm{\mu }\mathrm{s}$, while the estimated timescale according to the Petschek model is ${\tau }_{\mathrm{r}\mathrm{e}\mathrm{c}}\approx {\tau }_{\mathrm{A}-\mathrm{C}\mathrm{T}}=1.40\;\mathrm{\mu }\mathrm{s}$.

Similarly, the dipole angle shown in figure 5(b) represents the angle between the CT’s magnetic dipole direction and ITER’s toroidal magnetic field direction. During the CT’s motion in the ITER device, $\lambda =0$ could occur at eight positions, with two positions in the vacuum magnetic field region. If the parameters of ITER plasma are employed to calculate magnetic reconnection based on the previous calculation results, the Alfvén velocity of the ITER plasma is ${V}_{\mathrm{A}}=7308\;\mathrm{k}\mathrm{m}/\mathrm{s}$, corresponding to an Alfvén time of ${\tau }_{\mathrm{A}}=\dfrac{{r}_{\mathrm{C}\mathrm{T}}}{{{V}}_{\mathrm{A}}}=0.01\;\mathrm{\mu }\mathrm{s}$. For the magnetic reconnection process between the CT’s magnetic field lines and the ITER’s magnetic field lines, the estimated timescale for CT reconnection according to the Sweet-Parker current sheet model is ${\tau }_{\mathrm{r}\mathrm{e}\mathrm{c}}=\sqrt{{\tau }_{\mathrm{A}}{\tau }_{\mathrm{R}}}=2.41\;\mathrm{\mu }\mathrm{s}$, while the estimated timescale according to the Petschek model is ${\tau }_{\mathrm{r}\mathrm{e}\mathrm{c}}\approx {\tau }_{\mathrm{A}}=0.01\;\mathrm{\mu }\mathrm{s}$. If the parameters of the CT plasma in the ITER injection are employed to calculate magnetic reconnection, the Alfvén velocity is ${V}_{\mathrm{A}-\mathrm{C}\mathrm{T}}=20.33\;\mathrm{k}\mathrm{m}/\mathrm{s}$, corresponding to an Alfvén time of ${\tau }_{\mathrm{A}-\mathrm{C}\mathrm{T}}=\dfrac{{r}_{\mathrm{C}\mathrm{T}}}{{V}_{\mathrm{A}-\mathrm{C}\mathrm{T}}}=2.46\;\mathrm{\mu }\mathrm{s}$. The estimated timescale for magnetic reconnection according to the Sweet-Parker current sheet model is ${\tau }_{\mathrm{r}\mathrm{e}\mathrm{c}}=\sqrt{{\tau }_{\mathrm{A}-\mathrm{C}\mathrm{T}}{\tau }_{\mathrm{R}}}=45.70\;\mathrm{\mu }\mathrm{s}$, while the estimated timescale according to the Petschek model is ${\tau }_{\mathrm{r}\mathrm{e}\mathrm{c}}\approx {\tau }_{\mathrm{A}-\mathrm{C}\mathrm{T}}=2.46\;\mathrm{\mu }\mathrm{s}$.

When the Sweet-Parker current sheet model is used, the estimated magnetic reconnection timescale, obtained using CT parameters, is larger than that obtained using tokamak plasma parameters by one order of magnitude. When the Petschek model is used, the estimated magnetic reconnection timescale obtained using CT parameters is larger than that obtained using tokamak plasma parameters by two orders of magnitude. Based on the numerical simulation results in figure 5, using CT parameters to estimate the magnetic reconnection process between the CT and tokamak’s magnetic field lines may not be suitable. When tokamak plasma parameters are used, the estimated magnetic reconnection timescale in the Sweet-Parker current sheet model is larger than that in the Petschek model by two orders of magnitude. During the CT’s penetration through the tokamak plasma, the magnetic moment of the CT undergoes continuous oscillations around the toroidal magnetic field direction in the tokamak. Once the magnetic moment direction aligns with the toroidal magnetic field direction of the tokamak, a magnetic reconnection process may occur immediately, leading to the deposition of CT fuel particles at these points. Based on the observation of rapid particle deposition in the CT injection experiments on TdeV [12], a rapid magnetic reconnection process based on the Petschek model can be assumed to have occurred in TdeV.

The travel time of the CT in the vacuum magnetic field region is excessively long, and the frequent alignment of the CT and tokamak magnetic field lines may lead to a significant depletion of the fuel particles and kinetic energy in the CT. As discussed above, there are several viewpoints regarding the estimation of the magnetic reconnection timescale. In the design of the CT, magnetic reconnection processes are considered as a secondary effect compared to the magnetic field gradient force [1, 20, 41]. It is worth noting that the magnetic reconnection is crucial for the deposition of fueling particles by CT injection in fusion plasmas. However, no existing experiment has been reported with clear magnetic reconnection observation in CT fueling process. In our experiments, by adjusting the discharge parameters of the CT injector system, CT parameters can cover a wide range, which should be benefited for reducing the risk of ignoring magnetic reconnection process in initial CT design. The proposed CT dynamic model does not account for the magnetic reconnection process because the reconnection timescale and the exact process of particle release from CT to tokamak particles are still largely unknown. However, the events that the CT’s magnetic dipole moment direction aligns with the tokamak toroidal magnetic field during penetration may serve as references for future studies on magnetic reconnection between CT’s and background magnetic fields.

3.3   Optimization of initial injection velocity

In the context of CT penetration, we will focus on the dynamics within the HL-2A device, which shares similarities with the ITER device. Nevertheless, we will also provide CT parameters for both HL-2A and ITER. In the dynamic model of CT motion within tokamak discharges, as depicted in figure 4, the CT travels through the center of the tokamak plasma. It is worth noting that the initial CT velocity, denoted as ${V}_{\mathrm{C}\mathrm{T}0}$ and detailed in table 1, is estimated to provide sufficient directional kinetic energy for the CT to reach the center of the tokamak, in accordance with the commonly cited requirement for central tokamak fueling by CT injection. This requirement is expressed by the following equation:

where ${\rho }_{\mathrm{C}\mathrm{T}}$ denotes the mass density of the CT, ${V}_{\mathrm{C}\mathrm{T}0}$ denotes the CT’s initial velocity, and ${B}_{0}^{2}$ represents the magnetic field at the magnetic axis position of the target tokamak. Equation (38) stipulates that the kinetic energy density of the CT must surpass the magnetic energy density at the location where the CT needs to penetrate. This estimation, however, does not account for the drag experienced by the CT within the tokamak plasma. Additionally, this model assumes that the CT is injected from a location outside the fringe magnetic field region, where the magnetic field is nearly zero. However, in our computational model, we calculate the CT trajectory and penetration parameters starting from the normal toroidal magnetic field region, ${R}_{\mathrm{T}\mathrm{F}-\mathrm{i}} < R < {R}_{\mathrm{T}\mathrm{F}-\mathrm{o}}$ as stated in section 3.1. At the initial CT injection position in our model, the magnetic field strength is finite, as shown in figure 6, which provides information on the CT motion in the HL-2A device corresponding to the trajectory shown in figure 4(b). Figure 6(a) illustrates that the magnetic field at the starting position at $\dfrac{R}{{R}_{0}}=1.37$ is ${B}_{1}=1.46\;\mathrm{T}$. The CT velocity gradually decreases with time (figure 6(b)) at an increasing rate. Consequently, the rate of CT penetration gradually decreases with increasing the time and decreasing velocity, as shown in figure 6(c). The trend displayed in figure 6 may also be explained by figure 7, depicts the acceleration during the CT’s motion in the HL-2A device. During the CT’s motion in the tokamak, the radial components of the force acting on magnetic dipole in an external magnetic field and the MHD drag force are smaller than the magnetic field gradient force. The magnetic field gradient force primarily determines the depth of the CT fuelling. As shown in figures 4(b) and 7(c), the CT goes to the high field side of tokamak plasma due to its high velocity and moves towards the inner vacuum magnetic field region ($R < {R}_{0}-a$) between the plasma and the inner wall around $\tau =23$. As shown in equations (20) and (26), when the CT is within the regions of the inner vacuum magnetic field region without tokamak plasma, the coefficient ${Q}_{0}$ is 0. In this region, the MHD wave drag rapidly reduces to zero, as well as the CT is slowly deaccelerated by the outward force of magnetic field gradient.

Figure  6.  Motion analysis of CT perpendicular injection into HL-2A, corresponding to the CT trajectory shown in figure 4(b). The figure consists of the following subfigures: (a) relationship between the toroidal magnetic field of the tokamak and the normalized major radius, (b) variation of CT velocity with normalized time, (c) relationship between CT and time at different major radius positions.

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In the dynamic model of CT in tokamak discharges, the magnetic field at the starting position (${R}_{0}+a+ {\text{Δ}} x$) can be calculated as ${B}_{1}=1.46\;\mathrm{T}$. The magnetic field gradient force is the primary force acting on the CT during the penetration process. Therefore, neglecting the MHD wave drag and the magnetic dipole effect. If the CT with an initial injection velocity of ${V}_{\mathrm{C}\mathrm{T}0}$ is injected into the tokamak from a position with a magnetic field of ${B}_{1}$, its velocity decreases to 0, when the CT reaches the magnetic axis position (${R}_{0}$) of the tokamak plasma (position with magnetic field ${B}_{0}$). This enables penetration into the center and facilitating fuel particle deposition. This optimized CT velocity only for reaching the center of plasma is governed by the following equation:

The rough calculation formula for the initial velocity of CT in this case can be expressed as follows:

The fringe magnetic field region is not accounted for while evaluating CT penetration, and the effects of MHD wave drag and magnetic dipole effect in the bulk plasma region are also neglected.

Figure  7.  Analysis of radial acceleration experienced by CT injected in HL-2A, corresponding to the CT trajectory shown in figure 4(b) and the penetration process described in figure 6: (a) total radial acceleration of the CT, (b) ${C}_{0}\dfrac{\mathrm{s}\mathrm{i}\mathrm{n}({Y}_{2}-{Y}_{3})}{{Y}_{1}^{2}}$, proportional to ${C}_{0}$, arises from the force acting on a magnetic dipole in the tokamak toroidal magnetic field, (c) ${Q}_{0}{Y}_{4}$ due to the MHD wave drag force, (d) $\dfrac{{E}_{0}}{{Y}_{1}^{3}}$ due to the gradient of the tokamak toroidal magnetic field.

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As discussed earlier, the magnetic field at ${R}_{0}+a+ {\text{Δ}} x$ in the HL-2A tokamak is ${B}_{1}=1.46\;\mathrm{T}$, whereas at ${R}_{0}$, the magnetic field is ${B}_{0}=2.00\;\mathrm{T}$. According to the data in table 1, ${n}_{\mathrm{C}\mathrm{T}}=2.98\times {10}^{22}\;{\mathrm{m}}^{-3}$, and HL-2A operates with deuterium discharge. Consequently, the mass density of the CT can be determined as ${\rho }_{\mathrm{C}\mathrm{T}}=9.98\times {10}^{-5}\;{\mathrm{k}\mathrm{g}/\mathrm{m}}^{3}$ by considering the CT parameters. By substituting those values into equation (40), the optimized initial CT velocity, ${V}_{\mathrm{C}\mathrm{T}0}$, for central penetration is $122\;\mathrm{k}\mathrm{m}/\mathrm{s}$. The value of ${V}_{\mathrm{C}\mathrm{T}0}$ in table 1 is substituted with $122\;\mathrm{k}\mathrm{m}/\mathrm{s}$, and the remaining parameters are held constant in the numerical calculations. The resulting trajectory of the CT in HL-2A is shown in figure 8(a), and the calculated penetration parameters of the CT are presented in table 5. Through a similar procedure, analysis of the motion of the CT in ITER reveals that the magnetic field at position ${R}_{0}+a+ {\text{Δ}} x$ in the ITER tokamak is ${B}_{1}=3.19\;\mathrm{T}$, and at position ${R}_{0}$, the magnetic field is ${B}_{0}=5.30\;\mathrm{T}$. Based on the data in table 2, ${n}_{\mathrm{C}\mathrm{T}}=7.36\times {10}^{22}\;{\mathrm{m}}^{-3}$, and ITER operates with deuterium and tritium discharge. Considering these parameters, the mass density of the CT is ${\rho }_{\mathrm{C}\mathrm{T}}=3.08\times {10}^{-4}\;{\mathrm{k}\mathrm{g}/\mathrm{m}}^{3}$. By substituting the parameters into equation (40), the initial injection velocity of the CT is ${V}_{\mathrm{C}\mathrm{T}0}=215.13\;\mathrm{k}\mathrm{m}/\mathrm{s}$. Figure 8(b) illustrates the trajectory of the CT obtained by substituting ${V}_{\mathrm{C}\mathrm{T}0}$ in table 2 and utilizing the dynamic model for calculations, and table 5 displays the CT motion parameters.

Figure  8.  (a) Trajectory of the CT perpendicularly injected into the HL-2A tokamak with an optimized initial injection velocity, (b) trajectory of CT perpendicularly injected into the ITER tokamak with an optimized initial injection velocity. The solid red line represents the trajectory of the CT in the tokamak, and the blue arrows indicate the magnetic dipole orientations at normalized times separated by ${\text{Δ}} \tau =2$.

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Table 5 provides a summary of the CT penetration parameters for two distinct cases: Case I, involving the perpendicular injection of CT into the HL-2A tokamak with an optimized initial injection velocity, and Case II, concerning the perpendicular injection of CT into the ITER tokamak with an optimized initial injection velocity. A comparison of table 5 with tables 3 and 4 reveals some noticeable trends. Firstly, as the initial injection velocity of CT decreases, the time required for CT to penetrate the vacuum magnetic field region increases. Additionally, both the velocity and kinetic energy of CT consumed during the penetration process also increase. Equation (40) can be used to estimate the initial injection velocity of CT, neglecting the effects of MHD wave drag and magnetic dipole effect. Therefore, the radial minimum position where the CT velocity drops to zero is only an approximate value, closely related to the magnetic axis ${R}_{0}$. The aforementioned discussion presents an estimation method (equation (40)) for the initial injection velocity of CT. For precise control of the CT velocity becoming zero at the magnetic axis ${R}_{0}$, the proposed dynamic model for the CT trajectory in tokamak discharges can be employed for calculations. However, because this calculation process is simple, it has not been demonstrated herein. A small coefficient, $I=\mathrm{ }0.06$, has been used in all simulations presented in this paper, although it could be as high as 1. For HL-2A, ${\rho }_{\mathrm{p}}=-0.07$, slightly passing the center. For ITER, ${\rho }_{\mathrm{p}}=0.04$, which almost reaches the center.

Table  5.  Penetration parameters for the CT in the HL-2A and the ITER.

Cases	Tokamak	VCT0 (km/s)	nCT (m−3)	∆x (m)	${ {\text{Δ}} t}_{{R}_{0}+a}$ (μs)	${{ {\text{Δ}} V}_{\mathrm{C}\mathrm{T}}}_{{R}_{0}+a}$(km/s)	${ {\text{Δ}} E}_{\mathrm{k}}$ (J)	${t}_{\mathrm{p}}$ (μs)	${R}_{\mathrm{p}}$ (m)	${\rho }_{\mathrm{p}}$
Case I	HL-2A	122	$2.98\times {10}^{22}$	0.21	1.80	14.13	5.21	8.94	1.62	$-0.07$
Case II	ITER	$215.13$	$7.36\times {10}^{22}$	2.11	10.63	36.50	859.34	30.32	6.26	0.04

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4.   Tangential injection

As per reference [19], various theoretical and computational studies have predicted that tangential injection of CT can result in longer CT-plasma interaction time and potentially fewer disturbances to the tokamak magnetic field. This injection method offers several advantages. Firstly, it allows for more flexibility in controlling the fuel deposition points within the tokamak. Secondly, by selecting an appropriate injection angle and velocity, it becomes possible to finely tune the timing of the CT’s reconnection with the tokamak’s magnetic field and its penetration time. This may also help prevent the distortion and eventual tearing of tokamak magnetic field lines, as mentioned in reference [1]. Lastly, the substantial tangential momentum carried by CT particles can be harnessed to drive current or generate plasma toroidal rotation [4, 15, 34]. Consequently, in this section, we employ the dynamic model for CT trajectory in tokamak discharges to investigate the effects of injection angle on the trajectory for tangential CT injection into HL-2A and ITER devices. Unlike previous work, this analysis also accounts for the CT penetration through the vacuum magnetic field.

4.1   Tangential CT injection into tokamak

The tangential injections into both the HL-2A and ITER tokamaks are also studied in this paper, as illustrated in figure 9. While the simulation relies on normalized parameters, the methodology can be extended to any tokamak devices, such as the HL-3 and BEST tokamaks. The initial parameters used for numerical calculations, as presented in tables 1 and 2, remain unchanged, except for the modification of the injection angle. To illustrate the differences in CT trajectory and penetration parameters resulting from varying injection angles, we initially employ a geometric calculation method to determine an initial injection angle. In this context, the angle between the initial injection velocity direction of the CT and the positive $\overrightarrow{x}$-axis is defined as the injection angle. The blue lines depicted in figure 9 represent a line tangent to the magnetic axis (${R}_{0}$) along the direction of the initial injection velocity, starting from the injection point. If the CT is unaffected by any forces in HL-2A, this line represents the CT’s trajectory in the HL-2A tokamak. The angle between this line and the $\overrightarrow{x}$-axis, denoted as $\alpha$ ($\alpha ={\text{π}} -\beta$), represents the initial injection angle. Based on the geometry shown in figure 9, the parameter ${\beta }_{1}$, defined as $\mathrm{s}\mathrm{i}\mathrm{n}{\beta }_{1}=\dfrac{{R}_{0}}{{R}_{0}+a+ {\text{Δ}} x}$, has a value of 0.73 at the initial injection location, which is ${\alpha }_{1}={133} {\text{°}}$ for the CT tangentially injected into HL-2A. Similarly, $\mathrm{s}\mathrm{i}\mathrm{n}{\beta }_{2}=0.60$, corresponding to an initial injection angle of ${\alpha }_{2}={143} {\text{°}}$ for the CT tangentially injected into ITER. Notably, this geometric calculation does not take into account the subsequent motion of CT in tokamak.

Figure  9.  Schematic diagram (top view) of the tangential injection of CT into the HL-2A and ITER device. The blue lines and parameters represent the information of the HL-2A device, while the pink lines and parameters represent the information of the ITER device.

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To calculate the trajectory and extract penetration parameters of the CT in a tokamak with greater precision and accuracy, we employ a dynamic model for the CT trajectory during tokamak discharge. This study utilizes this model to determine the initial injection angle that allows the CT to reach the radial minimum position, where its velocity becomes zero, and experience reflection at the radial minimum position located on the magnetic axis (${R}_{0}$). This approach provides a more comprehensive understanding of CT behavior in tokamak plasmas. Figure 10 illustrates a scan plot depicting the relationship between the initial injection angle and penetration parameters for CT tangentially injected into HL-2A, as determined through model calculations. The graph clearly shows that when the initial injection angle of the CT into HL-2A is ${\alpha }_{1}={143} {\text{°}}$, the CT precisely reaches the radial minimum position at the magnetic axis (${R}_{0}$). At this point, the CT’s penetration parameters are $\dfrac{{R}_{\mathrm{p}}}{{R}_{0}}=1$ and ${\rho }_{\mathrm{p}}=0$. These results provide valuable insights for optimizing injection angles in fusion experiments. Figure 11 presents a scan plot illustrating the relationship between the injection angle and penetration parameters for CT tangentially injected into ITER. The results also indicate that the CT accurately reaches the radial minimum position (where the radial velocity becomes zero) at the location of the ITER magnetic axis (${R}_{0}$) when the initial injection angle of the CT into ITER is ${\alpha }_{2}={155} {\text{°}}$. Notably, for CT tangentially injected into HL-2A, the CT cannot reach the tokamak plasma region if the initial injection angle is less than ${119} {\text{°}}$ (${\alpha }_{1} < {119} {\text{°}}$). Similarly, for CT tangentially injected into ITER, the CT cannot reach the plasma region of ITER if the initial injection angle is less than ${134} {\text{°}}$ (${\alpha }_{2} < {134} {\text{°}}$). Therefore, the optimal injection angles for CT tangentially injected into HL-2A and the ITER tokamak determined through model calculations are ${\alpha }_{1}={143} {\text{°}}$ and ${\alpha }_{2}={155} {\text{°}}$, respectively. These findings provide a reference for optimizing CT trajectories and highlight the importance of considering the injection angle when analyzing fueling in fusion plasmas. As illustrated in figures 10 and 11, the penetration depth of the CT in the tokamak increases as the injection angle of the CT approaches $\alpha ={180} {\text{°}}$. Additionally, the time and velocity required for the CT to traverse the vacuum magnetic field region decrease. Thus, the CT achieves the deepest penetration when it is injected perpendicularly ($\alpha ={180} {\text{°}}$) into the tokamak.

Figure  10.  Dependence of the penetration parameters on the initial injection angles of the CT at an injection velocity of ${V}_{\mathrm{C}\mathrm{T}0}=200\;\mathrm{k}\mathrm{m}/\mathrm{s}$ within the HL-2A vacuum magnetic field region ${ {\text{Δ}} x}_{1}=0.21\;\mathrm{m}$. From top: (a) normalized time required to penetrate the vacuum magnetic field region, (b) velocity of the CT after penetrating the vacuum magnetic field region, (c) relative penetration depth $\dfrac{{R}_{\mathrm{p}}}{{R}_{0}}$, (d) penetration time ${\tau }_{\mathrm{p}}$, (e) relative penetration depth ${\rho }_{\mathrm{p}}=\dfrac{{R}_{\mathrm{p}}-{R}_{0}}{a}$, and (f) residual toroidal CT velocity $\dfrac{{V}_{\mathrm{p}}}{{V}_{\mathrm{A}}}$.

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Figure  11.  Dependence of the penetration parameters on the initial injection angles of the CT at an injection velocity ${V}_{\mathrm{C}\mathrm{T}0}=300\;\mathrm{k}\mathrm{m}/\mathrm{s}$ and the ITER vacuum magnetic field region ${ {\text{Δ}} x}_{2}=2.11\;\mathrm{m}$.

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4.2   CT trajectory in tokamak

Figure 12 depicts the trajectory of the CT tangentially injected into the HL-2A tokamak for three cases: Case I wherein the CT is injected parallelly into HL-2A at an initial injection angle of ${\alpha }_{1}={133} {\text{°}}$, Case II wherein the CT is injected parallelly into HL-2A at an initial injection angle of ${\alpha }_{1}={143} {\text{°}}$, and Case III wherein the CT is injected antiparallelly into HL-2A at an initial injection angle of ${\alpha }_{1}={143} {\text{°}}$. When the drag force is considered in the calculation, the injection angle ${\alpha }_{1}={133} {\text{°}}$, does not reach the center of the plasma, as indicated by the purple dashed line. However, when the CT is injected into HL-2A at the angle ${\alpha }_{1}={143} {\text{°}}$ with a parallel magnetic moment (initial CT velocity), the CT reaches a point at the magnetic axis (${R}_{0}$) where its radial velocity becomes zero and exhibits reflection, as indicated by the solid red line in figure 12. By contrast, in the case of antiparallel injection, the CT penetration depth is reduced [19].

Figure  12.  Trajectory of the CT injected tangentially into the HL-2A tokamak. The purple dashed line represents the trajectory of CT parallel injected into HL-2A with an initial injection angle of ${\alpha }_{1}={133} {\text{°}}$. The solid red line represents the trajectory of CT parallel injected into HL-2A with an initial injection angle of ${\alpha }_{1}={143} {\text{°}}$. The dashed blue line represents the trajectory of CT antiparallel injected into HL-2A with an initial injection angle of ${\alpha }_{1}={143} {\text{°}}$. The arrows indicate the magnetic dipole orientations at normalized times separated by ${\text{Δ}} \tau =2$.

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Table 6 summarizes the CT penetration parameters for CT tangentially injected into the HL-2A tokamak under three cases, corresponding to the trajectories shown in figure 12. These results underline that when utilizing the injection angles obtained from the dynamic model for the CT trajectory in tokamak discharge, the penetration parameters reach $\dfrac{{R}_{\mathrm{p}}}{{R}_{0}}=1$ and ${\rho }_{\mathrm{p}}=0$ at the reflection point. This point coincides with the magnetic axis (${R}_{0}$), where the radial velocity of CT decreases to zero. It is noteworthy that the computed injection angle from the model results in CT requiring less time and velocity to traverse the vacuum magnetic field region in HL-2A. Specifically, ${\alpha }_{1}={143} {\text{°}}$, compared with ${\alpha }_{1}={133} {\text{°}}$, results in CT consuming $147.61\;\mathrm{J}$ less kinetic energy during penetration through the vacuum magnetic field region. This suggests that conducting CT injection experiments in tokamaks using the injection angle obtained from the model may offer a promising approach with reduced CT kinetic energy and enhanced central fueling capability.

Table  6.  Penetration parameters of CT tangentially injected into the HL-2A tokamak.

Cases	Angle ${\alpha }_{1}$	Injection method	∆x (m)	${ {\text{Δ}} t}_{{R}_{0}+a}$ (μs)	${{ {\text{Δ}} V}_{\mathrm{C}\mathrm{T}}}_{{R}_{0}+a}$ (km/s)	${ {\text{Δ}} E}_{\mathrm{k}}$ (J)	${t}_{\mathrm{p}}$ (μs)	${R}_{\mathrm{p}}$ (m)	${\rho }_{\mathrm{p}}$	${V}_{\mathrm{p}}$ (km/s)
Case I	${133} {\text{°}}$	Parallel	0.21	1.71	30.55	232.37	5.72	1.83	0.44	168.89
Case II	${143} {\text{°}}$	Parallel	0.21	1.38	29.12	94.76	6.41	1.65	0	156.50
Case III	${143} {\text{°}}$	Antiparallel	0.21	1.40	31.50	102.39	6.35	1.68	0.08	168.89

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In figure 13, the purple and red lines respectively depict the trajectories of CT tangentially injected into the ITER tokamak with two different injection angles, ${\alpha }_{2}={143} {\text{°}}$ and ${\alpha }_{2}={155} {\text{°}}$. When CT is injected into ITER with an initial angle of ${\alpha }_{2}={155} {\text{°}}$, the reflection point (radial minimum position) precisely aligns with the magnetic axis (${R}_{0}$), resulting in significantly greater penetration depth compared to when CT is injected into ITER with an initial angle of ${\alpha }_{2}={143} {\text{°}}$. Table 7 summarizes the CT penetration parameters for CT tangentially injected into the ITER tokamak in two cases: Case I with CT injected parallelly with an initial injection angle of ${\alpha }_{2}={143} {\text{°}}$ and Case II with CT injected parallelly with an initial injection angle of ${\alpha }_{2}={155} {\text{°}}$ corresponding to the trajectories in figure 13. The results unequivocally demonstrate that CT injected with an angle of ${\alpha }_{2}={155} {\text{°}}$ achieves deeper penetration, a longer penetration time, and slightly lower relative penetration depth value compared to CT injected with an angle of ${\alpha }_{2}={143} {\text{°}}$. In particular, compared to ${\alpha }_{2}={143} {\text{°}}$, ${\alpha }_{2}={155} {\text{°}}$ results in the CT expending less kinetic energy, specifically CT consumes $975.08\;\mathrm{J}$ less kinetic energy during penetration through the vacuum magnetic field region.

Table  7.  Penetration parameters of CT tangentially injected into the ITER tokamak.

Cases	Angle (${\alpha }_{2}$)	Injection method	∆x (m)	${ {\text{Δ}} t}_{{R}_{0}+a}$ (μs)	${{ {\text{Δ}} V}_{\mathrm{C}\mathrm{T}}}_{{R}_{0}+a}$ (km/s)	${ {\text{Δ}} E}_{\mathrm{k}}$ (J)	${t}_{\mathrm{p}}$ (μs)	${R}_{\mathrm{p}}$ (m)	${\rho }_{\mathrm{p}}$	${V}_{\mathrm{p}}$ (km/s)
Case I	${143} {\text{°}}$	Parallel	2.11	10.41	82.86	13218	21.79	7.32	0.54	196.60
Case II	${155} {\text{°}}$	Parallel	2.11	8.42	47.16	3647.20	24.57	6.20	0	181.98

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Figure  13.  Trajectory of CT tangentially injected into the ITER tokamak. The purple line represents CT injected in parallel with an initial angle of ${\alpha }_{2}={143} {\text{°}}$, while the red line represents CT injected in parallel with an initial angle of ${\alpha }_{2}={155} {\text{°}}$. The arrows indicate the orientations of the magnetic dipole at normalized times separated by ${\text{Δ}} \tau =3$.

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For the perpendicular injection of the CT into the HL-2A device, figure 5(a) illustrates that there are six positions where the $\lambda =0$ condition is satisfied during the CT’s motion from the injection position to the radial minimum position. The first position that satisfies the $\lambda =0$ condition appears in the vacuum magnetic field region. The time required is ${ {\text{Δ}} t}_{{R}_{0}+a}=1.06\;\mathrm{\mu }\mathrm{s}$, and the kinetic energy required is ${ {\text{Δ}} E}_{\mathrm{k}}=1.65\;\mathrm{J}$ for the CT to penetrate the vacuum magnetic field region. For CT parallelly injected into HL-2A with an initial injection angle of ${\alpha }_{1}={143} {\text{°}}$, it takes a time of ${ {\text{Δ}} t}_{{R}_{0}+a}=6.41\;\mathrm{\mu }\mathrm{s}$ to move from the injection position to the magnetic axis. The calculation results show that during the CT’s motion in the HL-2A device, the $\lambda =0$ condition could occur in five positions, with the initial occurrence taking place in the vacuum magnetic field region. The time (${ {\text{Δ}} t}_{{R}_{0}+a}$) and kinetic energy (${ {\text{Δ}} E}_{\mathrm{k}}$) required for CT to penetrate the vacuum magnetic field region are $1.38\;\mathrm{\mu }\mathrm{s}$ and $94.76\;\mathrm{J}$, respectively. For the perpendicular injection or tangential injection (${\alpha }_{2}={155} {\text{°}}$), during the CT’s motion in the ITER device, the $\lambda =0$ condition could occur at eight positions, with two of them situated in the vacuum magnetic field region. It is worth noting that, with a decrease in the initial injection angle, the time and kinetic energy required for the CT to penetrate the vacuum magnetic field region increase, as shown in figures 11 and 12. Meanwhile, with the decrease of the initial injection angle, more positions satisfying the $\lambda =0$ condition may appear in the vacuum magnetic field region. This could result in an increased consumption of fuel particles carried by the CT in the vacuum magnetic field region. Under the same conditions of CT injection parameters and tokamak discharge parameters, perpendicular injection is more advantageous for CT to penetrate the vacuum magnetic field region than tangential injection. Perpendicular injection is more beneficial than tangential injection in reducing both the time required and the number of positions satisfying the $\lambda =0$ condition for CT to penetrate the vacuum magnetic field region, thereby reducing the consumption of fuel particles and kinetic energy. These findings highlight the utility of the dynamic model for CT trajectory in the tokamak discharge in designing optimal initial injection velocity and angle, leading to optimized initial injection conditions and reduced kinetic energy consumption of CT in the vacuum magnetic field region.

5.   Summary

In this study, we developed and applied a dynamic model for CT trajectory within a tokamak, accounting for the influence of the vacuum magnetic field region. This consideration enhanced our understanding of the injection scenario and improved the accuracy of penetration parameter calculations. Our findings highlighted the significant impact of the vacuum magnetic field region on CT penetration parameters. When CT is injected perpendicularly into ITER, the vacuum magnetic field region reduces the relative penetration depth by 0.13, compared to cases where this region was not considered. Similarly, for CT injected perpendicularly into the HL-2A tokamak, the vacuum magnetic field region causes a 0.08 reduction in relative penetration depth compared to cases without such consideration. Furthermore, we developed a computational method for determining the initial injection velocity of CT required to achieve central fueling in a tokamak. Based on this method, the necessary initial velocities for CT to achieve central fueling in the HL-2A and ITER tokamaks are ${V}_{\mathrm{C}\mathrm{T}0}=122\;\mathrm{k}\mathrm{m}/\mathrm{s}$ and ${V}_{\mathrm{C}\mathrm{T}0}=215.13\;\mathrm{k}\mathrm{m}/\mathrm{s}$, respectively. These initial injection velocities were validated using the dynamic model for CT trajectory within the tokamak discharge.

This study evaluated the relationship between the injection angle and CT penetration parameters within the tokamak. The simulation results revealed that the deepest penetration depth is achieved when CT is injected perpendicularly into the tokamak. Moreover, optimization calculations identified the optimal injection angles for tangential CT injection into HL-2A (${\alpha }_{1}={143} {\text{°}}$) and ITER (${\alpha }_{2}={155} {\text{°}}$). Injecting CT into the tokamak with the initial injection magnetic moment aligned with the initial velocity direction, as opposed to injecting it in the opposite direction, enables deeper penetration. Compared to CT evaluated through the geometrical estimation method, CT injected into ITER at an injection angle based on the dynamic model consumes significantly less kinetic energy (${E}_{{\mathrm{k}}_{143}}-{E}_{{\mathrm{k}}_{155}}=975.08\;\mathrm{J}$) that CT penetrated the vacuum magnetic field region. This highlighted the importance of considering the dynamic model for more accurate energy consumption estimations during CT injection.

The dynamic model for CT trajectory within tokamak discharges optimized the design of CT injection parameters and determined optimal initial injection conditions while reducing the kinetic energy consumption of CTs within the vacuum magnetic field region. This approach significantly reduced the construction cost of the CT injector. Although the proposed model lacks the precision required to calculate the fuel deposition position of CTs, it provides highly accurate trajectory information and represents a significant advancement because it accounts for the effects of the vacuum magnetic field region on CT trajectories and penetration parameters, ensuring that the model accurately represents real experimental conditions.