Coupled Electromagnetic–Thermal Modeling of HTS Transformer Inrush Current: Experimental Validation and Thermal Analysis

Abstract

The article presents a numerical model of a high-temperature superconducting (HTS) transformer rated at 13.8 kVA, equipped with windings made of 2G ReBCO tapes. The model was developed to analyze the coupled electromagnetic and thermal phenomena occurring during the inrush current period of transformer energization. It describes the dynamic processes of critical current exceedance, resistive zone formation, and local temperature rise within the superconducting tape structure under realistic operating conditions. The geometry of the ReBCO tape is represented with its active superconducting layer and metallic stabilizer layers. Temperature-dependent material properties of each layer, such as electrical resistivity, thermal conductivity, and specific heat capacity, are incorporated into the model. This approach enables a detailed analysis of the temperature distribution across all layers of the superconducting tape. The results indicate that the highest thermal stress occurs during the first inrush current peak, whose amplitude exceeds the critical current of the winding. At this stage, a distinct temperature rise is observed in the stabilizer layers, followed by gradual cooling in subsequent cycles of operation. The simulated current and temperature waveforms show good agreement with experimental measurements performed on a liquid-nitrogen-cooled transformer prototype. The developed model enables quantitative evaluation of local overheating risks, analysis of Joule loss distribution, and assessment of the influence of supply parameters and circuit impedance on the thermal stability of the system. Its application supports the optimization of HTS transformer design and provides valuable insight into the reliability of superconducting windings under transient inrush current conditions.

1. Introduction

The future of high-temperature superconducting (HTS) transformers is closely associated with the advancement in superconducting technologies, the development in cryogenic systems, the growing demands on power networks, and the ongoing energy transition [1]. HTS transformers are unlikely to replace conventional units on a large scale in the coming decade; however, they will continue to evolve in specialized applications. In the longer term (20–30 years), their role may become significantly more prominent. The development of HTS transformers strongly depends on advances in cryogenic technologies and the reduction in cooling-related costs. At the same time, the continuous improvement in superconducting wire properties enhances their implementation potential, increasing their resilience to magnetic fields, dynamic forces, and temperature fluctuations [2,3]. The HTS technology applied in transformers enables an almost complete elimination of resistive losses in the windings, with only the losses in the magnetic core and cryogenic system remaining. Consequently, the overall efficiency of HTS transformers exceeds that of conventional units, a factor that is becoming increasingly important in the context of the energy transition and the global need to minimize transmission losses. An additional advantage of HTS transformers is their capability to integrate with superconducting fault current limiters (SFCLs), enabling the development of multifunctional devices that combine voltage transformation with protection against fault conditions. Another important benefit of HTS conductors is their small cross-sectional area, which, when applied to windings, allows for a substantial reduction in the size and weight of transformers while preserving the rated power [4]. High investment costs, the susceptibility of HTS conductors to damage, and the operational complexity of cryogenic systems remain significant limitations that currently hinder widespread commercialization. Consequently, most existing HTS transformers are still pilot projects or demonstration units, tested in countries such as Japan, Korea, Germany, China, and the United States. In recent years, research on HTS transformers has primarily focused on the application of second-generation superconducting wires in high-power windings. Particular attention has been given to geometries that reduce alternating current (AC) losses and enhance operational reliability. Among the most promising approaches are Roebel-type cables, which, owing to their transposition, exhibit lower AC losses compared to parallel configurations. At the same time, concepts of REBCO tape multifilamentation are being explored, enabling a further reduction in losses under varying current waveforms [5]. Parallel arrangements of HTS tapes are also being investigated to improve cooling conditions, avoid non-simultaneous quenching of individual tapes, and mitigate current density inhomogeneities [6]. Considerable attention is also devoted to AC losses, transient phenomena, and distorted waveforms, including the presence of higher harmonics, which reflect the actual operating conditions of transformers in power networks. In many studies, coupled electromagnetic–thermal simulations are performed to predict the occurrence of local hot spots in windings and to mitigate their effects [7]. The concept of HTS transformers serving as fault current limiters is also evolving rapidly. Analyses have shown that the properties of superconductors enable the limitation of fault current peaks and the reduction in thermal stresses in the windings [8]. At the same time, research has been carried out on the impact of inrush current on HTS windings, highlighting the risk of overheating and the effectiveness of mitigation methods, such as the appropriate design of stabilizing layers or control of the switching phase. The inrush current in transformers is an unavoidable physical phenomenon associated with the sudden excitation of a transient state in an electrical circuit coupled with a magnetic circuit, and the influence of residual flux [9]. The waveform of this current is highly distorted and asymmetric, with a significant contribution of higher harmonics. Under certain conditions, the magnitude of this current, particularly in the first milliseconds after energization, may exceed the rated current by more than an order of magnitude. Although the inrush current typically lasts only a few seconds, it can lead to serious operational consequences. Within the transformer, significant electrodynamic forces may occur [10], acting on the windings and the core structure, which over time promote mechanical damage and accelerated insulation aging. Moreover, short-term but repetitive thermal shocks may cause local overheating and degradation of insulating materials. From the power system perspective, inrush current induces voltage dips and may trigger false operations of overcurrent or differential protection devices. It also adversely affects switching equipment, accelerating contact wear and reducing service life. Although the physical nature of inrush current in HTS transformers is similar to that in conventional units, the waveform and consequences differ fundamentally due to the specific properties of HTS conductors. In HTS transformers, the concern relates not only to electrodynamic forces and network disturbances but, most importantly, to the stability of the superconducting state. Superconducting windings in the cryogenic state are characterized by extremely low resistance, and their current-carrying capability is limited by a critical parameter: the critical current density $J_c$. The amplitude of inrush current in HTS transformers is often comparable to that of conventional units (ranging from several up to more than ten times the rated current $I_n$) [11]. However, the critical concern is that even a much lower current (only 2–$3\times I_n$) may rapidly cause both the critical current density $J_c$ and the critical temperature $T_c$ of the HTS windings to be exceeded. Therefore, in the case of HTS transformers, the key challenge is not the absolute amplitude of the inrush current, but rather its relation to the critical parameters of the conductor. Improperly controlled inrush current may cause local transitions of HTS conductor segments into the resistive state, accompanied by heat generation and a temperature rise, ultimately leading to a complete loss of superconductivity in the windings, thermal degradation of the conductors, and overloading of the cryogenic system. The inrush current phenomenon in HTS transformers is considerably more hazardous—even at lower amplitudes—than in conventional transformers. For this reason, both the design and operation of HTS transformers must consider methods for inrush current limitation, as well as detailed modeling of the coupled electromagnetic and thermal phenomena occurring in superconducting windings. Despite the progress in superconducting materials, a key limitation of HTS transformer technology remains the sensitivity of HTS conductors to thermal factors. One of the most important research challenges is the analysis of temperature rise in HTS windings under transformer operating conditions, particularly during transient phenomena such as inrush current, short-circuit current, or temporary overloads [12,13]. Research on thermal processes in superconducting windings is of critical importance for the following reasons:

• Enhancing the reliability of HTS transformers by improving the understanding of the conditions for superconductivity loss initiation and the dynamics of this phenomenon.
• Designing more efficient cryogenic cooling systems capable of compensating temporary thermal overloads.
• Optimizing winding design through the selection of appropriate geometries and improvements in HTS conductor architecture.
• Developing new methods for inrush current limitation that minimize the risk of exceeding the permissible temperature of HTS windings.

Conducting research in this area is, therefore, essential for the safe and efficient implementation of superconducting transformer technology in modern power systems.

Despite the growing interest in superconducting transformers, the scientific literature still lacks comprehensive studies dedicated to the detailed analysis of thermal phenomena occurring in HTS tapes during inrush current conditions [14]. Previous research has mainly focused on general aspects of superconducting operation and stability, largely overlooking the complex coupling between electromagnetic and thermal processes during transient states [15]. A significant research gap lies in the absence of dynamically coupled models capable of accurately describing the nonlinear phenomena occurring in superconducting windings. In most existing studies, electromagnetic and thermal effects are analyzed separately or by means of simplified models that do not account for the dependence of the critical current density on temperature and magnetic field, the nonlinear resistivity of stabilizing layers, or the variability of thermal conductivity in structural materials. Such an approach considerably limits the accuracy of predictions regarding local temperature rises and the moment of transition of the tape into the resistive state [16]. Another limitation of existing models is the lack of their experimental validation on real HTS transformer constructions. Consequently, it is difficult to assess their reliability and the practical applicability of simulation results for evaluating the thermal safety of windings [17,18]. Experimental validation therefore constitutes a crucial stage in the development of numerical models, allowing their adaptation to real operating conditions and the determination of thermal stability boundaries. In response to these shortcomings, this study presents an experimentally validated electromagnetic–thermal coupling model developed for a 13.8 kVA HTS transformer. The model enables the analysis of temperature rise in multilayer HTS conductors under inrush current conditions, taking into account the nonlinear properties of materials as well as cryogenic cooling conditions.

2. Object of the Study

The object of this study is a single-phase HTS transformer with a rated power of $13.8$ $\mathrm{k}$$\mathrm{V}$$\mathrm{A}$. The rated supply voltage of the transformer is 230 $\mathrm{V}$ at a frequency of 50 $\mathrm{Hz}$. The transformer steps down the voltage to 60 $\mathrm{V}$, and the rated current on the secondary side is 230 $\mathrm{A}$. The rated parameters of the transformer are summarized in

Table 1. Rated parameters of the HTS transformer.

Parameter	Value
Rated power	$13.8$ $\mathrm{k}$$\mathrm{V}$$\mathrm{A}$
HV/LV voltage	230 $\mathrm{V}$/60 $\mathrm{V}$
Rated frequency	50 $\mathrm{Hz}$
HV/LV current	60 $\mathrm{A}$/230 $\mathrm{A}$
Maximum magnetic flux density	$1.6$ $\mathrm{T}$
No-load current	$0.7$ $\mathrm{A}$
Short-circuit voltage	3.2%

.

The transformer operates with a warm core, meaning that the HTS windings are placed in a cryostat filled with liquid nitrogen (77 $\mathrm{K}$), while the magnetic core is located outside the cryostat at ambient temperature. The transformer core was manufactured with a mitred and stepped design, based on a 4, 3, 1 sheet-cutting arrangement with three steps. The geometry of the core is shown in Figure 1, and the cross-sectional area of the core limb is 0.008 m².

The core material is grain-oriented electrical steel, grade M150-30S, manufactured by Stalprodukt S.A. (Bochnia, Poland), in compliance with the EN 10107:2005 standard [19]. The sheet thickness is $0.3$ $\mathrm{m}$$\mathrm{m}$, with a silicon content of 3%. The maximum total loss at a frequency of 50 $\mathrm{Hz}$ and an induction of $1.5$ $\mathrm{T}$ is $0.97$ $\mathrm{W} {\mathrm{k}\mathrm{g}}^{\mathrm{-1}}$, while at $1.7$ $\mathrm{T}$ it reaches $1.5$ $\mathrm{W} {\mathrm{k}\mathrm{g}}^{\mathrm{-1}}$. The maximum magnetic flux density is $1.75$ $\mathrm{T}$ at a magnetic field strength of 800 $\mathrm{A} {\mathrm{m}}^{\mathrm{-1}}$. The magnetic hysteresis of the transformer core under rated operating conditions is shown in Figure 2.

The HTS windings of the transformer were manufactured in a concentric arrangement, with the secondary winding placed inside the primary winding, mounted on separate formers made of epoxy–glass composite. The total number of turns of the high-voltage winding (84 turns) and the low-voltage winding (44 turns) was divided into two sections with an equal number of turns, symmetrically distributed on both core limbs. The geometry of the windings is presented in Figure 3.

The transformer windings were fabricated from HTS tape manufactured by SuperPower Inc. (Glenville, NY, USA). The high-voltage windings were wound using SCS4050-AP tape with a minimum critical current of 87 $\mathrm{A}$ at 77 $\mathrm{K}$ under self-field conditions. The low-voltage windings were wound using SCS12050-AP tape with a minimum critical current of 333 $\mathrm{A}$. Both types of tape employed ReBCO superconductors. The structural design of the tapes is presented in Figure 4 [20].

The transformer operates with a ratio of the critical current of the HTS tapes to the rated current of the transformer equal to 1.45 for both the high-voltage and low-voltage windings. As a result, under rated load conditions, the instantaneous peak currents flowing through the transformer windings are only slightly below their critical values. The long-term current overload capability, which does not cause a loss of superconductivity in the windings, is 2.4%.

3. Experimental Setup and Results

The inrush current measurements were carried out using the setup shown in Figure 5. The supply was provided from a single-phase 230 $\mathrm{V}$, 50 $\mathrm{Hz}$ grid through an ATS-FAZ3-23 autotransformer (Metrel d.o.o., Horjul, Slovenia), with a regulation range of 0–260 $\mathrm{V}$ at 230 $\mathrm{V}$ nominal input and a rated power of $23.4$ $\mathrm{k}$$\mathrm{V}$$\mathrm{A}$. The inrush current was recorded indirectly by measuring the voltage drop across a resistive shunt of 1 $\mathrm{m}$$\Omega$, with an accuracy class of 0.5. Data acquisition was performed in real time using a National Instruments DAQ card installed in a PC-class computer and controlled by a dedicated application developed in the LabVIEW R2025a environment.

The energization of the unloaded transformer was performed at a voltage of 230 $\mathrm{V}$ using an electronic control system that synchronized the switching instant with the zero crossing of the supply voltage. In this way, the transformer was energized under the most unfavorable conditions, when the inrush current reaches its maximum values. De-energization of the transformer was preceded by gradually reducing the supply voltage to zero using the autotransformer, in order to eliminate the influence of unpredictable residual magnetism on the inrush current waveform.

The equivalent circuit diagram of the measurement system is presented in Figure 6, and the values of the circuit elements are summarized in

Table 2. Equivalent circuit parameters of the measurement setup.

Parameter	Value
Supply voltage E	230 $\mathrm{V}$
Supply frequency f	50 $\mathrm{Hz}$
Autotransformer resistance $R_t$	$0.8$ $\Omega$
Autotransformer reactance $X_t$	$40.212$ $\Omega$
Supply line resistance $R_n$	$0.011$ $\Omega$
Supply line reactance $X_n$	0 $\Omega$
Winding resistances $R_1,R_2$	$26.75$ $\Omega$
Leakage reactances $X_1,X_2$	$54.05$ $\mathrm{m}$$\Omega$
Core loss resistance $R_{fe}$	$737.8$ $\Omega$
Magnetizing reactance $X_{\mu }$	$431.9$ $\Omega$

.

The measured waveform of the inrush current is shown in Figure 7. Only the first current pulse exceeds the critical value of the superconducting transformer winding. This pulse reaches 258 $\mathrm{A}$, exceeding the critical current by 171 $\mathrm{A}$. The waveform of the first inrush current pulse is presented in Figure 8. The critical current is exceeded after $5.22$ $\mathrm{m}$$\mathrm{s}$ from the time of transformer energization and persists for $5.61$ $\mathrm{m}$$\mathrm{s}$. During this period, the conduction of the inrush current is taken over by the resistive layers of the superconducting tape, and the winding undergoes a heating process.

The measurement error may have significance in numerical and comparative analyses, as the recorded inrush current waveform serves as a reference point in the validation of simulation results. The accuracy of reproducing its shape, amplitude, and dynamics is crucial for assessing the correctness of the developed electromagnetic–thermal model and its capability to faithfully replicate the actual transient conditions in the superconductor. However, determining the total measurement error in the analyzed circuit is challenging, since the accuracy is influenced by numerous random and systematic factors associated with the shunt resistor, measurement leads, data acquisition card, environmental conditions, and data processing algorithms. In practice, these errors may overlap in a nonlinear and time-varying manner, allowing only for an approximate estimation of overall uncertainty. In the investigated system, the dominant source of error is shunt heating, which can change its resistance by up to approximately 0.5% under high current conditions.

4. Mathematical Model

Inrush current modeling has been extensively investigated for conventional transformers, for which numerous analytical, numerical, and hybrid models have been developed [21]. Despite the significant advancement of conventional models, the literature still lacks adequate approaches dedicated to superconducting transformers. HTS transformers introduce new physical mechanisms that must be taken into account in inrush current models [22]. In addition to the nonlinear magnetization characteristics of the core, the following aspects play a crucial role: the temperature- and magnetic field-dependent critical current of superconducting windings, the quenching phenomenon, and heat transfer processes under cryogenic conditions. These phenomena require a multiphysics approach that combines electromagnetic, thermal, and material modeling to accurately reproduce the dynamics of inrush currents in superconducting transformers. Therefore, although conventional transformer models may serve as a foundation, modeling their superconducting counterparts requires the application of different methods and extensions to incorporate the specific properties of superconductors and cooling conditions. The absence of such dedicated models represents a significant research gap, the closure of which is essential to ensure the safe and reliable operation of superconducting transformers in future power systems.

The mathematical model presented in this study describes the energization process of an unloaded single-phase HTS transformer connected to a sinusoidal voltage source. The corresponding equivalent electrical circuit diagram is shown in Figure 6.

In the model, the active component corresponding to iron losses was neglected ($R_{fe}=0$), and the following assumptions were made:

$$\begin{array}{c}\hfill L_d=L_{\mu },\end{array}$$

(1)

$$\begin{array}{c}\hfill R_{\mathrm{conn}}=R_s+R_n+R_t.\end{array}$$

(2)

where

• $R_{\mathrm{conn}}$—Total resistance of the circuit elements (wires, connectors) [$\Omega$];
• $L_d$—Differential (local) inductance of the magnetizing circuit, dependent on the operating point on the B–H magnetization curve [$\mathrm{H}$], as well as on the inductances of the circuit elements.

Two operating states of the transformer were considered: (a) the state in which the windings remain superconducting, and (b) the state in which the windings transition to the resistive state.

In modeling the inrush current, knowledge of the complete hysteresis characteristic of the transformer core material is essential [23]. Neglecting the actual hysteresis loop leads to either underestimation or overestimation of the calculated values, and consequently to an incomplete assessment of thermal and electromechanical stresses in the windings. Therefore, the integration of experimental hysteresis data into numerical models is a necessary step to accurately reproduce the inrush current phenomenon and obtain results consistent with measurements.

To reliably reproduce the inrush current waveform, the model implemented a hysteresis loop (Figure 2) obtained from measurements of a 13.8 kVA HTS transformer under rated operating conditions. The value and shape of the inrush current are strongly influenced by the magnetization characteristic in the saturation region of the transformer core. In most transformers, achieving deep core saturation experimentally is difficult or even impossible, which complicates the reconstruction of hysteresis loop fragments corresponding to maximum inrush current values. Therefore, in the saturation region, these fragments were modeled by fitting their shape to the characteristic shown in Figure 9 [24,25], which enabled the reconstruction of the complete hysteresis loop under saturation conditions, necessary for calculations within the given range of magnetic flux density and magnetic field strength.

4.1. Superconducting Tape Model

In the mathematical model, a simplified structure of the SCS4050-AP superconducting tape was assumed. The upper metallic layers, composed of copper and silver, were reduced to a single metallic layer $m_1$ with averaged parameters. Similarly, the lower metallic layers, including copper, silver, and Hastelloy, were replaced by a single metallic layer $m_2$, also with averaged parameters. A schematic of the simplified structure of the superconducting tape is presented in Figure 10. Previous studies have demonstrated that the cross-sectional areas of the resistive layers of the tape (Cu and Ag stabilizers, Hastelloy) significantly affect the dynamics of inrush current and the thermal load of the windings [26,27]. The averaging of layers $m_1$ and $m_2$ in the adopted model preserves the influence of the total resistive cross-section on the current waveform and heat balance during the resistive state of the winding.

4.2. Superconducting State

In the superconducting state of the HTS transformer windings, alternating current losses were neglected. It was assumed that the entire current flows through the superconducting layer, with the resistive layers of the tape disregarded, and that the resistance of the superconductor is zero, which allowed Joule losses to be omitted. The equivalent circuit diagram of the electrical system in this state is shown in Figure 11.

The basic voltage equation of the circuit shown in Figure 11 (corresponding to the superconducting state of the windings) can be expressed as

$$\begin{array}{cc}\hfill u_s\left(t\right)& =R_{\mathrm{conn}}{i}_m\left(t\right)+N{\displaystyle \frac{d\Phi \left(t\right)}{dt}},\hfill \end{array}$$

(3)

where

• $u_s\left(t\right)$—Supply voltage applied to the transformer winding at time t [$\mathrm{V}$];
• $i_m\left(t\right)$—Magnetizing current of the HTS transformer winding at time t [$\mathrm{A}$];
• $\Phi \left(t\right)$—Magnetic flux in the transformer core at time t [$\mathrm{Wb}$];
• $R_{\mathrm{conn}}$—Resistance of electrical connections and additional circuit elements (terminals, supply leads, connectors) [$\Omega$];
• N—Number of turns of the transformer winding [–];
• ${\displaystyle \frac{d\Phi \left(t\right)}{dt}}$—Time derivative of the magnetic flux, corresponding to the electromotive force induced in the winding [$\mathrm{V}$].

The value and settling process of the magnetic flux after transformer energization are significantly affected by both the magnitude and polarity of the residual flux [28]. To account for this effect in Equation (3), the following initial condition was adopted:

$$\begin{array}{c}\hfill \Phi \left(0\right)={\Phi }_{\mathrm{init}}.\end{array}$$

(4)

where

• ${\Phi }_{\mathrm{init}}$—Initial value of the magnetic flux in the transformer core, corresponding to the residual flux [$\mathrm{Wb}$].

The relationships between magnetic flux, magnetic flux density, and magnetic field strength are expressed as follows:

$$\begin{array}{cc}\hfill \Phi \left(t\right)& =AB\left(H\left(t\right)\right),\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill H\left(t\right)& ={\displaystyle \frac{N{i}_m\left(t\right)}{l_m}}.\hfill \end{array}$$

(6)

where

• A—Cross-sectional area of the transformer core through which the magnetic flux passes [${\mathrm{m}}^2$];
• $B\left(H\left(t\right)\right)$—Magnetic flux density in the core as a function of the magnetic field strength [$\mathrm{T}$];
• $H\left(t\right)$—Instantaneous magnetic field strength in the core [$\mathrm{A} {\mathrm{m}}^{\mathrm{-1}}$];
• N—Number of turns of the primary winding [–];
• $l_m$—Mean length of the magnetic path in the core [$\mathrm{m}$].

By transforming Equation (3), the following differential equation describing the magnetizing current of the HTS transformer was obtained, in which the inductance is a function of the instantaneous operating point on the magnetization curve:

$$\begin{array}{c}\hfill L_d\left(i_m\left(t\right)\right){\displaystyle \frac{d{i}_m\left(t\right)}{dt}}+R_{\mathrm{conn}}{i}_m\left(t\right)=u_s\left(t\right),\end{array}$$

(7)

while the local differential inductance is expressed as

$$\begin{array}{c}\hfill L_d\left(i_m\right)={\displaystyle \frac{N^{2}A}{l_m}}{\left.{\displaystyle \frac{dB}{dH}}\right|}_{H=\frac{N{i}_m}{l_m}}.\end{array}$$

(8)

where

• $L_d\left(i_m\right)$—Differential (local) inductance of the magnetizing circuit, dependent on the operating point on the B–H curve [H];
• N—Number of turns of the winding [–];
• A—Cross-sectional area of the core through which the flux passes [m²];
• $l_m$—Mean length of the magnetic path in the core [m];
• B—Magnetic flux density in the core material [T];
• H—Magnetic field strength in the core material [A/m];
• $\frac{dB}{dH}$—Differential permeability (local slope of the magnetization curve), equivalent to ${\mu }_{\mathrm{diff}}$ [H/m].

After the transformer is energized, the value of the current exciting the flux must initially be equal to zero, regardless of the instantaneous value of the voltage at the switching instant. Thus, the initial condition for Equations (7) and (8) takes the form

$$\begin{array}{c}\hfill i_m\left(0\right)=0.\end{array}$$

(9)

An important role in the modeling and analysis of the inrush current phenomenon is played by the condition of the transformer core entering magnetic saturation. The core enters the saturation region when the flux density exceeds the saturation flux density of the core material:

$$\begin{array}{c}\hfill B\left(H\left(t_{\mathrm{sat}}\right)\right)\ge B_{\mathrm{sat}}.\end{array}$$

(10)

where

• $B_{\mathrm{sat}}$—Saturation flux density of the core material [$\mathrm{T}$];
• $t_{\mathrm{sat}}$—Time instant at which the core reaches the saturation state [$\mathrm{s}$].

In this state, the magnetic permeability of the core drops sharply to values close to unity, i.e., similar to the permeability of air. From the electrical circuit point of view, the transformer then behaves almost like an air-core coil, which means that its effective inductance may decrease significantly:

$$\begin{array}{c}\hfill L_d\left(i_m\left(t\right)\right)={\displaystyle \frac{N^{2}A}{l_m}}{\left.{\displaystyle \frac{dB}{dH}}\right|}_{H=\frac{N{i}_m\left(t\right)}{l_m}}\underset{H\to H_{\mathrm{sat}}}{\to }{L}_{d,\mathrm{sat}}\ll L_{d,\mathrm{lin}}.\end{array}$$

(11)

where

• $H_{\mathrm{sat}}$—Magnetic field strength at which the core enters saturation [$\mathrm{A} {\mathrm{m}}^{\mathrm{-1}}$];
• $L_{d,\mathrm{lin}}$—Differential inductance in the linear region of the magnetization curve (before saturation) [$\mathrm{H}$];
• $L_{d,\mathrm{sat}}$—Differential inductance in the saturation region of the core (after reaching $B_{\mathrm{sat}}$) [$\mathrm{H}$].

4.3. Model of the Resistive State of the Windings

The tape geometry shown in Figure 10 was used to develop the model of the inrush current phenomenon in the resistive state of the HTS transformer windings. It was assumed that a sufficient condition for the loss of superconductivity is that the inrush current exceeds the critical current of the superconductor:

$$\begin{array}{c}\hfill \left|i\left(t_c\right)\right|>i_c\end{array}$$

(12)

where

• $t_c$—Time instant at which the superconducting layer transitions from the superconducting to the resistive state [$\mathrm{s}$].

After the condition (12) is met, resistance appears in the superconducting layer, increasing rapidly with rising temperature. In the model, it was assumed that the transient state is sufficiently short for the transition of the superconductor from the superconducting to the resistive state to be considered instantaneous. It was further assumed that after the loss of superconductivity, the entire current is redirected to the metallic layers $m_1$ and $m_2$, where it splits into two current components: $i_{m1}\left(t\right)$ and $i_{m2}\left(t\right)$. The equivalent circuit of the transformer in the resistive state of the windings is shown in Figure 12.

The circuit shown in Figure 12 can be described as follows:

$$\begin{array}{cc}\hfill u_s\left(t\right)& =L_d\left(i_m\left(t\right)\right){\displaystyle \frac{d{i}_m\left(t\right)}{dt}}+R_{\mathrm{conn}}{i}_m\left(t\right)+R_{m_1}\left(T_{m_1}\left(t\right)\right)i_{m_1}\left(t\right),\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill u_s\left(t\right)& =L_d\left(i_m\left(t\right)\right){\displaystyle \frac{d{i}_m\left(t\right)}{dt}}+R_{\mathrm{conn}}{i}_m\left(t\right)+R_{m_2}\left(T_{m_2}\left(t\right)\right)i_{m_2}\left(t\right),\hfill \end{array}$$

(14)

$$\begin{array}{c}\hfill i_m\left(t\right)=i_{m_1}\left(t\right)+i_{m_2}\left(t\right).\end{array}$$

(15)

where

• $R_{m_1}\left(T_{m_1}\left(t\right)\right)$—Resistance of the upper metallic layer ($m_1$), dependent on the temperature $T_{m_1}\left(t\right)$ [$\Omega$];
• $R_{m_2}\left(T_{m_2}\left(t\right)\right)$—Resistance of the lower metallic layer ($m_2$), dependent on the temperature $T_{m_2}\left(t\right)$ [$\Omega$];
• $T_{m_1}\left(t\right),T_{m_2}\left(t\right)$—Average values of the temperature fields $T_{m_1}(y,t)$ and $T_{m_2}(y,t)$ along the y-coordinate at time t; the functions $T_{m_j}(y,t)$ ($j=1,2$) are determined from the heat conduction model (Section 4.4).

The initial conditions for the above equations, corresponding to the time instant $t_c$ when the transformer windings transition to the resistive state, are expressed as follows:

$$\begin{array}{cc}\hfill i_{m_1}\left(t_c\right)& =i_c·{\displaystyle \frac{R_{m_2}\left(T_{\mathrm{ot}}\right)}{R_{m_1}\left(T_{\mathrm{ot}}\right)+R_{m_2}\left(T_{\mathrm{ot}}\right)}},\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill i_{m_2}\left(t_c\right)& =i_c·{\displaystyle \frac{R_{m_1}\left(T_{\mathrm{ot}}\right)}{R_{m_1}\left(T_{\mathrm{ot}}\right)+R_{m_2}\left(T_{\mathrm{ot}}\right)}}.\hfill \end{array}$$

(17)

The distribution of critical current is proportionall to the resistances of the layers at the cryogenic bath temperature $T_{\mathrm{ot}}$, where

• $t_c$—Time instant at which the superconducting layer transitions from the superconducting to the resistive state [$\mathrm{s}$];
• $i_c$—Critical current of the superconductor at temperature $T_{\mathrm{sc}}$ at time $t_c$ [$\mathrm{A}$];
• $i_{m_1}\left(t_c\right)$—Initial current in the upper metallic layer $m_1$ after entering the resistive state [$\mathrm{A}$];
• $i_{m_2}\left(t_c\right)$—Initial current in the lower metallic layer $m_2$ after entering the resistive state [$\mathrm{A}$];
• $R_{m_1}\left(T_{\mathrm{ot}}\right),R_{m_2}\left(T_{\mathrm{ot}}\right)$—Electrical resistances of the upper ($m_1$) and lower ($m_2$) metallic layers at the liquid nitrogen bath temperature $T_{\mathrm{ot}}$ [$\Omega$];
• $T_{\mathrm{ot}}$—Liquid nitrogen bath temperature [$\mathrm{K}$].

In the numerical modeling of the resistive state of the superconducting tape, the temperature-dependent electrical parameters of the structural materials of the SC superconducting tape, such as the temperature coefficient of resistance and the electrical resistivity, play a crucial role. These data are essential for correctly balancing the heat loss power and accurately reproducing the heating process of the layers in the resistive state. The characteristics of these parameters are presented in Figure 13. The temperature dependencies of the material properties were developed based on reliable literature sources and reference databases commonly used in cryogenic and superconducting applications [29,30,31,32]. The obtained datasets were implemented in the model as continuous interpolation functions, ensuring smooth and physically consistent variation of material properties throughout the entire analyzed temperature range (0–300 K). The resistance of the superconducting layer and stabilizing layers plays a dual role — it determines both the local Joule losses in the resistive state and the coupling between the electrical and thermal processes. This has a direct impact on the accuracy of reproducing the dynamics of the transition from the superconducting to the normal state, as well as the heating behavior of the metallic layers.

4.4. Heat Conduction Model

In the resistive state, heat removal from the superconducting conductor is realized through two-sided convective heat exchange with the liquid nitrogen bath ($T_{\mathrm{ot}}$) at the external metallic layers of the tape: the upper $m_1$ and the lower $m_2$. In the analysis, a one-dimensional heat conduction model across the tape thickness (with a total thickness d) was adopted. For each layer, heat conduction equations were formulated, taking into account temperature-dependent material parameters. In the metallic layers $m_1$ and $m_2$, an additional term was introduced to represent the heat generation $q_{J,m}(t,y,T_m)$ resulting from Joule losses due to current flow after the loss of superconductivity in these layers:

• Upper metallic layer $m_1$, $y\in [0,d_{m_1}]$:

$$\begin{array}{c}\hfill {\rho }_{m_1}{c}_{m_1}\left(T_{m_1}\right){\displaystyle \frac{\partial T_{m_1}}{\partial t}}={\displaystyle \frac{\partial }{\partial y}}\left(k_{m_1}\left(T_{m_1}\right){\displaystyle \frac{\partial T_{m_1}}{\partial y}}\right)+q_{J,m_1}(t,y,T_{m_1}).\end{array}$$

(18)

• Superconducting layer SC, $y\in [d_{m_1},d_{m_1}+d_{SC}]$:

$$\begin{array}{c}\hfill {\rho }_{SC}{c}_{SC}\left(T_{SC}\right){\displaystyle \frac{\partial T_{SC}}{\partial t}}={\displaystyle \frac{\partial }{\partial y}}\left(k_{SC}\left(T_{SC}\right){\displaystyle \frac{\partial T_{SC}}{\partial y}}\right).\end{array}$$

(19)

• Lower metallic layer $m_2$, $y\in [d_{m_1}+d_{SC},d]$:

$$\begin{array}{c}\hfill {\rho }_{m_2}{c}_{m_2}\left(T_{m_2}\right){\displaystyle \frac{\partial T_{m_2}}{\partial t}}={\displaystyle \frac{\partial }{\partial y}}\left(k_{m_2}\left(T_{m_2}\right){\displaystyle \frac{\partial T_{m_2}}{\partial y}}\right)+q_{J,m_2}(t,y,T_{m_2}).\end{array}$$

(20)

where

• y—Coordinate along the thickness axis of the HTS tape [$\mathrm{m}$];
• d—Total thickness of the composite tape [$\mathrm{m}$];
• $d_{m_1}$—Thickness of the upper metallic layer $m_1$ [$\mathrm{m}$];
• $d_{SC}$—Thickness of the superconducting (SC) layer [$\mathrm{m}$];
• $T_{m_1}(y,t)$—Temperature distribution in the upper metallic layer $m_1$ [$\mathrm{K}$];
• $T_{SC}(y,t)$—Temperature distribution in the superconducting (SC) layer [$\mathrm{K}$];
• $T_{m_2}(y,t)$—Temperature distribution in the lower metallic layer $m_2$ [$\mathrm{K}$];
• ${\rho }_{m_1},{\rho }_{SC},{\rho }_{m_2}$—Densities of layers $m_1$, SC, and $m_2$, respectively, [$\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$];
• $c_{m_1}\left(T\right),c_{SC}\left(T\right),c_{m_2}\left(T\right)$—Temperature-dependent specific heats of the respective layers [$\mathrm{J} {\mathrm{k}\mathrm{g}}^{\mathrm{-1}} {\mathrm{K}}^{\mathrm{-1}}$];
• $k_{m_1}\left(T\right),k_{SC}\left(T\right),k_{m_2}\left(T\right)$—Temperature-dependent thermal conductivities of the respective layers [$\mathrm{W} {\mathrm{m}}^{\mathrm{-1}} {\mathrm{K}}^{\mathrm{-1}}$];
• $q_{J,m_1}(t,y,T_{m_1}),q_{J,m_2}(t,y,T_{m_2})$—Volumetric Joule heat generation in metallic layers $m_1$ and $m_2$ [$\mathrm{W}/{\mathrm{m}}^3$], produced by current flow after the loss of superconductivity,
• t—time [$\mathrm{s}$];
• $T_{\mathrm{ot}}$—Liquid nitrogen bath temperature [$\mathrm{K}$].

In the model, it is assumed that once the critical current is exceeded, the main current is redirected to the metallic layers $m_1$ and $m_2$, while the superconducting layer (SC) ceases to conduct electrical current and therefore does not generate Joule losses. For each metallic layer $m_j$ ($j=1,2$), the local volumetric power density of Joule losses is defined as

$$\begin{array}{cc}\hfill q_{J,m_1}\left(t\right)& ={\rho }_{\mathrm{el},m_1}\left(T_{m_1}\right){\left({\displaystyle \frac{i_{m_1}\left(t\right)}{A_{m_1}}}\right)}^2,\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill q_{J,m_2}\left(t\right)& ={\rho }_{\mathrm{el},m_2}\left(T_{m_2}\right){\left({\displaystyle \frac{i_{m_2}\left(t\right)}{A_{m_2}}}\right)}^2.\hfill \end{array}$$

(22)

where

• ${\rho }_{\mathrm{el},m_j}\left(T_{m_j}\right)$—Electrical resistivity of the metallic layer $m_j$ ($j=1,2$), temperature-dependent [$\Omega \mathrm{m}$];
• $A_{m_j}$—Effective cross-sectional area of the metallic layer $m_j$ [m²];
• $q_{J,m_j}\left(t\right)$—Volumetric power density of Joule losses in the metallic layer $m_j$ [W/m³].

At the interfaces between layers $m_1$–SC and SC–$m_2$, continuity of temperature and heat flux was imposed, corresponding to the conditions of energy balance. Temperature continuity requires that at the contact between two layers, the temperature must be identical on both sides:

• At the $m_1$–SC interface, for $y=d_{m_1}$:

$$\begin{array}{c}\hfill T_{m_1}(d_{m_1},t)=T_{SC}(d_{m_1},t).\end{array}$$

(23)

• At the SC–$m_2$ interface, for $y=d_{m_1}+d_{SC}$:

$$\begin{array}{c}\hfill T_{SC}(d_{m_1}+d_{SC},t)=T_{m_2}(d_{m_1}+d_{SC},t).\end{array}$$

(24)

Continuity of heat flux requires that the conducted heat flux along the y-axis is identical on both sides of the layer interface:

• At the $m_1$–SC interface, for $y=d_{m_1}$:

$$\begin{array}{c}\hfill k_{m_1}\left(T_{m_1}\right){\left.{\displaystyle \frac{\partial T_{m_1}}{\partial y}}\right|}_{y=d_{m_1}}=k_{SC}\left(T_{SC}\right){\left.{\displaystyle \frac{\partial T_{SC}}{\partial y}}\right|}_{y=d_{m_1}}.\end{array}$$

(25)

• At the SC–$m_2$ interface, for $y=d_{m_1}+d_{SC}$:

$$\begin{array}{c}\hfill k_{SC}\left(T_{SC}\right){\left.{\displaystyle \frac{\partial T_{SC}}{\partial y}}\right|}_{y=d_{m_1}+d_{SC}}=k_{m_2}\left(T_{m_2}\right){\left.{\displaystyle \frac{\partial T_{m_2}}{\partial y}}\right|}_{y=d_{m_1}+d_{SC}}.\end{array}$$

(26)

where:

• $\frac{\partial T_{m_1}}{\partial y},\frac{\partial T_{SC}}{\partial y},\frac{\partial T_{m_2}}{\partial y}$—temperature gradients along the y-axis in the respective layers [$\mathrm{K} {\mathrm{m}}^{\mathrm{-1}}$].

At the external surfaces of the superconducting tape, boundary conditions for convective heat exchange with the liquid nitrogen bath at temperature $T_{\mathrm{ot}}$ were applied.

• Upper surface of the $m_1$ layer ($y=0$):

$$\begin{array}{c}\hfill -k_{m_1}\left(T_{m_1}\right){\left.{\displaystyle \frac{\partial T_{m_1}}{\partial y}}\right|}_{y=0}=h_1\left(T_{s1},\Delta T_1\right)\left(T_{s1}-T_{\mathrm{ot}}\right),T_{s1}=T_{m_1}(0,t).\end{array}$$

(27)

• Lower surface of the $m_2$ layer ($y=d$):

$$\begin{array}{c}\hfill k_{m_2}\left(T_{m_2}\right){\left.{\displaystyle \frac{\partial T_{m_2}}{\partial y}}\right|}_{y=d}=-h_2\left(T_{s2},\Delta T_2\right)\left(T_{s2}-T_{\mathrm{ot}}\right),T_{s2}=T_{m_2}(d,t).\end{array}$$

(28)

Physically, this means that at both external boundaries (the upper and lower surfaces of the tape), heat conduction along the y-axis is balanced by convective heat exchange with the cryogenic bath. The parameters $h_1$ and $h_2$ denote effective heat transfer coefficients, which may depend on the liquid nitrogen flow conditions, local temperature gradients, and the boiling regime (nucleate or film boiling).

4.5. Recovery of the Windings to the Superconducting State

The recovery of the transformer winding to the superconducting state occurs when the temperature in the superconducting layer decreases below the critical value and the magnetizing current does not exceed the critical current of the superconductor, which is temperature-dependent. To unambiguously formulate the thermal condition, the following assumption is introduced:

$$\begin{array}{c}\hfill T_{sc}^{max}\left(t\right)=\underset{y\in [d_{m_1},d_{m_1}+d_{SC}]}{max}{T}_{sc}(y,t),\end{array}$$

(29)

where $T_{SC}^{max}\left(t\right)$ denotes the maximum value of the temperature field in the superconducting layer at time t. The condition for recovery to the superconducting state can thus be expressed as

$$\begin{array}{c}\hfill T_{sc}^{max}\left(t\right)<T_c\mathrm{oraz}\left|i_m\left(t\right)\right|\le i_c\left(T_{sc}^{max}\left(t\right)\right).\end{array}$$

(30)

where

• $T_c$—Critical temperature of the superconductor [$\mathrm{K}$];
• $i_c\left(T_{SC}^{max}\left(t\right)\right)$—Critical current of the superconductor, dependent on the maximum temperature of the SC layer [$\mathrm{A}$].

Once condition (30) is satisfied, the entire current is redirected to the superconducting layer, while the instantaneous currents in the upper metallic layer $m_1$ and the lower metallic layer $m_2$, i.e., $i_{m1}\left(t\right)$ and $i_{m2}\left(t\right)$, together with the corresponding Joule losses in the stabilizers, vanish. The system then reverts to the previously described model corresponding to the superconducting state.

5. Numerical Implementation

Numerical calculations were performed in the MATLAB R2025a environment. In modeling thermal phenomena in the multilayer superconducting tape, it is necessary to account for several material parameters that determine the dynamics of heat conduction and heat transfer in the entire system, namely: heat capacity, specific heat, thermal conductivity, heat transfer coefficient, and material density. Considering the temperature dependence of these parameters is essential for accurately reproducing thermal processes, as it enables the determination of local conditions of conduction, accumulation, and heat exchange under real operating conditions. This, in turn, allows for a more precise assessment of the risk of exceeding the critical temperature and the subsequent loss of the superconducting state. The temperature-dependent characteristics of the above parameters, implemented in the thermal model, are presented in Figure 14, Figure 15 and Figure 16. The thermal properties of the constituent materials were derived from well-established literature sources and cryogenic materials databases widely used in the modeling of superconducting systems [30,31,33,34,35,36]. These data were implemented in the numerical model as continuous interpolation functions to ensure smooth transitions between temperature regions and physically consistent behavior throughout the entire temperature range of 0–300 K.

The accuracy of the numerical model strongly depends on the precision of the adopted material parameters and the way their temperature dependence is implemented. The specific heat $c\left(T\right)$ plays a crucial role in correctly representing the thermal inertia of individual layers within the tape. Underestimation of this parameter leads to an unrealistically fast thermal response of the model, whereas overestimation may cause artificial smoothing of temperature curves and a reduction in peak values. The thermal conductivity $k\left(T\right)$ determines the rate of heat propagation along and across the tape. An inaccurate representation of this dependence results in errors when determining temperature gradients between the metallic and superconducting layers. The heat transfer coefficient $h\left(T\right)$ affects the efficiency of heat removal from the tape surface to the coolant (e.g., liquid nitrogen). It is one of the parameters with the greatest uncertainty, as it strongly depends on flow conditions, surface state, and local turbulence. An inappropriate value of this coefficient may lead to significant computational errors.

Consequently, the accuracy of reproducing thermal phenomena in the HTS tape is closely related to the quality of the material data and to the proper consideration of their temperature dependence. Therefore, in the developed model, all key parameters were implemented as continuous functions, which minimizes discretization errors and ensures physical consistency between the analyzed phenomena.

6. Simulation Results

A numerical analysis was performed for the first five inrush current pulses of the transformer. For the first two pulses, good agreement was obtained between the numerical simulation results and the experimental measurements carried out on the real transformer. The waveforms of both currents are presented in Figure 17. The differences between simulation and experiment become noticeable starting from the third pulse and gradually increase in the subsequent waveforms.

The measurement results show that only the first inrush current pulse exceeds the critical current of the SCS4050-AP tape, which is equal to 87 $\mathrm{A}$. This pulse reaches a maximum value of approximately 255 $\mathrm{A}$, corresponding to about 2.9 times the critical current. The duration of this exceedance is approximately $6.2$ $\mathrm{m}$$\mathrm{s}$. During this period, the transformer winding enters the resistive state, and the current is conducted through the metallic layers of the tape, resulting in increased Joule losses and a temporary reduction in the critical current value.

The temperature variations of the individual tape layers are shown in Figure 18. The loss of the superconducting state in the tape leads to a rapid temperature rise in the stabilizers (Ag and Cu), which reach peak values above 82 $\mathrm{K}$. The superconducting layer heats up more slowly, reaching a maximum of approximately 80 $\mathrm{K}$.

After the first pulse ends, a gradual decrease in the temperature of all layers is observed; however, the stabilizers remain at an elevated level for some time (about 78 $\mathrm{K}$ after 40 $\mathrm{m}$$\mathrm{s}$). The second inrush current pulse reaches a maximum value comparable to the critical current of the tape. Considering the reduction in the critical current with increasing temperature, this pulse may cause a short-term loss of the superconducting state and additional heating of the resistive layers.

The subsequent inrush current pulses are characterized by significantly smaller amplitudes (below 87 $\mathrm{A}$), which do not exceed the critical current. Their thermal impact is negligible, as reflected in the stabilization of the temperature waveforms of the individual tape layers.

7. Discussion

The numerical analysis and comparison with experimental results demonstrated that the proposed model accurately reproduces the dynamics of the initial inrush current pulses of the HTS transformer. The agreement obtained for the first two pulses confirms that the nonlinear properties of the core and the initial condition associated with residual magnetization were correctly accounted for in the model. However, the discrepancies observed from the third pulse onward suggest that the simplified approach does not fully capture the thermal phenomena and their influence on the electrical parameters of the tape, which become increasingly significant due to the cumulative heating effects in the stabilizers and the superconducting layer.

The analysis of the temperature distribution in the ReBCO tape layers confirms that the highest thermal load occurs in the stabilizing layers (Ag and Cu), which reach values above 82 $\mathrm{K}$. The superconducting layer heats up more slowly, reaching a maximum of approximately 80 $\mathrm{K}$ due to its specific thermal properties, while the Hastelloy substrate acts as a thermal buffer, stabilizing the temperature distribution across the entire structure. The persistence of elevated stabilizer temperatures even after the first pulse has decayed indicates a relatively slow thermal relaxation to the cryogenic bath. This phenomenon increases the risk of additional heating in the resistive layers during subsequent pulses, particularly when the critical current value is reduced as a result of prior heating of the tape.

The observation that only the first pulse significantly exceeds the critical current is consistent with the typical dynamics of the inrush current phenomenon, where the highest amplitude is associated with an unfavorable switching angle and the presence of residual flux in the core. The subsequent pulses, characterized by lower amplitudes, remain below the critical value and therefore do not introduce additional thermal disturbances.

It should be emphasized, however, that temperatures exceeding 82 $\mathrm{K}$ in the stabilizers pose a significant risk of overheating and may lead to the uncontrolled development of resistive zones. Although the ReBCO layer heats up more slowly, the combined effect of cumulative thermal processes creates a risk of reducing the critical current below the safe operating level. This may result in more frequent short-term transitions to the resistive state, even at currents close to the rated values. An additional risk factor is the relatively slow heat exchange with the cryogenic bath, which is evident in the persistence of elevated stabilizer temperatures for tens of milliseconds after the first pulse. Such conditions promote the accumulation of Joule losses during subsequent current surges, which, in the long term, may contribute to the degradation of HTS tape parameters or even cause permanent damage.

8. Conclusions

The combined experimental investigations and numerical simulations confirmed that the proposed model accurately reproduces the dynamics of the initial inrush current pulses in the HTS transformer. The close agreement between simulation and experimental results for the first waveforms indicates that the nonlinear properties of the magnetic core and the initial conditions related to magnetic remanence were appropriately accounted for. In contrast, the discrepancies observed in subsequent pulses suggest a significant impact of thermal phenomena, which were not fully represented in the simplified model. These findings highlight the necessity of extending the model to incorporate detailed electromagnetic–thermal coupling in order to achieve a comprehensive description of the processes occurring in superconducting transformers.

The analysis of the temperature distribution within the HTS tape layers revealed that the metallic stabilizers experience the highest thermal loads, while the superconducting layer exhibits slower heating and the substrate functions as a thermal buffer that stabilizes the system. The observed slow thermal relaxation following the current pulse indicates the potential for cumulative heating effects during subsequent surges, which constitutes a considerable risk factor for the operational stability of the transformer. The most critical operational threat is associated with the first inrush current pulse, during which the winding may temporarily transition into a resistive state, with current conduction transferred to the stabilizing layers.

In the context of ensuring the long-term operational stability of HTS transformers, efficient cooling of superconducting tapes is of paramount importance. This is typically achieved through direct immersion in a liquid nitrogen bath, which, owing to its high heat capacity and nucleate boiling mechanism, provides effective heat removal. Nevertheless, the inherently limited thermal conductivity of LN² highlights the necessity of implementing additional supporting measures. In this regard, nanomaterials with exceptionally high thermal conductivity, such as graphene, carbon nanotubes, Cu–CNT composites, and boron nitride-based insulations, offer considerable promise. These materials can serve as supplementary heat transport pathways, thereby reducing the overall thermal resistance of the system, mitigating temperature gradients within the windings, and accelerating thermal relaxation. The use of graphene coatings, Cu–CNT stabilizers, and BN-based insulations not only enhances the safety margin against resistive zone propagation but also improves the thermo-mechanical compatibility of the winding components.

In summary, the results of both experimental investigations and numerical simulations confirm the necessity of incorporating electromagnetic–thermal coupling into HTS transformer models and emphasize the potential benefits of applying nanomaterials to enhance their thermal stability. Extending existing models to account for the specific properties of nanomaterials represents an important step toward the development of a new generation of HTS transformer prototypes with improved reliability and cooling efficiency.

The main contributions of this study can be summarized as follows:

• Development and validation of a numerical model of the HTS transformer;
• Demonstration of close agreement between simulation results and experimental measurements for the initial inrush current pulses;
• Identification of thermal mechanisms within the HTS tape structure and their influence on winding stability;
• Indication of the highest operational risk associated with the first inrush current pulse;
• Proposal of employing nanomaterials with high thermal conductivity as a solution to enhance cooling efficiency and thermal stability of HTS transformers.

9. Future Prospects of Graphene and CNT Coatings in HTS Transformer Design

One of the key conclusions presented in this study is the necessity to improve the efficiency of heat removal from superconducting tapes in HTS transformers. The limited thermal conductivity of liquid nitrogen and the potential accumulation of heat within the stabilizing layers indicate the need for materials with exceptionally high thermal conductivity. In this context, carbon-based nanomaterials such as graphene and carbon nanotubes (CNTs) gain particular importance, as their unique physicochemical properties can significantly enhance heat transfer efficiency under both boiling and convective cooling conditions. The application of graphene coatings on the surfaces of HTS tapes in direct contact with the cryogenic coolant may, therefore, represent one of the most promising approaches for designing next-generation superconducting transformers with improved thermal stability.

The mathematical model of the HTS transformer developed in this work was designed, among other purposes, to enable further research on the use of superconducting tapes coated with CNT or graphene layers, aimed at improving thermal conductivity and mitigating local overheating during inrush current conditions. This model provides a foundation for future analysis of the influence of nanostructured coatings on temperature distribution, cooling efficiency, and the long-term thermal stability of HTS transformers. The phenomenon of thermal superconductivity in nanomaterials, particularly in carbon nanotubes (CNTs), was initially predicted in theoretical models and later confirmed experimentally. Theoretical investigations continue to provide a crucial framework for analyzing heat transport in high-conductivity materials [37,38,39,40,41]. In recent years, experimental studies have demonstrated that coatings based on graphene and its derivatives can significantly enhance heat transfer efficiency during boiling. Graphene oxide (GO) layers deposited on copper substrates via electrophoretic deposition or hydrothermal methods increase the density of active nucleation sites, improve the boiling heat transfer coefficient (BHTC), and raise the critical heat flux (CHF) by several tens of percent compared with unmodified surfaces [42]. More recent investigations have revealed that hybrid coatings combining CNTs and GO exhibit superior wettability and thermal stability, resulting in further improvements in boiling performance [43]. Additionally, graphene coatings applied within microchannels have been shown to intensify flow boiling by stabilizing bubble dynamics and reducing the frequency of dryout phenomena [44]. The latest review highlights that graphene-based coatings and their derivatives represent one of the most effective strategies for enhancing heat transfer in boiling systems, with properties that make them particularly attractive for cryogenic and superconducting applications [45].