# Integrating PV Systems with Non-Standard Grid Voltages

José Antonio Badri, Jordi-Roger Riba, Antoni Garcia, Santi Trujillo, Albert Marzàbal

► To cite this version:

José Antonio Badri, Jordi-Roger Riba, Antoni Garcia, Santi Trujillo, Albert Marzàbal. Integrating PV Systems with Non-Standard Grid Voltages. The 26th European Conference on Power Electronics and Applications, GDR SEEDS France & EPE Association, Mar 2025, Paris, France. 10.34746/epe2025-0018. hal-05073988

HAL Id: hal-05073988

https://utc.hal.science/hal-05073988v1

Submitted on 19 May 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Integrating PV Systems with Non-Standard Grid Voltages: The Need for Low-Induction Transformers

José Antonio Badri¹,², Jordi-Roger Riba¹, Antoni Garcia¹, Santi Trujillo², Albert Marzàbal²

¹DEPARTMENT OF ELECTRICAL ENGINEERING, UNIVERSITAT POLITÈCNICA DE CATALUNYA

Rambla Sant Nebridi 22
Terrassa, Spain

Tel.: +34 93 739 83 65

E-Mail: jose.antonio.badri@upc.edu; jordi.riba-ruiz@upc.edu; antoni.garcia@upc.edu

URL: https://mcia.upc.edu/en

²RESEARCH DEPARTMENT, SALICRU S.A.
Avinguda de la Serra, 100

Santa Maria de Palautordera, Spain

Tel.: +34 93 848 24 00

E-Mail: joseantonio.badri@salicru.com; santi.trujillo@salicru.com; albert.marzabal@salicru.com

URL: https://www.salicru.com

## Acknowledgements

This research was funded by the Generalitat de Catalunya, grant numbers 2021 DI 080 and 2021 SGR 00392. The authors are grateful to SALICRU for their technical and material support in carrying out this research.

## Keywords

«grid integration», «inrush current», «magnetic saturation», «modelling», «pv inverter», «simulation», «transformer», «design», «electrical machine».

## Abstract

Installing grid-connected photovoltaic (PV) systems is challenging in regions with non-standard grid voltages, as PV inverters are designed for standard levels. To overcome this limitation, transformers can be used, but their inrush current often trips inverter protection. This paper explores this issue, emphasizing the need for specially designed low-induction transformers.

## Introduction

As a result of global decarbonization policies, the promotion of renewable energy sources has

expanded beyond large-scale wind and solar farms to include small-scale residential users, particularly through the adoption of residential solar photovoltaic (PV) systems. Interestingly, the main driver of this development has been homeowner investment in rooftop PV systems, rather than utility-scale PV projects. Today, rooftop PV accounts for the majority of installed capacity [1].

In Spain alone, the number of residential PV installations surged from 424 in 2014 to 111,795 in 2023, with a remarkable 217,248 installations in 2022 alone [2]. This trend is echoed across the European Union, with a significant annual increase in residential PV installations [3].

Commercial residential PV inverters, whether three-phase or single-phase, are designed for standardized European distribution grid voltages of 3x400V, providing single-phase voltages of 230V. Some areas, particularly isolated residential regions, still use non-standardized grid voltages. This is due to the historical evolution of electrical grids. In the specific case of Spain, up until the 1950s, the use of a 127/220 V three-phase system was common. Later, between the 1960s and 1970s, the system transitioned to 220/380 V to meet the growing need for increased power delivery, allowing for 1.73 times more power. Finally, in the early 2000s, to standardize voltages across the European Union, the current 230/400 V system was adopted. This voltage
upgrade process was conducted similarly in other countries, such as France, Germany, Portugal, and Italy, throughout the 20th century, culminating in the harmonized grid used in Europe today. Despite this modernization, some distribution lines operating at 127/220 V still exist in Spain, primarily in very isolated rural areas. Thus, it is expected that other European countries facing similar challenges may also have remnants of outdated systems in remote regions.

While the impact of such cases is minimal compared to the total number of PV residential installations, advances in engineering have made it possible to address this issue at a highly competitive cost, without the need to update the supply line to standardized voltage levels. For PV applications, autotransformers are often preferred over traditional transformers. By sacrificing the galvanic isolation provided by standard transformers, autotransformers can deliver the same power requirements at a reduced cost. Given that European inverter standards require 3x400V, incorporating autotransformers becomes the most cost-effective solution, enabling the use of commercially available PV inverters while keeping the overall system affordable.

When a transformer is energized on its primary side, an inrush current peak occurs. This phenomenon happens regardless of whether the primary is connected to the distribution line or to the output of a solar inverter. However, residential solar inverters are not designed to handle this current peak, and as a result, overcurrent protection may be triggered.

Over the years, various methods have been proposed to mitigate these inrush currents, such as managing residual flux, using grounding resistors, and implementing series compensators [4]. In PV installations, it has been explored to use the PV system itself to create a pre-flux at the transformer, which counteracts the remanent flux and thus reduces the inrush current [5]. However, these techniques may not completely reduce the inrush current [6] and are typically used in large power installations. In the case of residential PV installations, where there is fierce competition among several manufacturers to reduce costs and be more competitive, these innovative methods to reduce inrush current are not seen.

Since solar inverters can operate in different modes (e.g., no generation, partial generation from PV panels and battery systems, or zero grid injection mode), a commercial inverter was first studied under various conditions, using a standard autotransformer designed for common use. This

autotransformer was not specifically designed for this purpose, and it is expected to cause the inverter to fail due to the activation of the overcurrent protection [7], providing valuable insight into the issues at hand. This underscores the importance of closely examining the autotransformer design to ensure the proper functioning of the inverter.

After evaluating the performance of the inverter with a standard autotransformer, we propose using a reluctance network to design an optimized autotransformer capable of handling the power inverter without causing inrush current damage. Methods based on reluctance networks present a viable alternative to finite element methods (FEM) for solving electromagnetic problems. Reluctance networks operate in the time domain and assume that the magnetic field intensity *H* and the flux density *B* are uniformly distributed within each reluctance element. Despite their simplicity, reluctance networks can be highly effective, especially when combined with hysteresis models [8].

Compared to mathematical hysteresis models without a physical basis, hysteresis models with a physical foundation, such as the Preisach and Jiles-Atherton models, are more applicable in engineering. These models more accurately describe the magnetization process under any magnetic excitation in both steady-state and transient conditions. Both the Preisach and Jiles-Atherton models use the magnetic field *H* as an independent variable, enabling the simulation of the hysteresis and saturation characteristics of ferromagnetic materials. [9], [10]

A significant advantage of the Jiles-Atherton model is its ability to represent the physical phenomenon of hysteresis through a first-order differential equation using a set of five parameters directly related to the physical properties of the ferromagnetic core.

In this study, we employ a reluctance model in which the magnetic permeability is modeled using the Jiles-Atherton approach to predict inrush currents. This model is then used to design a transformer that meets specific inrush current requirements.

## Magnetic Hysteresis

The ferromagnetic material of a transformer core exhibits nonlinear behavior, characterized by magnetic hysteresis, also known as the BH curve. The magnetic flux density (*B*) is proportional to the transformer voltage, while the magnetic field
strength (H) corresponds to the magnetizing current. The transformer core will exhibit different BH curves depending on the magnetic flux flowing through it, as shown in Fig. 1.

<figure><img src="image_2.png" /><figcaption>Fig. 1: Hysteresis loops of the core material at various magnetic flux densities.</figcaption></figure>

When the transformer is de-energized (H = 0), it remains at the remanence point (B<sub>rem</sub>) of the BH curve, which can be either positive or negative. This point represents the intrinsic magnetization level of the transformer [6]. As shown in Fig. 1, the specific induction B work point (B<sub>peak</sub>) determines the remanence value.

When a transformer is first energized, the remanence and coercive points are zero. However, after the initial energization, subsequent transformer stops and restarts will result in non-zero remanence and coercive points due to hysteresis. To return to the initial state, the core would have to be heated to the Curie temperature to remove any magnetization from the ferromagnetic domains. However, this is rarely done in industrial applications.

## Inrush Current

When a transformer is connected through the primary side with the secondary open, it behaves as a simple inductance. In steady state, the linked flux ψ in the core is in quadrature with the applied voltage, so there is a 90° phase shift between the voltage and the magnetic flux, as stated in Faraday's law,

$$e(t) = -\frac{d\psi(t)}{dt} \qquad (1)$$

$$e(t) = E_m \sin(\omega t) \qquad (2)$$

where E<sub>m</sub> is the maximum value of the steady-state electromotive force.

Consequently, when the voltage is zero, the corresponding steady-state value of the magnetic

flux should be at its maximum (either negative or positive).

In practice, when a transformer is connected, it takes a few instants for the magnetic flux to reach its steady state.

If the transformer is connected at the moment when the voltage is zero, the linked flux wave starts from the same origin as the voltage waveform. The linked flux value at the end of the first half cycle of the voltage waveform ψ' can be calculated from (1) and (2) as,

$$\psi' = \frac{E_m}{\omega} \int_{0}^{\pi} \omega \sin(\omega t) dt = \psi_m \int_{0}^{\pi} \sin(\omega t) d\omega t = 2\psi_m \qquad (3)$$

Where ψ<sub>m</sub> is the maximum value of the steady-state flux. Thus, the magnetic linked flux may be twice the nominal steady-state magnetic linked flux, causing the magnetic core to enter the saturation region. In this core-saturated state, the magnitude of the permeability could be similar to the absolute permeability μ<sub>0</sub> [11]. After the maximum steady-state flux is reached, the core becomes saturated and generating more flux requires a very high current. This high peak current drawn by the transformer primary is known as the inrush current or magnetizing inrush current.

However, taking into account the magnetic hysteresis and in particular the remanence, the maximum flux to be expected during energization is given by,

$$B_{\text{energization}} = 2B_{\text{peak}} + B_{\text{rem}} \qquad (4)$$

where B<sub>peak</sub> is the maximum or peak value of the nominal magnetic flux density. Note that if B<sub>energization</sub> < B<sub>knee-point</sub>, the transformer will always operate non-saturated, thus minimizing inrush current effects.

Based on this principle, analytical empirical approaches such as the Bertagnolli or Specht formulas have been developed to predict the inrush current [12]. However, not only the magnetic characteristics of the core affect the inrush current, but also the type of load [4].

## Core Modelling

The magnetic core of the transformer of length l (m), cross section S (m²) and magnetic permeability μ (H/m) can be modelled using a reluctance network [13].
$$ \Re = \frac{l}{\mu S} \qquad (12) $$

$$ \Im = \Re\varphi \qquad (13) $$

where $\Re$ (H⁻¹) is the reluctance, $\Im$ (A) is the magnetomotive force and $\varphi$ (Wb) is the magnetic flux.

A key aspect of a reluctance network is the definition of magnetic permeability, as it captures the intrinsic nonlinearity of ferromagnetic materials. The modeling of ferromagnetic hysteresis has evolved significantly, progressing from simple magnetization curves to advanced hysteresis models. Among these, the Jiles-Atherton model, has been applied as it is one of the most widely used and recognized approaches in the field nowadays [8].

Thus, from the transformer's supplied voltage, electromagnetic relationships can be established, allowing the reluctance network to be analyzed and solved:

$$ e(t) = v(t) - i(t)R - L\frac{di(t)}{dt} \qquad (14) $$

$$ e(t) = NS \frac{dB(t)}{dt} = N \frac{d\varphi(t)}{dt} \qquad (15) $$

$$ \Im = H(t)l = Ni(t) \qquad (16) $$

$$ B(t) = \mu H(t) \qquad (17) $$

Fig. 2 shows the reluctance circuit model of a three-phase transformer based on a previous work [8].

<figure><img src="image_3.png" /><figcaption>Fig. 2: Reluctance circuit model of a three-phase transformer.</figcaption></figure>

## Studied Case

This study examines a real-world case of an isolated, old farmhouse located in a rural area of Tarragona (Spain), which has been repurposed into a children's camp facility with plans to install a 30 kW grid-connected PV system. The supply grid operates at 3x230V, a mid-20th-century standard in Spain, with the house supporting both three-phase loads at 3x230V and single-phase

loads at 133V. Due to its isolation, the grid operator has not updated the grid voltage to current standard values, and updating the voltage of the grid is currently not an option.

This is a significant problem because commercial solar inverters for residential use are designed for a grid voltage of 3x400V (three-phase) or 230V (single-phase). Designing an inverter to work on a 3x230V grid is not feasible due to its rarity, both in terms of effort and cost. Therefore, the proposed technical solution is to provide two three-phase autotransformers (230/400V) to allow the use of existing PV inverters at 3x400V. Fig. 3 shows the proposed technical solution.

<figure><img src="image_4.png" /><figcaption>Fig. 3: Proposed technical solution to install a 3x400 V solar inverter in a 3x230 V system.</figcaption></figure>

## Experimental Measurement

Prior to the installation of the equipment in the customer's home, a series of experimental measurements were carried out in order to reproduce the real conditions in the test area (see Fig. 4).

The autotransformer analyzed is a 30 kVA YNA0 3x230V/3x400V + N 50 Hz, designed for 75/43.3 A. This autotransformer is commonly used in PV installations due to its cost-effectiveness compared to conventional transformers. The autotransformer core is made of M-235 steel with a designed induction peak value of 1 T.

The 30 kW solar inverter can manage both PV and backup power simultaneously. An electronic load (Cinergia EL50, Argentona, Barcelona, Spain) is used to simulate PV generation. The load is simulated using a three-phase resistor bank. Inrush current measurements are performed using a power analyzer (LMG670 ZES ZIMMER, accuracy 0.015% + 0.01% of range, Oberursel,
Frankfurt, Germany) and an oscilloscope (Yokogawa DLM2024, 2.5 GSamples/s and 200 MHz, Nakacho, Musashino, Tokyo, Japan).

<figure><img src="image_5.png" /><figcaption>Fig. 4: Experimental setup</figcaption></figure>

## Test 1: Autotransformer under no load conditions

To capture the inrush current of the autotransformer alone, it was supplied from the 3x400V side at nominal voltage value while the 3x230V side remained open. This setup allowed for an isolated study of the core's behavior. However, it is important to note that, according to the theory described in the aforementioned Inrush Current Section, the maximum inrush current occurs when the voltage phase angle is 0° at the connection time instant. In this study, without a phase angle controller, dozens of tests were conducted to ensure the identification of the maximum inrush current. Some of the inrush current measurements are shown in Fig. 5.

<figure><img src="image_6.png" /></figure>

a)

<figure><img src="image_7.png" /></figure>

b)

<caption>Fig. 5: Inrush current measurements with the autotransformer connected to the grid under no load conditions. a) Inrush current of 98 A. b) Inrush current of 120 A. c) Inrush current of 140 A.</caption>

c)

Of the various tests performed, the maximum measured inrush current was 140A (peak).

## Test 2: Inverter connected to the grid

The power test bench used in this study was limited to the standardized 3x400V grid voltage. To adapt the test to the real case (see Fig. 3), the inverter was supplied directly from the 3x400V grid. As shown in Fig. 6, the output of the inverter was connected to the 400V/230V three-phase autotransformer, which fed a 3x230V resistor bank. The PV panels and the Energy Storage System (ESS) supplied part of the energy demand of the resistor bank, while the rest was supplied by the grid. The load current was set to the nominal current of the inverter to test the worst case scenario.
<figure><img src="image_8.png" /><figcaption>Fig. 6: Tested solution with a pure resistive load connected to the commercial inverter through a three-phase autotransformer. The inverter was connected to the grid.</figcaption></figure>

The inrush current peak occurs in the primary side of the autotransformer, which is connected to the output of the solar inverter, potentially affecting the inverter. However, in this scenario, the PV inverter's overcurrent protection never tripped.

As in the previous section, the circuit breaker was switched on and off several times to generate several inrush current peaks at the moment of connection. Note that the expected inrush current now includes not only the magnetizing current component but also the load current. As mentioned above, the inrush current is influenced by the type of load.

As shown in Fig. 7, several tests were performed to capture the worst initial voltage phase angle and consequently the higher inrush current, which was 310A (peak).

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 7: Inrush current measurements with pure resistive load and the autotransformer connected to the output of the inverter. a) 310 A. b) 260 A</figcaption></figure>

## Test 3: Inverter not connected to the grid

This test evaluates the inverter when it is connected to the autotransformer, where a pure resistive load is supplied solely by the PV panels and battery system. This scenario simulates a situation where 100% of the energy is supplied by the PV system, with no grid supply (see Fig. 8).

<figure><img src="image_11.png" /><figcaption>Fig. 8: Tested solution with a pure resistive load connected to the commercial inverter through a three-phase autotransformer. The inverter was not connected to the grid.</figcaption></figure>

In this scenario, although the inrush current levels were similar to those observed previously in Test 2, the inverter failed due to the activation of its overcurrent protection, resulting in a shutdown of the PV system. Here, the overcurrent was clearly caused by the autotransformer inrush phenomenon.

However, for solar inverters, a sudden drop in grid voltage can also cause inrush current. This is due to the delay in the inverter's digital control
system. A sudden voltage drop in the grid can cause a significant voltage drop at the output of the L or LCL filters, which are used to attenuate the current harmonics generated by the switching of the semiconductors [7]. Obviously, different factors can lead to the same result. To accurately diagnose the cause, a thorough understanding of the system's behavior is essential.

This problem does not occur when the grid is connected because the grid absorbs the current peak, not the inverter. So, this is the worst case scenario for the inverter.

## Proposed Solution

It was observed that the inrush current from the transformer triggered the overcurrent protection of the PV inverter, as shown in the previous section. The problem was attributed to the inrush current during the energization of the load-side autotransformer.

There are two possible solutions, i.e. to develop an inverter capable of operating at 3x230V or to adapt the autotransformer to reduce its inrush current. Given the high cost of producing a unique inverter for a single case, this option is not feasible. Therefore, adapting the autotransformer to reduce its inrush current peak is the most viable solution. To ensure system reliability, the most efficient solution is to design an autotransformer to limit the inrush current.

The operating point of a transformer's magnetic induction is directly related to the maximum inrush current, as described in (4). Lowering the operating induction has several design implications: it increases the core size (volume) and the number of turns, which, in turn, requires more winding material and leads to higher winding losses. On the other hand, since core losses depend on both the volume and the induction point, it is not immediately evident whether they will increase or decrease. Therefore, when comparing two transformers with the same specifications except for their induction levels, a detailed analysis is necessary to determine which one has the better efficiency. Regarding manufacturing costs, a low-induction transformer requires more core and winding material, resulting in a higher cost compared to a transformer with a higher induction level.

Therefore, a reluctance model of the autotransformer is used to predict the maximum inrush current. By sweeping the induction operating point, it is possible to determine which operating point is compatible with the normal

operation of the PV inverter. To ensure safe operation, the maximum inrush current of the autotransformer is adjusted to the maximum nominal current of the inverter.

## The core reluctance model

Fig. 9 shows the developed MATLAB Simulink reluctance model of the autotransformer using a reluctance network.

<figure><img src="image_12.png" /><figcaption>Fig. 9: Simulink model of the autotransformer using a reluctance network.</figcaption></figure>

To validate the model, the autotransformer operating conditions were simulated according to the experimental tests performed in the previous sections. It is worth noting that the reluctance model accurately reproduced the peak inrush current, confirming that the worst case condition corresponding to an inrush current of 310 A was considered.

It is noted that the results presented in Fig. 10 are based on an autotransformer designed to operate with an induction of 1 T. To validate the reluctance model, simulations were conducted to explore how to limit the inrush current by reducing the induction of the autotransformer to 0.7 T (see Fig. 11).

<figure><img src="image_13.png" /></figure>

a)
<figure><img src="image_14.png" /><figcaption>Fig. 10: Results of the Simulink model of the autotransformer operating at 1 T. a) no load. b) purely resistive load.</figcaption></figure>

<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /><figcaption>Fig. 11: Results of the Simulink model of the autotransformer operating at 0.7 T. a)</figcaption></figure>

Autotransformer at no load. b) Autotransformer supplying a purely resistive load.

The results presented in Fig. 11 clearly show that when the autotransformer is redesigned to operate at 0.7 T, the inrush current is greatly reduced.

## Discussion on the Applicability of the solution

For a specialized transformer manufacturer, the design and production process for a custom autotransformer, like the one discussed in this work, would take less than a week. Consequently, the proposed design methodology is as straightforward as any other transformer design. However, special attention must be given to the inrush current value in this context.

In this study, the autotransformer's design parameters were manually adjusted to minimize inrush current. Once the methodology is validated, it can be extended to an optimization-based approach. This optimization method would be particularly valuable for industrial applications, where cost minimization is critical, and the inrush current becomes an additional design constraint.

Other simple solutions, such as upgrading the entire house's loads to the standardized 230/400V by installing a transformer to convert the 3x230V grid to a 3x400V line, were rejected by the customer due to the high cost of replacing all the loads. In comparison, the installation of two autotransformers is significantly more cost-effective.

Another interesting option could be installing a series inductance to reduce the inrush current. However, to achieve this, relatively large inductances may be required, and despite being feasible, this approach is rejected from a cost perspective. Adding an extra element would be more expensive than simply installing the autotransformer. What is commonly done in power electronics devices is to use the leakage inductance of the primary winding as the desired inductance, which helps reduce costs. However, the leakage inductances are far from those needed to effectively reduce the inrush current, making this idea unfeasible for this application. Despite this, the simpler solution remains to reduce the induction of the autotransformer itself.

## Conclusion

There are still areas where grid voltage differs from the standardized 3x400 V value, challenging the installation of grid-connected photovoltaic
(PV) systems. These unique conditions complicate matters because PV inverter manufacturers design their equipment for standardized voltages. It is not feasible to design and manufacture specific inverters for these unique cases, much less for residential use. The only practical solution is to use transformers at both the input and output of the PV inverter. However, because commercial PV inverters for residential applications cannot handle the inrush phenomena inherent in transformer core magnetization, not all transformers are compatible with PV inverters. By modeling the magnetic core of the transformer, this paper has shown that it is possible to predict the maximum inrush current based on the transformer characteristics and design criterion. The main conclusion of this work is the need for specially designed low induction transformers to be paired with residential PV inverters. The continuation of this work is to test a transformer with reduced inrush current that combines high quality steel laminations with appropriate core dimensions and winding design to evaluate the behavior of the PV inverter in field tests.

## References

[1] D. E. H. J. Gernaat, H.-S. De Boer, L. C. Dammeier, and D. P. Van Vuuren, “The role of residential rooftop photovoltaic in long-term energy and climate scenarios,” 2020, doi: 10.1016/j.apenergy.2020.115705.

[2] “APPA Renovables,” 2023. [Online]. Available: https://www.appa.es/.

[3] G. Martinopoulos, “Are rooftop photovoltaic systems a sustainable solution for Europe? A life cycle impact assessment and cost analysis,” 2019, doi: 10.1016/j.apenergy.2019.114035.

[4] A. A. Ahmed and H. A. Abdelsalam, “Mitigation of transformer-energizing inrush current using grid-connected photovoltaic system,” *Int. J. Electr. Power Energy Syst.*, vol. 79, pp. 312–321, 2016, doi: 10.1016/j.ijepes.2016.01.005.

[5] H. A. Abdelsalam and A. Ahmed, “Utilizing the Grid-Connected Photovoltaic System for Reducing Transformer Inrush Current,” pp. 1–6, 2014.

[6] G. de J. Martínez-Figueroa, S. Bogarra, and F. Córcoles, “Smart Switching in Single-Phase Grid-Connected Photovoltaic Power Systems for Inrush Current Elimination,” *Energies*, vol. 16, no. 20, 2023, doi: 10.3390/en16207211.

[7] Z. Li, R. Zhao, Z. Xin, J. M. Guerrero, M. Savaghebi, and P. Li, “Inrush Transient Current analysis and suppression of photovoltaic grid-connected inverters during voltage sag,” *Conf. Proc. - IEEE Appl. Power Electron. Conf. Expo. - APEC*, vol. 2016-May, pp. 3697–3703, 2016, doi: 10.1109/APEC.2016.7468402.

[8] J. A. Badri, J. R. Riba, A. Garcia, S. Trujillo, and A. Marzàbal, “Transformer modelling considering power losses using an inverse Jiles-Atherton approach,” *Int. J. Electr. Power Energy Syst.*, vol. 154, no. August, 2023, doi: 10.1016/j.ijepes.2023.109461.

[9] F. Preisach, “Über die magnetische Nachwirkung,” *Zeitschrift für Phys.*, vol. 94, no. 5, pp. 277–302, 1935, doi: 10.1007/BF01349418.

[10] D. C. Jiles and D. L. Atherton, “Theory of ferromagnetic hysteresis,” *J. Magn. Magn. Mater.*, vol. 61, no. 1–2, pp. 48–60, Sep. 1986, doi: 10.1016/0304-8853(86)90066-1.

[11] C.-K. Cheng, T.-J. Liang, J.-F. Chen, S.-D. Chen, and W.-H. Yang, “Novel approach to reducing the inrush current of a power transformer,” *IEE Proc. - Electr. Power Appl.*, vol. 151, no. 3, p. 289, 2004, doi: 10.1049/ip-epa:20040223.

[12] N. Chiesa, “Power Transformer Modeling for Inrush Current Calculation,” Norwegian University of Science and Technology, 2010.

[13] J.-R. Ruiz and A. G. Espinosa, “A novel parametric model for AC contactors,” 2008, doi: 10.1109/TMAG.2008.2000544.