Assessment of Non-Linear Modeling of Ladle Furnace Transformer Using Finite Element Analysis

Abstract

This paper assesses a non-linear model of a three-phase Ladle Furnace Transformer, on slow front transients under no-load conditions. The model is designed to maintain accuracy and reduce complexity in estimating equivalent circuit parameters using three methods: analytical calculations, finite element analysis, and test measurement. The results reveal that analytical and finite element methods show discrepancies lower than 1%. Tests measurement, on the other hand, shows discrepancies higher than 5%, when compared to ones obtained from analytical and finite elements methods. Such discrepancy is particularly high in the estimation of leakage inductances and capacitances, and it is attributed mainly to differences between the transformer design and its actual assembly. Additionally, there are inherent inaccuracies in test procedures and instrumentation errors. The proposed model does not require difficult-to-obtain parameters and incorporates the non-linearity of magnetizing inductance, contributing to more accurate simulations. This simplified model is suitable for analyzing slow front transients and can be integrated into future studies addressing vacuum circuit breaker switching in electric arc furnace power systems, contributing to performance improvements in industrial applications. Additionally, the methodology for parameter determination can be applied to conventional power transformers, highlighting its versatility.

1. Introduction

Power transformers are critical devices that connect energy suppliers to consumers [1]. Their failure can lead to severe operational disruptions and substantial costs associated with service interruptions and equipment replacements [1,2]. Literature indicates that voltage regulators, windings, and bushings are the main components susceptible to damage in power transformers, accounting for 50% of total failures. Furthermore, dielectric failures are significant contributors to these incidents [2,3]. The main causes of such failures are electrical disturbances, such as switching overvoltage, overvoltage, line flashovers, lightning, and insulation failures [2,4].

Different types of power transformers have been designed based on their specific applications [5]. Electric Arc Furnace (EAF) and Ladle Furnace (LF) transformers are vital in steel mill electric systems, which are responsible for melting and refining metal materials to produce steel. Failure of these devices leads to high repair or replacement costs and significant revenue losses during operational interruptions [6,7]. Moreover, they operate under more severe conditions than conventional power transformers due to the non-linear nature of the load and the disturbances they encounter, including switching transients [4,8,9,10]. These transient waveforms can repeatedly affect the transformer windings, deteriorating insulation and potentially leading to failure, due to cumulative dielectric damage [11].

To protect three-phase furnace transformers from damage caused by switching transients, it is essential to develop a specific and straightforward model for investigating these phenomena. This model will be used in future simulations aimed at creating effective solutions.

Transformer modeling using equivalent circuits has matured over the years and is widely employed in practice [5,12,13]. Conventional linear models use simplified equivalent circuits featuring linear inductances and resistances to represent magnetization, leakage flux and losses within the transformer [5]. These impedances are typically quantified through test measurement. While these models provide a basic understanding of transformer operation, they do not capture the non-linearities and dynamic responses. Despite the inherent limitations due to their linear nature, these models continue to hold significance in the recent literature [14,15,16,17]. Mbetmi (2024) use conventional linear models as foundational frameworks for developing more sophisticated models aimed at studying failure mechanisms in transformers [14].

Power transformer modeling, particularly considering the frequency range, has been the subject of extensive research [18,19,20,21]. Each frequency range requires distinct models, which are composed of numerous parameters that can be difficult to determine. Despite advancements in transformer modeling for a wide frequency range, there remains a gap in the literature regarding the implementation of a simple model for slow front transient. There is a challenge in accurately obtaining the equivalent circuit parameters. The difficulty involved in determining these parameters arises from the need for the transformer detailed design documentation. Such limitations may reduce the precision in computing various switching transients, resulting in inaccuracies in EAF and LF three-phase transformer simulations. Therefore, it is essential to determine equivalent circuit parameters and develop a simple model that incorporates non-linearities and other important parameters.

In this paper, a simplified non-linear model is presented and applied to a three-phase Ladle Furnace Transformer, specifically designed to address slow front transients during no-load conditions. The equivalent circuit parameters are derived through three different methods: analytical calculation, finite elements analysis (FEA), and test measurement. The results demonstrate that the parameters with the greatest discrepancies among the three calculation methods do not affect the system’s response to slow front transients. Additionally, the importance of accurately obtaining the non-linear magnetizing inductance curve is highlighted in this study. The proposed simplified model yields results comparable to the comprehensive model recommended by CIGRE [18,19,20,21]. The main contribution of this work is the development of a simple model suitable for slow front transient analysis, which can be integrated into future studies of vacuum circuit breaker switching in electric arc furnace power systems.

In Section 2, a brief description of the studied equipment is made. Section 3 presents calculations of transformer parameters based on analytical calculation, finite elements analysis, and test measurement. In Section 4 comparisons are presented from the results obtained previously and then a simplified non-linear transformer model suitable for the frequency range under study is proposed. The conclusions are presented in Section 5.

2. Proposed Model

According to CIGRE, an exact representation of a transformer is challenging. Therefore, it proposes an appropriate model for each of the frequency ranges shown in

Table 1. Origin and Frequency Ranges of Transients [18].

Origin	Frequency Ranges	Type of Transient
Ferroresonance	0.1 Hz to 1 kHz	Low frequency
Load rejection	0.1 Hz to 3 kHz	Low frequency
Fault clearing	50 Hz to 3 kHz	Low frequency
Line Switching	50 Hz to 20 kHz	Slow front
Lightning Overvoltage	10 kHz to 3 MHz	Fast front
Switching in GIS	100 kHz to 50 MHz	Very fast front

[18].

The simplest model, here identified as the conventional linear model of any transformer, comprises basic impedance as presented in Figure 1 [5,12,13]. This model is normally used for frequencies up to 3 kHz [19]. However, it does not include capacitances and non-linear parameters, and also it does not account for core saturation effects. As a result, this model has limitations and does not accurately represent the transformer’s response during low frequency ranges.

In the field of power transformers, the challenging of furnace transformers comprises repetitive transient events. Overvoltage occurs several times each day due to switching events in high inductance power systems, controlled by vacuum circuit breakers. These switching events happen when the arc furnace electrodes are raised, leaving the transformer unloaded [4,8,9,10]. Figure 2 shows this system modeled using Matlab/Simulink R2016a [22].

The switching events occur within the frequency range of 50 Hz to 20 kHz, and therefore can be characterized as slow fronts events, as outlined in Table 1. Therefore, the transformer model must effectively capture these phenomena by incorporating relevant parameters [18,19], as well as important construction aspects intrinsic to furnace transformers. It will be used by the CIGRE Group II model, which is suitable for frequencies ranging from 50 Hz to 20 kHz, as shown in Figure 3 [18,19,20,21]. While this model is well-recognized for its comprehensiveness, it requires knowledge of various equivalent circuit parameters to implement it in simulation software. These parameters can be determined according to the unique characteristics of the equipment under investigation.

Table 2. Information of LF Transformer.

Variable	Value
Rated power	33 MVA
Rated frequency	60 Hz
Winding connection	Yd1
Primary voltage	69 kV
Secondary voltages	437…329 V (13 TAPs)
No-load current at high voltage	0.83 ARMS at 13° TAP
No-load losses	33 kW at 13° TAP
Core specification	H110-27 oriented grain silicon steel (Aperam) [23]
Rated operating magnetic flux density	1.67 T
Core cross-section area	0.24522 m2

outlines significant transformer characteristics that are used to quantify the parameters. The H-B saturation curve is depicted from Aperam H110-27 oriented grain silicon steel [23]. Figure 4 provides a front view of the transformer core. Figure 5 depicts the top view of the core and windings. These figures intricately detail core and winding dimensions that are critical for transformer modeling employing finite element analysis, as FEMM 4.2 software [24].

3. Methodology for the Calculation of Parameters

To define LF transformer parameter, three different methods are used: analytical calculations, finite elements analysis (FEA), and test measurement. The first method involves calculating the parameters per phase of the transformer’s equivalent circuit through analytical calculations. The second method requires additional information on transformer dimensions, which is provided by the designer’s specifications. The last method employed involves tests conducted at the factory, including assessments of no-load loss, load losses, insulation resistance, and power factor [25,26]. The calculations outlined in this paper will be conducted at the higher voltage level (TAP13).

3.1. Analytical Calculation of Impedances

3.1.1. Leakage Impedances

The magnitude of the leakage impedance is influenced by the shape, size, and positioning of the coils. The primary-referred leakage inductance is determined by Equation (1). The vacuum permeability is represented by µ₀, D_m denotes the average diameter of the insulation between the windings, and N_HV refer to the quantities of turns in high voltage (HV) winding. All other variables are depicted in Figure 6 [27,28].

$$L_{SH}=\left(1-{\displaystyle \frac{c}{\pi ·h}}\right)·{\displaystyle \frac{{\mu }_0·\pi ·D_m·N_{HV}^2}{h}}·\left({\displaystyle \frac{a_2+a_3}{3}}+g_2\right)·{10}^{-4} [H]$$

(1)

3.1.2. Ohmic Resistance

The ohmic resistances at 75 °C can be analytically calculated according to Equations (2) and (3). This equation considers the total length of the coil and the approximation of the resistance model. The variable $\rho$ represents the electrical resistivity of copper at 75 °C (0.020967 Ω·mm²/m). The variables l_LV and l_HV denote the average lengths of the low voltage (LV) and high voltage (HV) conductors. N_LV and N_HV refer to the quantities of turns in LV and HV windings. S_LV and S_HV represent the gauges of LV and HV conductors [28]. To enhance precision, dimensions of 0.8 mm were omitted due to the absence of sharp corners in the wires [29].

$$R_{LV}={\displaystyle \frac{\rho ·l_{LV}·N_{LV}}{S_{LV}}}[\mathsf{\Omega }]$$

(2)

$$R_{HV}={\displaystyle \frac{\rho ·l_{HV}·N_{HV}}{S_{HV}}}[\mathsf{\Omega }]$$

(3)

3.1.3. Iron Loss Resistance

The relative core losses resistance can be determined by referencing to the magnetic losses data (W/kg) and magnetic induction B provided in the silicon steel supplier’s catalog, as shown in Figure 7 [23]. Equation (4) is used to calculate the HV resistance per phase (R_m). The variable V represents the voltage and P_iron denotes the iron loss.

$$R_m={\displaystyle \frac{V^2}{\raisebox{1ex}{$P_{iron}$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}} [\mathsf{\Omega }]$$

(4)

3.1.4. Magnetizing Impedance

The magnetizing inductance L_m varies in accordance with the core saturation curve presented in Figure 4 and can be computed using Equation (5). Variable N represents the ratio of N_HV to N_LV, µ_r is the relative permeability derived from core saturation curve, A is the core’s cross-sectional area, and l symbolizes the magnetic path length [27,28].

$$L_m={\displaystyle \frac{N^2·{\mu }_r·{\mu }_0·A}{l}} [H]$$

(5)

The average magnetic path for each transformer phase is described in Equations (6) and (7). Figure 2 shows the variables used in these equations. Due to the smaller average magnetic path of phase B compared to phases A and C, the magnetization current of phase B is reduced, resulting in a higher magnetization inductance. The magnetization inductance curve for each transformer phase can be calculated using Equation (5) at Matlab [22], as illustrated in Figure 8.

$$l_{a\_TOTAL}=l_{c\_TOTAL}=l_a+{\displaystyle \frac{l_b·l_c}{l_b+l_c}}=\left(h+2·l\right)+\left[{\displaystyle \frac{h·\left(h+2·l\right)}{h+\left(h+2·l\right)}}\right]$$

(6)

$$l_{b\_TOTAL}=l_b+{\displaystyle \frac{l_a·l_c}{l_a+l_c}}=h+\left[{\displaystyle \frac{\left(h+2·l\right)·\left(h+2·l\right)}{\left(h+2·l\right)+\left(h+2·l\right)}}\right]$$

(7)

3.1.5. Capacitance

Capacitances can be analytically calculated according to Equation (8) [27,28]. It is crucial to consider during the calculations the coil high (h_coil) and equivalent radius of each winding (r_{eq_LV} and r_{eq_HV}), a combination of paper and oil between coils, along with the electrical permittivity of each material (ε_oil = 2.2 and ε_paper = 4.4). The high voltage–ground (HV-G) capacitance in TAP 13 can be determined similarly by factoring in the distance between the HV and REG windings, given that the latter coil is grounded. The low voltage–ground (LV-G) capacitance calculation involves summing the capacitance between LV and HV (with HV now grounded) and LV and tank, as there are two capacitances in parallel to ground.

$$C_{HV-LV}=2·\pi ·{\epsilon }_0·{\epsilon }_{eq\_material}·{\displaystyle \frac{h_{coil}}{ln\left(\frac{r_{eq\_LV}}{r_{eq\_HV}}\right)}} [F]$$

(8)

3.2. Transformer Modeling Using Finite Element Analysis

Finite element analysis enables the modeling of a transformer by computer simulating its electromagnetic behavior and physical properties. One commonly used software for this purpose is FEMM (Finite Element Method Magnetics) [24], which is limited to simulations in two dimensions. Adjustments are necessary to account for variations in the core dimensions along the transformer’s depth to approximate its original three-dimensional shape. Notably, irregularities such as protrusions and cooling channels were addressed to prevent discrepancies between real and modeled section areas. Adjustments were made to column and crown widths to preserve the core’s cross-sectional area and the transformer’s average magnetic path. Moreover, design adjustments were unique to each parameter.

3.2.1. Leakage Impedances

Leakage impedance, which is associated with the air volume between windings, required adaptations, when compared to the original transformer design. The width of the limb and crown remained at the design dimensions of 580 mm to preserve the area with air inside the core as originally planned, while the depth was adjusted to 442 mm.

To calculate the leakage impedance per phase, a no-load current I_{A_HV} of 0.9 A is applied in phase A of the HV winding. It reflects the transformer’s nominal operating point (B ~ 1.67 T). All other currents in phases B and C of HV winding and the three phases of LV winding are null.

The leakage inductance L_SH can be calculated using Equation (9), based on the method proposed by Saraiva (2010). This work employs a no-load condition with single-phase energization [30]. This approach integrates the magnetic field energy (MFE) stored only in the air and windings, excluding the core.

$$L_{SH}={\displaystyle \frac{2·MFE}{I_{A_{HV}}^2}}\left[H\right]$$

(9)

While short-circuit simulations may offer a direct representation of the transformer’s operational conditions, the choice of the no-load condition in this study is justified as it allows for the consideration of the effects of magnetic saturation in the leakage inductance. A planar representation of magnetostatic simulation developed in the software FEMM [24] is employed for this purpose, as illustrated in Figure 9. The colors within the transformer core illustrate the gradient of magnetic flux density (B), as indicated by the color scale shown on the right. This visualization helps identify areas of higher and lower magnetic flux within the core. The designated green areas, including the air regions and windings, display the integrated magnetic field energy (MFE) value.

3.2.2. Ohmic Resistance

Due to the resistance varying with wire length, modifications were incorporated in the FEMM model, which operates in 2D and focuses on planar calculations while overlooking the coil head. To address this limitation, an equivalent depth was applied in the FEMM model, corresponding to half the total cable length and accounting for the cable’s path from entry to exit. This resulted in a depth of 1299 mm for HV and 1547 mm for LV. The column width did not influence this computation.

The ohmic resistance values remained constant across the transformer’s operational states. To investigate this aspect, a current close to its nominal no-load values of 1 A was applied to HV phase A in one simulation, while a current corresponding to the transformation ratio (100 A) was applied in LV phase A in another, with currents for phases B and C of both HV and LV being set to zero. FEMM simulations are depicted in Figure 10 and Figure 11. The value displayed in the results box indicates circuit properties for the high-voltage and low-voltage sides, respectively. The resistance value is represented as Voltage/Current.

Additionally, copper’s resistivity varies with temperature (α²⁰ °C = 3.93 × 10⁻³C⁻¹), impacting the ohmic resistance value. Equations (10) and (11) are utilized to calculate the resistance and then convert it to 75 °C.

$$R={\displaystyle \frac{\rho ·l·N}{S}}$$

(10)

$$R_{75°C}=R_{20°C}·\left[1+{\alpha }_{20°C}·\left(∆T\right)\right]=R_{20 °C}·1216$$

(11)

3.2.3. Magnetizing Inductance

The magnetizing inductance L_m per phase of the transformer was determined under the same operating conditions as the test measurement, at a single operating point. This involved applying a current I_{A_HV} of 0.712 A to phase A of the HV winding while keeping the other phases null. It is important to note that magnetizing inductance is not a constant value; rather, it forms a curve that depends on the saturation of the core.

The core depth and width used in this scenario were 567 mm and 432.5 mm, respectively. The distance between the windings and the core was preserved to maintain unaltered leakage effects.

The calculation of magnetizing inductance, excluding the portion related to leakage inductance (which is negligible during no-load operation), is detailed in (12). Flux linkage λ is obtained in circuit properties result box in FEMM simulation, as illustrated in Figure 12.

$$L_m={\displaystyle \frac{\lambda }{I_{A HV}}}\left[H\right]$$

(12)

3.2.4. Capacitance

The capacitance values between the HV-LV, HV-G, and LV-G windings were determined using the Electrostatic mode of FEMM. A depth of 1460 mm, a Problem Type of Planar, and relative electrical permittivity values of ε_oil = 2.2, ε_paper = 4.4, ε_coils = 1 and εcore = 1 were employed. The geometric model was adjusted, providing a top view representation as shown in Figure 13. Capacitance values were calculated by integrating the stored energy E in the green region. By applying voltage V of 10,000 V to phase A of the HV winding in pink, the capacitance values between HV-G per phase were determined by Equation (13). Discrepancies between these results and theoretical calculations arise from the latter’s simplifications, neglecting edge effects.

$$C={\displaystyle \frac{2·E}{V^2}}$$

(13)

3.3. Tests Parameters Measurement

This section presents the methodology for determining the equivalent circuit parameters of the transformer from a series of test measurement. This method aligns with the practical application of the transformer [25,26]. Figure 14 illustrates the transformer that was tested in the factory. Subsequent subsections will detail each test, outlining the setups and data acquisition processes.

3.3.1. Load Loss Test

During testing with the LV winding short-circuited, a voltage is applied to the HV winding until it reaches the nominal current. This procedure ensures that the current flowing through the LV winding matches its nominal full load value [25,26].

The measured power during this test represents the losses in the winding, which consist of ohmic losses, leakage, and stray losses. The ohmic loss component P_r is determined using the HV resistance R_HV and LV resistance R_LV values obtained from electrical resistance tests on the windings, adjusted to 75 °C, as shown in Equation (14). Additional losses P_a were accounted for as the sum of leakage losses and stray losses, as these components were challenging to separate during the test measurements. Subsequently, the short circuit losses P_sc were calculated by Equation (15).

$$P_r=\left(I_{HV}^2·R_{HV}\right)+\left(I_{LV}^2·R_{LV}\right)\prime$$

(14)

$$P_{sc}=P_r+P_a$$

(15)

Assuming a minimal magnetizing current, the leakage inductances per phase X_SH were computed according to Equation (16), utilizing the turns in the high voltage N_HV and low voltage N_LV windings, along with the measured current I_SC and voltage V_SC from the short circuit test. The measured values for TAP 13 are detailed in

Table 3. LF Transformer Parameters Obtained on Load Loss Test.

Measured Values During Test
VSC_1Ø (V)	4327
ISC (A)	292.86
Pr (W)	151,051
Pa (W)	22,660
Psc (W)	173,711
Calculated Values
NHV/NLV	91
RHV (Ω)	0.197
RLV (Ω)	0.00004
LHV (H)	0.01131
LLV (H)	1.3661 × 10−6

.

$$X_{SH}=X_{HV}+{\left({\displaystyle \frac{N_{HV}}{N_{LV}}}\right)}^2·X_{LV}={\displaystyle \frac{\raisebox{1ex}{$Q_{a\_3\varnothing }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}{I_{sc\_1\varnothing }^2}}={\displaystyle \frac{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.·\sqrt{{\left(\sqrt{3}·V_{sc\_3\varnothing }·I_{sc\_3\varnothing }\right)}^2-P_{a\_3\varnothing }^2}}{I_{sc\_1\varnothing }^2}}$$

(16)

3.3.2. No Load Loss Test

To conduct the no-load loss test, the nominal RMS voltage is applied at LV within a maximum allowable deviation of 3, in accordance with technical standards. The HV winding remains open [25,26]. Subsequently, the no-load power and the RMS no-load currents in the three phases are measured.

Considering a negligible voltage drop in the small series impedance due to the significantly lower no-load current compared to the nominal value (I₀ = 0.06% to 0.1% depending on the TAP), the iron loss resistance R_m and magnetizing reactance X_m per phase can be computed by Equations (17) and (18). Here, I₀ represents the average of the RMS current measurements per phase in the three phases, V_{0_1Ø} denotes the single-phase voltage, and P₀ represents the sum of the corrected no-load power measured in the three phases using the three-wattmeter method. Since the measurements were taken on the LV side, these impedances need to be referred to the HV winding. The measured values obtained from tests conducted on TAP 13 are succinctly summarized in

Table 4. LF Transformer Parameters Obtained on No Load Loss Test.

Measured Values During Test
V0_1Ø (V)	438
I0 (A)	50.87
P0 (W)	28,942
Calculated Values
NHV/ NLV	91
Rm_HV (Ω)	164,673.5
Lm_HV (H)	209.8

.

$$R_m={\displaystyle \frac{{V_{0\_1\varnothing }}^2}{\raisebox{1ex}{$P_0$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}} [\mathsf{\Omega }]$$

(17)

$$X_m={\displaystyle \frac{{V_{0\_1\varnothing }}^2}{Q_{0\_1\varnothing }}}={\displaystyle \frac{V_0^2}{\sqrt{{\left(V_0·\raisebox{1ex}{$I_0$}\!\left/ \!\raisebox{-1ex}{$\surd 3$}\right.\right)}^2-{\left(\raisebox{1ex}{$P_0$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\right)}^2}}} [\mathsf{\Omega }]$$

(18)

3.3.3. Power Factor Test

According to IEEE standards, the insulation power factor test is useful for identifying potential insulation issues. However, it is questionable for accurately determining the insulation power factor and, consequently, capacitances. The test reveals substantial variations in the insulation power factor with temperature, which can be erratic and difficult to correct [26].

This standard indicates that to determine the winding capacitance, measurement tests shall be conducted by applying 10 kV, 60 Hz voltage to the short-circuited three HV phases. The three LV phases were also short-circuited, but without voltage application. Capacitance values were measured for each phase, and to obtain a single-phase parameter, the total capacitance value was divided by 3.

4. Results and Discussion

Table 5. Comparison of LF Transformer Parameter.

	Tests Measurement	Analytical Calculation	Finite Element Analysis (FEA)
RHV (Ω)	0.197	0.1927	0.1928
RLV (Ω)	0.00004	3.929 × 10−5	3.93 × 10−5
LHV (H)	0.0113	0.0104	0.0103
LLV (H)	1.36 × 10−6	1.26 × 10−6	1.25 × 10−6
Rm_HV (Ω)	164,673.5	175,670	−
CHV_LV (F)	2.75 × 10−9	2.61× 10−9	2.625 × 10−9
CHV_G (F)	2.18 × 10−9	2.30 × 10−9	2.29 × 10−9
CLV_G (F)	2.8 × 10−9	3.03 × 10−9	3.04 × 10−9

summarizes the parameter values of an LF transformer at TAP 13 obtained through the three methods discussed previously, while

Table 6. Error of LF Transformer Parameters Obtained in Different Methods.

	FEA vs. Tests Measurement	FEA vs. Analytical Calculation
RHV (Ω)	2.13%	−0.05%
RLV (Ω)	1.75%	−0.03%
LHV (H)	8.25%	0.96%
LLV (H)	8.1%	0.8%
Rm_HV (Ω)	-	-
CHV_LV (F)	−4.5%	−0.57%
CHV_G (F)	−5.05%	0.43%
CLV_G (F)	−8.57%	−0.33%

displays the discrepancies between them.

An important observation is that magnetization inductance is a non-linear parameter that varies depending on the current magnitude. Phase B exhibited higher magnetizing inductance compared to the other phases due to its shorter magnetic path, as illustrated in Figure 8. Figure 15 highlights the disparity between the magnetization inductance curves of phases A and C, comparing those obtained through analytical calculation and FEMM simulation. To generate the magnetization curve using the transformer model in FEMM, an interactive simulation was developed using Matlab and FEMM software’s [22,24]. Currents ranging from 0.02 to 13.68 A were applied to the HV winding. The available test measurement presents a limitation since it is not possible to determine this curve, but only a value at a specific operating point. In this case, the calculated L_{m_HV} is 209.8 H.

The discrepancies observed between the methods of determining the equivalent circuit parameters through analytical calculations and finite element analysis are less than 1%. However, discrepancies higher than 5% arise when comparing the values obtained from FEMM and measurement testing, particularly regarding capacitances and leakage inductances.

The differences in leakage inductance values can be attributed to winding end region simulation, variations in the winding design and assembly processes. Additionally, the additional losses P_a measured during load testing account for the sum of leakage losses and stray losses. Therefore, when calculating leakage inductance, only the leakage losses should be considered [26].

Regarding capacitance, the IEEE standard [26] indicates that the accurate determination of the insulation power factor is questionable, which in turn affects capacitance measurements. This standard highlight substantial variations in the capacitance measurements with temperature, which can be erratic and difficult to correct [26].

Figure 16 presents a comparison of switching transients of the unloaded LF transformer when modeled using finite element method and factory test measurements to obtain the equivalent circuit parameters. It is evident that there are significant differences in the results between these methods, warranting further investigation into their underlying causes. The observed differences can be attributed to the modeling approach used for magnetizing inductance. In Figure 17, it becomes clear that the primary contributor to the error in the transient voltage comparison between the FEMM simulation and factory tests is the treatment of the magnetizing inductance. The model based on factory test parameters assumes a constant value for the magnetizing inductance, which does not accurately reflect the non-linear behavior of the transformer core during transient conditions. In contrast, the FEMM-based method accounts for the non-linear variation in the magnetizing inductance with the magnetic flux density, leading to more accurate transient response predictions.

As shown in Figure 18 and Figure 19, variations in other equivalent circuit parameters, specifically capacitance between HV and LV, are also significant in the study of slow front transients during no-load transformer operation. This can be attributed to the fact that the induced voltage is related to the rate of change in magnetic flux, and these variables are closely connected to flux variations under these operating conditions. However, the capacitance between low voltage and ground, along with the leakage inductances of HV and LV—which exhibit greater errors as described in Table 5—do not significantly influence the waveform shapes. Therefore, a simplification of the three phase transformer model is proposed, as illustrated in Figure 20, to effectively address the operational conditions being studied.

To validate the simplified model, Figure 21 illustrates the results comparing this simplified model with the complete model suggested by CIGRE. Finite element analysis is used to calculate the magnetizing inductance curve and the HV to LV capacitance. Resistance loss was determined by analytical calculation. Other parameters are omitted in this model. Both models exhibit similar overvoltage results in phase A during the switching of a vacuum circuit breaker in a high-impedance system with an unloaded transformer.

The proposed simplified model overcomes the challenges associated with obtaining accurate parameters by excluding those that are difficult to measure with high accuracy. A comparative analysis illustrating the main contributions of this proposed model with the conventional model is presented in

Table 7. Main Contributions.

References	Model Type	Consideration of Non-Linearities	Ease of Parameter Acquisition	Suitability for Slow Front Transients
[14,15,16,17]	Conventional Linear Model	X	≈	X
[18,19,20,21]	CIGRE Models	√	X	√
This work	Simplified and non-linear model	√	√	√

, where the symbols X, ≈, and √ indicate that each contribution is not observed, there is little contribution, and the contribution is observed, respectively. Calculating leakage inductances and capacitances analytically or using finite elements requires detailed knowledge of winding designs, including dimensions and materials. While these parameters can be obtained through tests measuring, discrepancies are observed between the results and those obtained via analytical calculations or finite element methods. These discrepancies arise from differences between the transformer design and its actual assembly, as well as inherent inaccuracies in tests procedures and instrumentation. Furthermore, the importance of the non-linear magnetizing inductance curve in the model has been shown. Equivalent circuit parameter computation from test measurement often assumes a constant magnetizing inductance, which can introduce significant error. By focusing on the most relevant parameters, the simplified model reduces complexity while maintaining the accuracy necessary for simulating slow front transients in electric arc furnace power systems.

5. Conclusions

This paper presents a simplified non-linear model of a three-phase Ladle Furnace Transformer, specifically designed to address slow front transients of a no-load condition. Equivalent circuits electrical parameters are estimated using three different methods. The results from analytical calculations and finite element analysis exhibit similar values. However, parameters computation from test measurements shows greater discrepancies, especially on leakage inductances and capacitances. Moreover, this method is limited in its ability to obtain the magnetization inductance non-linear curve, which is crucial to this model.

To overcome the challenges associated with obtaining accurate parameters, a model has been proposed that omits parameters that have discrepancies higher than 5%, such as leakage inductances and capacitances. The calculation of these parameters, whether analytically or through finite element analysis, requires detailed knowledge of winding designs, including dimensions and materials. In contrast, parameter computation from test measurements frequently results in considerable discrepancies, complicating the equivalent circuit parameter determination process. The proposed simplified model effectively addresses these issues by considering only the crucial parameters for the study, and it aligns closely with the comprehensive model suggested by CIGRE. Moreover, this model incorporates the non-linear magnetizing inductance curve, contributing to more accurate simulations, especially for slow front transients during no-load conditions.

This research contributes by presenting a simplified non-linear model for the Ladle Furnace Transformer, developed using Simulink/Matlab simulation software. This model is particularly suitable for slow front transient analysis and can be integrated into future studies vacuum circuit breaker switching in electric arc furnace power systems, ultimately contributing to performance improvements in industrial plants. Furthermore, this study indicates that the three methods for estimating equivalent circuit parameters are valid for accurately determining the necessary parameters to assess transient performance, as long as the non-linear magnetizing inductance is considered.

Although the focus lies on LF transformers, the methodology established for parameter determination can also be applied to conventional power transformers. Future research may explore the use of the simplified non-linear transformer model to simulate electrical systems, assessing the impacts of switching transient disturbances on furnace transformers, and subsequently proposing targeted design modifications based on the findings.