1. Introduction
Power transformer is a critical component in electricity distribution systems. Nowadays, the medium-frequency (MF) transformer has become an important part of many power conversion systems. The high frequency reduces the volume of magnetics but risks increasing the core and winding loss densities, which cause deteriorated thermal performance and potentially lead to challenges, such as insulation damage, shortened lifespan, and even malfunctions or explosions [
1]. In this regard, the thermal limit of temperature rise on structural parts is the major concern restricting the design of MF transformer [
2].
Due to the generated heat, temperature will increase largely if the transformer is left thermally isolated. Depending on the amount of heat to be dissipated, different cooling systems are used to maintain the temperature at an expected level. Amongst others, the oil natural air natural (ONAN) approach cools the device internally and externally with the natural movement of oil and air driven by buoyancy force [
3]. Inside the oil tank, the natural circulation of oil acts as both an electrical insulator and a medium for heat transfer. ONAN design manifests itself with simple structure without active driving devices. While the oil enhances the insulation, the temperature rise is the main concern for ONAN transformers.
The prediction of temperature rise of transformer typically includes the electromagnetic modelling for loss determination and thermal-fluid analysis [
4]. Existing techniques for electromagnetic and thermal-fluid analysis can be broadly categorized into four groups, i.e. experimental method, artificial intelligence (AI), analytical method, and numerical simulation [
5]. The direct measurement is straightforward and accurate but time-consuming and costly. Hence, its application is mostly limited to model validation rather than design optimization. The AI method, such as fuzzy information granulation combined with wavelet neural network [
6], generalizes the nonlinearity between design/operating parameters and temperature without requiring the physical knowledge of system. However, the intrinsic drawbacks, such as extensive training, overfitting, and low adaptiveness remain unsolved for real applications [
7,
8,
9].
The analytical method has been widely used to calculate the AC effect and system losses due to the simplicity. Chen et al. [
10] proposed an analytical model to calculate the core loss under both sinusoidal and non-sinusoidal excitations. An equivalent circuit model using impedance networks was proposed in [
11] to simulate the skin and proximity effect of planar transformers. An improved analytical model incorporating material characteristics and geometrical structures was proposed in [
12] to calculate the eddy current losses in each winding of transformer. Based on the loss calculation, varieties of thermal models have been developed to predict the temperature response of transformers. Physical models use simple first-order differential equations to describe the thermal behavior and top-oil temperature. This principle was adopted by both the IEC [
13] and the IEEE [
14] loading guides. Some thermal models extended from loading guides have also been investigated by using either constant or variant impedances [
15,
16,
17,
18]. Additional efforts have also been made to address the dependence of thermal impedance of key components to environmental factors, such as temperature and moisture [
19]. It should be noted that the accurate description of fluid circulation, which determines the heat dissipation condition to be essential to the thermal analysis of oil-immersed power transformers. In this regard, the thermal-hydraulic network model has been widely used to characterize the circulation of oil and the temperature rise [
20,
21,
22]. However, the fluid pattern in an oil-immersed transformer system is quite complicated. Despite the simplicity, the physical model may be difficult to simulate the detailed thermal-hydraulic behavior accurately, especially when the system geometry and flow behavior are complex, as reported in [
22].
Numerical methods allow a refined representation of the system geometry and physics, thus manifest themselves with the ability of in-depth performance analysis for power devices [
23,
24,
25]. In light of this, numerical methods have been widely used for the design optimization of power transformers [
22,
26,
27,
28]. A comparative study of two-dimensional (2D) and three-dimensional (3D) computational fluid dynamics (CFD) model was carried out and it was shown that the 3D model, albeit computationally intensive, was needed to simulate the non-uniform flow and temperature distribution accurately [
29]. To this end, a 3D finite element modelling (FEM)-based thermal model was used in [
30] to determine the loss and temperature rise of a high-frequency transformer. However, the inter-dependence of multiple physics and key parameters requires the coupling analysis of electromagnetic and thermal-fluid behavior to maintain a high modelling accuracy. In this regard, the coupled numerical analysis is required to achieve more accurate solutions. A thermal-fluid coupled analysis was performed in [
31] to compute the temperature distribution in a 31.5 MVA/110 kV ONAN transformer, where the losses were experimentally determined. Further, the 3D coupled magneto-thermal-fluid analysis was carried out in [
32,
33] to study the feature of multiple physics in the power transformer. A quasi-3D coupled numerical method that combines the 3D core loss and velocity simulation with the 2D fluid-thermal analysis was proposed in [
34] to analyze the temperature rise of transformer. In spite of the improved accuracy, the coupled numerical analysis increases the complexity and occupies too much computing resources, thus it is not favorable in a design program that commonly takes several iterations to achieve the final solution. Hence, a trade-off has to be made to achieve sufficient modelling accuracy while keeping the computing cost at an expected level as well. An attempt was shown in [
35], where the dimensionless least squares and upwind FEM were combined to simulate the thermal-fluid field in an oil-immersed transformer to improve the computing efficiency of coupling analysis.
In this paper, a coupled, semi-numerical model is proposed for the electromagnetic-thermal-fluid analysis of MF ONAN transformers. An analytical model that is based on spatial distribution of flux density and AC resistance factor is exploited for electromagnetic analysis to calculate the core and winding losses. A 3D numerical model that is based on CFD technique is then used in conjunction with the analytical model to predict the temperature rises of different system components. The electromagnetic and the thermal-fluid behaviors are solved iteratively in a close-loop manner to address the temperature-dependent properties and achieve a converged solution. Load experiments on two transformer prototypes with different conductor types and physical geometries are performed to validate the proposed method. The proposed method contributes to improving the modeling of the MF transformer in the following points. Firstly, the refined core loss modeling well addresses the flux inhomogeneity stemming from the nonlinearity of magnetic materials and the difference of magnetic paths, thus the core loss can be estimated with higher fidelity. Secondly, the AC effects are modeled in a multi-layer manner in seeking to determine the total winding loss more prudently. Thirdly, the proposed semi-numerical method avoids the time-consuming 3D electromagnetic numerical calculation, while it keeps a detailed modeling of both fluid flow and temperature distribution, thus it can be expected to better manage the trade-off between accuracy and complexity. Due to the improved electro-magnetic modeling, the proposed method also has good potential to be further simplified by replacing the numerical thermal model with the analytical model, so as to ease the real-time application.
The rest of paper is organized, as follows.
Section 2 introduces the fundamentals of electromagnetic analysis for core and winding loss determination.
Section 3 presents the 3D CFD model and the overall iterative framework of the proposed method.
Section 4 describes the experimental setup and tests. The validation and discussion are presented in
Section 5, while
Section 6 draws the key conclusions.
2. Electromagnetic Modeling
The accurate estimation of losses, which depends on high-fidelity electromagnetic modelling, is critical, especially in the thermal point of view. Instead of using the complicated numerical method, this section describes an analytical method for the determination of core loss and winding loss, which are respectively based on the spatial distribution of flux density and AC factor.
2.1. Core Loss
The total core loss is composed of hysteresis loss, eddy current loss, and residual loss. At medium/high frequency operations, amorphous and nanocrystalline materials exhibit lower hysteresis loss and they are commonly employed in medium/high frequency transformer applications. While separating the core loss into three sources needs extensive efforts on experiments and coefficient extraction, the empirical method requires much less measurements, thus it is more attractive in real applications.
Depending on the flux density, the core loss in per unit volume occurred could be approximated by the well-known empirical Original Steinmetz Equation (OSE) for purely sinusoidal excitation [
36]:
where
K,
α, and
β are Steinmetz constants that depends on the material properties,
f is the operating frequency in Hz, and
B is the flux density. The sinusoidal excitation is applied on the low-voltage (LV) side in this study, which well fits to the assumption in OSE model. In the case of arbitrarily shaped excitations, several models have been developed for core loss modeling, including the modified Steinmetz equation (MSE), general Steinmetz equation (GSE), and improved generalized Steinmetz equation (iGSE), which are all based on the OSE [
37].
It should be noted the flux density in Equation (1) is assumed to be the average of flux inside the core. Due to the nonlinearity of magnetic materials and different magnetic paths, however, the flux is not distributed homogenously depending on the material characteristics and core configuration.
E-E core is used in this paper, and half of the core geometry is shown in
Figure 1. The amorphous alloy core is assembled by lacing thin ribbon layers together in seeking to attenuate the eddy current effect. Despite the better flux distribution of this structure, a different magnitude of flux density can still be observed in each ribbon due to the different magnetic paths of flux. To capture the inhomogeneous spatial distribution, the non-uniform variation of the flux density is taken into account by dividing the core into
n equal segments with the thickness of dx. The flux density for the
i-th segment and the total flux can be expressed as:
where
Rt, and
φt denotes the total reluctance and flux of the core, while
Ri and
φi represents the corresponding parameter of the
i-th magnetic segment.
The reluctance of the
i-th magnetic segment can be calculated as:
where
li is the length of the magnetic path,
ε is the material permeability, and
Ai is the cross-sectional area of the
i-th magnetic path. For the E-E core in use, the following equation can be drawn:
Combining Equations (2)–(5), the following relationship in terms of the reluctance and flux density can be drawn:
where
Bi is the flux density of the
i-th magnetic segment.
It is explicit from Equations (1) and (6) that the actual core loss varies spatially according to this non-uniform distribution of flux density, i.e. higher core loss allocation occurs in inner locations. To this end, the proposed model scrutinizes the spatial distribution of flux density thus contributes to improving the modeling accuracy of core loss.
2.2. AC Effect and Winding Loss
The ohmic losses in the windings are calculated by:
where
J is current density and
σ is electric conductivity.
As eddy current effects become more prominent at higher frequency, the effective AC resistance of winding increases as a consequence of the skin and proximity effects. To take eddy current effect into consideration, two well-known analytical models that are applicable to foil and round conductors, respectively, are used in this paper. The classical Dowell’s equation [
38] gives the AC resistance of the foil-type winding, assuming that foil conductors occupy the entire height of the core window. The AC factor of the
j-th layer (
Frn) can be calculated by:
where Δ is the penetration ratio,
dfoil the foil thickness,
δ the skin depth, and
ξ1 and
ξ2 the skin and proximity effect respectively.
Dowell’s equation for foil conductors can also be applied on round-type conductors by introducing the porosity factor (
η), which is the ratio of the height occupied by conductors to the window height. In addition, in 1990, Ferreira proposed a new solution for round-type wire using the Kelvin–Bessel functions by considering the orthogonality between skin and the proximity effect. However, this formula overlooked the porosity factor. To take Dowell’s porosity coefficient into account and to give a more accurate solution, Bartoli modified Ferreira’s formula and proposed a new model as [
39]:
The multilayer configuration of windings are taken into account using Equations (8) and (9), so that the AC resistance for each layer can be calculated more prudently. With the conductor geometry and winding arrangement, the total AC resistance can be derived by accumulating all individual layer resistances:
Dowell’s equation is applicable when the transformer is subject to sinusoidal excitation. In the case of non-sinusoidal excitation, the current can be decomposed into harmonics with Fourier transform, while the loss for each harmonic is calculated by multiplying the harmonic current with the AC resistance under corresponding frequency. The total winding loss can then be determined by summing all of the harmonic losses.
It should be noted that the resistivity of the conductor copper is also temperature-dependent, suggesting that the winding loss is sensitive to the result of thermal modelling. The resistivity of copper can be calculated empirically as:
where
φ0 is the resistivity at
T0 and
τ is the temperature coefficient of copper, which is equal to 0.004041.