A Novel Fast FEA No-Load Loss Calculation Method for Stacked Core Three Phase Distribution Transformers

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2.1. XY-Plane Modeling

General equation governing the problem is magnetostatic Maxwell equation written as follows ^[10].

(1)

where H, B and r are respectively magnetic field intensity vector, magnetic flux density vector and reluctivity tensor. Common model of the reluctivity tensor is ^[10]:

(2)

where r_xx, r_yy and r_zz are non-linear function of B. Z axis is considered perpendicular to the surface of laminations of the core. r_xx and r_yy are assumed independent of z direction.

Since the transformer core is made of many high permeable sheets (maximum thickness 0.3 mm), separated from each other by the insulation coating with a thickness of less than 10 μm, the magnetic field can only be considered in sheets ^[10]. The normal magnetic flux density is negligible and usually less than 4 mT ^[17]. In this paper, the direction of magnetic flux density assumed to be perpendicular to the core sheets’ surface in the whole core structure, except for the joints, in which it is assumed to be zero. This assumption, significantly, reduces the problem complexity and the original 3-D problem can be substituted by some 2-D analytical from models and the core loss estimation becomes independent the number of pockets in the assembled core. With this assumption, Equation (1) can be rewritten as follows:

(3)

Figure 2. xy-plane model of a three phase stacked core

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As it can be seen in Equ. (3), B_x(x,y) and B_y(x,y) are independent of the z direction. Therefore, only the largest pocket of the transformer core is simulated in xy-plane model. The xy-plane loss is calculated by integrating the spatial loss distribution on main pocket over the whole core structure. Therefore, core loss evaluation is independent of the number of the pockets, unlike the conventional methods. Thus, computation speed is increased by a factor of the number of pockets. Figure 2 shows xy-plane model of three phase stacked core.

The imposed voltages on the transformer coils have a sinusoidal form, under the standard test condition. Flux boundary conditions are used to guarantee the sinusoidal waveform for the applied voltage. Using Faraday’s equation, the total flux in the cross section of the limbs is a given value at each instant of time as follows ^[10].

(4)

where V, , n and s are sinusoidal imposed voltage, total flux, normal vector of limb cross section and area of limb cross section, respectively.

The presence of core stacking holes can increase the core loss up to 3% ^[17]. The holes distort the flux wave shapes and result in higher harmonics and so higher losses ^[8]. For example, Figure 3 shows effect of a hole on distortion of flux density across sheet width. The holes are modeled in this simulation for more accurate results unlike ^[10].

Figure 3. Distortion of flux density in the presence of a stacking hole

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Figure 4. Magnetic flux density distribution at 2 instant of time

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FEA program solves 2-D model of main pocket and returns the magnetic flux density distribution in the xy-plane at any instant of time. In Figure 4 the magnetic flux density distribution at two different instants of time is shown.

To calculate the magnetic flux density variation with respect to time at any element of transformer core, similar to ^[10], 12 equal space instants of time are assumed. The xy-plane model is solved for these 12 instants with their specific flux boundary conditions. For example, Figure 5 shows magnetic flux density waveforms at P point in the rolling direction and the cross rolling direction as indicated in Figure 2.

Figure 5. Magnetic induction waveform at point P indicated in Figure 2

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2.2. Z-Plane Modeling

Due to presence of air gaps at the joints, magnetic flux travels between the overlapping laminations of yokes and limbs and when these laminations are saturated, flux crosses the air gaps ^[10]. This flux distortion causes additional localized losses. This additional loss cannot be evaluated from xy-plane models. A suitable method is introduced to solve this problem in modeling the sheets in the z-plane in the vicinity of the joints ^{[2, 4, 10, 14]}. In Figure 6, the magnetic flux density behavior for Single Step Lap (SSL) and Multi-Step Lap (MSL) structures, using CRGO steel while, operating under a magnetic flux density of 1.7 Tesla is shown.

Figure 6. Magnetic flux distribution at area of joints due to presence of air gaps, a) Single Step Lap, b) Multi-Step Lap

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The variation of magnetic flux density in the SSL and MSL joints (with different steps) along the line L is plotted and shown in Figure 7. As it is shown in Figure 7, the flux distortion takes place only in a distance of several centimeters around the air gaps. Therefore the application of z-plane model is limited to this area and it is independent of the overall structure of transformer core.

In MSL joints, the magnetic flux density in the air gaps reduces to about zero in comparison with the SSL joints. The magnitudes of core sheets saturation are decreased; however the area of the saturated region is increased as shown in Figure 7. However, usually MSL joints are known to reduce the loss in the core joints.

For several magnetic flux densities, 2-D structure of Figure 6 is solved and the extra localized joint loss is calculated.

Figure 7. Magnetic induction along line L from Figure 3 a) SSL structure, b) MSL structure

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To calculate total z-plane loss, magnetic induction waveforms at the L-joints and T-joints are evaluated from Solved xy-plane models. Then ALJL is integrated along the length of joints as follows.

(5)

where ALJLden, B_m(x) are the density of additional localized joint loss over the area of joints and maximum magnetic induction at the length of x along joints.

Total z-plane loss can be obtained by multiplying ALJLden by the area of joints cross section.

Figure 8. Gaussian curve fitting for modeling non-linear behavior of M5 CRGO Steel

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3. Curve Fitting of Reluctivity

In numerical modeling of the non-linear behavior of the transformer core material, often a mathematical relation between relative permeability and the flux density is required. Different fitting methods, such as the hyperbolic equations ^[18] are presented in past, in the literature. In this paper, Gaussian curve fitting with 3 terms is used as Equ. (6) ^[19].

(6)

where is relative permeability. a_i, b_i and c_i are parameters which are determined with curve fitting algorithm. Gaussian method has a powerful ability for modeling the extremely non-linear behavior of material permeability. For example, Figure 8 shows the measured data and the fitted curve for M5 CRGO Steel.

4. Core Loss Model for Non-Sinusoidal Waveforms

As shown in Figure 5, the instantaneous magnetic flux density waveforms in the elements of transformer core demonstrate a non-sinusoidal variation. Therefore, in this modeling the equations related to the sinusoidal waveforms cannot be used. In recent years, accurate equations are developed to model the core loss when a non-sinusoidal flux density waveform exists in the magnetic materials. In this paper, to calculate the losses in each element of the transformer core, Equ. (7), as proposed by ^{[15, 16]}, used to determine the losses:

(7)

where maximum flux density, the sheet thickness, its conductivity, its mass density and the time period are represented by b_m, a, σ, ρ and T respectively. K_h, K_exc and n_a,b,c are the constant coefficients of the magnetic material. These coefficients can be obtained using the specific loss of the magnetic material in specific frequencies and specific magnetic flux densities.

Using Equ. (7), spatial distribution of core loss of xy-plane is shown in Figure 9. Stacking holes are ignored in Figure 9 for a better visualization.

Figure 9. Spatial distribution of core loss of xy-plane model in upper half of 3-phase 3-limb core

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5. Verification of Calculations

To verify proposed method, this model is implemented on 12 commercial size three-phase three-limb distribution transformer samples and the results are compared with the factory measurements. For every distribution transformer, several numbers of a specific size are tested and the average core loss is used as the measured loss. The selected samples have various core structures, rated powers, rated voltages and the design magnetic flux density. In Table 1 some samples of the results are shown. The average accuracy of the new method was 0.28% and its standard deviation was 3.41%. Thus, compared to the conventional method, such as ^[10], this method increases the speed of calculation by a factor of 8 while the average error increases only by 0.08%.

Table 1. Comparison of the Proposed Method Outputs for Various Commercial Size Transformers, Against the Measurements

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