Advanced 3D electromagnetic transient-thermal steady coupled FEM simulation of a LV busbar of a rectifier transformer

Abstract

The design of industrial rectifier transformers for use in the field of aluminum production requires the evaluation of the effects of current harmonics in the waveforms, in terms of electromagnetic fields interacting with the transformer’s constituent materials. The waveforms of the currents flowing through the windings and high-current output connections of transformers are highly distorted waveforms due to a high presence of harmonics that contribute significantly to the overall losses developed within the system causing overstress of the insulation materials. The evaluation of the electromagnetic effects of these harmonics cannot be approached by means of a time-harmonic steady-state study of the problem, but a time transient must be simulated for a more accurate evaluation.

Keywords

3D finite element method (FEM)transient thermal coupled aluminium rectifier harmonics

1. Introduction

The aim of this work is to investigate the effects of non-sinusoidal currents in service conditions in terms of eddy losses and temperature distributions on the low voltage (LV) busbars and tank walls of a smelter rectifier transformer used for aluminium production. This result has been achieved by combining a magnetic 3D FEM transient coupled with a 3D FEM thermal static study.

One of the most common numerical approaches consist in use of running the harmonic spectrum of non-sinusoidal currents; then performing a set of electromagnetic sinusoidal steady-state 3D FEM simulation, one for each harmonic current; and finally perform the post-processing of the results in terms of the numerical combination of the total losses distribution. However, this approach, requires a considerable number of repetitive solutions, and omits two fundamental aspects, the first being the number of harmonics to be considered, that must be at least seven, and moreover, far more significant, the simplified approximation of the behavior of highly non-linear materials due to sinusoidal steady-state solver.

In this paper a method using the transient 3D FEM approach is used allowing to get rid of the issues presented above, in particular, the truly representation of the current waveshape as in service and the proper numerical treatment of non-linear material [1,2].

2. Theoretical background

In the modern context Maxwell’s equations refers to a set of four equations that describe the properties and the interrelations of electric and magnetic fields [3]. As follows: Gauss’ Law $\begin{eqnarray}{\nabla}\cdot D={\rho}.\end{eqnarray}$ (1) Gauss’ Law for magnetism $\begin{eqnarray}{\nabla}\cdot B=0.\end{eqnarray}$ (2) Faraday’s Law $\begin{eqnarray}{\nabla}\times E=-{\displaystyle \frac{{\delta}B}{{\delta}t}}.\end{eqnarray}$ (3) Ampère–Maxwell Law $\begin{eqnarray}{\nabla}\times H=\frac{{\delta}D}{{\delta}t}+J.\end{eqnarray}$ (4) The fifth can be obtained from the above equations taking the divergence of (4) and using the result on (1). $\begin{eqnarray}{\nabla}\cdot J=-\frac{{\delta}{\rho}}{{\delta}t}.\end{eqnarray}$ (5) Which satisfied the following constitutive equations: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle B={\mu}H\end{eqnarray}$ (6) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle D={\varepsilon}E\end{eqnarray}$ (7) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle J={\sigma}E.\end{eqnarray}$ (8) The low frequency electromagnetic under the conditions of low frequency, considering constitutive relations, can be rewritten as: $\begin{eqnarray}{\nabla}\times H={\sigma}E.\end{eqnarray}$ (9) The basic equations related to the low frequency electromagnetic field includes the field differential equations and the constitutive Eqs (6, 7, 8), nowadays with the development of computational electromagnetics and relative applications, a series of electromagnetic field analysis methods based on various “potential sets” have been developed.

The A − V − A , potential set herein after described and adopted in our calculation is a basic eddy current analysis method in which A is magnetic vector potential and V is electric scalar potential respectively [4–6]. A − V − A method can easily solved general complex problems such as non-linearity, multiply connected regions, multi-subdomains like our case. However, it still has disadvantages: the usage of vector potential in the eddy current free region requires a large amount of memory and a long CPU time [7].

The aim of this paper is to confirm, with nowadays hardware and software, the feasibility of a time dependent solution using a time transient solution.

The derivations of governing equations for A − V − A are of typical significance and the derivation of governing equations for the corresponding regions is shown below.

Definition of vector potential $\begin{eqnarray}B={\nabla}\times A.\end{eqnarray}$ (10) Governing equations in conducting region: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle H=\frac{1}{{\mu}}{\nabla}\times A\end{eqnarray}$ (11) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\nabla}\times E=-\frac{{\delta}}{{\delta}t}({\nabla}\times A)\end{eqnarray}$ (12) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle E=-\frac{{\delta}A}{{\delta}t}-{\nabla}V^{\prime }\end{eqnarray}$ (13) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\nabla}\cdot J=0\end{eqnarray}$ (14) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\nabla}\cdot \left(-{\sigma}\left(\frac{{\delta}A}{{\delta}t}+{\nabla}V^{\prime }\right)\right)=0\end{eqnarray}$ (15) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle {\nabla}\times \frac{1}{{\mu}}{\nabla}\times A+{\sigma}\left({\displaystyle \frac{{\delta}A}{{\delta}t}}+{\nabla}V^{\prime }\right)=0.\end{eqnarray}$ (16)

3. Magnetic model

The 3D FEM electromagnetic model represents a portion of the geometry, LV busbars, place of high current, tank of the rectifier transformer. The study focuses on the portion of the geometry particularly exposed to high-intensity electromagnetic fields. To reduce computational time, all the possible symmetries were adopted.

3.1. Geometry

The simplified model includes part of the tank wall, cover and floor, cover flanges, internal sealing plates (see Fig. 1), connection bars, and interconnection bars between the transformer and rectifier (see Fig. 2), thus omitting those components of the transformer that are negligibly affected by high field.

Fig. 1.

E.M Model geometry – front view.

Fig. 2.

E.M. model Geometry – LV Busbars details – front view.

3.2. Materials

The materials physical characteristics 𝜎 and 𝜇 relate the basic field quantity. The modelling of the electrical and magnetic properties of the material adopted under standard e non-standard condition is the basic guarantee for performing effective electromagnetic analysis. Therefore, both advanced method of computation and advanced material model are required [7].

Fig. 3.

Mild steel – FE360, approximated magnetic characteristic.

Conductive materials such as aluminum and copper, which are intended for the conduction of high currents, as well as magnetic materials that are also conductors with linear and non-linear magnetic characteristics, see Fig. 3 (e.g. stainless steel), are included in the model. The characteristics are shown in the table below.

Table 1

Materials electromagnetic properties

Material ID	Electrical conductivity	Relative permeability	Mass density
	[S/m]	#	[kg/m³]
20MN23AL	1388900	1.001	7800
AL 6061-T6	25062657	1.000	2700
Copper	47303690	1.000	8933
FE360	9523810	non-linear	7830

3.3. Currents setup

The transformer, in service, is connected to a rectifier which, in its operating principle, works with highly distorted currents, typically with square form waveshape (Fig. 4). An harmonic analysis deal with the presence of harmonics at multiple frequencies of the fundamental (e.g. the fifth, the seventh, the eleventh, up to fiftieth). In addition, in case the numerical analysis of the harmonic spectrum shows a DC component of the current superimposed on periodic sinusoidal, variable, currents, then the analysis requires a DC static solution in addition to the simulation of sinusoidal steady performed for each harmonic. On the contrary a transient approach directly allows the usage of the LV busbar currents as shown below.

Fig. 4.

LV busbar currents.

Fig. 5.

Model solution mesh. On the right the detail of highlighted section.

3.4. Time transient solver settings

In the following table a summary of the configuration parameters and the solution time required by the simulation are shown in Table 2, the number of tetrahedra reflects the refinement of the mesh shown in Fig. 5.

Table 2
Electromagnetic FEM model info

Number of tetrahedra # 1,333,053

No. of non-zeros # 79,836,205

Allocated RAM memory Gb 3.2

Time transient [ms] 30

Time step [ms] 0.2

Total solution Time [hr] 174.3

3.5. Fields

Figure 6 shows the flux density distribution over the rectifier transformer tank wall; each part of the figure shows a different time frame (left column, top to bottom, 10, 18, 26; right column, top to bottom 14, 22, 30 ms).

Fig. 6.

|B| smoothed shaded plot at 10-14-18-22-26-30 ms - Scale from 0.0 T to 0.75 T – front view.

Figure 7 shows the current density distribution over the rectifier transformer LV busbar; each part of the figure shows a different time frame (left column, top to bottom, 10, 18, 26; right column, top to bottom 14, 22, 30 ms).

Fig. 7.

|J| smoothed shaded plot at 10-14-18-22-26-30 ms - Scale from 0.0 A/m² to 1.5 × 10⁶ A/m² – side view.

In Fig. 8 the instantaneous losses in the transformer case over the period considered, 10–30 ms are depicted.

Fig. 8.

Ohmic loss smoothed shaded plot at 10-14-18-22-26-30 ms - Scale from 0.0 W/m³ to 2 × 10⁵ W/m³ – front view.

It can be noticed the different periodicity of the losses corresponding to the variability of current distribution in the busbars (see Fig. 7). This, together with the non-sinusoidal waveform of the currents are the reasons of adopting the transient approach.

Figure 9 shows the instant losses in the tank wall ranging from 100 W to 900 W. The distorted waveform of the losses is given due to the spatial distribution of losses in the wall, the highly non-linear behaviour of the material and the non-sinusoidal shape of the imposed currents.

Fig. 9.

Time-transient ohmic losses on tank wall from 10 to 30 ms.

The losses in the other components like plates, flanges, cover, tank side wall, floor have the same typical transient pattern as in the tank, with instant values ranging from 0 to 450 W for the plates, and from 0 to 10 W for the other components.

Table 3 reports the average values of the losses calculated in the various components of the model in the time transient solution over the considered time period (10–30 ms).

Table 3

Time average Ohmic losses on the model components over the 10 ms–30 ms time transient period

Component ID	Time average Ohmic losses
#	[W]
Plate 1	17.9
Plate 2	27.4
Plate 3	27.6
Plate 4	25.6
Plate 5	17.4
Plate 6	17.6
Side wall	1.1
Wall	299.9
Flange	2.2
Floor	2.9
Cover	2.6

4. Thermal model

The thermal problem is coupled to the electromagnetic one via the environment that uses the losses calculated during the electromagnetic transient as input for the calculation of the equations governing the thermal regime in order to provide the temperature distribution as the final result.

Due to the different time constant of the electromagnetic and thermal phenomena, milliseconds for the electromagnetic problem vs tens of seconds for the thermal problem, the average losses are evaluated over one period.

The geometry of the thermal model (see Fig. 10) is similar to the geometry of the magnetic model minus, for example, the external busbars of the rectifier or some other components that are not relevant to the thermal simulation and have therefore been disabled in the thermal model.

Fig. 10.

Thermal model Geometry – LV Busbars details – front view.

4.1. Theoretical background

The magneto-thermal coupled analysis considering the heat emission, heat conduction, and temperature dependent of the magnetic characteristic can be carried out by the following procedure: at the first step the magnetic field analysis and the eddy current are analysed using 3D FEM by the basic Eq. (16).

Then in the thermal analysis the following heat conduction equation is used: $\begin{eqnarray}\displaystyle {\nabla}\cdot ({\lambda}{\nabla}T)+Q={\rho}c{\displaystyle \frac{{\delta}T}{{\delta}t}} & & \displaystyle\end{eqnarray}$ (17) where T is the temperature, 𝜆 is the thermal conductivity, Q is the rate of the internal heat generation, 𝜌 is the material density, c is the specific heat and t is the time. $\begin{eqnarray}\displaystyle q_{c}=h(T-T_{a}) & & \displaystyle\end{eqnarray}$ (18) where T is the temperature and T_a is the ambient temperature. The thermal properties of the material used are shown in the Table 4 below.

Table 4

Materials thermal properties

Material	Specific heat capacity	Thermal conductivity	Mass density
ID	[J/kg C°]	[W/m C°]	[kg/m³]
20MN23AL	500	16.2	7800
AL 6061-T6	896	167	2700
Copper	397	393	8933
FE360	440	84	7830

Table 5 shows the solution time of the thermal simulation and the mesh characteristics.

Table 5

Thermal FEM model information

Number of tetrahedra	#	1,333,053
No. of non-zeros	#	8,118,138
Total solution time	[hours]	0.2

To the model has been assigned, as boundary conditions, the temperatures of the oil, in the transformer tank, and of the environment, surrounding air, as specified in the design phase.

Table 6

Temperature boundary conditions

Temperature	[°C]
Max ambient	40
Tank oil	60

As mentioned above, the distribution of losses in the geometry, averaged over the observation period, is used as input for the thermal problem. The coupling between the two simulations software takes place automatically, unmediated by the operator; the same mesh is used in the thermal model as in the electromagnetic model, and the quantities calculated in the mesh are shared tetrahedron by tetrahedron; the thermal simulator then reads the data required from the electromagnetic simulator solution, using the temperature boundary conditions shown in Table 6.

Fig. 11.

Time average ohmic losses in EM solver (left) vs Input source EM heat density in thermal solver (right) - Scale from 0.0 W/m³ to 1e5 W/m³ – front view.

From Fig. 11, which shows the thermal density due to electromagnetic losses via the shaded plot, one can see that there is a good correspondence in the shared values [8]. Figure 12. Represent the temperature distribution over the portion of the tank.

Fig. 12.

Temperature shaded plot. Scale from 40 °C to 70 °C - front view.

Table 7 reports the average values of the temperatures calculated in the various components of the model in the thermal coupled solution over the considered time period (10–30 ms).

Table 7

Average temperature on the models components over the 10 ms–30 ms time transient period

Component ID	Average temperature
#	[°C]
Plate 1	56.9
Plate 2	57.3
Plate 3	57.3
Plate 4	57.1
Plate 5	56.9
Plate 6	56.8
Side wall	56.0
Wall	56.8
Flange	53.9
Floor	53.7
Cover	48.2

5. Conclusion

A 3D transient-thermal steady coupled electromagnetic model has been created and solved to verify the effects of current harmonics in transformer’s internal LV busbars, portion of the transformer plates and flanges, composed by various metallic materials and the rectifier/transformer connection busbars. The results obtained, confirm the chance to use this methodology in the case of EM/thermal coupled problems using non-linear materials, keeping solution times compatible with the needs of design. Moreover, the results have a high level of accuracy if compared with other methods based on steady state simulations that lead to high approximation.

5.1. Future development

The design data used in this paper have been approximated by generic numerical parameters.

A refined model will be investigated to explore the usage of the resources needed for the time transient. Hence, it will be possible to get a more accurate comparison among the traditional and the proposed approach with the aim to reduce the solution time.

A further possible development will be the verification of the experimental data by means of simulations, in advance, based on CFD (Computational Fluid Dynamics). This way, make possible to improve the quality of the results obtained.

Disclaimer

Due to confidentiality issues, it was not possible to use a real design of the model and therefore the model was simplified adopting customized design data in order to explore the phenomena. Due to this reason the comparison of the model to real case is subject to limitations.

Footnotes

Acknowledgements

The authors have no acknowledgments.

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