## Off-core magnetic flux paths in power transformers. Modeling and applications

##### Doctoral thesis

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##### Date

2020##### Metadata

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- Institutt for elkraftteknikk [1634]

##### Abstract

Zero sequence (ZS) characteristics of transformers are of great importance in some circumstances including unbalanced and DC biased operations. Any unbalanced condition in the voltage supplying a transformer or in the load to be supplied can result in ZS currents assuming that at least one of the windings is Y-connected and grounded. An example of DC biasing conditions is a transformer subjected to geo-magnetically induced current (GIC) because of geo-magnetic disturbances of the earth’s magnetic field due to solar activities. In the case of 3-phase, 3-legged transformers, the ZS flux takes the path off the core including the oil gaps, structural steels and the tank walls.
The tank is made from mild steel, which is several hundred times more magnetically permeable than the oil. This means that the tank’s presence will disturb the flux distribution in the oil gap depending on its relative proximity to the active part. The flux reaching the wall will enter and exit through continuously distributed points making it complicated to determine the wall’s effective height and thickness. In addition, the losses in the tank may increase the temperature and thus affect the electric conductivity of the tank steel, leading to a negative feedback on the losses and positive feedback on the magnetic penetration depth.
Although there have been extensive advancements in the core and the windings modeling, the off-core flux path has not been properly represented in the dual circuit models of 3- phase, 3-legged power transformers. Differences between the ZS impedances seen from different windings of a transformer, as reported in the experimental results, have not been considered in the models presented so far. As repeatedly reported in the literature, there has been a lack of information on the electromagnetic properties of the tank steel including the electric conductivity and magnetic characteristics. As mentioned, the temperature rise in the tank affects both the ZS impedance and the losses; however, thermal models presented in the literature primarily address the oil and the winding temperature rises with much less focus on the tank. Furthermore, considerations on the tank design such as the use of magnetic shunts, magnetic shields, and choosing from plain and corrugated walls that influence the off-core flux path, have not been discussed properly in the previous works. In cases when the tank losses are not of interest, the tank can be represented with proper boundary conditions. Which of the flux-normal or flux-tangential boundary conditions is more accurate in the representation of the tank has also not been discussed in the previous research works.
The main contribution of the current PhD thesis is to address the abovementioned challenges and research gaps in the off-core flux path modeling. The thesis has introduced an electromagnetic/thermal model for the off-core flux path. The electromagnetic model consists of linear inductances (representing the oil gap) and non-linear R-L branches (representing the tank). The non-linear branches are Cauer-like equivalent circuits obtained from finite difference approximation of the magnetic diffusion equation. The thermal model is based on the thermal network method assuming that the temperature drop across the tank thickness is negligible. Hence, the tank is considered as a thermal node connected to nonlinear thermal resistances representing convective and radiative heat transfer from the tank. All the equations governing the electromagnetic and thermal equivalent circuits are implemented in MATLAB-SIMULINK. However, to simulate target applications including transformers under unbalanced conditions and transformers subjected to the GIC, the equivalent circuits are drawn in ATPDraw and are solved with the EMTP-ATP program.
Difficulties in the calculation of the model parameters are discussed in detail using finite element analysis (FEA) of simple 2D models of the off-core flux path. The FEA provides an advantage that the different parameters can be thoroughly studied without influence from unknown factors. For example, the temperature impact may not be considered adequately when studying the model parameters based only on experimental results. In addition, the significance of the tank walls, cover and bottom as well as the temperature can be separately studied under controlled case studies. As investigated in the thesis, the wall and bottom are significant for the off-core flux path; however, the cover can be neglected when its distance to the core is large and typically more than 2.5 times the distance from the wall to the core. At high off-core flux, when the wall becomes magnetically saturated, the open-circuit ZS impedance starts decreasing due to the reduction in the magnetic permeability. However, the temperature rise in the wall will affect the electric resistivity, thus having an increasing impact on the impedance. In some cases such as the corrugated walls (where the wall resistance is relatively higher than the plain walls), the temperature impact may keep the open-circuit ZS impedance almost constant.
It is shown that the effective tank height, the wall thickness and electric conductivity and the linear inductances (representing the oil gap) need to be tuned in order to reproduce the impedance and losses measured on a 3-phase 3-legged 300 kVA sample transformer. These tuning factors are identified by fitting the circuital model to either test or FEM results. As demonstrated in the thesis, there is no need to re-tune the parameters when the temperature of the tank elements varies in a practical range as observed in the actual operations.
Characterization of the electromagnetic properties of the tank steel in terms of the magnetization curves and the losses is performed using extensive magnetic measurements as a part of the PhD work. As discussed in the thesis, classical eddy current loss is the dominant component for the tank steels thicker than 4 mm. However, the excess losses must also be considered in the thinner walls, which are normally used in corrugated tank walls.
In cases the tank losses are not of interest, and in order to reduce the model complexity, lossless variations of the proposed model can be used. In the lossless variations, the tank is excluded and proper boundary conditions are set instead. As investigated, the corrugated walls as well as the walls equipped with magnetic shunts can be assumed as a flux-normal boundary condition. The plain walls as well as the walls equipped with magnetic shields can be assumed as a flux-tangential boundary condition. The bottom as a base of transformer is made from relatively thick steel that can be considered as a flux-tangential boundary condition.
The difference between ZS impedances seen from HV and LV depends on the behavior of the tank walls. As discussed in the thesis, the ZS impedance difference can be small in corrugated walls as well as walls equipped with magnetic shunts, but it is large for plain walls and even larger in cases where the walls are equipped with magnetic shields.