3-elements to estimate stray losses in transformer

JOURNAL OF APPLIED COMPUTER SCIENCE Vol. 16. No 1 (2008), pp. 89-100

3D Finite Element Estimation of Stray Losses in Three-Phase Transformers

M.A. Venegas Vega1, R. Escarela Pérez2, T. Niewierowicz3 1Arteche Transformadores y Tecnología

Ant. Carr. México-Querétaro km 73.5 Tepeji, Hidalgo, México (e-mail: [email protected])

2Universidad Autonoma Metropolitana- Azcapotzalco, Departamento de Energia

Av. San Pablo No. 180, Col. Reynosa, C.P. 02200, México, D.F. (e-mail: [email protected])

3Instituto Politécnico Nacional, SEPI, ESIME-Zacatenco C.P. 07738, México, D.F. (e-mail: [email protected])

Abstract. Stray losses and flux leakages of a three-phase three-limb transformer are determined and analyzed using 3D finite element analysis. The problem is not current driven but voltage fed. Thus, a circuit-field problem is established and solved. The operating condition analyzed is the load-loss test. Since low excitation voltages are excited during this test, the transformer works under unsaturated conditions. As a result, the transformer is modelled using a time-harmonic approach. The transformer geometry is truly three dimensional, forbidding the use of conventional 2D models. The clamp plates and transformer tank are conveniently modelled with surface impedance boundary conditions.

1. Introduction

Prediction of electromagnetic phenomena in structural metallic parts of power transformers is an important step for their correct design, where optimal reduction of local overheating due to leakage magnetic fluxes is sought. Numerical determination of losses in massive conductors has been treated by several authors [1]-[4], where the losses have been determined through the use of 2D and 3D finite element approaches. For instance, references [1] and [2] tackled the problem of losses in the regions near the high-current bushings of transformers. However, the problem has been solved by withdrawing specific portions from the whole geometry.

M.A. Venegas Vega, R. Escarela Pérez, T. Niewierowicz 90

The purpose of this work is to determine and analyse the losses produced by leakage fluxes in a three-phase three-limb transformer, using a 3D time-harmonic finite-element formulation of the full geometry. The operating condition considered here is the three-phase load-loss test which injects impedance voltage in one side of the transformer windings while keeping short-circuited the other side. Since the skin depth of the transformer massive conductors is much smaller than any of their defining dimensions, surface impedance boundary conditions can be used to avoid explicit meshing of these volume regions. This not only results in requiring smaller computational resources but also allows the use of magnetic scalar potentials at all volume regions, further reducing computational costs.

2. Model of the transformer

Fig. 1 shows the geometry of a three-phase three-limb distribution transformer. The tank is not shown but can be visualized as a box enclosing the transformer geometry. The nominal and geometry data of the transformer are presented in Table I and in Fig. 2. The following constraints must be incorporated to properly model the load-loss test. The high voltage (HV) windings are connected to a three-phase external source whereas the low voltage (LV) windings are short circuited. The voltage source is feeding impedance voltage, that is, the nominal voltage times the nominal short circuit impedance (14.3% of the nominal voltage for the transformer considered in this work, see Table 1). The conductivity of the tank and clamps is numerically given by 5x106 S/m. The core relative permeability was chosen as 20000 since it is a typical value for non-saturated steel.

Fig. 1. Complete transformer geometry: windings, core and clamps of transformer.

The tank is not shown

3D Finite Element Estimation of Stray Losses in Three-Phase Transformers 91

Table 1. Data of the transformer

Quantity Value Rated power 31.5 MVA Frequency 50Hz Rated Voltages HV/LV 132/33kV Rated Currents HV/LV 138/318 A Number of turns HV/LV 1000/433

Fig. 2. Geometry dimensions of the transformer

Due to the geometry and excitation symmetry that exists in the transformer,

a quarter of the geometry is only modeled to reduce the numerical size of the finite element model (Fig. 3). External boundary conditions are imposed in the tank as impedance surfaces while parallel and perpendicular flux conditions are considered in the two remaining faces as shown in Fig. 4.

Tank

HV

LV

LV

LV

LV

LV

LV

HV

HV

HV

HV

1520

130

560

560

560 130

10901090

HV

HVHVHV

150

150150

All dimension in mm.

Core

Windings

LVLVLV


Fig. 3. Three-dimensional finite element mesh

Fig. 4. Surface impedances and boundary conditions in the transformer The finite element model consists of three-dimensional elements that include

tetrahedrons, prisms and cubes. the surfaces of impedance are also employed in the clamps (Fig. 4) the windings of the transformers are connected to external sources of voltage by means of external resistances as depicted in Fig. 5.

Fig. 5. Connections of windings, voltage sources and external resistances during the load-loss test


3. Electromagnetic formulation

The electromagnetic field is formulated using a time-harmonic model. This representation means that the ferromagnetic materials show linear behavior, avoiding, this way, costly time-stepping simulations. The assumption of linearity is admissible since the load-loss test is performed with low-excitation voltages. Thus, the transformer is operating in the linear regions of the BH curves of the transformer ferromagnetic materials. The low-frequency time-harmonic representation of electromagnetic fields can naturally incorporate impedance surfaces as boundary conditions.

The model consists of three different regions: coils, transformer core and air. Tank and frames are not explicitly included as they are represented with impedance surfaces. This substitution is allowed because the skin depth of the ferromagnetic materials considered is smaller than the dimensions of the tank and frame. Explicit consideration of the tank and frames is possible but they would have to be meshed, leading to a huge mesh that in turn implies high computational costs. If the reduced magnetic scalar potential is employed, transformer windings do not need to be meshed. It is also assumed at this stage that transformer windings are filamentary, that is, the current density is uniform over the cross section of the winding conductors. The transformer core is modeled using the total magnetic scalar potential. The regions of eddy currents use a semi-analytic formulation that uses the total magnetic scalar potential. A brief description of the electromagnetic formulation using magnetic scalar potentials is given in the next paragraphs.

The intensity of magnetic field in eddy current free regions can be obtained as the contribution of two components as follows:

φ∇−= sHH (1)

where Hs is the magnetic field produced by the density currents JS when the existence of material media is disregarded and the problem region is considered unbounded. It is calculated from Biot-Savart's law

Ω

Ω∇×= dR

JsH s1

41π

(2)

The magnetic potential φ in (1) is called the reduced magnetic scalar potential. It accounts for the presence of magnetic materials. The divergenceless property of the magnetic field density ( )0B =⋅∇ and the constitutive relationship between the magnetic field intensity and density ( )HB µ= can be combined with (1), resulting in the following Poisson type equation:

sHµφµ ⋅∇=∇⋅∇ (3)

Equation (3) also fulfills Ampere’s law since ss JH =×∇ and ( ) 0HH s =−×∇ .


Interface boundary conditions between magnetic media 1 and 2 can be written as:

( ) ( ) 0221112 =∇−∇⋅+−⋅ φµφµµµ nsHn (4)

for the continuity condition of the normal component of B . The continuity of the tangential component of H is automatically enforced with any continuous φ [7].

The main problem of the φ formulation is its poor numerical accuracy in

highly permeable materials, where sH and φ∇ have similar magnitudes since B is small there.

Cancellation errors in regions with high permeabilities (free of currents of sources) can be avoided with the use of the total magnetic scalar potential ψ.

The magnetic field intensity in these regions is then given by

ψ−∇=H (5)

Combining the divergenceless property of B and the constitutive relationship between H and B , the following Laplace type equation is obtained:

0=∇⋅∇ ψµ (6)

Interface boundary conditions between magnetic media 1 and 2 can be written as:

( ) ( )2211 ψµψµ ∇⋅+∇⋅ nn (7)

Similarly, the continuity condition of the tangential component of H is fulfilled with a continuous ψ. Scalar potentials are very attractive when calculating 3D magnetic fields since the computational times are largely reduced. The core of the transformer is modeled using this formulation [5].

The whole transformer is then modelled by coupling (3) and (6) at boundaries where interface boundary conditions are enforced:

n-H

n-

∂∂

=⋅+∂∂ ψµµφµ 211 sn (8)

ψφ ∇×−=×∇×− + nHnn s (9)

Where n is a normal vector to the surface. The subscript n in the magnetic field intensity indicates normal component. Thus, the magnetic field Hs is only calculated in the interface of the regions 1 and 2 [5] for solving (7).

The theory shown in this section has been coded in MEGA [8] a general purpose electromagnetic software which has 3D graphics capabilities for displaying of results. This software has also been tested through benchmark problems that have been accepted by the finite-element community.


4. Surface impedances The transformer has structural metallic parts such as the tank and frames that

are exposed to leakage fields. As a result, losses are generated in these massive conductors. Direct modeling of these regions is possible with the use of vector potentials. However a great deal of elements is required to mesh these zones since the skin depth of the metallic parts is small when compared with their geometry dimensions. This means that the magnetic field does not penetrate deep into the conductor. So, direct modeling of eddy current regions not only requires lots of elements but more degrees of freedom are needed for the vector potential representation. The analytical solution of flux penetration and currents into plane-faced boxes is well known [9] and can be used to avoid explicit meshing of massive conductors. A finite element implementation of the approach has been proposed in [6] and [4]. It basically uses the analytic solution of magnetic field as surface impedances in the faces of the elements that coincide with interfaces of conducting and non conducting regions.

The decay of magnetic field inside a massive conductor is therefore modeled using the distribution of field penetrating a conductive semi-infinite slab. The current per unit width [6], it is related with the current density in the surface by:

( ) 2/1 jJI S −= δ (10)

where δ is the skin depth and given by:

ωσµδ

2= (11)

If the conducting region is facing a total magnetic scalar potential region, the governing equation of the non-conducting region is (5). Finite element discretization of (5) gives surface integral terms related of H×n. The integrals on interfaces of eddy current region can be put in terms of the tangential component of Ψ with the aid of the analytic solution, leading to the following boundary condition

212

in o

jψ µψµ

∂∇

∂−

= (12)

This condition is only applied at interfaces of regions of thin eddy currents. This way, explicit meshing of eddy current regions and use of vector potentials has been avoided. 5. External circuits

The electromagnetic field of low-frequency transformers is produced by

currents circulating through its windings. However, these currents are not known


a priori because the transformer is fed from voltage sources during the load-loss test. Hence, the electromagnetic equations are not current driven but voltage excited. It was already mentioned that transformer windings are considered filamentary. Hence, uniform current densities circulate through their traverse sections, neglecting skin and proximity effects.

For a non-current driven problem, it is necessary solve a circuit-field coupled problem. The key to solve this sort of problems lies in the voltage equations at the terminals of windings:

tIRV

∂

∂+−=

λ (13)

where R is the total resistance of the coil, λ represents the magnetic flux linkages of a winding and I is the winding current. Flux linkages are calculated from the magnetic potentials yielding a linking route between field and circuit equations. This way, additional equations are generated for each winding connected to external circuits, which are added and solved simultaneously with the finite element system. 6. Load-Loss test

After establishing boundary conditions and connecting external circuits to the transformer, it can be proceeded to obtain a finite-element solution for a specified frequency. The system of simultaneous equations has complex form, that is, the finite-element global matrix, as well as the forcing vector, has complex numbers as entries. This in turn means that the solution of potentials is also complex.

The power absorbed during the load-loss test consists basically of Joule-type losses (I2R) in winding conductors, losses produced by eddy currents in winding conductors (neglected in this work) and stray losses in structural metallic parts (main objective of this work). Losses in the transformer core are disregarded as well, because the impedance voltage is small (compared with rated voltage), and the magnetic flux is being forced into air leakage paths.

Transformer impedance voltage [5] is the necessary value to make nominal current circulate through the transformer windings, when one of the transformer sides is short-circuited and the other one is fed with impedance voltage. It is usually expressed as percentage of the transformer rated voltage and equal to the per-unit leakage impedance value times 100. This result is easily obtained from the classical T equivalent circuit when the secondary side is short-circuited and the excitation branch is neglected. Here, the short-circuit impedance nearly equals the total leakage impedance.

Thus, during the load-loss test of a three phase transformer, the losses are given by:

strayRIload PPP += 23 (14)


where: Pload: Load losses (W) PI

2R: Joule Losses (W)

Pstray: Stray losses (W) Notice that losses, produced by eddy currents and proximity effects, have

been neglected inside the transformer windings. The stray losses are considered here as those taking place in metallic structural parts of the transformer. They are produced by the time-harmonic leakage fluxes reaching these massive conductors.

Fig. 6 shows the distribution of magnetic field in the transformer core and windings at ωt = 0o, ωt = 120o and ωt = 240o. It can be immediately seen that the magnetic flux is being expelled from the core legs and therefore forced into air leakage paths. It is interesting to note that the magnetic fluxes are approximately 90 degrees behind the voltage sources according to Faraday’s law ( E jωλ= in the frequency domain). This means that voltage is reaching its peak value when the current is approaching or leaving a zero value. Flux linkages are nearly in phase with currents. For instance, Fig. 6 shows that the currents on the left core leg windings are peaking while voltage is zero. All these statements are approximate since the presence of tank, core and winding resistances introduce small phase shifts. The numerical results of the short circuit currents and voltages are presented in Table 2.

Fig. 6. Distribution of the magnetic field during the test of short circuit


Table 2. Voltages and currents during the short circuit test

PHASE

Voltage (HV,RMS) [V]

Current (LV,RMS) [A]

Current (HV,RMS) [A]

A 10910.9 ∠0º

318.5943973 ∠91.27109483º

137.804226 ∠-88.8610295º

B 10910.9 ∠120º

317.6049701 ∠-28.72440114º

137.8780422 ∠151.2842585º

C 10910.9 ∠120º

318.5518172 ∠-148.6828969º

137.7490197 ∠31.44052071º

The total power supplied by the voltage sources can now be determined from the calculated voltages and currents of Table 2. Similarly, the I2R losses of the transformer windings can be obtained. Losses in massive conductors are semi-analytically calculated using the surface impedance approach of Section 4. Input powers, winding losses and stray losses are reported in Tables 3 and 4. Fig. 7 shows the distribution of losses in the frames and tank of the transformer.

Table 3. Input power supplied by the sources and losses in the primary

and secondary windings of the transformer

PHASE

Real power supplied by

voltage source [W] Losses, LV [W] Losses, HV [W]

A 29914.52928 13794.16702 19407.7721 B 33747.70946 13708.632 19392.29524 C 37818.18498 13790.51752 19428.67148

Total 101480.4237 41293.31655 58228.73882

Table 4. Losses in structural parts

Structural Parts Losses [W] Tank 1767.49744

Plates Clamps 192.4532 Total 1959.95064

The total three-phase power supplied by the source is 101480.4237 W, which equals the sum of I2R and stray losses: 101482.006 W. This is a simple check that verifies the consistency of the finite-element code.


Fig. 7. Distribution of surface losses in the tank and frame during the test of short circuit

7. Conclusions

A 3D finite-element model was developed to determine the magnetic field and losses of a three-phase three-limb transformer. The load-loss test was simulated to this end, making possible the use of a time-harmonic linear model. The determination of stray losses in the tank and frame of the transformer was achieved using scalar potentials and surface impedance boundary conditions. This resulted in a relatively small 3D finite-element mesh with less degrees of freedom than the required by vector potentials. The numeric model was able to incorporate external circuits through a circuit-field coupling approach. The numerical results show that the balance of powers during the test is completely fulfilled, verifying the consistence of the computations. Currents, voltages and losses also concur with expected experimental results.

References [1] Olivares J.C., Escarela-Perez R., Kulkarni S.V., de León F., Venegas-Vega M.A.:

2D finite-element determination of tank wall losses in pad-mounted transformers, Journal of Electric Power Systems Research (Elsevier), 71 (2004) 179-185.


[2] Olivares J.C., Cañedo J., Moreno P., Driesen J., Escarela-Perez R.: Experimental Study to Reduce the Distribution-Transformers Stray Losses Using Electromagnetic Shields, Journal of Electric Power Systems Research (Elsevier), 63 (2002) 1-7.

[3] Guérin C., Tanneau G., Meunier G.: 3D Eddy Current Losses Calculation in Transformer Tanks Using the Finite Element Method, IEEE Trans. Magn., Vol. 29, (1993) 1419-1422.

[4] Holland S., O’Connell G., Haydock L.: Surface Impedance for 3D Non-linear Eddy Current Problems Application to Loss in Transformer, IEEE Trans. Magn., Vol. 32, (1996) 808-811.

[5] Rodger D., Eastham J.F.: A formulation for low frequency eddy current solutions, IEEE Trans. Magn., Vol. MAG 19, (1983) 2443-2446.

[6] Rodger D., Leonard P.J., Lai H.C., Coles P.C.: Finite element modeling of thin skin depth problems using magnetic vector potential, IEEE Transactions on Magnetic, Vol. 33, (1997) 1299-1301.

[7] Binns K.J., Lawrenson P.J., Trowbridge C.W.: The Analitycal and Numerical Solution of Electric and Magnetic Fields, John Wiley and Sons, 1992.

[8] MEGA V6.27: USER’S MANUAL, Applied Electromagnetic Research Centre, Bath University, UK, 2000.

[9] Carter G.W.: Electromagnetic Field and its Engineering Aspects, Second Edition. Longman, London, 1967.

3-elements to estimate stray losses in transformer

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