Electric Power Systems Research 78 (2008) 18141818

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Electric Power Systems Research

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3D nite-element determination of stray losses in power transformer

Livio Susnjica,, Zijad Haznadarb, Zvonimir Valkovicc

a Faculty of Engineering, Vukovarska 58, 51000 Rijeka, Croatiab Faculty of Electrical Engineering and Computing, Unska 3, 10000 Zagreb, Croatiac Polytechnic of Zagreb, Konavoska 2, 10000 Zagreb, Croatia

a r t i c l e i n f o

Article history:Received 16 March 2007Received in revised form 3 August 2007Accepted 10 March 2008Available online 22 April 2008

Keywords:Stray lossesFinite-element analysesEddy currentsPower transformer

a b s t r a c t

This paper presents a three-dimensional (3D)nite-element (FE) anain the tank walls and yoke clamps of a three-phase 40MVA powermodel is used to compute the magnetic leakage eld in the case of atransformer. Three cases are analyzed to study the impact ofmodelinFE context in estimation of their losses. The load loss test was carriedto validate the simulation.

2

1. Introduction

Accurate predictions of strayand their reduction mechanismtransformer design. The stray loshysteresis effects inside the yokeportion of stray losses in carbonlosses [1], whereas hysteresis lo30% of total stray losses. Several alosses of the power transformer, nlosses [2,3]. To determine stray(3D) nite-element (FE) analysis o3D geometry discretization of ahuge number of nodes and elemdimensions aremeasured inmetetank walls and yoke clamps whecurrent is measured in millimeteof nodes and elements and to avoid using very demanding com-putational resources, the computation of eddy current losses by3D FE is made using a different model of the tank and yoke clampsin the context of the nite element [46]. The objective of thisresearch is to investigate the impact of different modeling for thetank walls and yoke clamps on th

Corresponding author at: Faculty of Enneering, Vukovarska 58, 51000 Rijeka, Crofax: +385 51 651416.

E-mail address: livio.susnjic@riteh.hr

ouslyof tlateepenedandaneticcircuof tf regntederfors ar

The transformer FE model is shown in Fig. 1, and the relevantdata are given in Table 1. Fig. 2 shows a sketch of the transformercross-section. The tank walls and clamps thickness are 10mmand 25mm, respectively. BH data of carbon steel material usedfor the tank and clamps are given in Table 2. The rated ampere-

0378-7796/$ see front matter 2008 Edoi:10.1016/j.epsr.2008.03.009e computed eddy current losses.

gineering, Department of Electrical Engi-atia. Tel.: +385 51 651435;

(L. Susnjic).

turns are prescribed for the appropriate coils of each phase. Theampere-turns balanced equation corresponding to the short circuitcondition, in phasor form is

IH(NH + NR) + ILNL = 0 (1)Balance of the ampere-turns can be assumed for the coils

wound on the same leg. With the exception of the coils region,

lsevier B.V. All rights reserved.losses of a power transformers are necessary for improvingses arise from eddy current andclamps and tank walls. The mainsteel plates are the eddy currentsses comprise between 25 anduthors have considered the strayot taking into account hysteresislosses, a full three-dimensionalf thewhole structure is required.power transformer requires a

ents because overall transformerrs and there exist regions such asre the penetration depth of eddyrs. In order to reduce the number

This paper summarizes previresults for simulation by usemethod. First, yoke clamp pmodeled with skin depth indwith the linear surface impthe non-linear surface impeleakage eld based on magnlated for a transformer shortThe magnetic non-linearityconsidered. The inuence ocomputed losses is also prese3D V9.3 [9]) was used to ppaper. The computed resultexperimental ones.

2. Numerical analysislysis of eddy current losses generatedtransformer. The time harmonic FEshort circuit condition of the powerg tankwalls and yoke clamp plates inout on an experimental transformer

008 Elsevier B.V. All rights reserved.

works [7,8] and in addition giveshe non-linear surface impedances and unshielded tank walls aredent shell elements, rather thance method, and in the end withce method. The electromagneticscalar potential has been calcu-it condition with a rated current.he transformer core material isulating coil tapping position on. A commercial FE software (Fluxm the simulation shown in thise discussed and compared with

L. Susnjic et al. / Electric Power Systems Research 78 (2008) 18141818 1815

Fig. 1. The transformer FE model (tank is not shown).

Table 1Transformer data

Symbol Quantity Value

S Rated power 40MVAf Frequency 50HzVH/VL Rated voltages 11015%/21kVIH/IL Rated currents 209.9/1100ANL/NH/NR Number of turns 152/677/120+120 Clamp plate and tank conductivity 5106 S/m

the sub-regions of the calculation domain are dened with thetotal scalar magnetic potential formulation. Reduced potentialdescribed the coils region. The calculation of the magnetic eldfrom the Biot-Savarts law allows for the exclusion of the coils fromthe nite element mesh.

The transformer is reconectable on the high voltage (HV) side.The current in the coils and the number of turns correspondingto a tapping position are given in Table 3. Estimation meth-ods for computing the eddy current losses are briey presentedas follows.

Fig. 2. The transfor

Table 2BH data of carbon steel material used for the tank and clamps

H (A/m) B (T) H (kA/m) B (T)

16 0.051 1.0 1.25033 0.1 1.26 1.35084 0.238 1.59 1.436119 0.324 2.0 1.504219 0.527 2.52 1.551288 0.642 3.97 1.631483 0.89 6.25 1.720619 1.01 7.82 1.763788 1.13 9.80 1.8

2.1. Skin depth independent shell elements

The thin steel plates can bemodeled bymeans of surface region,given frequency, permeability and conductivity of the material.Shell elements independent in terms of skin depth have been usedfor calculatingeddycurrent losses. The tangential componentof themagnetic eld in a thin plate of thickness e through the depth oftheplate in the z-direction is described analytically by the followingexpression:

Ht(z) = 1sh(ae)[H1tsh

(ae

2+ az

)+ H2tsh

(ae

2 az

)](2)

where a= (1+ j)/, H1t and H2t are the eld values on both sidesof the plate and is the skin depth.The volume current densityvariation has a tangential compon

J(z) = ash(ae)

[H1tsh

(ae

2+ az

)

Eddy current loss per surface uni

P =e/2

e/2

12

J(z)2 dz

where is material conductivity.

2.2. Linear surface impedance met

Surface impedance links the cotangential on the thin steel surfacthe electric eld E:

o of tHs:

s de

sitynxE = Zsnx(nxH)For linear material it is a ratithe tangential magnetic eld

Zs = EsHs =1 + j

The surface current density i

K = nx HsThe steel platepower lossdenmer cross-section.

by

P = 0.5Re(Zs)Hs2

Table 3Coils data

Tapping position Turns number (L

15% 152/677/00 152/677/120+15% 152/677/120+12ent only, and is described by

H2tsh(

ae

2 az

)](3)

t in the plate is

(4)

hod

mponent of the magnetic eld He to the tangential component of

(5)

he tangential electric eld Es and

(6)

ned as

(7)

(surfacedensity) inW/m2 is given

(8)

V/HV/RC) Current (A) (LV/HV/RC)

1100/247.1/01100/209.9/209.9

0 1100/182.4/182.4

1816 L. Susnjic et al. / Electric Power Systems Research 78 (2008) 18141818

2.3. Non-linear surface impedance method

According to [5], the surface impedance for non-linear material(permeability depends onmagnetic ux density) over a large rangeof elds (from low to high elds), and is given by the followingformula:

Zs = kw(Hs)Zsl + (1 kw(Hs))Zsnl (9)where Zsl and Zsnl are the surface impedances for the linear andnon-linear material, respectively, and kw is the weighting function.

Weighting function kw(Hs) is:

kw(Hs) = 11 + k(Hs/Hk)(10)

Hk corresponds to the value of the magnetic eld at the knee of theBH curve, and k is the coefcient to be chosen. The magnetic eldreaches the tank wall and clamp plates in a mostly normal direc-tion. In this case, the electric eld is mostly tangential, remainingunchanged through an interface and is assumed to be mostly sinu-soidal. The value of coefcient k equals 1 in this case of sinusoidalelectric eld [5].

The steel plate power loss density in W/m2 is calculated by (8).

3. Results

Calculations on a three-phase, three-limb transformer ratedat 40MVA and 110/21kV have been performed. The computedresults by skin depth independent shell elements and by linear sur-face impedance method are shown in Figs. 3 and 4, respectively.These results correspond to the regulating coil tapping position 0.Figs. 3a and 4a shows the relative tank permeability (rt) depen-

Fig. 3. Permeability dependence of the: (aand varied relative permeability of the yofor both xed (rt = 500) and varied relaindependent shell elements).

Fig. 4. Permeability dependence of the: (aand varied relative permeability of the yofor both xed (rt = 500) and varied relatiimpedance method).

dence of the tank loss values for bpermeability (rcl = 500) and vayoke clamps (rcl =rt). Figs. 3b apermeability (rcl) dependence oparametrically given (rt = 500)of the tank (rt =rcl). The variafor the tank and yoke clamps is100 to 1000, with an incremenrent losses computed by modelwith skin depth independent sheimpedance method are in closeis obvious that the losses dependity. The clamp plates loss dependas well as on the permeability ofa higher prescribed permeabilityleakage eld in the clamp plate area, and as a consequence reducedyoke clamps losses. A similar conclusion applies to the tank loss.The skin depths variation for linear analyses is from 3.18mm to1mm, for the chosen relative permeability of the steel from 100

sses computed by simulation with thee method, for different regulating coilin Table 4. It could be seen that theear surface impedance analyses are at

es (W) Tank (W) Total losses (W)

10,700 14,50819,680 25,20236,476 44,406) tank loss value for both xed (rcl = 500)ke clamps and (b) yoke clamps loss valuetive permeability of the tank (skin depth

to 1000. The eddy current lonon-linear surface impedanctapping positions, are givenlosses obtained with non-lin

Table 4Computed losses

Tapping position Clamp plat

15% 38080 5522+15% 7930) tank loss value for both xed (rcl = 500)ke clamps and (b) yoke clamps loss valueve permeability of the tank (linear surface

oth parametrically given clampsried relative permeability of thend 4b shows the relative clampsf yoke clamps loss values for bothand varied relative permeabilitytion of the relative permeabilitysimultaneous in the range fromtal value of 100. The eddy cur-ing tank walls and yoke clampsll elements and the linear surfaceagreement. From Figs. 3 and 4 iton the chosen steel permeabil-

s on the permeability of the platethe tank. It has been shown thatof the tank results in a reduced

L. Susnjic et al. / Electric Power Systems Research 78 (2008) 18141818 1817

Fig. 5. Distribution of eddy current densimpedance method).

Fig. 6. Surface power density on the clamethod).

least 30% higher than those with lpendent skin depth shell elementthe surface eddy current densitypower loss density on the clampPower loss density distribution oin Fig. 7. Distribution of the magnthe symmetry plain outside the cin Fig. 8. Maximum value of the le

4. Experimental validation

According to IEEE Std. C57.12.9should be measured at a load cu

the tank inner surfaces (non-linear surface

ity on the tank wall (non-linear surface

Fig. 7. Power loss distribution onimpedance method).mp plate (non-linear surface impedance

inear surface impedance or inde-s. Fig. 5 shows the distribution ofon the tank wall. Distribution ofplate surface is shown in Fig. 6.

n the inner tank surface is shownetic induction or leakage eld onore (in the coils regions) is shownakage eld is 0.22T.

0, power transformer load lossesrrent equal to the rated current

Fig. 8. Distribution of magnetic induction (leakage eld) on symmetry plain in thecoils region (max. value is 0.22T).

Fig. 9. The transformer during itsmanufacturing (Koncar Power Transformers Ltd.).

1818 L. Susnjic et al. / Electric Power Systems Research 78 (2008) 18141818

Table 5Discrepancy between computed eddy current and measured stray losses

Tapping position Total losses I2R Winding eddycurrent losses

Tank and clampsstray losses

Tank and clamps eddycurrent losses

Discrepancy eddycurrent vs. stray losses

15% 209,500 169,500 19,200 20,800 14,508 30.2%0 210,100 155,600 21,100 33,400 25,202 24.5%+15% 228,900 146,200 27,600 55,100 44,406 19.4%

for the corresponding regulating coils tapping position. The loadloss test is accomplished by short-circuiting the secondary wind-ing and applying a reduced voltage to the primary winding, i.e. thevoltage necessary to cause a rated load current to ow. A 40MVAtransformer was used to investigate stray losses (Fig. 9). The powerabsorbed in the short-circuit test consists of the I2R and eddy cur-rent losses in the winding, and stray losses in constructive steelparts. Stray losses in the constructive steel parts are obtained bysubtracting I2R losses and eddy current losses in the winding fromthe power obtained in the load test [10]:

Pstray = Pload Pi2R Pec (11)where Pload is the load losses (W); Pi2R the I

2R losses inwinding (W)and Pec the winding eddy current losses (W).

The stray losses obtained by (11) are treated asmeasured losses.The winding eddy current losses are calculated analytically byknown distribution of the magnetic leakage eld, calculated previ-ously. The magnetic leakage eld inside windings are calculated asa 2Daxisymetric eld [11]. Themethoddescribed inRef. [12] is usedto estimatewinding eddy current losses. Thewinding eddy currentlosses depend on the tapping position of the regulating coils, e.g.for a 0 tapping position there are 21.1 kW. Table 5 shows the mea-surement values of total load losses and I2R in windings, calculatedwinding eddy current losses, stray losses and calculated eddy cur-rent losses in the tank walls and clamp plates. Also, stray lossesfor different tapping positions are comparedwith the eddy currentlosses computed bymodeling clamnon-linear surface impedance mthe computed and the experimendiscrepancy results due to approplates geometry and in not takinFor the three methods mentioneeling clamp plates geometry is ntight the coils) are not included. Mhigh in permeability and liable tocausing eddy current losses.

5. Conclusion

In this paper, power transformclamps and unshielded tankwalls

The permeability of the tank and clamps has a signicant inuenceon eddy current losses. The losses obtainedwith non-linear surfaceimpedance analyses are at least 30% higher than those with linearsurface impedance or independent skin depth shell elements. Straylosses are a function of many factors including the physical geom-etry of the cores and coils, the voltage class of the transformer, andthe material used in the tank and clamps construction. The com-putedvaluesof lossesby3DFEanalysis donotmatchclosely the testvalues. The comparisonbetween the computed and the experimen-tal results shows the discrepancywhich arise due to approximationin the modeling clamp plates geometry (where the brackets werenot included) and in not taking hysteresis losses into account. Thetappingposition is showntohavea strong inuenceon theobtainededdy current losses values.

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3D finite-element determination of stray losses in power transformerIntroductionNumerical analysisSkin depth independent shell elementsLinear surface impedance methodNon-linear surface impedance method

ResultsExperimental validationConclusionReferences