Harmonic Domain Modelling of Transformer Core Nonlinearities Using the Digsilent Power Factory Software

8/14/2019 Harmonic Domain Modelling of Transformer Core Nonlinearities Using the Digsilent Power Factory Software

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Electrical Power Quality and Utilisation, JournalVol. XIV, No. 1, 2008

Harmonic Domain Modelling of TransformerCore Nonlinearities Using the DIgSILENTPowerFactory Software

Wojciech WIECHOWSKI1)

, Birgitte BAK-JENSEN2)

,Claus LETH BAK2), Jan LYKKEGAARD1)

1) Danish TSO Energinet.dk; Denmark

2) Aalborg University, Denmark

Summary:T pp ntt t t f pnttn n vtn f nalready existing algorithm that allows for calculating saturation characteristics of single-p p tnf. T t b f t t t n 1993. N talgorithm has been implemented using the DIgSILENT Programming Language (DPL)as an external script in the harmonic domain calculations of a power system analysis toolPFt [10]. T t v b n nt n n-ppower transformer. A theoretical analysis of the core nonlinearities phenomena in single

and three-phase transformers is also presented. This analysis leads to the conclusion thatthe method can be applied for modelling nonlinearities of three-phase autotransformers.

1. INTRODUCTION

In harmonic analysis studies, power transformers can beeither modelled as linear power system components, or usingmore sophisticated models that take into account the iron corenonlinearity and its property of distorting currents resultingin an injection of harmonic currents into the network. Ifthe transformer core is to be modelled as an impedance, it is

sufcient to model the iron core as a shunt linear inductanceand a resistance, connected in parallel. In order to model thenonlinear behaviour of the core, voltage dependence of anequivalent inductance and resistance shall be also included.Presented in this paper modelling of nonlinear cores of powertransformers is a part of a larger project, where an existingcomputer model of the western grid of the Danish TSOEnerginet.dk is to be adjusted for harmonic analysis purposes.Besides all linear power system components, special focusis attached to moidelling fo the nonlinearities present on thetransmission level: converters of HVDC links and powertransformers/autotransformers.

In order to minimize the core size, power transformersare usually designed to operate just below the knee point.Therefore, when the voltage increases above a certain value,the transformer magnetising current becomes distorted; evenat the nominal operating voltage the no-load current THDcan be as high as 20 % [4]. The harmonic contribution from

power transformers can be most likely noticed in a grid withmany transformers installed, during low load hours, whenthe load current is low and the voltage rises [4].

DIgSILENT PowerFactory enables to precisely modeltransformer core nonlinearity in the time domain (EMT)[8]. In the frequency domain it is not possible to do it so far.Since the frequency domain is more convenient for harmonicanalysis than EMT, an attempt was made to model transformercore nonlinearity in the harmonic domain calculations ofPowerFactory. Several ways of modelling the iron corenonlinearities exist, described for instance in [5, 9 or 2].

These methods require that the dynamic hysteresis loops ofthe material are known. This is not always possible, especiallyif the nonlinearities of older transformers are of interest. Inthis project the only data available for the autotransformersare typical test reports. Therefore, a different method has

been found, where the core nonlinearities can be determinedusing a simpler set of input data [7]. For computation, themethod requires only Vrms Irmscurves and the no-loadlosses at fundamental frequency. These data are present inthe test reports. The method was implemented as an externalscript of DIgSILENT PowerFactory using the DIgSILENTProgramming Language (DPL). The implementation of themethod and its verication is illustrated. It is also shownthat the method is applicable for modelling nonlinearitiesof power autotransformers.

2. THEORY

In an iron-core power transformer, the ux and the uxdensityBwithin the transformer core is proportional to the

voltage u1 applied across the primary winding, according to theFaradays law. In the sinusoidal case it can be written [1]:

(1)

1

1

m

m m

UU sin t dt cos t cos t

= = =

(2)

1mU sin t dt

BS S

= =

(3)

where Sis the cross-section of the core.The resultant magnetizing current Imneeded to produce

this sinusoidal ux is related to the magnetic eld strength

1 1m

du U sin t

dt

= =

Key words:autotransformers,harmonic analysis,nonlinearities,simulation software,transformer cores

Wojciech Wiechowski et al.: Harmonic Domain Modelling of Transformer Core Nonlinearities...


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4 Power Quality and Utilization, Journal Vol. XIV, No 1, 2008

Hby a constant value, which is the ratio of number of turnsNto the total length of the magnetic path l:

NH I

l=

(4)

The magnetic ux densityBis related to the magnetic eld

strengthHby the magnetic permeability of the steel used inthe transformer core:

B H= (5)

Equation 5 shows that if the magnetic permeability islinear andBis sinusoidal, the magnetic eld strengthH(andwhat follows, the magnetizing currentIm) will be sinusoidal.If the transformer operates in saturation, is no longerlinear and the magnetic eld strength H is nonsinusoidal,

because the product of Hwith the magnetic permeability must still produce the sinusoidal flux density B, seeFigure 1. If the magnetizing currentImis allowed to ow freely(low impedance pathZfor all harmonics), after decomposing itto Fourier components, it will contain the fundamental frequencyand all the odd harmonics: 3rd, 5th, 7th,9thetc [1].

3. The algoriThm aNd iTs

imPlemeNTaTioN

In order to determine the magnetizing current harmonics,a special algorithm has been implemented [7]. Initially thistechnique was proposed in [3] for estimating the magnetizinginductanceLm. Reference [7] extended this algorithm with

computation of the nonlinear resistance RFe (see Figure2a). The main advantage of this algorithm is that both theequivalent resistanceRFeand inductanceLmare calculateddirectly from the conventional transformer no-load test data:Vrms Irms curve and the no-load losses at fundamentalfrequency. The nonlinear resistance (piecewise linear u-iRcharacteristic) accounts for two effects: linear eddy currentloss and nonlinear hysteresis loss. This resistanceis foundfrom the measured total no-load losses. Calculated in thisway the resistive part of the total, measured no-load currentcan be subtracted geometrically. This allows for a calculationof the current owing through the nonlinear inductance(piecewise lineari

Lcharacteristic). Next, the resulting, real

and imaginary harmonic components are geometrically addedand modelled as a current source. During harmonic load owcalculations, this current source will inject harmonic currentsinto the investigated network (Figure 2b).

The algorithm requires that the following assumptionsand simplications are made:

winding impedances are ignored, supplying voltage and resulting ux are sinusoidal, the piecewise resistance and piecewise inductance are

both symmetric.The algorithm has been implemented in a dedicated simulation

software DIgSILENT PowerFactory (PF). PowerFactory allowsfor creating automatic simulation tasks and build miscellaneousmodels of power system components, as the program possessesits own programming language DPL (DIgSILENT ProgrammingLanguage). The syntax of this language is similar to C++. Thislanguage was used to implement the described algorithm. The

basic idea was that the whole algorithm was programmed asseveral DLP subroutines and a main DPL controlling script.The structure of the algorithm implemented in PF is shownin Figure 3, and individual components of the algorithm aredescribed in the following sections. Special emphasis is put

Fig. 1. Typical relations between main ux , supplying voltage U and

no-load (magnetizing) currentI(without hysteresis effects) [6]

Fig. 2. Equivalent diagram of a power transformer: RHV, RLV winding

resistances,LHV,LLV Leakage inductances; a)RFe magnetizing resistance

hysteresis losses and eddy current losses in the core,Lm magnetizing

inductance; b) In the harmonic domain, nonlinearities are modelled as current

source;Im magnetizing current


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on the implementation details, and calculated, ready-to-useequations.

3.1. rtn mn

The main script Main (see Fig. 3), when executedrecalculates all input data stored in the Data matrix (Um,Im) to the peak values and stores them in the matrix of

results Results. Letter m denotes the number of availablemeasurements stored in Data. Next, the script automaticallyconnects a current source to the HV transformer bus. Thenit initiates conventional load-ow calculations. As a resultof the load-ow, the three-phase voltages at the transformer

bus are obtained. Next, the script Main executes in turn allthe subroutines, beginning with subroutine volt_comp.

3.2. sbtn vt_p

This subroutine compares the calculated terminal voltageswith the voltages stored inData matrix and returns the rownumberfof the closest voltage value. If the bus voltage is

higher than the highest voltage in the input data, the subroutinereturns the max number of rowsf = m. This means that thedistorted magnetizing current will be constant for highervoltage values. This shall not be problematic during normaloperation, since the voltage in no-load tests is usually variedup to 1.1 Un. When Subroutinevolt_comp is accomplished,main script executes the next subroutine iR.

3.3. sbtn r

This subroutine calculates the first linear and nextpiecewise linear elements of the nonlinear resistance curveR. It is done in the following way. For the rst segment, i.e.

up to the rst measured voltage, the peak current and the rstlinear segment of the resistance curve is calculated directlyfrom input data:

1 11 rms r rms

P V I

=

(6)

1

1

1

2

r

PI

V=

(7)

1

1

1

r

VR

I=

(8)

The power measured at higher voltage values is a productof the sinusoidal voltage and nonsinusoidal current Im (seeFig. 4). It can be seen that if linear resistance determined by (8)is used, the resultant current would be sinusoidal, regardlessof the amplitude of the applied voltage. The peak value of thiscurrent would be much lower than in reality. Therefore, in orderto assess the real current and the next segments of the piecewiselinear curve, the power denition must be used:

( ) ( )2

0

2r

P v i d

=

(9)

It is sufcient to calculate the current shape for of aperiod because the voltage is assumed sinusoidal. Therefore,the resultant current will contain only odd harmoniccomponents and therefore, will be symmetrical.

The applied voltage can be written as v(a) = Vk sin(a).The total no-load power loss:

( )

( )

( )

1

2

1

1

1

1

10

1

2

2

1

2

k

k

k

k

k

k k r

k k

k r

k

V sinV sin d

R

V sin V

P V sin I dR

V sin V V sin I d

R

+

= + + +

+

Fig. 3. Detailed structure of the implemented algorithm

(10)



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The angles a1, a2, ..., ak1are calculated from aj = arcsin(Vj/Vk),j= 1, 2, ..., k1 and stored at each recursive step in thematrix of phase angles Angles. The computation using (10)must be done recursively segment by segment. The part of(10) that is underlined with a single line shall be recomputedat each recursive step, because the voltage Vkmust be updated.Part of (10) underlined with double line must be updated andcomputed recursively for each segment up to the last segmentwhere reaches/2. For instance to calculate (10) for k= 3, therewould be 3 unknowns:R2,Ir2andR3, and only one equation.

Therefore, rst it is necessary to compute (10) for k= 2. Thenthe only unknown isR2so it can be computed. Equation 10 can

be rewritten in the form from which it is easy to determine theunknown value of resistance:

k k

k

k

r r

k r k

k k r

b bP a R

R P a= + =

(11)

Components ark and brkwere computed and are givenbelow in a ready-to-use fashion:

( )

( ) ( ) ( ) ( )( )

2

1 1

1

1

1 1

1

2 22k

k

r

k r k k k

V cos sin

Ra

V I cos cos

+ =

(12)

( ) ( ) ( )

( )

( ) ( ) ( ) ( )( )

2

1 1

1

1 1

22

k

k kk kk

k kr

k k k k

cos sin cos sinV

b

V V cos cos

+ =

(13)

Next, the subroutine iRmust calculate the currentIr2 from:

1

1

k k

k k

r r

k

V VI I

R

= +

(14)

WhenR2 andIr2 is known, the calculations can be startedagain, now for k= 3, to determine R3. This procedure isrepeated for each segment (each voltage in theData matrix)except for the rst one, i.e. for k 2. ResistanceRand current

peakIrkvalues are stored in the matrixResults.

3.4. sbtn iL

The ilcurve is calculated by the Subroutine iL. Peakux values kare related to the peak voltage values Vkbythe expression [7]:

k

k

V

=

(15)

where Itrmsk is the total measured rms no-load current,Irrmskis the determined in previous section rms resistivecurrent component andIlrmskis the inductive rms currentcomponent to be computed. For the rst segment in the ilcurve subroutine iLcalculates the assumed linear inductanceusing equations:

1 12l l rmsI I = (16)

1

1

V

=

(17)

1 1

1 1

l l

VL

I I

= =

(18)

For segments k 2, assuming k(a) = ksin(a), rmsdenition for the current will be:

1 2 1

1

1

1

11

0

22

2

1

2

k

k

k

l

k

k

l rms

k k

l

k

Isin

d dsinL

LI

sinI d

L

+ + + + =

+

(19)

This equation is analogous to (10), i.e. it must be computedsegment by segment. This equation can be rewritten in the

form:

( )2

0

1 10

lk lk lk

k k

a b c xL L

+ + + =

(20)

For this quadratic equation it is assumed [7] that alk> 0,blk> 0 and (clk+x0) < 0. Therefore, the linear segments ofthe inductanceLkare calculated from:

( )2 0

2

4

lk

k

lk lk lk lk

aL

b b a c x

= + +

(21)

Components aLk, bLk, cLk,x0were computed and are givenbelow in a ready-to-use fashion:

Fig. 4. Sinusoidal voltage applied to the piecewise linear v-IRcurve and the

resultant, approximated nonlinear current


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( ) ( ) ( ) ( )

( ) ( )

1 1

2

0 2

11

2 2

2

k k k k

k

k k

cos sin cos sin

xL

+

=

(22)

( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )

2 2

1 1 1

1 12

k n n n

lk

k n n n

xa

cos cos

+ =

(23)

( ) ( ) ( ) ( )

( ) ( )

1 1

1

1

2 2

2

n n n n

n n

cos sin cos sin

x

+

=

(24)

( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )

1

1

1

1 1

2

2

n )

n )

Lk n n

lk

Ln n n

I cos cos

b

I

=

(25)

( ) ( ) ( )( )12 1nlk L n nc I =

(26)

and the peak current Ilkmust be also computed at eachsegment from:

( )1

11

k kl l k k

k

i iL

= +

(27)

4.5. sbtn iRiLrms

This subroutine calculates the rmsvalue of the distortedresistive current component Irkrmsand later, the rmsvaluedistorted inductive componentILkrms, at each recursive step.The rmsvalue of the resistive current Irkrmsis found fromthe rmsdenition:

( )

2

2 2

0

2rmsI i d

= (28)

1

2

1

1

1

1

2

10

2

2 1

2

22

1

2k

k

k

k

k

r rms r

k k

r

k

V sind

R

V sin V I I d

R

V sin V I d

R

+

= + + +

+

(29)

Equation 29 is given below in a ready-to-use fashion:

( ) ( ) ( )

( ) ( )

( ) ( ) ( ) ( )

( )

( )

( )

( ) ( ) ( )

( )

( ) ( )

( )

( )

( )

( )

1

1 1

21 1 1

2

1

1 1

2

2

1

2

1 1

2

2

1 12

2

2 2

2

2 22

2 2

2

k

n

n n

,k ,k ,kk

n ,k n ,k

k

n,k n ,k n ,k n ,k n

r rms

k k n n ,k

rn n n,k

n n

r rn n

cos sinV

R

cos sin

V

cos sinR

I

V V V cosI

R R cos

V VI I

R R

+ +

+

+

= +

+ ( ) ( )( )1n n

(30)

If sinusoidal voltage is applied to the parallel connection oflinear inductance and resistance, the resistive current will be

in phase with the applied voltage and inductive current will beshifted by 90. In case of nonlinear resistance and inductance,the resistive and reactive currents will be distorted, i.e.they will contain harmonics but still the adequate harmoniccomponents will be orthogonal with respect to each other[7]. Based on this assumption, the subroutine computes theinductive part of the current subtracting geometrically thecomputed above resistive rms component form the totalmeasured rms no-load current.

Determined in angle domain inductive and resistivecurrent components are stored in separate vectors of thelength 256 points each. Next, a DFT function available in

DPL is applied to the vectors, and the resultant harmonicmagnitudes and phase angles are geometrically addedand assigned to a current source. This current source shallinject the calculated harmonic currents into the investigatednetwork during harmonic load ow studies.

4. VeriFicaTioN

The implemented algorithm has been veried rst with theexample data provided in the original paper [7]. The resultswere identical as in [7], which means that the algorithm

has been implemented correctly. This verication processis presented in detail in [10]. In this paper an additionalverication has been performed, against own, real-worldharmonic measurements performed on a single-phasetransformer. The no-load test was performed. The voltage has

been changed in the range form 0.1 p.u. up to 1.3 p.u. and thefollowing values have been measured: sinusoidal supplyingvoltage Urms, no-load current Irms, harmonic distortion ofthe no-load current, harmonics 3rd, 5th, 7th, 9th, 11th, 13th,no-load lossesP0. Next, the measured no-load current andno-load losses were used as input data for the implementedalgorithm. Determined piecewise linear resistance R and

inductanceLare shown in Figure 5. The resultant inductiveand resistive current components for saturated core (U=1.3 p.u.) are shown in Figure 6. Figure 7 shows the contentof no-load current harmonics versus exciting voltage. It can



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be seen that the variation of the no-load current harmonicsversus exciting voltage measured and determined using thealgorithm is comparable, especially for a heavily saturatedcore. However higher order harmonics show better agreementwith the measured values than lower order harmonics.

5. Three-Phase TraNsFormers

5.1. T-b t-p tnf

Now, a case where three, single-phase transformers arecombined, to create a three-limb transformer but with allthree windings supplied with the same sinusoidal (zero-sequence) voltage U1 is considered, see Figure 8a. Theresultant uxes will have the same direction in all three legs,and therefore they will be pushed-out of the ferromagneticcore and run in the air, oil, tank walls, etc. The magnetic

permeability of the air or oil air is much lower than thepermeability of the ferromagnetic core, and therefore theresultant zero-sequence uxairwill be much lower than the

positive/negative sequence ux . This in turn means that

the zero-sequence equivalent core inductanceLmis muchsmaller than the positive/negative sequence inductanceLm,where the voltages and resultant uxes, shifted with respectto each other by 120 deg run entirely in the magnetic core,see Figure 8b. Moreover the zero-sequence inductanceLm

probably may be assumed linear, because the ux practically

depends on the linear air/oil permeability, and the ux will bereduced because of the higher equivalent reluctance, whichwill cause the core to be less saturated. For the same reasonthe zero-sequence core losses may be ignored.

Therefore, justied would be the assumption that forthree-leg transformers, the magnetizing current caused by thezero-sequence, sinusoidal voltage is also sinusoidal. Besides,it can be concluded that even if the magnitudes of supplyingvoltages are the same in both cases 8a and 8b, the magnetizingcurrent in case 8a will be larger than in case 8b. If the uxairwould run entirely in the air, the resultant equivalentinductance would have similar value like the leakage

inductance of the primary windingLHV. In reality the uxesstill enter the core and therefore the inductanceLmis severaltimes higher than the leakage inductance of the windingsLHV


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Fig. 7. No-load current harmonics versus exciting voltage. Dotted line

measured values, continuous line calculated using the algorithm

voltage will create a three-phase sinusoidal ux that will runentirely in the core. The consequence is that the ux will

pass only through the material that is characterized by a highand nonlinear value of permeability, and the magnetizingcurrent will be nonlinear in the same way like in the case ofa single-phase transformer (ignoring core asymmetries thatintroduce coupling between sequences).

5.2. efft f nn nntn

The analysis presented above was made with theassumption that the primary winding is connected ingrounded star, so the impedance for all harmonics isZ. In thatcase the magnetizing current will contain all odd harmonicsincluding that of the zero-sequence, which are the 3rd, the9th, 15th, etc. If the grounding of the starpoint is removed,the impedance Z for all zero-sequence harmonics would

become innitely large, which means that the triplen currentharmonics will be blocked, see Figure 9. Since (5) must bealways true, the blocked triplen current harmonics will causethe magnetic ux and the voltage at the winding terminalsto be distorted. This is a very undesirable phenomenon. Thezero-sequence ux components will be pushed out of the core(this will result in the lower magnitude of the ux, due tothe higher reluctance) and this ux will in turn induce backzero-sequence voltage in the primary winding.

Therefore, a power transformer connected in ungroundedWye (with no delta connected winding) if exposed tosaturation will produce zero-sequence harmonic voltages.This voltage distortion can be modelled as a voltage sourceconnected across the transformer terminals, see Figure 9.

In this gure, the equivalent diagrams for the fundamentaland higher frequencies of the positive, negative and zero-sequence harmonics are shown, for two transformer windingarrangements: Yyand Yd. When looking at the equivalentdiagram, the advantage of including a winding connectedin delta becomes apparent. If the starpoint of the primarywinding is isolated, winding connected in delta will providethe only path for zero-sequence currents generated by thecore. Therefore, the presence of a delta winding will preventthe transformer from distorting the supplying voltage withzero-sequence harmonics. A power system can usually copewith relatively high levels of current distortion. Bearing inmind that the measure of power quality in a power systemis actually the voltage quality [4], it can be emphasized thatthe distorted current does not have the direct inuence onthe power quality it must be rst coupled by the systemimpedanceZ, and then, by the created harmonic voltage drops

Fig 8. Three-phase transformer supplied with a) purely zero-sequence

voltage U1 and its equivalent circuit, b) three-phase positive sequence

voltages UA, UB, UC, and its equivalent circuit



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it will affect the power quality. All the harmonics for which

the diagrams are valid are created within the transformer,due to the saturation. It shall be also stated that the 4 or5-legged transformers will behave principally like single-

phase transformers the ux will always have a return paththrough the core. There can be small differences noticed foreach phase due to the core asymmetries.

5.3. rttn n n t t f

three-phase transformers

The described in this paper algorithm works for single-phase transformers. However, under certain circumstancesit can be used also for modelling the nonlinearities of three-

phase power transformers. From the theoretical analysisarises that the following restrictions regarding the no-loadtests shall apply: The three-phase transformer must be supplied through

a star connected winding YN, The starpoint of this primary winding shall be grounded

(YN) because otherwise all triplen harmonics will beblocked,

The voltage source that is used for supplying thetransformer must be connected in grounded star YN.

Delta connected winding shall not be used as primarybecause the triplen harmonics will circulate in the winding

and will not be present in the measured current, Also none of the secondary windings shall be connectedin delta. This can be deducted from the equivalentdiagram shown on Figure 9. The zero-sequence harmoniccurrents originating in the core will split and ow intothe supplying winding Yn, any other winding connectedin grounded star yn terminated with a load impedanceconnected in YN, or any winding connected in delta.

It does not matter if the transformer core is 3, 4 or 5-legged,because the supplying voltage and the resulting ux in thecore is sinusoidal and purely positive sequence. No zero-sequence ux will appear due to the core nonlinearity becausethe supplying winding is connected in grounded star.

5.4. T ttnf

The main difference between a regular transformerand an autotransformer is that the autotransformer has the

primary and secondary windings connected in the way thatthe voltage drop on the primary winding rises the potentialof the secondary winding, see Figure 10.

Therefore, the secondary voltage to ground is the sumof the voltage drops on the primary and secondary coils.By raising the voltage, the power that can be transferredthrough the transformer is higher. However, this operationremoves the galvanic insulation between the windings. Thereis also a difference for the dccomponent. The dccurrentis not transferred through a regular transformer while in aautotransformer it will split and the major part will closethrough the resistance of the primary winding and theremaining part can ow into the secondary side, dependingon the dc(zero-sequence) resistance of the loads. The no-load testing procedure performed on an autotransformer

does not differ from the procedure applied when testinga regular transformer. If the transformer is supplied withnominal voltage, the resultant ux density in the core hasalso its nominal value, whichever winding is used as the

primary. The core nonlinearity of power autotransformerscan be modelled using the described algorithm if theycomply with the restrictions described for regular three-

phase power transformers. The autotransformers installed inthe Energinet.dk power system are YNynconnected and someof them have an additional tertiary winding. However, thiswinding is normally not used and therefore, it is connectedin open delta. Consequently it will not affect the ow ofharmonic currents because no current can circulate andthey cannot produce opposing ux. Consequently it can beconcluded that if three-phase power autotransformers complywith the restrictions described in section V.C, the algorithmcan be used for modelling their core nonlinearities.

6. summary aNd coNclusioNs

It is possible to implement the algori thm in thePowerFactory simulation software,

The algorithm has been veried rst against the originaldata provided in [7] and the agreement was perfect,

which proves that the algorithm has been correctlyimplemented,

The comparison of the results with own measurementson a small single phase transformer has shown that

Fig. 9. Equivalent diagrams of transformers connected in a) yY and

b)yD

Fig. 10. Process of creating the equivalent diagram of an autotransformer,

which is useful for harmonic analysis


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11

higher order harmonics show better agreement with themeasured values than lower order harmonics,

A difference in the measured and calculated 3rdharmoniccontent in the no-load current, for low exiting voltage(0.8 0.9 p.u.) has been noticed but the reason for thishas not been revealed,

Based on the performed theoretical analysis it can bestated that the algorithm initially valid for single-phasetransformers can be used for modelling nonlinearities ofthree-phase autotransformers, however verication withmeasurements would be desirable,

The algorithm allows assessing the levels of harmonicdistortion due to the aggregated effect of powerautotransformers on the harmonic levels in the Danishtransmission grid.

7. FUTURE WORK

Tat the most important future task shall be to verify the

implemented algorithm with precise measurements on full-size power autotransformers.

ackNowledgemeNT

The authors wish to gratefully acknowledge thevaluable assistance of M. Pller and E. Zimmermann fromDIgSILENT GmbH.

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of Dynamic Hysteresis Loops and its Application to a SeriesFerroresonant Circuit.Proc. IEE 117, January 1970, pp. 234240.

10. Wiec how ski W.: Harmonics in Transmission Power Systems.PhD dissertation, Aalborg University, Denmark, 2006, ISBN978-87-89179-69-8.

Wojciech Wiechowskireceived the M.Sc. degree from Warsaw University ofTechnology in 2001 and the Ph.D. degree from AalborgUniversity, Denmark in 2006. From 2001 to 2002 heworked for HVDC SwePol Link as a Technical Executor. In theperiod from 2002 to 2006 he was with the Institute of EnergyTechnology, Aalborg University, rst as a PhD Student andlater as an Assistant Professor. Since 2006 he has been

employed in the Planning Department of the Danish TSO Energinet.dk. His

current responsibilities include various power system analyses related to theplanning of the transmission network with extensive use of long AC cablelinks and wind power generation. He is a Senior Member of IEEE.

Birgitte Bak-Jensenwas born in Grenaa, Denmark, on August 27, 1961. Shereceived the M.S. degree in electrical engineering and thePh.D. degree in modeling of high-voltage componentsfrom Aalborg University, Aalborg East, Denmark, in1986 and 1992, respec-tively. From 1986 to 1988, she waswith Electrolux Elmotor A/S, Aalborg East, Denmark,as an Electrical Design Engineer. She is an Associate

Professor in the Institute of Energy Technology, Aalborg University, whereshe has worked since August 1988. Her elds of interests are modelingand diagnosis of electrical components in power systems, power quality,

dispersed generation and stability in power systems with large integrationof wind power.

Claus Leth Bakwas born in rhus in Denmark, on April 13, 1965. Hegraduated from High School in rhus and studied atthe Engi-neering college in rhus, where he receivedthe B.Sc. with honours in Electrical Power Engineeringin 1992. He pursued the M.Sc. in Electrical PowerEngineering with specialization in High VoltageEngineering at the Institute of Energy Technology

(IET) at Aalborg University (AAU), which he received in 1994. Afterhis studies he worked with Electric power transmission and substationswith spe-cializations within the area of power system protection at the

NV Net company. In 1999 he got employed as an assistant professor atIET-AAU, where he is holding an associate professor position today. Hismain research areas include corona phenomena on overhead lines, powersystem transient simulations and power system protection. Main teachingareas are high voltage technology, power system analysis and simulation,relay protection and electrophysics. He works as a consultant engineer forelectric utilities, mainly in the eld of power system protection and powersystem analysis.

Jan Lykkegaardwas born in Denmark in 1968. He received the M.Sc.degree in electrical engineering in 1994 from Instituteof Energy Technology, Aalborg University. From 1995to 1999 he was employed at Elsam. Since 1999 he hasbeen working in the Substation Department of the Danish

Transmission System Operator Energinet.dk. His mainresponsibilities include transmission transformers, shuntreactors and various R&D tasks.


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