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Hindawi Publishing CorporationAdvances in Power ElectronicsVolume 2013, Article ID 249146, 8 pageshttp://dx.doi.org/10.1155/2013/249146

Research ArticleTransformer Model in Wide Frequency Bandwidth forPower Electronics Systems

Carlos Gonzalez-Garcia and Jorge Pleite

Electronic Technology Department, Carlos III University of Madrid, Avenida de la Universidad 30, Legans, 28911 Madrid, Spain

Correspondence should be addressed to Carlos Gonzalez-Garcia; [email protected]

Received 1 June 2012; Revised 30 October 2012; Accepted 30 November 2012Academic Editor: Francesco Profumo

Copyright 2013 C. Gonzalez-Garcia and J. Pleite. is is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

e development of the smart grids leads to new challenges on the power electronics equipment and power transformers. euse of power electronic transformer presents several advantages, but new problems related with the application of high frequency voltage and current components come across. us, an accurate knowledge of the transformer behavior in a wide frequency rangeis mandatory. A novel modeling procedure to relate the transformer physical behavior and its frequency response by means of electrical parameters is presented. Its usability is demonstrated by an example where a power transformer is used as lter and voltage reducer in an AC-DC-AC converter.

1. Introduction

e future trends on the power delivery are focused onthe development of a smart grid. is concept implies new challenges as interconnection or treatment of renewableenergy resources and the application of technologies such astelecommunication or powerelectronicswhose products andsolutions should play an important role in the future of themedium voltage distribution.

e power transformer is widely spread in electricalsystems providing basic functionalities such as voltage insu-lation and voltage adaptation. As an essential element onthe energy distribution, it must be involved in the new smart grid perspective, and therefore new roles regardingits manufacturing, maintenance, use, and operation must betaken into account. One of these roles consists in the analysisof its behavior in a wide frequency bandwidth, not limited tothe 50 or 60 Hz power frequency.

Considering the transformer maintenanceprogram,mostof routine tests to assess the transformer condition are basedon power frequency (50/60 Hz) measurements (load and noload losses, capacitance, and Tan-Delta). However, the FRAtest measures the impedance between two terminals of thetransformer in the 20 Hz to 20 MHz bandwidth. It has been

extensively proven that thisprocedureallows the detection of failures that are not visible for other techniques [1].

Considering the power delivery, the conventional line-frequency transformers are not able to deal with powerquality problems (e.g., sags, swells, icker, and harmonics).To solve this task, there have been some attempts (althoughstill under development) for the installation of additionalequipment as power electronics converter that operates athigher switching frequencies in the medium voltage grid.It is also applied in the LV grid due to the advances insemiconductor technology (faster switching actions, higherblocking voltages, and higher power densities) and the devel-opment of new magnetic materials with low loss densities athigher operating frequencies [2, 3]. e application of PETsis extensible to traction applications [4] or Plug-in ElectricalVehicle (PEV) charging stations [5].

is solution implies replacing the typical powerfrequency distribution transformer by a high frequency switched one, commonly named as PET or Intelligent Solid-State Transformer (ISST) consisting in a Medium Frequency Transformer (MFT) and one or several power electronicconverters. As a result, fast switching actions are applieddirectly to transformer terminals, so that overall magnetic volume is reduced and more compact converter designs are

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2 Advances in Power Electronics

200

100

0

100

200 Frequency (Hz)

L o w

f r e q .

C HV Xindependent

H.F.W.M

Ld Xcoupled

Ln X coupled

M e d . f

r e q .

V . H

. F .

H i g

h f r

e q . p

h I

H i g

h f r

e q . p

h I I

C H V X a n

d

C L V X c o u p

l e d

100 101 102 103 104 105 106 107

100 101 102 103 104 105 106 107

102

104

106

m o d u l u s

( d b

s )

p h a s e

( )

F 1: Identi cation of the most signi cant bandwidths in anend-to-end open circuit typical response.

Test equipment

+

+

(50 Ohmns)

Power transformer (device under test)

F 2: Electromagnetic eld distribution at low frequency band-width. Mainly magnetic on the core.

Test equipment


+

+

(50 Ohmns)

F 3: Electromagnetic eld distribution at medium frequency bandwidth. Mainly electric eld along the windings .

reached [6]. However, new problems regarding losses oroverheating and subsequent insulation damages and loss of transformer life expectancy could come across. erefore, anaccurate knowledge of the transformer in a wider frequency bandwidth is needed to tackle with new analysis.

(50 Ohmns)

Test equipment


+

+

F 4: Electromagnetic elddistributionathigh frequencyphaseI bandwidth. Coexistence of electric eld along the windings , andmagnetic eld on core 1 and core-dielectric 2 .

(50 Ohmns)

Test equipment

+

+

Winding (device under test)

F 5: Electromagnetic elddistributionathigh frequencyphase

II bandwidth. Coexistence of electric

and magnetic

,

eldexclusively along the measured winding.

Transformer models have been used to interpret andanalyze the transformer behavior, but they have been com-monly limited to a narrow bandwidth centered in the powerfrequency.

In this paper, a novel procedure to obtain transformermodels, that are able to represent the transformer behaviorin a wide frequency bandwidth, is presented and applied toa 25 kVA 16 kV/420V YNYn distribution transformer. eaccuracy of the model is evaluated comparing the simulation

and actual responses, obtained by an FRA test.Finally, the transformer model is used as part of an AC-DC-AC power electronic converter. e exact knowledge of the frequency behavior allows taking advantage of the l-tering characteristics associated with the leakage inductanceand shunt capacitance of the windings, saving the use of theexternal C lter in the nal stage of the 3-phase setup.

2. Modeling Procedure

e proposed solution to obtain a wide frequency model isbased on a novel modeling proceduredeveloped in [7] whosemain features are as follows.

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Advances in Power Electronics 3

0

1 2

3 4 5

Rln U Rln V Rln W

Ni HV U Ni HV V Ni HV W

Rld URld V Rld W

Ni LV a Ni LV b Ni LV c

+

+

+

++

+

F 6: Magnetic circuit based on reluctances and magnetomotive forces for representing the magnetic eld behaviour in the core andcore-dielectric at low and high frequency phase I bandwidths.

+

+

+

0

1

2Rle 2

Rle 1

RlaNi e2

Ni e1

Ni e Rle

F 7: Magneticcircuitbased onreluctancesandmagnetomotiveforces for representing the magnetic eld behaviour in the windingat high frequency phase II bandwidth.

(i) e input data are obtained by the Frequency Re-sponse measured in a wide frequency bandwidth onthe FRA test.

(ii) e resulting model is based on a parametric struc-ture developed using the Magnetic-Electric Dual-ity Principle and Energy Storage Concept. esesynthesizing tools allow the interpretation of theelectromagnetic phenomena occurring not only onthe magnetic core as other solutions proposed butalso on the leakage path and dielectric media alongthe winding. Subsequently, the physical behavior isconcentrated on an electrical circuit.

(iii) e structure is built in a modular way and consti-tutedby plugging in different andindependent blocks

representing only the physical constitution of any internal element. e vector group connection of thewindings (Star or Delta) is externally implementedon the model and is independent of the physicalsubmodels or blocks.

(iv) e value of the parameters is calculated using an adhoc optimization algorithm. It searches the solutionthat minimizes the difference between simulated andactual measured data. e algorithm is separately applied to the different blocks building the completemodel and responsible for the response in a spe-cic frequency bandwidth of the entire measurementsweep.

As a result, the modeling procedure follows three stages:

the structure design (stage 1, Section 2.1), the algorithmdesign for calculation of the value of the parameters(stage 2, Section 2.2), and experimental validation (stage 3,Section 2.3).

2.1. Structure Design. e design of the structure is the rststage on the modeling procedure and it follows, for its part,other three steps.

e rst one is the qualitative analysis of the electromag-netic eld at different frequency bandwidths in the FRA test,divided in the partial bandwidths of Figure 1 and namedas low frequency bandwidth, medium frequency bandwidth,and high frequency phase I and II bandwidths.

e constitution of the eld at the low frequency band-width is depicted in Figure 2, where the measurement setupis included and the magnetic ux paths are represented by the 1 and 2 solid lines. e medium frequency bandwidthis dened as the range where the constitution of the analyzedeld is mainly electrical (representedwith the E dashed lines)and distributed along a structure dened along the windings(not only in the excited one but also in the rest due to themagnetic coupling) in the fashion depicted in Figure 3 withdashed lines.

e electromagnetic effects in the high frequency phase Iand phase II bandwidths dened in the modeling procedureare represented in Figures 4 and 5, respectively.

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Rd W

Ld W

Ln W

Ci W

W

Rd V

Ld U

Rd URe1

Le1

Ce1

U La

Ra

Ideal transformer

ideal transformerLd V

Ln V

Ln U

Ci V

C

Ci U

b

V1

2

1

2

1 2 122 11 2

1

a C W

U

a

b

V

CeRe

Le1: 1

+ 2

2

+ 3

3

+

+

1+ +

+

+

+

F 8: Complete electric circuit. Interwinding capacitances Ci U, Ci V, and Ci W added.

F 9: 25 kVA 16.125 kB/420 V YNyn distribution transformermeasured and modeled.

e second step designing the structure consists oncombining the obtained results in a magnetic circuit thatsimulates the same magnetic eld distribution.

e referenced magnetic eld in the core ( 1 in Figure 4)and in the core-dielectric ( 2 in Figure 4) for low and highfrequency phase I bandwidths can be represented by thewell-known magnetic circuit of Figure 6. e parametersRln X (where X is one of the U, V, or W phase of the 3phase transformers indistinctly) represent the reluctances of the magnetic paths in the core and therefore depending onthe core geometry and its permeability . e parametersRld X represent the reluctances of the magnetic paths in

0100

200

Frequency (Hz)

Frequency (Hz)

200

100

100 101 102 103 104 105 106 107

100 101 102 103 104 105 106 107

F 10: Actual versus simulated measurement. YNyn congura-tion end-to-end open circuit, phase U HV.

the core-dielectric and depending on the winding geometry and mainly the dielectric permeability . e sources NiHV X and Ni LV X represent the magnetomotive forcesof the HV and LV windings, respectively. Using the samerepresentation, the magnetic circuit depicted on Figure 7represents the magnetic eld distribution for the denedhigh frequency phase II bandwidth, represented in Figure 5,where Rla represents the reluctance of the magnetic path of the dielectric surrounding the turns of the winding. Rlerepresents thesame reluctance but fora groupof turns in the

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0

100

200

Frequency (Hz)

Frequency (Hz)

200100

100 102 104 106

100 102 104 106

F 11: Actual versus simulated measurement. YNyn con gura-tion end-to-end open circuit, phase V HV.

winding. Finally, Nie represents the magnetomotive force of a group of turns in the winding.e third step designing the structure consists in obtain-ing the nal equivalent electric circuit using the well-established Magnetic-Electric Duality Principle to convertthe magnetic circuit into electric one and the Energy StorageConcept, adding and elements to represent the lossesand electric eld eect, respectively. e resulting threesubmodels are

(i) LFCM submodel representing the magnetic core, atthe low frequency bandwidth of Figure 1;

(ii) HFCM submodel representing the core anddielectricmedia at the high frequency phase I bandwidth of Figure 1;

(iii) HFWM submodel representing the winding at themedium frequency and high frequency phase IIbandwidths of Figure 1.

e complete transformer model in Figure 8 is composedof the models representing the three previous submodels

(for simplicity only U phase for HV winding is completely represented with 2 groups of turns) and the capacitors CiX that represent the electric energy storage on the dielectricmedia between HV and LV windings on the generic X phase.

e YNd con guration is represented in Figure 8 to illustratehow easily the vector group is implemented only connectingtheoutput terminalsof thewindings. Any of these submodelsis in charge of a speci c bandwidth de ned in Figure 1.

2.2. Algorithm Design. e second stage of the proposedmodeling procedure consists in designing the mathematicalalgorithm in order to t the actual measured response by the simulation of the model. In this case, the optimization

algorithm is completely developed in [7] and its workingprinciple is summedup in theobjective function given by (1).

= == measured 2

submodel measured 2 = 0,(1)

where represents the total error when comparing thesimulated frequency response of the model and the actualresponse of the measuredtransformer, represents a generic

, , or parameter of a submodel, is the indexrepresenting each frequency point in a [ , bandwidth,measured represents the complex impedance measured in

the frequency, submodel() is the complex impedance,depending of the

,

, and

parameters of the submodels

and the angular pulsation for a frequency .2.3. Experimental Validation. e third and last stage in theproposed modeling procedure consists in the experimental validation of the resulting model. e model is approvedif it is able to simulate the FRA measurements. For thiscontribution, a 25 kVA, 16.125 kV/420V YNyn two-windingthree-phase distribution transformer (Figure 9) has beenmeasured with an FRA commercial device and modeled by implementing the mathematical algorithm in mathemati-cal/simulationso ware. e calculatedparameters areshownin Table 1 where they are split using the submodel division.

Figures 10 and 11 show the comparison between theactual measurement andthe simulated response of the modelimplemented in MATLAB Simulink.

e modeling procedure is able to produce generalmodels with the following features.

(i) e transformer behavior in a wide frequency band-width is related with the parameters of the model.

(ii) e submodels are obtained based on general andcommon physical principles, and particularizationsfor speci c transformers are avoided. However, if particularization needed to be considered, it could besolved with a concrete analysis of the electromagnetic

eld, as shown in Section 2.1.(iii) e vector group connection is independent of the

electrical structure that represents the physical behav-ior of the transformer. As a result, any possible vectorgroup can be modeled by changing the terminalconnections of the model. is same conclusion canbe applied for any FRA measurement setup.

(iv) Modeling of -windings transformer is straightfor-ward. Only the addition of number of windingsubmodels is needed.(v) Modeling of the autotransformer is straightforward.

Only an external series connection of series andmutual windings is needed.

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AC-DC-AC PWM converter

AB

C

AB

C

ABC

ABC

A

B

C

ABC

A B C

AB

C

A

BC

WU

V 45

6

Rectier

abc

PWMIGBT inverter

g VnUn

Uini

ViniWini

Wn

CnBn

A n

AiniBiniCini

Trans. model

a

bc

c

A B C

1

2

3

25 kV, 60 Hz10 MVA

25 kV/600 V50 kVA

L1

Measure50 kW

380 V rms50 Hz

Scope

Pulses

mDiscrete

PWM generator

Voltage regulator

+ +

+

+

+

+

ab inverter

ab inverterDiscrete,

= 2 006 s.Ideal switch

1

1

ab

ab

bcbc

cacaabc

ref (pu)Uref abc inv abc (pu)

d-ref (pu)LC lter

F 12: AC-DC-AC converter.

0.15 0.16 0.17 0.18 0.19 0.2

Time (s)

1000500

0500

1000

0.15 0.16 0.17 0.18 0.19 0.2

Time (s)

600200

200

600

0.15 0.16 0.17 0.18 0.19 0.2

Time (s)

30

10

10

30

F 13: Upper (le ) trace: PWM voltage; upper (right) trace: 3 phase to phase symmetrical voltages when lter is used; lower trace: 3phase to phase symmetrical voltages when transformer model is used.

e capability of the resulting models can be exploitedfor sensitivity analysis, FRA traces interpretation, and imple-mentation of simulation systems. at allows knowingexactly the transformer behavior in a wide frequency band-width in order to be used in the design process of power elec-tronics converters as it is shown in the following Section 3.

3. AC-DC-AC Converter Application

e DC-AC conversion needs a lter in order to obtain asinusoidal waveform from a PWM voltage waveform, avoid-ing the high frequency components. is task usually can besolved with the addition of an lter on the laststage oftheconverter. However, that solution implies an economic costand total volume increase (critical for transport applications)due to the addition of and components. Moreover,

an additional adaptation stage could be needed in order toincrease/reduce the voltage amplitude of the inverter.

e use of a PET could solve at the same time thelteringand adaptation necessities. However, an accurate transformermodel,as theonepresented in previous paragraphs, isneededto assure a complete comprehensionof itsfrequencyresponseand therefore the behavior (losses, voltage regulator, and soon) when working as part of a DC-AC converter.

In thefollowingexample, the transformer model is imple-mented and simulated as part of an AC-DC-AC converter(Figure 12) based on the power electronic model namedpower_bridges.mdl in MATLAB SimPowerSystems.

e converter consists in a rst stage where a DC voltageof 1000 volts is obtained. e second stage consists in aPWM inverter to obtain 3 phase to phase PWM voltages.One of the three voltages in the 3-phase converter, named as

ab_inverter, has been plotted in the upper side of Figure 13.

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T 1: Value of the parameters of the experimental model.

Phase U Phase V Phase WL.F.C.M sub-model

LFB (Hz) [12000] [12000] [12000]Number of points 201 201 201Ln X (H) 183.1504 399.8045 185.3597C HV (nF) 0.7452 0.6648 0.7356Rn X (M) 2.400 12.124 2.3

H.F.C.M sub-modelHFPIB (kHz) [22.89] [22.89] [22.89]Number of points 17 17 17Ld X (H) 4.0227 5.5128 5.0427Rd X (k ) 11.690 10.154 11.690

H.F.W.M sub-model (all phases HV side)HFPIIB (kHz) [10335] [10335] [10335]Number of points 239 239 239Le , (mH) 3.9218 3.8539 3.4211Ce , (nF) 1.0838 1.0602 1.0721

Re , (k ) 7.412 7.895 7.296Le , (mH) 3.9414 4.0031 3.9612Ce , (nF) 2.5919 2.5219 2.5528Re , (k ) 5.339 5.762 5.391C HV (nF) 0.7452 0.6648 0.7356

Ci X parametersBandwidth (Hz) [4083.4810023.8] [4083.4810023.8] [4083.4810023.8]Number of points 39 39 39Ci X (nF) 0.1186 0.1170 0.1181

Using lterUsing transformer model

Frequency (Hz)101 102 103

100

10 1

10 2

10 3

10 4

F 14: Spectrum of the ab voltages for and transformersimulation.

eltering andadaptation stageshavebeensolvedusingtwo alternatives. e rst one is an lter (pointed by reddashed lines in Figure 12) used to get symmetrical 3 phase tophasesinusoidal voltages ab, bc,and ca. esecondoneis

a power electronic transformer that hasbeensimulated usingthe presented model.

e plots ofFigure13 showthe PWM voltage signal in theupper side (the input of the or PET) and the comparisonof theltered voltage signals using the lter (results in themiddle side of Figure 13) and the model of the PET (resultsin the lower side of Figure 13).

It has been possible to simulate the PET because itsbehavior is completely represented till 500 kHz using themodel shown in Section 2. e analysis of the model (com-pletely represented in Figure 8) and its simulated response(showed in Figures 10 and 11) demonstrate that the leakageinductance (Ld X in Table 1) and the shunt capacitance of the windings ( HV in Table 1) conform an lter whose values give a resonant frequency of 2600 Hz. at means that

the transformer lter can be used as the element needed inthe inverter, saving the use of the external component.

isconcept is developed in [8]where thebases for designing lters integrated in the transformer is explained. e

accurate agreement between actual and simulated responsesuntil 500 kHz demonstrates that the simulation bandwidthis enough to cover the 2600 Hz behavior when the cut-off frequency of the lter appears.

e use of the transformer in the complete systemin spiteof the lter offers not only the ltering capability but alsothe voltage adaptation. For this reason, the 3 phase to phase voltages ab, bc, and ca (represented in the lower side of Figure 13) present a sinusoidal shape with 18 Volts rms in

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spiteof the383Volts rmsobtained when theexternal LC lteris used.

elteringcapabilityof the transformer is demonstratedin the comparison of Figure 14. e frequency spectrum of the ab phase to phase sinusoidal voltage is shown for bothcases, when the and the transformer model are used.

Both graphics are fairly equal, which demonstrates that thetransformer ltering capability can be considered as accurateas the lter.

Finally, the efficiency and voltage regulation of the PETcan be both calculated and simulated. e calculation canbe done because the values of the resistive parametersrepresenting the losses for the completefrequency bandwidthin the model are known, as represented in the example of Table 1 in the Rn X, Rd X, Re parameters. As a result, anefficiency of 93.74% and a voltage regulation of 94.7% werecalculated a er simulation.

4. Conclusions

enewdevelopmentof smartgrids involves theuse ofpowerelectronic equipment and new challenges in the operationof the power transformers. erefore, the knowledge of itsbehavior in a wide frequency bandwidth, not limited to thepower frequency (50/60 Hz), is a necessity.

A novel modeling procedure to obtain the transformermodel in a wide frequency bandwidth is presented. emain feature of the model is its capability of interpretingthe physical behavior by means of electrical parameters. atallows developing simulations, design revision, or sensitivity analysis in order to take advantages of the capabilities of the transformer in power electronics design or avoid risky situations as additional losses, overheating, or overvoltagesdue to high frequency components.

Acronyms

FRA: Frequency response analysisHFCM: High frequency core modelHFWM: High frequency winding modelLFCM: Low frequency core modelPET: Power electronic transformers.

Con ic of n eres s

e authors would like to clarify that there is not nancialgain and/or commercial interest related with MEGGER and MATLAB registered trademarks. e only reason thesenames have been included in the paper is for the sake of precision and to determine, in the more exact way, theprocedure to simulate the model and measure the prototypetransformer.

References

[1] International Electrotechnical Commission, Power Transform-ers-Part 18: Measurement of Frequency Response, vol. 60076,International Electrotechnical Commission, Geneva, Switzer-land, 2012.

[2] Z. Wang and K. Yu, e research of Power Electronic Trans-former (PET) in smart distribution network, in Proceedingsof the International Conference on Power System Technology:Technological Innovations Making Power Grid Smarter (POW-ERCON 10), pp. 17, October 2010.

[3] J. M. Carrasco, L. G. Franquelo, J. T. Bialasiewicz et al., Power-electronic systems for the grid integration of renewable energy sources: a survey, IEEE Transactions on Industrial Electronics, vol. 53, no. 4, pp. 10021016, 2006.

[4] D. Dujic et al., Laboratory scale prototype of a power elec-tronic transformer for traction applications keywords powerelectronic transformer topology, in Proceedings of the 14thEuropean Conference on Power Electronics and Applications(EPE 11), vol. 1, pp. 110, September 1 2011.

[5] P. S. Moses, M. A. S. Masoum, and K. M. Smedley, Harmoniclosses and stresses of nonlinear three-phase distribution trans-formers serving plug-in electric vehicle charging stations, inProceedings of the IEEE PES Innovative Smart Grid Technologies(ISGT 11), pp. 16, January 2011.

[6] M. Kang, P. N. Enjeti, and I. J. Pitel, Analysis and design of

electronic transformers for electric power distribution system,IEEE Transactions on Power Electronics, vol. 14, no. 6, pp.11331141, 1999.

[7] C. G. Garca, Procedimiento de modelado basado en el anlisisde la respuesta en frecuencia y aplicacin en transformadorestrifsicos de potencia para su caracterizacin y diagnstico.volume 1 [Ph.D. thesis], 2012.

[8] V. Valdivia, J. Pleite, P. Zumel, and C. Gonzalez, Improvingdesign of integrated magnetics for power electronics convert-ers, Electronics Letters, vol. 44, no. 11, pp. 693694, 2008.

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