Measurement of Three-Phase Transformer Derating and Reactive Power Demand Under Nonlinear Loading...

IEEE TRANSACTIONS ON POWER DELIVERY 1

Measurement of Three-Phase TransformerDerating and Reactive Power Demand

under Nonlinear Loading ConditionsEwald F. Fuchs, Fellow, IEEE, Dingsheng Lin, and Jonas Martynaitis

Abstract—The measurement of real and apparent power der-ating of three-phase transformers is important for transformersfeeding nonlinear loads. This paper presents a new digital data-ac-quisition method for measuring derating and reactive power de-mand of three-phase transformers under full or partial load con-ditions. The accuracy requirements of the instruments employed(potential, current transformers, shunts, voltage dividers, optocou-plers volt- and current meters) are addressed. Application exam-ples demonstrate the usefulness of this new digital data-acquisitionmethod.

Index Terms—Harmonics, nonsinusoidal operation, reactivepower demand, real and apparent power derating, transformerlosses.

I. INTRODUCTION

MEASURING the real and apparent power derating ofthree-phase transformers is desirable because additional

losses due to power quality problems (e.g., harmonics, dc exci-tation) can be readily limited before any significant damage dueto additional temperature rises occurs.

Measuring transformer losses from the input power minus theoutput power in real time is inaccurate because the losses are thedifference of two large values; this approach results in maximumerrors in the losses of more than 60% for high-efficiency

% transformers [1]. The usually employed indirect methodconsisting of no-load (iron-core loss) and short-circuit (copperloss) tests [2] cannot be performed on-line while the transformeris partially or fully loaded.

IEEE Recommended Practice C57.110 computes the trans-former derating based on for various harmonics , which isderived from the dc winding resistance and the rated load loss[3]. Kelly et al. [4] describe an improved measuring technique ofthe equivalent effective resistance as a function of frequency

of single-phase transformers, which allows the direct calcula-tion of transformer loss at harmonic frequencies fromHz up to 100 kHz. This equivalent effective resistance takes into

Manuscript received April 23, 2004; revised June 9, 2005. This work wassupported in part by the U. S. Department of Energy, Office of Energy Manage-ment, under Contract 19X-SK205V, and in part by the EPRI under Contract No.2951-07 and 4887-01. Paper no. TPWRD-00187-2004.

E. F. Fuchs is with the University of Colorado, Boulder, CO 80309 USA(e-mail: [email protected]).

D. Lin is with Ansoft Corporation, Pittsburgh, PA 15219 USA (e-mail:[email protected]).

J. Martynaitis is with the Department of Electrical and Lighting Engineering,Kaunas University of Technology, Kaunas LT-44244, Lithuania (e-mail:[email protected]).

Digital Object Identifier 10.1109/TPWRD.2005.858744

account the total losses of the transformer: the copper losses plusthe iron-core losses. Based on the fact that the iron-core lossesdo not depend on harmonic currents, but depend on harmonicvoltages (amplitudes and phase shifts) [5], the total losses deter-mined by [4] are not accurate. In addition, temperature-depen-dent operating conditions cannot be considered in [4]. Mocci[6] and Arri et al. [7] present an analog measurement circuitto directly measure the total losses for single- and ungroundedthree-phase transformers. However, employment of many PTsand CTs in the three-phase transformer measuring circuits [7]decreases the measurement accuracy.

This paper presents a direct method for measuring thederating of three-phase transformers while transformers areoperating at any load and any arbitrary conditions. The mea-suring circuit is based on potential transformers (PT) currenttransformers (CT), shunts, voltage dividers or Hall sensors[8], A/D converter and computer (microprocessor). Using acomputer-aided testing (CAT) program [1], [9] losses, effi-ciency, harmonics, derating and wave shapes of all voltages andcurrents can be monitored within a fraction of a second. Themaximum errors in the losses are acceptably small and dependmainly upon the accuracy of the sensors used.

II. APPROACH

A. Three-Phase Transformers in or Y-Y UngroundedConnection

For - or ungrounded Y-Y connected three-phase trans-formers (see Figs. 1, and 2 for using shunts, voltage dividersand optocouplers), one obtains with the application of the two-wattmeter method at the input and output terminals by inspec-tion

(1)

The term representsthe instantaneous iron-core loss, and

is the instantaneous copper loss.

B. Three-Phase Transformers in - Connection

Fig. 3 illustrates the application of the digital data-acquisitionmethod to a - connected transformer bank.

0885-8977/$20.00 © 2006 IEEE

2 IEEE TRANSACTIONS ON POWER DELIVERY

Fig. 1. Eight-channel CAT circuit for accurate �-� or ungrounded Y-Yconnected three-phase transformer loss monitoring using PTs and CTs.

Fig. 2. Eight-channel CAT circuit for accurate �-� or ungrounded Y-Yconnected three-phase transformer loss monitoring using shunts, voltagedividers, and optocouplers.

For ungrounded - connected three-phase transformers, thecurrents of the Y side must be referred to the line currents of the

side, as shown in Fig. 3. The loss of the transformer is given by

(2a)

Fig. 3. Eight-channel CAT circuit for accurate �-Y three-phase transformerloss monitoring using PTs and CTs.

where and are input line currents of phases and, respectively; and are the

two-wattmeter secondary currents; and are the outputline-to-neutral voltages of phases and respectively.

Because the neutral of the Y-connected secondary winding(N) is not accessible, the secondary phase voltages are measuredreferred to the neutral of the Y-connected PTs (see Fig. 3).This does not affect the accuracy of loss measurement, whichcan be demonstrated below. The output power is

(2b)

where denotes the neutral of the Y-connected secondarywinding. Because , the measured output powerreferred to the neutral of PTs is the same as that referred tothe neutral of the secondary winding.

III. ACCURACY REQUIREMENTS FOR INSTRUMENTS

The efficiencies of high-power electrical apparatus suchas single- and three-phase transformers in the kVA and MVAranges are 97% or higher. For a 15 kVA three-phase transformerwith an efficiency of 97.02% at rated operation, the total lossesare W, copper and iron-core lossesW and W at V and A,respectively, as listed in Table VII(a).

FUCHS et al.: THREE-PHASE TRANSFORMER DERATING AND REACTIVE POWER DEMAND 3

TABLE IINSTRUMENTS AND THEIR ERROR LIMITS FOR FIGS. 1 AND 2

The determination of the losses from voltage and current dif-ferences as described in Fig. 1-where differences are calibrated-greatly reduces the maximum error in the loss measurement.The series voltage drop and exciting current at rated operationreferred to the primary of the 15 kVA, 240 V/240 V three-phasetransformer are V, A, re-spectively (Table VII(a)). The instruments and their error limitsare listed in Table I. In Table I, ,and stand for the relevant calibrated voltmeters and am-meters. Because all voltage and current signals are sampled viaPTs, CTs (or optocouplers) and current shunts, the error limitsfor all instruments are equal to the product of the actually mea-sured values and their relative error limits, instead of full-scaleerrors, as shown in Table I. All error limits are referred to themeter side.

In Fig. 2, the voltage divider combined withan optocoupler emulates the function of a PT withouthysteresis. The optocoupler can alter the amplitude of a signaland provide isolation without affecting the phase shift of thesignal as it is corrupted by PTs. The current shunt andoptocouplers emulate that of a CT withouthysteresis and parasitic phase shift. The prime indicates that

is about of the same magnitude as , this is accomplishedby the adjustment of the amplifier gain(s) of the optocoupler(s).

The line-to-line voltage is measured with the maximum errorof (taking into account the maximum errors of and volt-meter)

(3)

The difference current is measured with the maximum error of

(4)

Therefore, the loss component in (1) is measuredwith the maximum error of

(5)

TABLE IIMEASURED IRON-CORE AND COPPER LOSSES OF 9 kVA Y–Y-CONNECTED

TRANSFORMER (V = 416V )

The series voltage drop is measured with the maximum errorof

(6)

The output current is measured with the maximum error of

(7)

and the loss component in (1) is measured withthe maximum error of

(8)

Thus, the total loss is measured with the maximum error of

% (9)

The above error analysis employs PTs and CTs. If these de-vices generate too-large errors because of hysteresis, voltage di-viders and shunts combined with optocouplers can be used, asindicated in Fig. 2, [10]. A similar error analysis using shunts,dividers and optocouplers leads to the same maximum error inthe directly measured losses, provided the same standard max-imum errors (0.1%) of Table I are assumed. The factor 2 in (9) isemployed because loss components in (5) and (8) are only halfof those in (1).

IV. COMPARISON OF DIRECTLY MEASURED LOSSES WITH

RESULTS OF NO-LOAD AND SHORT-CIRCUIT TESTS

A computer-aided testing program (CATEA) [9] is used tomonitor the iron-core and copper losses of three-phase trans-formers. The nameplate data of the tested transformers are givenin the Appendix.

The results of the on-line measurement of the iron-core andcopper losses for a Y-Y connected 9 kVA three-phase trans-former (consisting of three 3 kVA single-phase transformers,Appendix A.1) are given in Table II for sinusoidal rated line-to-line voltages of 416 V, where the direct (on-line) measurement


TABLE IIIMEASURED DATA OF 4.5 kVA TRANSFORMER BANK #1

data are compared with those of the indirect (separate open-cir-cuit and short-circuit tests) method.

The iron-core loss of the indirect method is larger than thatof the direct method because the induced voltage of the formeris larger than that of the latter. The copper loss of the indirectmethod is smaller than that of the direct method because theinput current of the former (which is nearly the same as theoutput current) is smaller than that of the latter.

V. APPLICATIONS

A. 4.5-kVA Three-Phase Transformer Bank #1 FeedingFull-Wave Rectifier

A 4.5 kVA, 240 V/240 V, - -connected three-phase trans-former (Fig. 1) consisting of three single-phase transformers(bank #1, Appendix A.2) is used to feed a full-wave diode rec-tifier (see Appendix A.5) with an LC filter connected across theresistive load (see Figs. 6-5 of [11]). In Table III, measured dataare compared with those of linear load condition. The measuredwave shapes of input voltage , exciting current ,series voltage drop and output current of phaseA are shown in Figs. 4(a), (b) for linear and nonlinear load con-ditions. The total harmonic distortion -factor and har-monic components of the output current are listed in Table IV.

Table III compares measured data and shows that the trans-former is operated at nonlinear load with about the same lossesoccurring at linear load (261.3 W). With the apparent power de-rating definition

(10)

the derating at the nonlinear load of Table III is 99%. The realpower delivered to the nonlinear load is 91.4% of that suppliedat linear load.

B. 4.5 kVA Three-Phase Transformer Bank #2 SupplyingPower to Six-Step Inverter

A 4.5 kVA transformer bank #2 (Appendix A.3) suppliespower to a half-controlled six-step inverter (Appendix A.6),which in turn powers a three-phase induction motor. The motoris controlled by adjusting the output current and frequency ofthe inverter. The transformer is operated at rated loss at variousmotor speeds. Rated loss of bank #2 is determined by a linear

(a)

(b)

Fig. 4 (a) Measured wave shapes of 4.5 kVA three-phase transformer bank #1feeding linear load (see rms values of Tables III and IV). (b) Measured waveshapes of 4.5 kVA three-phase transformer bank #1 feeding full-wave dioderectifier load (see rms values of Tables III and IV).

TABLE IVOUTPUT CURRENT HARMONIC COMPONENTS, THD ,

AND K-FACTOR [Fig. 4(b)]

(resistive) load at rated operation. The iron-core and copperlosses are measured separately and are listed in Table V(a).

Measured wave shapes of input voltage, exciting current, se-ries voltage drop and output current of phase A are shown inFigs. 5(a), (b) for linear and nonlinear conditions. The outputcurrent includes both odd and even harmonics due to the half-controlled input rectifier of the six-step inverter. Dominant har-monics of input voltage and output current are listed for differentmotor speeds in Table V(b). The total harmonic distortion ofthe input voltage and output current as well as the -factor arelisted in Table V(c) for the speed conditions of Tables V(a), (b).

C. 15 kVA Three-Phase Transformer Supplying Power toResonant Rectifier

A 15 kVA, 240 V/240 V, - connected three-phase trans-former (Appendix A.4) is used to supply power to a resonantrectifier [12] (Appendix A.7). The transformer is operated withthe resonant rectifier load at the same total loss generated by a


TABLE V(a)MEASURED IRON-CORE AND COPPER LOSSES OF 4.5 kVA,�-� CONNECTED

TRANSFORMER BANK #2 FEEDING A SIX-STEP INVERTER AT VARIOUS

MOTOR SPEEDS [FIG. 5(b)]

(a)

(b)

Fig. 5 (a) Measured wave shapes of 4.5 kVA three-phase transformer bank #2feeding linear load [see rms values of Tables V(a), (b), and (c)]. (b) Measuredwave shapes of 4.5 kVA three-phase transformer bank #2 feeding half-controlledsix-step inverter [see rms values of Tables V(a), and (b), (c)].

three-phase linear (resistive) load. Measured data are comparedin Table VI. Measured wave shapes of input voltage, excitingcurrent, series voltage drop and output current of phase A are de-picted in Figs. 6(a), (b). The fundamental phase shift betweenoutput transformer line-to-line voltage and phase current

of the resonant rectifier is 67.33 , and the output displace-ment factor (within transformer phase) is therefore

. The fundamental phase shift between outputline-to-line voltage and phase current for the linear resistive

TABLE V(b)INPUT VOLTAGE (IN rms VOLTS) AND OUTPUT CURRENT (IN rms AMPERES)

HARMONICS OF PHASE A [Figs. 5(a) AND (b)]

TABLE V(c)MEASURED THD-VALUES ANDK-FACTOR [FIG. 5(a) AND (b)]

load is 30.95 , and the output displacement factor (within trans-former phase) is, therefore, . Note thatthe wave shapes of the output currents of Figs. 6(a), (b) are aboutsinusoidal.

If the transformer with the resonant rectifier load is operated(see Table VI) at about the same loss as linear load (233.9 W),the output current of the transformer with the resonant recti-fier load is 26.94 A, which corresponds to the copper loss of192.2 W ( W W). Therefore, the transformer ap-parent power derating of the transformer for the nonlinear loadis 99.7%. The real power supplied to the nonlinear load is 78.5%of that of the linear load.

D. 15 kVA Three-Phase Transformer Bank Absorbing PowerFrom a PWM Inverter

The same transformer of Section V-C absorbs power froma PWM inverter [12] (Appendix A.8) and supplies power to aresistive load (case #1) and to a utility system (case #2). Thetransformer losses are also measured when supplying a linearresistive load fed from sinusoidal power supply (linear load).All measured data are compared in Table VII(a).


(a)

(b)

Fig. 6 (a) Measured wave shapes of 15 kVA three-phase transformer feedinglinear load (see rms values of Table VI). (b) Measured wave shapes of 15 kVAthree-phase transformer feeding resonant rectifier (see rms values of Table VI).

TABLE VIMEASURED DATA OF 15 kVA THREE-PHASE TRANSFORMER WITH

RESONANT RECTIFIER LOAD [FIG. 6(a) AND (b)]

Measured wave shapes of input voltage, exciting current, se-ries voltage drop and output current of phase A are shown inFigs. 7(a)–(c). The total harmonic distortion -factorand harmonic amplitudes of the transformer output current arelisted in Table VII(b).

Fig. 8 summarizes the total harmonic distortion , ap-parent power (kVA) derating, and real power (kW) derating foruncontrolled (a, b), half-controlled (c) and controlled (d, e) con-verter loads of transformers. In particular the graphs of Fig. 8can be identified [11] as follows:

a) 25 kVA single-phase pole transformer [13], [17], [14],[18], [15] feeding uncontrolled full wave rectifier load;

TABLE VII(a)MEASURED DATA OF 15 kVA THREE-PHASE TRANSFORMER CONNECTED

TO PWM INVERTER [FIG. 7(a)–(c)]

b) 4.5 kVA three-phase transformer feeding uncontrolled fullwave rectifier load;

c) 4.5 kVA three-phase transformer feeding half-controlledrectifier load;

d) 15 kVA three-phase transformer absorbing power fromPWM inverter (14 kW);

e) 15 kVA three-phase transformer feeding resonant rectifierload (8 kW).

VI. DISCUSSION OF RESULTS AND CONCLUSIONS

A. Discussion of Results

A new approach for the measurement of the derating of three-phase transformers has been described and applied under non-sinusoidal operation. It extends the measurement approach ofsingle-phase transformers [1], [9], [12], [13], [17], [14], [18],[15] to three-phase transformers [11], [15], [16].

The apparent power (kVA) derating, (10), of three-phasetransformers is not greatly affected by the . Even for

values of about 70%, derating is about 99%.The real power (kW) derating is greatly affected (see Fig. 8)

by the current wave shape generated by solid-state converters, inparticular by the phase shift of the fundamental current compo-nent. Therefore, inverters and rectifiers should be designed suchthat they supply and draw power, respectively, at a displacement(power) factor of about 1.

Three-phase transformers have similar derating properties assingle-phase transformers [1], [9], [12], [13], [17], [14], [18],[15].

The maximum error in the directly measured losses is about15%, which compares favorably with the maximum error ofmore than 60% [1] for loss measurement based on the differencebetween input and output powers as applied to high-efficiency

% transformers.Transformers of the same type may have significantly dif-

ferent iron-core losses as measured in Table III (261.3 W) andTable V(a) (220.6 W).

Small transformers (kVA-range) have relatively small wirecross-sections resulting in small skin-effect losses. Large trans-formers (MW-range) have aluminum secondary windings withrelatively large wire cross-sections resulting in relatively largeskin-effect losses. For this reason substation transformers canbe expected to have larger apparent power derating than the


(a)

(b)

(c)

Fig. 7(a) Measured wave shapes of 15 kVA three-phase transformer feedinglinear load [see rms values of Tables VII(a) AND (b)]. (b) Measured waveshapes of 15 kVA three-phase transformer fed by a PWM inverter (case #1)[see rms values of Tables VII(a) AND (b)]. (c) Measured wave shapes of 15kVA three-phase transformer fed by a PWM inverter (case #2) [see rms valuesof Tables VII(a) AND (b)].

TABLE VII(b)OUTPUT CURRENT HARMONICS, THD ANDK-FACTOR [FIG. 7(b) AND (c)]

ones measured in this paper. Unfortunately, transformers in theMW-range cannot be operated in a laboratory under real-lifeconditions. Therefore, it is recommended that utilities sponsorthe application of the method of this paper and permit on-sitemeasurements.

Fig. 8. Total harmonic distortion of current (THD ), apparent power (kVA)derating, and real power (kW) derating for uncontrolled (a, b), half-controlled(c) and controlled (d, e) converter loads.

B. Comparison With Existing Techniques

The maximum error in the directly measured losses is about15% (using potential and current transformers), which comparesfavorably with the maximum error of more than 60% [1] (em-ploying shunts and voltage dividers) for loss measurement basedon the difference between input and output powers as applied tohigh-efficiency % transformers.

The technique of [4] uses the pre-measured transformer ef-fective resistance as a function of frequency to calculatetransformer total losses for various harmonic currents. Thismethod can be classified as an indirect method because thetransformer losses are obtained by computation, instead ofdirect measurement. In addition, the approach of [4] neglectsthe fact that the iron-core losses are a function of the harmonicphase shift [5], in other words the values are not constantfor any given harmonic current amplitude but vary as a functionof the harmonic voltage amplitude and phase shift as well.Finally, temperature-dependent operating conditions, for ex-ample, cannot be considered in [4]. For the above reasons, themethod of [4] must be validated by some direct measurements,such as the method presented in this paper.

The method of Mocci [6] and Arri et al. [7] has not been prac-tically applied to three-phase transformers. The presented mea-surement circuit for three-phase transformers [7] uses too manyinstrument transformers (e.g., 9 CTs and 9 PTs), and therefore,the measured results will be not as accurate as those based onthe measurement circuits of this paper, where only 4 CTs and 5PTs are used as shown in Fig. 3.

APPENDIX

1) 9 kVA Three-Phase Transformer Bank: Y-Y connectedthree-phase transformer bank with V of rated line-to-linevoltagesconsistsof threesingle-phasetransformers.Thetypenumbers of the three single-phase transformers are J7065, J7610,and J7065 (bank #1), and the nameplate data of which are:

Powerformer Dry-type transformerCAT. no.: 211-101, 3 kVA, single-phase, 60 Hz;High voltage 240/480 V; Low voltage 120/240 V.

2) 4.5 kVA Three-Phase Transformer Bank #1: Single-phase transformers with the nameplate data listed in A.1 are


used in this bank #1 and are connected in such way that the ratedline-to-line voltages are V. Only the two high-voltagewindings of each single-phase transformer are employed asprimary and secondary, and the rated apparent power of thethree-phase transformer bank is therefore 4.5 kVA.

3) 4.5 kVA Three-Phase Transformer Bank #2: Single-phase transformers with the nameplate data listed in A.1 havingthe type numbers J7605, J7606, and J7605 are used in this bank#2, and are connected in such way that the rated line-to-linevoltages are V. Only the two high-voltage windingsof each single-phase transformer are employed as primary andsecondary, and the rated apparent power of the three-phasetransformer bank is therefore 4.5 kVA.

4) 15 kVA Three-Phase Transformer Bank: 15 kVA,- connected bank consists of three 5 kVA trans-

formers. The original rated apparent power of the single-phaseunits is 10 kVA. The two high-voltage windings were usedas primary and secondary, and the rated power of the bank istherefore 15 kVA. The nameplate data are:

Westinghouse Type: EP transformerFrame no.: 179, 10 kVA, single-phase, 60 Hz;High voltage: V Low voltage: V.

5) Three-Phase Diode Bridge: A,V.

6) Half-Controlled Three-Phase Six-Step Inverter: GeneralElectric AC adjustable frequency drive

Model no.: 6VGAW2007CIInput: Output:Volts: 230 HP: 7.5, kVA: 8.8;Hertz: 50/60 Volts: variable;Amps: 22 Hz: 6–60;Phase 3 Amps ac: variable;

Phase 3.7) Controlled Three-Phase Resonant Rectifier

[12]: University of Colorado

ID no.: 5484Input: Output:Volts: 340–600 ac 20 kW;Hertz: 12–60 Volts: 380 dc;Amps: 45 Amps: 60 dc.Phase 3

8) Controlled Three-Phase PWM Inverter [12]: Universityof Colorado

ID no.: 5485Input: Output:Volts: 360 dc 20 kW;Amps: 70 dc Volts: 240 ac;

Amps: 80 ac;Power factor: 0.7 (lead) to 1.0 p.u.;Hertz: 50/60.

REFERENCES

[1] E. F. Fuchs, D. Yildirim, and T. Batan, “Innovative procedure for mea-surement of losses of transformers supplying nonsinusoidal loads,” inProc. Inst. Elect. Eng., Gen., Transm. Distrib., vol. 146, IEE Proceed-ings online no. 19 990 684, Nov. 1999.

[2] Electrical Transmission and Distribution Reference Book, 1964. West-inghouse Electric Corporation.

[3] IEEE Recommended Practice for Establishing Transformer Capabil-ities when Supplying Nonsinusoidal Load Currents. IEEE Std. IEEEC57.110-1998.

[4] A. W. Kelly, S. W. Edwards, J. P. Rhode, and M. Baran, “Transformerderating for harmonic currents: A wideband measurement approach forenergized transformers,” in Proc. Industry Applications Conf., 30th Ind.Appl. Soc. Annual Meeting, vol. 1, Oct. 8–12, 1995, pp. 840–847.

[5] M. A. S. Masoum and E. F. Fuchs, “Derating of anisotropic transformersunder nonsinusoidal operating conditions,” Elect. Power Energy Syst.,vol. 25, pp. 1–12, 2003.

[6] F. Mocci, “Un nuovo methodo per la determinazione della potenza assor-bita dai doppi bipoli,” L’Energia Elettrica, no. 7–8, pp. 323–329, 1989.

[7] E. Arri, N. Locci, and F. Mocci, “Measurement of transformer powerlosses and efficiency in real working conditions,” IEEE Trans. Instrum.Meas., vol. 40, no. 2, pp. 384–387, Apr. 1991.

[8] LEM USA, Inc., Current and Voltage Transducer Catalog, 3rd ed., Mil-waukee, WI.

[9] D. Lin, E. F. Fuchs, and M. Doyle, “Computer-aided testing of electricalapparatus supplying nonlinear loads,” IEEE Trans. Power Syst., vol. 12,no. 1, pp. 11–21, Feb. 1997.

[10] S. Rieman, “An optically isolated data acquisition system,” IndependentStudy, Dec. 1997.

[11] W. M. Grady, E. F. Fuchs, D. Lin, and T. D. Stensland, “The potentialeffects of single-phase power electronic-based loads on power systemdistortion and losses, Volume 2: Single-phase transformers and induc-tion motors,” Electric Power Research Institute (EPRI), Palo Alto, CA,Tech. Rep. 000 655, Sep. 2003.

[12] D. Yildirim, “Commissioning of 30 kVA variable-speed direct-drivewind power plant,” Ph.D. dissertation, Univ. Colorado, Boulder, CO,May 1999.

[13] E. F. Fuchs, D. Yildirim, and W. M. Grady, “Measurement of eddy-cur-rent loss coefficient P , derating of single-phase transformers, andcomparison with K-factor approach,” IEEE Trans. Power Del., vol. 15,no. 1, pp. 148–154, Jan. 2000.

[14] D. Yildirim and E. F. Fuchs, “Measured transformer derating and com-parison with harmonic loss factorF —Approach,” ibid, vol. 15, no. 1,pp. 186–191, Jan. 2000.

[15] E. F. Fuchs, “Transformers, liquid filled,” Encyclopedia Electr. Electron.Eng., 2000. paper no. 934C.

[16] E. F. Fuchs, Y. You, and D. Lin, “Development and validation of GICtransformer models,” Final Rep. Contract 19X-SK205V, Jun. 1996.

[17] T. Batan, “Discussion of ‘Measurement of eddy-current loss coefficientP - derating of single-phase transformers, and comparison withK-factor approach’,” IEEE Trans. Power Del., vol. 15, no. 4, pp.1331–1333, Oct. 2000.

[18] E. F. Fuchs, D. Yildirim, and W. M. Grady, “Corrections to ‘Measure-ments of eddy-current loss coefficient P - derating of single-phasetransformers, and comparison with K-factor approach,” IEEE Trans.Power Del., vol. 15, no. 4, p. 1357, Oct. 2000.

Ewald F. Fuchs (F’90) received the Dipl.-Ing. degree in electrical engineeringfrom the University of Stuttgart, Stuttgart, Germany, and the Ph.D. degree inelectrical engineering from the University of Colorado, Boulder, in 1967 and1970, respectively.

Currently, he is a Professor of Electrical Engineering at the Universityof Colorado.

Dingsheng Lin received the B.Sc. and M.Sc. degrees in electrical engineeringfrom the Shanghai University of Technology, Shanghai, China, in 1982 and1987, respectively.

Currently, he is a Senior Research and Development Engineer with AnsoftCorporation, Pittsburgh, PA. He was promoted to Associate Professor of Elec-trical Engineering with Shanghai University of Technology in 1991. His maininterests are the design and optimization techniques of electrical machines andelectromagnetic field calculation.

Mr. Lin received the third prize of the Chinese National Award of Scienceand Technology in 1987, and two second prizes of the Shanghai City Award ofScience and Technology in 1986 and 1989.

Jonas Martynaitis received the Dipl.-Eng. degree from Kaunas PolytechnicInstitute, Lithuania, in 1969, and the Ph.D. degree in lighting engineering andelectronics from the Moscow Institute of Energy, Moscow, Russia, in 1980.

Currently, he is an Associate Professor with the Department of Electrical En-gineering, Kaunas University of Technology, Lithuania. During the 1980-1981academic year, he was with the University of Colorado at Boulder performingresearch.

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