Influence of voltage harmonics on transformer no-load loss measurements and calculation of magnetization curves

Electric Power Systems Research 146 (2017) 43–50

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Influence of voltage harmonics on transformer no-load loss measurements and calculation of magnetization curves Jayanth R. Ramamurthy a,∗ , Nicola Chiesa b , Hans K. Høidalen b , Bruce A. Mork a , Nils M. Stenvig a , Adam C. Manty a a b

Michigan Technological University, 1400 Townsend Dr, Houghton, MI 49931, USA Norwegian University of Science and Technology, Trondheim, NO-7491, Norway

a r t i c l e

i n f o

Article history: Received 6 October 2016 Received in revised form 13 December 2016 Accepted 14 January 2017 Keywords: Harmonics Magnetization No-load losses Total Harmonic Distortion (THD) Transformer Zero-Crossing

a b s t r a c t This paper investigates the voltage distortion phenomenon during no-load testing of transformers, its influence on no-load loss calculations, and a novel measurement technique that is unaffected by harmonics for obtaining transformer magnetization curves. Owing to economic reasons, no-load loss correction for distorted test waveforms has been addressed to some extent in existing standards, while not much has been done on the aspect of calculation of magnetization characteristics. This is important for parameter estimation and transformer modeling in power system transient simulations. Results from two test cases are presented, one is based on factory testing with 290-MVA, three-phase, three-legged power transformer and second is a more comprehensive analysis using a laboratory test setup with a 22-kVA single-phase transformer. Application of loss correction equations and limits based on existing testing standards was evaluated and found to result in overcompensation, while calculation of magnetization curves based on existing methods resulted in error up to 20%. Whereas, application of proposed measurement technique based on voltage “Zero-Crossing” detection is shown to result in negligible error. The proposed measurement technique uses the same input signals as in a standard no-load test procedure. Hence, it can be easily implemented in parallel with existing instrumentation. © 2017 Elsevier B.V. All rights reserved.

1. Introduction Transformers represent an important link in generation, transmission, distribution and utilization of electrical energy in power systems. Before leaving the production test facility, a series of tests are performed to inspect and certify the performance of a transformer in accordance with IEEE or IEC standards [1–4]. No-load or open-circuit testing of transformers is part of the routine test series and the outcome of the no-load test is typically a series of data points at different excitation voltage levels, with rms current and active power as shown in Table 1. Even though no-load losses represent a small fraction of the transformer throughput, they are present as long as the transformer is energized and can contribute significantly to system-wide energy losses [5,6]. Hence, accurate measurement of no-load losses is of crucial importance. Another application of transformer test results is for parameter estimation and modeling in power system simulations such as electromagnetic transients program (EMTP) representing low-frequency or slow-front transient phenomenon

∗ Corresponding author. E-mail address: [email protected] (J.R. Ramamurthy). http://dx.doi.org/10.1016/j.epsr.2017.01.022 0378-7796/© 2017 Elsevier B.V. All rights reserved.

[7–9]. In the last few years, EMTP simulations have gained lot of importance in predicting dangerous transient conditions involving the power system network, devices, and equipment. However, the first step is to reduce uncertainty in parameter estimation. Short circuit test results are used to identify leakage reactance and winding resistance of transformers; these parameters behave linearly, hence, their estimation is fairly simple. On the other hand, no-load test data is used to characterize the core losses and magnetizing inductance, which is intrinsically non-linear, thus their estimation is more challenging [10–14]. An effect of non-linearity of the core is the development of harmonics primarily in the current [15,16], however, due to the source and line impedance of the test circuit, the resulting voltage drop is affected and hence the voltage applied at the terminals of the transformer during the no-load test is distorted. Both IEEE and IEC recognize this limitation and propose correction formulas for moderate test voltage distortion within limits of 5% and 3% respectively for correction of measured no-load losses to a sinusoidal excitation voltage basis [1–4]. Further, IEEE standard C57.123 [3] recommends the terminal voltage Total Harmonic Distortion (THD) should be within 15% to avoid test waveforms with multiple zero-line crossings as it may result in additional hysteresis losses. However, the present standards do not allow any compensation for the excitation

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Table 1 290-MVA three-phase transformer factory test report data.

of interest such as active power and flux-linkage can be obtained using numerical integration as below in (5) and (6).

Main data

[kV]

[MVA]

[A]

Coupling

HS LS

432 16

290 290

388 10,465

YN d5

Open-circuit

E0 [kV, (%)]

[MVA]

I0 [%]

P0 [kW]

LS

12 (75) 14 (87.5) 15 (93.75) 16 (100) 17 (106.25)

290 290 290 290 290

0.05 0.11 0.17 0.31 0.67

83.1 118.8 143.6 178.6 226.5

Short-circuit

[kV]

[MVA]

ek, er [%]

Pk [kW]

HS/LS

432/16

290

14.6, 0.24

704.4

P=

1 T

T vex (t)iex (t)dt

(5)

vex (t)dt

(6)

0

T (t) = 0

current, as the 5% and 3% limit enforced on the magnitude of loss correction guarantees that the effect of voltage harmonics on the magnitude of rms value of the excitation current is small to cause it to exceed the manufacturer’s guaranteed value [3]. While this may not pose any financial implications, it can introduce significant error in the calculation of transformer magnetization curves. As will be demonstrated in this paper, it is possible to violate the 5% and 3% loss correction limits even for waveforms with relatively low harmonic content (THD < 15%). Secondly, even for situations when the above recommended loss correction limits are met, application of IEEE and IEC correction formulas may result in corrected losses that deviate from the case with nearly sinusoidal voltage excitation [17,18]. Hence, objective of this paper is (1) to present a systematic investigation on the nature of terminal voltage distortion experienced during no-load testing of transformers, (2) to quantify the magnitude of IEEE and IEC recommended loss correction with varying levels of excitation and distortion, and (3) to demonstrate application of voltage “Zero-Crossing” detection as an enhanced method for calculation of magnetization curves.

2.2. No-load loss correction The problem of measuring no-load losses under influence of non-sinusoidal excitation is not unique to transformers and is a much broader issue encountered with design and application of magnetic materials [20–22]. However, most of the approaches proposed require access to magnetic and physical properties of the core and use empirical relations; hence, they are not practical for application in a factory testing environment. Secondly, very few references [23–25] actually address the problem of correction of no-load losses to a sinusoidal excitation basis for transformers. The approach recommended by present IEEE and IEC standards [1–4], based on the average-responding voltmeter method is used. In this method, the test voltage is adjusted based on the average-responding voltmeter to obtain hysteresis losses; however, readings of both true rms and average-responding voltmeters are necessary to correct the eddy current loss component to a sinusoidal basis using (7) and (8) for IEEE and IEC, respectively. Magnitude of loss correction can be calculated using (9). IEEE : P0 =

Pm ,k = P1 + kP2

IEC : P0 = Pm (1 + d), d = 2. Voltage distortion, no-load loss correction, and magnetization curves

%correction =

2.1. Measurement of voltage distortion during no-load test If vex (t), represents the instantaneous excitation voltage waveform applied at the terminals of the transformer, with time period T and Vk represents rms magnitude of harmonic components, with k = 2. . .N representing a finite number of harmonics, the following parameters (1)–(4) can be used to quantify distortion:

  T  1 True rms : Vrms =  vex 2 (t)dt

(1)

T

0

 

Average rms : Vavg−rms =

2 T

␯ex (t) dt 0

Total Harmonic Distortion : THD = Form Factor : FF =

Vrms Vavg−rms



T/2

  N   Vk2   k=2 V12

 · √ 2 2

(2)

(3)

(4)

Harmonic decomposition can be performed using Fast Fourier Transform (FFT) with integer power of 2 samples and whole number of cycles to avoid spectral leakage effects [19]. Other quantities



Vrms

2

Vavg−rms Vavg−rms − Vrms Vavg−rms

|P0 − Pm | x100% Pm

(7)

(8) (9)

where P0 represents the corrected value and Pm represents measured losses. The IEEE formula (7) also makes provision to account for separation of hysteresis and eddy current losses as P1 and P2 in per unit values respectively. If actual values are not available, IEEE recommends using 0.5 per unit for each. Both the IEEE and IEC formulas are equivalent when the Form Factor of the waveform approaches unity, as in the case of pure sinusoidal waveforms. As the Form Factor increases, the equations begin to deviate. If waveform distortion causes the magnitude of loss correction to exceed 5% and 3% for IEEE and IEC respectively, the test is subject to agreement between the manufacturer and purchaser [3,4]. 2.3. Transformer magnetization curves The third aspect of this paper deals with the calculation of transformer magnetization curves, which is critical for modeling transformers in power system transient simulation programs like EMTP and also harmonic load flow studies [7,26–28]. In order to describe the nonlinear characteristic of a transformer iron core, various approaches can be used [29–31]. However, the fundamental step involves calculation of the single-valued anhysteretic magnetization curve. Methods to convert rms quantities from the no-load test data into piece-wise nonlinear magnetization characteristic are described in Refs. [26,27] and is referred to as the “CONVERT” routine in EMTP [26] and “SATURA” routine in Alternative Transients

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Fig. 1. Voltage “Zero-Crossing” detection method, (a) proposed logic circuit with integrator, triggering unit and two sample and hold blocks, (b) conceptual illustration of voltage, no-load current and flux linkage waveforms at voltage Zero-Crossing instance.

Fig. 2. Waveforms and harmonics for 3-phase, 290-MVA transformer factory test case, at 85%, 100% and 112% excitation voltages, (a) phase C voltage waveform, (b) voltage harmonics, (c) phase C current waveform, (d) current harmonics, (e) phase C flux linkage, (f) flux linkage harmonics.

Program (ATP) [27]. The basic idea behind these widely used methods is to obtain the peak flux by scaling each rms voltage data point using (10) √ peak (j) =

2Vavg−rms (j) , j = 1, 2...N 2f

(10)

where j denotes each data point available from the test report and f denotes frequency of the voltage waveform. However, the scaling of rms current to peak current values beyond the last point

representing the linear portion of the magnetization curve is not straightforward, owing to its nonlinear characteristic. Different approaches have been proposed in literature [10–13,26,27] which include both iterative methods and deriving analytical expression for slope of the magnetization curve as a function of the rms value of current. However, the questionable assumption in the above widely used algorithms is the sinusoidal nature of the applied voltage. Secondly, the core-loss component of the rms excitation current is assumed to be only of fundamental frequency and is subtracted

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orthogonally from the no-load current to obtain the magnetization current. Further, the rms quantities available from test reports provide only an average-rectified rms reading of voltage and rms current but detail nothing about the harmonic content of the waveforms. As a consequence, the voltage distortion problem can result in a reduced level of accuracy in the calculation of magnetization curves when such standard methods are used. 2.4. Voltage “Zero-Crossing” detection method The proposed method in this paper is based on the simple realization that when the voltage applied across the transformer magnetizing branch reaches a Zero-Crossing, the no-load line current becomes primarily inductive and flux-linkage reaches a peak value. Thus, magnetizing inductance L is defined by (11) and (12).

Fig. 3. Voltage THD curve plotted as a function of excitation voltage (3-ph 290-MVA transformer, factory test—a reference case).



(t) = L=

v(t)dt

(11)

| ipeak v(t)=0

(12)

peak

As the inductive component of the no-load line current remains in-phase with the flux-linkage at the voltage ZeroCrossing instance, the peak values of flux-linkage and current for a given excitation level can be calculated using linear interpolation between instantaneous excitation voltage samples j and j + 1 delimitating a voltage Zero-Crossing data point using (13)–(15). zero − cross : vex (j)vex (j + 1) ≤ 0 ipeak = iex (j) − peak = (j) −

 i (j + 1) − i (j) ex ex vex (j + 1) − vex (j)

 (j + 1) − (j) vex (j + 1) − vex (j)

(13)

vex (j)

(14)

vex (j)

(15)

A simple logic circuit composed of an integrator, a triggering unit, and two sample and hold blocks can be used to realize this method, as shown in Fig. 1. Input quantities for the logic circuit described in Fig. 1(a) are instantaneous voltage and current, the same signals used by rms voltmeter and ammeter in the standard no-load test configuration. The new logic can therefore be built in parallel to existing instrumentation. In case of heavily distorted voltage waveform with multiple zero-line crossings, several techniques available in literature [32] can be used to compensate for frequency and phase errors; however, to keep the proposed logic circuit simple and practical, a passive low-pass filter block with cut-off frequency in the order of few kHz is suggested as an optional pre-filtering step.

Fig. 4. Schematic of laboratory test setup with 22-kVA single-phase transformer.

may be observed from Fig. 2(a) that even at 100% rated excitation, the voltage waveforms begin to have some minor high frequency distortion; however, the voltage waveform still retains two zeroline crossings per period. Increasing the excitation to 112% causes the voltage waveform to become heavily distorted with multiple zero-line crossings and peaky in nature while the flux becomes flat-topped as can be observed in Fig. 2(e). As the delta connected low voltage side blocks circulation of any third harmonic in the voltage, the fifth harmonic (250 Hz) emerges as the dominant component in Fig. 2(b). However, the excitation current (Fig. 2(d)) still retains a large circulating third harmonic (150 Hz) component. To further quantify the distortion, voltage THD curves were calculated as shown in Fig. 3, for the three phase test voltages applied across each phase of the transformer. The THD computation was performed using (3) with N = 63 chosen as the total number of harmonics. While the voltage THD curve stays relatively flat at lower excitation voltages in Fig. 3, it reaches 15% at 100% rated excitation voltage. As excitation voltage is increased to 112%, THD increases exponentially to 40%. This provides an adequate measure to demonstrate the severity of harmonic voltage distortion problem that can be encountered in an actual factory test environment.

3. Factory test report—a reference case 4. Single-phase laboratory case study In order to first identify the problem of voltage distortion, a 50-Hz, 290-MVA, three-phase transformer with factory test data shown in Table 1 was chosen and some additional testing was performed. During the no-load test, the transformer was energized from the delta connected low voltage side. The applied voltage was varied systematically from 10% to 112% of rated excitation based on average-responding voltmeter. However, in addition to the rms meters that are used in general during standard testing, the applied phase voltages and phase current for one phase (phase C) was recorded at each excitation level with a high sampling rate of 100 kHz using a digital oscilloscope. This approach removed the limitation of standard no-load test procedures and helped identify the problem of voltage distortion. The processed waveforms up to 40 ms for applied phase C voltage, phase C current, and phase C flux linkage, at 85%, 100% and 112% excitation and corresponding Fourier harmonic analysis is shown in Fig. 2. It

4.1. Laboratory test setup A more comprehensive laboratory test with a 22-kVA, 220/220 V, 50-Hz, single-phase transformer as the test object, with setup as shown in Fig. 4 was used for further investigation. A 400 V, 50-Hz test source regulated with a three-phase variac was used to apply single-phase excitation voltage with secondary of the transformer left open circuited. It was ensured that the rating of the variac chosen was fairly large compared to the tested transformer in order to keep the impact of source impedance to a minimum. Hence, it is assumed that any voltage distortion at the transformer test terminals is caused by the voltage drop in the line impedance of the test circuit due to the flow of nonlinear magnetization current. In order to investigate the effect of high levels of voltage distortion, variable line impedance was inserted in series with the

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Fig. 5. Waveforms and harmonics for single-phase, 22-kVA laboratory transformer, at 85%, 100%, 135% excitation voltages with additional reactive line impedance X = 14 , (a) terminal voltage waveform, (b) voltage harmonics, (c) phase current waveform, (d) current harmonics, (e) flux linkage, (f) flux linkage harmonics.

Fig. 6. Nature of terminal voltage distortion, snapshot at 125% excitation voltage, (a) effect of varying resistive impedance, (b) effect of varying reactive impedance.

variac. This additional impedance was then varied systematically from 0  (no additional impedance) to 9, 14, 19, and 24 , both resistive (using a standard resistive load-cart) and inductive (using a continuously variable air-gap inductor) respectively. A matrix of test measurements was then obtained for a series of excitation voltages applied at the transformer terminals in the range of 45–155% of rated voltage (based on average-responding voltmeter) and 8 levels of additional impedance (4 resistive and 4 inductive). For each test measurement, voltage and current waveforms were recorded at locations indicated in Fig. 4 with passive multiplier voltage probes and clamp-on style current probes for a time period of 40 ms and high sampling rate of 100 kHz using a digital oscilloscope. Active

power and flux linkage for each test measurement was calculated using (5) and (6).

4.2. Laboratory test results—voltage distortion and loss correction A case with additional reactive impedance of X = 14 , is chosen to illustrate the test results obtained at excitation levels of 85%, 100% and 135% of rated voltage. The terminal voltage, current and flux-linkage waveforms for a period of 40 ms with their respective harmonic content are shown in Fig. 5. It may be observed that even with X = 14  inserted, the voltage waveform at 100% rated excitation voltage in Fig. 5(a) is clean without any high frequency

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Fig. 7. Voltage THD and %loss correction, (a) voltage THD resistive, (b) voltage THD reactive, (c) %loss correction resistive impedance, (d) %loss correction reactive impedance.

Fig. 8. Comparison of measured and corrected no-load losses, (a) R = 24 , (b) X = 24 .

distortion compared to the factory test case, owing to the more stable voltage source and better filtered laboratory measurements. However, as excitation level is increased to 135%, voltage distortion due to third harmonic (150 Hz) component increases. For an excitation voltage of 125%, the effect of adding resistive and reactive impedance in the no-load test circuit is illustrated in Fig. 6. As resistive voltage drop component is in-phase with the excitation current, it causes waveforms in Fig. 6(a) to be shifted to the right with increasing levels of resistance. The inductive voltage drop component on the other hand will be phase-shifted by /2 with respect to the excitation current. However, the excitation current in the test circuit is also lagging the applied voltage; hence, addition of reactive impedance results in an in-phase voltage drop with the source voltage and the effect of harmonics is amplified. This explains the peaky nature of the voltage waveforms shown in Figs. 5(a) and 6(b) due to addition of reactive impedance in the test circuit. Thus, for a given excitation voltage addition of reactive impedance results in waveforms with higher Form Factor and reduced average rms voltage. Voltage THD curves along with %loss correction are plotted in Fig. 7 with varying levels of excitation voltage and test circuit impedance. It may be observed from Fig. 7(a) and (b) that for the case with no additional impedance (0 ) the voltage THD remains less than 5% even at 155% of rated excitation voltage. This confirms the basic assumption that the source impedance of the laboratory test circuit is strong enough and any

voltage distortion resulting at the transformer terminals is due to the flow of nonlinear magnetization current through additional line impedance. Secondly, even with low harmonic content (THD < 15%) for reactive impedance test cases in Fig. 7(b) at 100% rated excitation voltage, the IEEE and IEC loss correction limits of 5% and 3% are violated in Fig. 7(d). Comparison of measured and corrected losses for R = 24  and X = 24 , is shown in Fig. 8. While the actual no-load losses in Fig. 8(a) and (b) are expected to occur between the measured and corrected 0  impedance curves, the application of IEEE and IEC loss correction equations causes the compensated 24  impedance curves to shift further below. At 135% of rated excitation this error in overcompensation is around 2.5% for the case with R = 24  in Fig. 8(a) and around 6% in Fig. 8(b) at 120% rated excitation for the case with X = 24 .

4.3. Laboratory test results—magnetization curves The input rms data obtained for the laboratory test case is shown in Fig. 9(a) and (b). Magnetization curves calculated are shown in Fig. 9(c) and (d). As addition of resistive impedance results in lower distortion, the error in Fig. 9(c) and zoom-in of Fig. 9(e), is small. However, addition of reactive impedance results in an error of about 20% in the calculation of peak magnetizing current at a flux-linkage of 1.35 Wb-T in Fig. 9(d) and zoom-in shown in Fig. 9(f). Application

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Fig. 9. Input data and magnetization curves for 22-kVA transformer based on SATURA/CONVERT routine, (a) input Vrms-Irms curves with resistive impedance, (b) input Vrms-Irms curves with reactive impedance, (c) magnetization curves for resistive impedance, (d) magnetization curves for reactive impedance, (e) zoom-in of (c) for resistive magnetization curves, (f) zoom-in of (d) for reactive magnetization curves.

Fig. 10. Calculation of magnetization curves for 22-kVA transformer using voltage “Zero-Crossing” detection method, (a) with varying resistive impedance, (b) with varying reactive impedance.

of the proposed voltage “Zero-Crossing” detection logic and resulting magnetization curves are shown in Fig. 10(a) and (b) for the resistive and reactive impedance cases respectively. A summary of %error in the calculation of magnetization curves is shown in Table 2. It may be observed that the error in the calculation of magnetization curves is almost negligible with application of the proposed “Zero-Crossing” method.

5. Conclusions The phenomenon of voltage distortion during no-load testing of transformers and its influence on no-load loss measurements and calculation of magnetization curves has been investigated using

extensive laboratory and factory test measurements. As illustrated for the large three-phase transformer factory test case, voltage THD can easily reach the 15% THD limit recommended in IEEE standard C57.123 at 100% of rated excitation voltage. Increasing the excitation voltage to 112% causes a steep increase of voltage THD values in all three phase voltages up to 40%. This demonstrates the severity of voltage distortion problem that can be encountered in a factory test environment. Similar voltage distortion effects were reproduced using a 22-kVA single-phase laboratory transformer with peaky voltages and flat-topped flux waveforms, by inserting reactive impedance in the no-load test circuit. However, the variation in voltage THD curves with increasing excitation levels was more gradual in the laboratory test case as the source impedance used

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Table 2 %error in peak current calculation of magnetization curves.

in the factory test (typically synchronous generator and step-up transformer) presents a not so stiff source. The IEEE and IEC loss correction limits of 5% and 3% were violated even for low level of harmonics (THD < 15%) in the laboratory test case. Moreover, the corrected no-load losses were overcompensated by 2.5% and 6% for the cases with 24  resistive and reactive impedances, respectively. This error can be much higher as voltage distortion and excitation level increases. As a consequence of the voltage distortion problem, application of existing methods resulted in an error of 20% in the calculation of peak magnetization current at a peak flux-linkage of 1.35 Wb-T. If curve fitting techniques are used to extrapolate the magnetization curve at higher excitation voltages, this error will increase exponentially. However, application of the proposed voltage “Zero-Crossing” detection method resulted in a more accurate calculation of magnetization characteristics completely unaffected by voltage distortion. Further, the proposed logic circuit can be easily implemented in a factory test environment, and integrated in parallel with existing no-load test instrumentation. Acknowledgements This work was performed as part of collaboration between Michigan Technological University (MTU), Houghton, MI 49931, USA and Norwegian University of Science and Technology (NTNU), Trondheim, NO-7491, Norway, and was supported in part by KMB research projects “Transformer Performances” and “EMT Transients.” References [1] IEEE Standard Test Code for Liquid-Immersed Distribution, Power, and Regulating Transformers, IEEE Standard, 2015, C57.12.90. [2] Standard IEEE Test Code for Dry-Type Distribution Power Transformers, IEEE Standard, 2011, C57.12.91. [3] IEEE Guide for Transformer Loss Measurement, IEEE Standard, 2010, C57.123. [4] Power Transformers Part 1: General, International Standard IEC 60076-1, Edition 3.0, 2011. [5] The J&P Transformer Book, in: M.J. Heathcote (Ed.), 13th ed., Newnes, Burlington, MA, 2007, pp. 42–43. [6] P. Rohan, T. Kalliomaa, Transforming Revenue—ABB’s Technology Cuts Transformer Losses, ABB Review 2|15, Report # 9AKK10103A2880, 2015. [7] J.A. Martinez-Velasco, Power System Transients: Parameter Determination, CRC Press, Boca Raton, FL, 2010, pp. 192–196. [8] B.A. Mork, F. Gonzalez, D. Ishchenko, D.L. Stuehm, J. Mitra, Hybrid transformer model for transient simulation—part I: development and parameters, IEEE Trans. Power Deliv. 22 (January (1)) (2007) 248–255. [9] B.A. Mork, F. Gonzalez, D. Ishcheko, D.L. Steuhm, J. Mitra, Hybrid transformer model for transient simulation—part II: laboratory measurements and benchmarking, IEEE Trans. Power Deliv. 22 (January (1)) (2007) 256–262.

[10] N. Chiesa, H.K. Hoidalen, Analytical algorithm for the calculation of magnetization and loss curves of delta-connected transformers, IEEE Trans. Power Deliv. 25 (July (3)) (2010) 1620–1628. [11] S.G. Abdulsalam, W. Xu, W.L.A. Neves, X. Liu, Estimation of transformer saturation characteristics from inrush current waveforms, IEEE Trans. Power Deliv. 21 (January (1)) (2006) 170–177. [12] S. Prusty, M.V.S. Rao, A direct piecewise linearized approach to convert RMS saturation characteristic to instantaneous saturation curve, IEEE Trans. Magn. 16 (January (1)) (1980) 156–160. [13] W.L.A. Neves, H.W. Dommel, On modeling iron core nonlinearities, IEEE Trans. Power Syst. 8 (February (1)) (1993) 417–422. [14] D.G. Ece, H. Akcay, An analysis of transformer excitation current under nonsinusoidal supply voltage, IEEE Trans. Instrum. Meas. 51 (October (5)) (2002) 1085–1089. [15] S.V. Kulkarni, S.A. Kaparde, Transformer Engineering—Design, Technology, and Diagnostics, CRC Press, Boca Raton, FL, 2013, pp. 44–49. [16] F.C. De La Rosa, Harmonics and Power Systems, CRC Press, Boca Raton, FL, 2006, pp. 35–37. [17] E. So, R. Arseneau, E. Hanique, No-load loss measurements of power transformers under distorted supply voltage waveform conditions, IEEE Trans. Instrum. Meas. 52 (April (2)) (2003) 429–432. [18] R. Arseneau, E. So, E. Hanique, Measurement and correction of no-load losses of power transformers, IEEE Trans. Instrum. Meas. 54 (April (2)) (2005) 503–506. [19] B.P. Lathi, R.A. Green, Essentials of Digital Signal Processing, Cambridge University Press, New York, NY, 2014. [20] S. Barg, K. Ammous, M. Hanen, A. Ammous, An improved emperical formulation for magnetic core losses estimation under non-sinusoidal excitation, IEEE Trans. Power Electron. 32 (March (3)) (2017) 2146–2154. [21] L.K. Rodrigues, G. Jewell, Model specific characterization of soft magnetic materials for core loss prediciton in electrical machines, IEEE Trans. Magn. 50 (November (11)) (2014) 1–4. [22] A.J. Batista, J.C.S. Fagundes, P. Viarogue, An automated measurement system for core loss characterization, IEEE Trans. Instrum. Meas. 48 (April (2)) (1999) 663–667. [23] T. Tran-Quoc, L. Pierrat, Correction of the measured core losses under distorted flux, IEEE Trans. Magn. 33 (March (2)) (1997) 2045–2048. [24] R. Arseneau, W.J.M. Moore, A method for Estimating the Sinusoidal Iron Losses of a Transformer From Measurements made with Distorted Voltage Waveforms, IEEE Trans. Power Appar. Syst. PAS-103 (November (10)) (1984) 2912–2918. [25] D. Slomovitz, Correction of power transformer no-load losses measured under nonsinusoidal voltage waveforms, IEE Proc. 136 Part C (January (1)) (1989) 42–47. [26] H.W. Dommel, Electromagnetic Transients Program Reference Manual (EMTP Theory Book), Bonneville Power, Portland OR, 1992. [27] ATP Rule Book, Leuven EMTP Center, Leuven, 1987. [28] J. Pedra, F. Cyrcoles, L. Saiz, R. Lypez, Harmonic nonlinear transformer modeling, IEEE Trans. Power Deliv. 19 (April (2)) (2004) 884–890. [29] S.E. Zirka, Y. Moroz, R.G. Harrison, N. Chiesa, Inverse hysteresis models for transient simulation, IEEE Trans. Power Deliv. 29 (April (2)) (2014) 552–559. [30] A. Rezaei-Zare, Enhanced transformer model for low- and mid- frequency transients—part I: model development, IEEE Trans. Power Deliv. 30 (February (1)) (2015) 307–315. [31] N. Chiesa, H.K. Hoidalen, Hysteretic iron-core inductor for transformer inrush current modeling in EMTP, in: 16th Power System Computation Conference, Glasgow, UK, 2008. [32] R.W. Wall, Simple methods for detecting zero crossing, in: The 29th Annual Conference of IEEE Industrial Electronics Society (IECON’03), Roanoke, VA November 02–November 06, 2003.