Harmonic Impact on Distribution Transformer No-Load Loss

Harmonic Impact on Distribution Transformer No-Load Loss

Themistoklis D. Kefalas, Member, IEEE, and Antonios G. Kladas, Member, IEEE

Abstract

The losses in European Union distribution transformers are estimated at about $33 \mathrm{TW} \cdot \mathrm{h} /$ year, whereas reactive power and harmonic losses add a further $5 \mathrm{TW} \cdot \mathrm{h} /$ year. The reduction of distribution transformer no-load loss is particularly important as the ratio of no-load to load losses is nearly three. In this paper, the no-load operation of wound-core transformers under sinusoidal and distorted supply-voltage conditions is investigated. For that purpose, a 2-D nonlinear transient finite-element analysis taking into account hysteresis has been developed. The hysteresis model is based on a modified Jiles-Atherton representation, and the proposed analysis is compared to experimental data.

Index Terms—Finite element methods, industrial power system harmonics, magnetic anisotropy, magnetic cores, magnetic hysteresis, magnetic losses, nonlinear magnetics, power transformers, soft magnetic materials, transient analysis.

NOMENCLATURE

$\alpha, a \quad$ Parameters of the Langevin function.
$\dot{\mathbf{A}} \quad$ Time derivative of the magnetic vector potential vector.
$\alpha_{H} \quad$ Parameter of the hysteresis model.
$A_{z} \quad z$ th component of the magnetic vector potential.
$B_{p} \quad$ Peak flux density.
D Potential-current coupling stiffness matrix.
DF Deterioration factor.
$f \quad$ Frequency.
G Inductive damping matrix.
$H_{\mathrm{AN}} \quad$ Anhysteretic component of the magnetic field.
$H_{H} \quad$ Hysteretic component of the magnetic field.
$H_{\mathrm{HS}} \quad$ Parameter of the hysteresis model.
$H_{\mathrm{HW}} \quad$ Irreversible part of the hysteretic component.
$H_{\text {RET }} \quad$ Reversible part of the hysteretic component.
I Time derivative of the electric current vector.
$J_{z} \quad z$ th component of the current density.
$L\left(\lambda_{\mathrm{AN}}\right) \quad$ Langevin function.
$M_{S} \quad$ Saturation magnetization.
$M_{x}, M_{y} \quad$ Components of magnetization.
$N \quad$ Number of turns of the excitation coil.
$S \quad$ Lamination cross-sectional area.

^[1]$\langle V\rangle$ Average rectified voltage.
$\gamma_{H} \quad$ Parameter of the hysteresis model.

I. INTRODUCTION

ACCORDING to Strategies for development and diffusion of Energy Efficient Distribution Transformers (SEEDT), the losses in European Union (EU-27) distribution transformers are estimated at about $33 \mathrm{TW} \cdot \mathrm{h} /$ year. Furthermore, reactive power and harmonic losses add a further $5 \mathrm{TW} \cdot \mathrm{h} /$ year. Thus, the total losses of distribution transformers in EU-27 might total to about $38 \mathrm{TW} \cdot \mathrm{h} /$ year. In comparison, the energy consumption of Athens subway (ATTIKO METRO S.A.) in 2006 was more than 400 times less, i.e., $0.09 \mathrm{TW} \cdot \mathrm{h} /$ year.

Transformer losses can be resolved into two components, namely, no-load and load losses. No-load loss occurs from the energy required to retain the continuously varying magnetic flux in the core, and it is invariant with load on the transformer. Load loss arises mainly from resistance losses in the conducting material of the windings, and it is varying with the second power of the load. One might argue that the load loss of a typical distribution transformer is more important than the noload loss. This stems from the fact that the load loss at rated load is six times more than the no-load loss in the case of transformers constructed of grain-oriented steel and 13 times more in the case of transformers constructed of amorphous steel [1]. Nevertheless, a distribution transformer is in circuit every day for 24 h and has a no-load loss load factor of $100 \%$, irrespective of the load delivered. The load loss, on the other hand, will vary over a wide range, even for the same average load, since load losses are proportional to the square of load. As a result, the continuous loss of energy to the magnetic core exceeds the loss of energy to the conductive material of the transformer, even in the cases where the core is constructed of amorphous steel.

Due to the loading characteristics of EU-27 distribution transformers, the ratio of no-load to load losses is close to three. Consequently, the reduction of distribution transformer no-load losses has recently become the top priority of many countries [1].

No-load loss is sensitive to the distortion of the supplyvoltage waveform [2]. The deterioration of the supply-voltage waveform is caused by industrial and domestic loads, e.g., switch-mode power supplies and fluorescent lamps that inject current harmonics in the power grid. Researchers developed passive networks for the reduction of the ac input-current total harmonic distortion (THD) [3], [4], techniques for the minimization of undesired low-order harmonics at the input

current of asymmetrically loaded six-pulse rectifiers [5], and ac-dc converters with low-line-current distortion [6]. More recently, cascaded multilevel inverters were developed for effective control of the input-current and output-voltage waveform, diminishing in this manner harmful harmonic components [7], [8], and active power filters and static var compensators using multilevel inverters for the compensation of load currents with a high-harmonic impact [9].

This paper deals with the distorted supply-voltage waveforms of the power grid that arise from the aforesaid situations, i.e., without multiple zero crossings (no minor loops) and only odd higher harmonics [10], in order to evaluate in particular the harmonic impact on wound-core distribution transformer no-load operation. Transformers are classified, depending on their magnetic core type, stack core, or wound core. Many researchers have published their results from investigations of stacked-type transformer cores [11], [12]. As a counterpart, there are a few published works in the case of wound-type transformer cores, and many issues still remain unclear since the stack- and wound-core topologies are completely different. The authors have developed recently numerical techniques based on the finite-element (FE) method, specifically formulated for the no-load loss and flux-distribution evaluation of woundcore distribution transformers under sinusoidal supply-voltage conditions [13], [14]. This paper extends the aforementioned numerical technique in order to include the case of distorted supply-voltage conditions commonly encountered in the power grid. A transient analysis is necessary for determining the harmonic impact on wound-core distribution transformer noload operation. The analysis developed herein integrates the hysteresis phenomena in the FE method. The hysteresis model that the authors used is based on previous formulations given in [11] and [15]-[17]. Nevertheless, the hysteresis model and the respective FE formulation have been modified, as previous attempts were constrained to the modeling of stacked magnetic cores [11], ferrite pot cores [16], and rectangular Epstein frames [17]. The simulated results are compared to the experimental results obtained by a no-load loss and local-flux-density measurement apparatus [18].

The purpose of this paper is to point out to the reader the importance of distribution transformer no-load loss, as well as the harmonic impact on transformer no-load operation, and to provide transformer manufacturers a no-load loss analysis numerical tool with which they can evaluate the technically and economically optimum distribution transformer, working under either sinusoidal or distorted supply-voltage conditions.

II. Brief Description of Wound-Core Transformers

In order to transmit and distribute electrical energy over large distances economically, it is necessary to minimize Joule losses in the transmission lines by using a high voltage. The required increase and decrease of the voltage are performed by the electrical transformer that, in its simplest form, consists of two coils of conductive wire wound around a magnetic core of soft iron. In practice, two types of magnetic core are used, namely,

Fig. 1. Typical wound core.

Fig. 2. Wound-core transformer. (a) One-phase core type. (b) One-phase shell type. © Three-phase shell type.
the stack core and the wound core. A typical wound core, shown in Fig. 1, is composed of hundreds of long continuous strips of sheet steel. The main advantages include reduction of joints and the use of the grain direction of the steel for the flux path. Wound cores are commonly used for distribution transformers with ratings that range from a few kVA to a few MVA. Oneand three-phase wound-core distribution transformers are constructed by using one or more wound cores assembled about a preformed electrical winding coil, as shown in Fig 2.

III. FE Analysis for No-Load Loss Evaluation

The FE method has been used extensively by many researchers for the accurate field analysis of electromagnetic devices. A 2-D FE analysis approach used for the analysis of 3-D magnetic components was developed in [19], an FE analysis to validate a new winding design for planar integrated magnetics has been carried out in [20], and a coupled electrothermal

numerical analysis of planar and medium-frequency power transformers was presented in [21] and [22], respectively. Also, the FE method was used for determining the transient behavior and short-circuit forces of stack-core power transformers in [12] and [23], respectively. The main advantage of the FE method over analytical methodologies like the one developed in [24] for the behavior modeling of magnetizing currents in a dc-dc converter is the accurate computation of the local distribution of field quantities such as flux density. This is necessary in the case of wound cores where it is experimentally verified that the flux-density distribution is nonuniform [13], [14].

The authors developed an FE methodology in [14] that is particularly formulated for wound cores in order to evaluate accurately the no-load losses under sinusoidal excitation. The method is based on the combination of the experimentally evaluated local specific core losses with the FE-computed peak-flux-density distribution of the wound core. The local specific core losses are expressed as a function of peak flux density, and the computation of the peak-flux-density distribution of the wound core is performed using the FE method. The accurate representation of the wound core with low computational cost was achieved by considering the iron-laminated material as a homogeneous medium and by developing an elliptic anisotropy model that is specifically formulated for wound cores [14]. The elliptic anisotropy model takes into account the directional dependence of the $B-H$ characteristic due to the iron laminations and the grain orientation of the magnetic steel. The material modeling of the wound core with the proposed technique results in significant computational effort reduction, not only in the case of 2-D FE analysis but also in the case of 3-D FE analysis [14].

In this paper, the numerical methodology developed in [14] is extended by considering the time evolution of flux density. For this to be achieved, a nonlinear transient FE analysis is developed, which integrates the hysteresis phenomena. In this manner, the technique developed in [14] can be applied not only to sinusoidal supply-voltage cases but also to distorted supplyvoltage conditions.

IV. HYSTERESIS MODEL

The hysteresis model is similar to the Jiles-Atherton model, but the independent variable is the flux density $B$ instead of the magnetic field $H$ [15]. In this way, the hysteresis model can be integrated in a 2-D FE analysis, where $B$ in each element can be directly evaluated by the curl of magnetic vector potential $A$. Simplicity, computational efficiency, and ease of implementation are its main advantages, in contrast to other hysteresis models like Preisach [11]. According to [15], $H$ is partitioned into two components, namely, the anhysteretic $H_{\mathrm{AN}}$ and the hysteretic $H_{H}$, as follows:

$$H=H_{\mathrm{AN}}+H_{H}$$

Furthermore, $H_{H}$ is divided into two parts, i.e., one associated with the irreversible part of domain-wall motion $H_{\mathrm{HW}}$, and the other one associated with the reversible part of domain-wall
motion $H_{\text {RET }}$. Thus, it follows that the magnetic energy of the hysteretic component $H_{H}$ is given by

$$\int_{\Delta B} H_{H} d B=\int_{\Delta B} H_{\mathrm{HW}} d B-\int_{\Delta B} H_{\mathrm{RET}} d B$$

The anhysteretic component $H_{\mathrm{AN}}$ is obtained by solving (3), where $M_{S}$ is the saturation magnetization, $L\left(\lambda_{\mathrm{AN}}\right)$ is the Langevin function of $\lambda_{\mathrm{AN}}$ given by (4) and (5), and $\alpha$ and $a$ are the parameters of the Langevin function

$$\begin{aligned} H_{\mathrm{AN}} & =B / \mu_{0}-M_{S} L\left(\lambda_{\mathrm{AN}}\right) \\ L\left(\lambda_{\mathrm{AN}}\right) & =\operatorname{coth}\left(\lambda_{\mathrm{AN}}\right)-1 / \lambda_{\mathrm{AN}} \\ \lambda_{\mathrm{AN}} & =\left[H_{\mathrm{AN}}(1-\alpha)+B\left(\alpha / \mu_{0}\right)\right] / a \end{aligned}$$

The irreversible part $H_{\mathrm{HW}}$ and the reversible part $H_{\mathrm{RET}}$ of the hysteretic component $H_{H}$ are given by (6) and (7), respectively, where $I_{D}$ is a variable that takes the +1 and -1 values when $\Delta B>0$ and $\Delta B<0$, respectively, and $H_{\mathrm{HS}}, a_{H}$, and $\gamma_{H}$ are the parameters of the hysteresis model

$$\begin{aligned} H_{\mathrm{HW}} & =H_{\mathrm{HS}} L\left(\lambda_{H}\right), \quad \lambda_{H}=\left(H_{H}+I_{D} H_{\mathrm{HS}}\right) / a_{H} \\ H_{\mathrm{RET}} & =I_{D} \gamma_{H} d H_{H} / d B \end{aligned}$$

The differential equation of hysteresis is obtained by substituting (6) and (7) into (2) and isolating $d H_{H} / d B$

$$\frac{d H_{H}}{d B}=\frac{H_{\mathrm{HS}} L\left(\lambda_{H}\right)-H_{H}}{I_{D} \gamma_{H}}$$

The application of the Euler scheme to (8) yields the following nonlinear equation, from the solution of which the hysteretic component $H_{H}$ is obtained:

$$I_{D} \gamma_{H} \Delta H_{H}=\left(H_{H S} L\left(\lambda_{H}\right)-H_{H}\right) \Delta B$$

V. TRANSIENT 2-D FE ANALYSIS

In 2-D FE analysis, Poisson’s equation is solved, which is a function of magnetic vector potential and reluctivity. When hysteresis is taken into account, numerical difficulties arise due to the discontinuity of reluctivity when $B$ is equal to zero. The aforesaid problem is tackled by considering the constitutive equation for ferromagnetic materials [11], [17]

$$\mathbf{B}=\mu_{0}(\mathbf{H}+\mathbf{M})$$

Combining the aforementioned equation and Ampere’s law yields the following equation for the 2-D problem:

$$\nabla \cdot v_{0} \nabla A_{z}+\left(\partial M_{y} / \partial x-\partial M_{x} / \partial y\right)+J_{z}=0$$

Considering that the two components of magnetization are a function of $B$ [17], i.e., $M_{x}=f\left(B_{x}, B_{y}\right)$ and $M_{y}=$ $f\left(B_{x}, B_{y}\right)$, the differential of $\mathbf{M}$ is given by

$$\left[\begin{array}{c} d M_{x} \\ d M_{y} \end{array}\right]=\left[\begin{array}{ll} \partial M_{x} / \partial B_{x} & \partial M_{x} / \partial B_{y} \\ \partial M_{y} / \partial B_{x} & \partial M_{y} / \partial B_{y} \end{array}\right]\left[\begin{array}{c} d B_{x} \\ d B_{y} \end{array}\right]$$

By using (12) and the 2-D vector potential, the second lefthand term of (11) is given by

$$\begin{aligned} \frac{\partial M_{y}}{\partial x}-\frac{\partial M_{x}}{\partial y} & =\frac{\partial M_{y}}{\partial B_{x}} \frac{\partial}{\partial x}\left(\frac{\partial A_{z}}{\partial y}\right)-\frac{\partial M_{y}}{\partial B_{y}} \frac{\partial}{\partial x}\left(\frac{\partial A_{z}}{\partial x}\right) \\ & -\frac{\partial M_{x}}{\partial B_{x}} \frac{\partial}{\partial y}\left(\frac{\partial A_{z}}{\partial y}\right)+\frac{\partial M_{x}}{\partial B_{y}} \frac{\partial}{\partial y}\left(\frac{\partial A_{z}}{\partial x}\right) \end{aligned}$$

Also, the relationship between the differentials of magnetization and flux density can be expressed by means of a tensor [16], [17]. Using the Euler scheme to represent the differentials yields

$$\left[\begin{array}{l} \Delta M_{x} \\ \Delta M_{y} \end{array}\right]=\frac{|\Delta \mathbf{M}|}{|\Delta \mathbf{B}|}\left[\begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]\left[\begin{array}{l} \Delta B_{x} \\ \Delta B_{y} \end{array}\right]$$

where $\theta$ is the rotation angle between $\Delta \mathbf{M}$ and $\Delta \mathbf{B}$. By multiplying and dividing the right-hand side of (14) with $|\Delta \mathbf{B}|$, it follows that

$$\left[\begin{array}{l} \Delta M_{x} \\ \Delta M_{y} \end{array}\right]=\left[\begin{array}{cc} \eta & -\xi \\ \xi & \eta \end{array}\right]\left[\begin{array}{l} \Delta B_{x} \\ \Delta B_{y} \end{array}\right]$$

where $\eta$ and $\xi$ are given by

$$\eta=\frac{\Delta \mathbf{M} \cdot \Delta \mathbf{B}}{|\Delta \mathbf{B}|^{2}} \quad \xi=\frac{|\Delta \mathbf{M} \times \Delta \mathbf{B}|}{|\Delta \mathbf{B}|^{2}}$$

From (12) and (15), it follows that

$$\left[\begin{array}{ll} \frac{\partial M_{x}}{\partial B_{x}} & \frac{\partial M_{x}}{\partial B_{y}} \\ \frac{\partial M_{y}}{\partial B_{x}} & \frac{\partial M_{y}}{\partial B_{y}} \end{array}\right]=\left[\begin{array}{cc} \eta & -\xi \\ \xi & \eta \end{array}\right]$$

The problem’s solution is obtained by discretizing (18), which is derived by substituting (13) and (17) into (11)

$$\nabla \cdot\left(\left[\begin{array}{cc} v_{0}-\eta & \xi \\ -\xi & v_{0}-\eta \end{array}\right] \nabla A_{z}\right)+J_{z}=0$$

The coupled field-circuit global system of equations is given by (19), where $\mathbf{D}$ is the potential-current coupling stiffness matrix, $\mathbf{G}$ is the inductive damping matrix, and $\dot{\mathbf{A}}$ and $\dot{\mathbf{I}}$ are the time derivatives of the magnetic vector potential and electric current vectors, respectively. To solve the time-dependent system (19), the Euler backward numerical integration technique is used, and the nonlinearities are taken into account by using the Newton-Raphson iterative scheme

$$\left[\begin{array}{cc} \mathbf{0} & \mathbf{0} \\ \mathbf{G} & \mathbf{L} \end{array}\right]\left[\begin{array}{l} \dot{\mathbf{A}} \\ \dot{\mathbf{I}} \end{array}\right]+\left[\begin{array}{cc} \mathbf{S} & \mathbf{D} \\ \mathbf{0} & \mathbf{R} \end{array}\right]\left[\begin{array}{l} \mathbf{A} \\ \mathbf{I} \end{array}\right]=\left[\begin{array}{c} \mathbf{0} \\ \mathbf{V} \end{array}\right]$$

The important advantage of the proposed methodology is that it provides similar accuracy to the other techniques, existing in the literature [25], while involving faster convergence due to the adoption of the modified Jiles-Atherton model. However, such techniques involve time-stepping procedures for waveformdistortion consideration and require important computation resources.

Fig. 3. Experimental setup.

Fig. 4. Block diagram of the experimental setup.

VI. EXPERIMENTAL SETUP

The experimental setup used for evaluating the harmonic impact on wound-core no-load loss is shown in Fig. 3. The block diagram of the experimental setup is shown in Fig. 4.

A 23 -turn excitation coil was supplied from a single- and three-phase supply via a variable transformer (Regulac, type RQ 25-M, input: 240 V 50/60 Hz, output: $0-275 \mathrm{~V}, 25 \mathrm{~A}$ and Regulac, type RK8-G3M, input: 415 V 50/60 Hz, output: $0-476 \mathrm{~V}, 8 \mathrm{~A}$ ) in order to magnetize the wound cores with a distorted supply-voltage waveform. A programmable ac and dc power supply (California Instruments MX30) was used so as to magnetize the wound cores with a sinusoidal waveform.

The voltage across the excitation coil terminals was captured using an active differential voltage probe. The specific voltage probe provides safe means of measuring floating potentials by converting the high input differential voltage ( $\leq 1400 \mathrm{~V}$ peak) into a low voltage ( $\leq 5 \mathrm{~V}$ ). The 3-dB frequency of the voltage probe is 18 MHz , the rejection rate on common mode at 50 Hz is 90 dB , the dc output offset voltage is 0.15 mV , and the attenuation factor is equal to 205.

A current probe based on the Hall effect is used for capturing the no-load current. It provides a galvanic isolation between the primary circuit (high power) and the secondary circuit (electronic circuit), and very good linearity ( $<0.2 \%$ ). The current measurement range is $\pm 36 \mathrm{~A}$ peak, and the 1-dB frequency of the probe is 150 kHz .

Fig. 5. Current and active differential voltage probes.
The voltage and current probes were designed and manufactured by the authors at the Laboratory of Electrical Machines, National Technical University of Athens. The circuit analysis of the probes was carried out by using OrCAD-PSpice. Both probes are shown in Fig. 5.

The output of the probes was connected to a noise-rejecting shielded BNC I/O connector block (BNC-2110) via passive probes (TP6060). A noise-rejecting shielded cable (SHC68-68EP) connects the data-acquisition (DAQ) device directly to the BNC-2110 connector block.

The DAQ device used was National Instruments (NI) 6143. NI 6143 has eight differential channels, an ADC resolution of 16 bit, a maximum sampling rate of 250000 samples/s per channel analog input, and a $\pm 5 \mathrm{~V}$ analog-input-signal range. Finally, the DAQ device was placed into a PCI slot of a desktop PC (processor: $1.8-\mathrm{GHz}$ Intel Core 2 Duo, memory: 1-GB DDR2).

Three virtual instruments (VIs) were created with the use of LabVIEW software. The purpose of the first VI was the realtime measurement of the excitation coil voltage and no-load current waveforms, fast Fourier transform (FFT), and peak, rms, and THD values. The second VI was used for capturing the voltage and no-load current waveforms and FFT, for two periods of the fundamental frequency, and for the maximum sampling rate of the DAQ device. The third VI was used for manipulating the acquired voltage and current data in order to compute the no-load loss.

The repeatability error of the experimental setup is $\pm 0.3 \%$, whereas the absolute error is within $\pm 0.5 \%$. Thus, the specific experimental setup permits the evaluation of the harmonic impact on no-load loss, even in the cases of distorted supplyvoltage waveforms of low THD.

The experimental setup was also used for local-flux-density measurements in order to evaluate the FE-computed fluxdensity distribution of wound cores. Two-turn search coils wound around the total width of the steel sheet were inserted in order to determine the peak-flux-density distribution along the limb, yoke, and corner of the wound cores. A $0.1-\mathrm{mm}$-diameter solderable enameled copper wire was used for the search coils. The overall loss of the core did not change noticeably after the coils were wound in position, showing that the flux distribution

Fig. 6. Percentage of higher voltage harmonics plot of deformed line-to-line and line-to-neutral supply voltages.
did not change much due to the search coils being introduced. The voltages induced in the search coils were captured by directly connecting the search coils into the NI6143 DAQ card inputs. Necessary calculations and analysis of the captured data were carried out by using LabVIEW software.

VII. ReSults and DiSCuSSION

Seven wound cores of different geometric parameters and core materials, i.e., a conventional grain-oriented (M4 0.27 mm ) and a high-permeability (M-OH 0.27 mm ) grainoriented electrical steel, have been tested at peak flux densities up to 1.8 T . Both line-to-line and line-to-neutral voltages were used during the no-load test in order to evaluate the effect of their different harmonic contents. The no-load loss and the spectral information of the voltage and no-load current, up to the 29th harmonic component, were obtained by the VIs described in Section VI. The results are presented in terms of the deterioration factor (DF) that is defined as the ratio of the no-load loss due to a distorted voltage waveform to the no-load loss due to a pure sinusoidal voltage, for the same peak flux density [10]. The peak flux density $B_{p}$ is given by (20), where $\langle V\rangle$ is the average rectified voltage, $f$ is the frequency, $N$ is the number of turns of the excitation coil, and $S$ is the lamination cross-sectional area

$$B_{p}=\langle V\rangle / 4 f N S$$

The phase of the voltage harmonics relative to the fundamental, as well as their amplitude, influences the no-load loss [10]. Figs. 6 and 7 show the harmonic contents of the distorted line-to-line and line-to-neutral supply voltages, which were used as input for the nonlinear FE transient analysis.

One-half of the wound-core geometry is modeled due to symmetry. Fig. 8 shows the geometry and the respective fluxdensity vector plot of one of the seven wound cores tested. The output of the nonlinear FE transient analysis is the no-load current waveform. The mean power, i.e., no-load loss, is easily obtainable by the voltage and the theoretically determined noload current waveforms.

The simulated and experimental no-load currents for sinusoidal, distorted line-to-line, and distorted line-to-neutral

Fig. 7. Phase-angle plot of deformed line-to-line and line-to-neutral supply voltages.

Fig. 8. Wound-core geometry and flux-density vector plot (wound-core width: 190 mm , electrical steel: $\mathrm{M}-\mathrm{OH} 0.27 \mathrm{~mm}$ ).

Fig. 9. Wound-core simulated and experimental current waveforms under sinusoidal voltage excitation.
voltage excitations are shown in Figs. 9-11, respectively. The peak flux density is 1.8 T , and $f=50 \mathrm{~Hz}$.

Table I presents a comparison between the experimental and calculated no-load loss, DF, and rms values of no-load current for the sinusoidal and distorted voltage waveforms examined. The deterioration percentages of no-load loss due to distorted line-to-line and line-to-neutral voltages are $3.46 \%$ and $1.51 \%$, whereas the rms values of no-load current, an important transformer design constraint, increase by $37.3 \%$ and $27 \%$, respectively.

Fig. 10. Wound-core simulated and experimental current waveforms under deformed line-to-line voltage excitation.

Fig. 11. Wound-core simulated and experimental current waveforms under deformed line-to-neutral voltage excitation.

TABLE I
Simulated Versus Experimental Results

A comparison of the harmonic content of the experimental and simulated no-load currents under sinusoidal, deformed line-to-line, and deformed line-to-neutral supply-voltage waveforms is shown in Figs. 12-17. The simulated results are in relatively good agreement with the experimental ones. Nevertheless, in the case of deformed line-to-line and line-to-neutral excitations, there is a large discrepancy in the experimental and simulated phase angles of the 17 th, 19 th, and 23 rd and the 19 th, 25 th, and 27 th harmonics, respectively. These differences explain the deviance between the experimental and simulated no-load current waveforms shown in Figs. 10 and 11.

Fig. 12. Amplitude plot of experimental and simulated no-load currents under sinusoidal supply-voltage excitation.

Fig. 13. Phase-angle plot of experimental and simulated no-load currents under sinusoidal supply-voltage excitation.

Fig. 14. Amplitude plot of experimental and simulated no-load currents under deformed line-to-line supply-voltage excitation.

Also, Figs. 12-17 show the effect of the distorted supplyvoltage waveform on the no-load current harmonic content. A comparison of Figs. 12, 14, and 16 shows that the amplitude of the fundamental, third, and fifth harmonics of the no-load current under deformed-voltage-waveform excitation is larger than that of the no-load current under sinusoidal waveform

Fig. 15. Phase-angle plot of experimental and simulated no-load currents under deformed line-to-line supply-voltage excitation.

Fig. 16. Amplitude plot of experimental and simulated no-load currents under deformed line-to-neutral supply-voltage excitation.

Fig. 17. Phase-angle plot of experimental and simulated no-load currents under deformed line-to-neutral supply-voltage excitation.
excitation. Nevertheless, this is not the case for the rest of the harmonics. Also, the impact of deformed line-to-line excitation on no-load current is more significant than that of deformed line-to-neutral excitation.

To sum up, the theoretical and experimental results presented in this paper clearly illustrate that the distorted supply-voltage

waveform of the grid caused by nonlinear loads results in a notable increase of wound-core distribution transformer noload loss and a significant increase of the magnetizing noload current. The deterioration of no-load loss and magnetizing current depends not only on the amplitude of voltage harmonics but also on the phase of voltage harmonics relative to the fundamental.

VIII. CONCLUSION

This paper has described the significance of distribution transformer no-load loss, as well as the harmonic impact on no-load operation, and has introduced a methodology based on FE hysteresis model, enabling one to predict satisfactorily the harmonic impact on wound-core distribution transformer noload operation. The originality of the proposed method relies on the modified Jiles-Atherton hysteresis model adopted, necessitating reduced execution time with respect to other similar techniques.

The methodology introduced in this paper, if implemented at the design stage of distribution and power transformers, may provide great services in reducing the no-load losses due to distortion of the supply voltage.

Future research will focus on the modeling of eddy currents in the windings of the transformers, caused by current harmonics, using a coupled thermal-electromagnetic analysis.

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Themistoklis D. Kefalas (M’09) was born in Greece in 1977. He received the Electrical Engineering Educator degree from the School of Pedagogical and Technological Education, Athens, Greece, in 1999 and the Diploma and the Ph.D. degree in electrical engineering from the National Technical University of Athens, Athens, Greece, in 2005 and 2008, respectively.
He is currently with the School of Electrical and Computer Engineering, National Technical University of Athens. His research interests include transformer and electric machine modeling and optimization.
Dr. Kefalas is a member of the Technical Chamber of Greece.

Antonios G. Kladas (S’80-A’99-M’02) was born in Greece in 1959. He received the Diploma in electrical engineering from the Aristotle University of Thessaloniki, Thessaloniki, Greece, in 1982 and the D.E.A. and Ph.D. degrees from Pierre and Marie Curie University (Paris 6), Paris, France, in 1983 and 1987, respectively.
He was an Associate Assistant with Pierre and Marie Curie University from 1984 to 1989. From 1991 to 1996, he was with the Public Power Corporation of Greece, where he was engaged in the System Studies Department. Since 1996, he has been with the School of Electrical and Computer Engineering, National Technical University of Athens, Athens, Greece, where he is currently a Professor. His research interests include transformer and electric machine modeling and design, as well as the analysis of generating units by renewable energy sources and industrial drives.
Dr. Kladas is a member of the Technical Chamber of Greece.