Practical Experiments with a Ready-Made Strategy for Energizing a Suitable Pre-Magnetized Three-Column Three-Phase Dy Transformer in Unloaded State for Inrush Current Computations

Abstract

This article presents the results of an experimental verification of three-phase Dy transformer dynamics under no-load conditions. This study is motivated by previous ferroresonance analyses where the occurrence of inrush currents has been observed. The measurements covered all available electrical quantities in a transient state (12 measured and 3 additionally computed waveforms) during the device’s start-up under no-load conditions, as well as in a long-term steady state. A detailed analytical analysis is carried out for the obtained comprehensive set of measurement results. As a result of the conducted research, the mathematical model of the pre-magnetized three-phase Dy transformer is modified. Particular attention is paid to the issue of residual magnetism of the transformer core and its consideration in further research. The original strategy for energizing a three-column three-phase Dy transformer with a suitable pre-magnetization of its columns and original control switching system with a given/set value of the initial phase in the supply voltage is put to the test. The evolution of the induced inrush phenomenon up to the quasi-steady state under given (forced) conditions is documented (currents, voltages and the dynamics of changes taking place in the core (hysteresis loops)). This article represents a continuation of ongoing work on the study of transient states (dynamics of transformer inrush currents). At present, the Dy three-phase transformer is analyzed because of the requirements of industrial operators.

1. Introduction

Electrical transformers are crucial components in systems for the transmission and distribution of electrical power. Distribution transformers are particularly important. They play a vital role in converting electrical energy from one voltage level to another for efficient and safe distribution [1,2]. There is a long list of harmful events and factors occurring in the power system that might affect this critical component. Transformer transient states occurring due to lightning and switching as well as other disturbances such as short circuits, no-load connections or ferroresonance phenomena always lead to overvoltages or overcurrents [3,4,5]. Since overvoltages and overcurrents are inevitable in transformer operation, two issues must be taken into account. Transformers remaining in service must be systematically monitored on account of ongoing parameter deterioration. On the other hand, operating scenarios potentially harmful to these devices must be carefully investigated. Conclusions drawn from this research must be taken into account in the design and manufacture of the next generations of transformers.

The transformer inrush phenomenon occurring as a result of switching on the transformer in a no-load state constantly intrigues scientists and engineers [6,7,8,9,10]. This current, which is characterized as being almost entirely unidirectional, rises abruptly to its maximum value in the first half of the cycle after the transformer is energized. This transient current lasts a few milliseconds; afterwards, it slowly decays until the normal steady-state magnetizing conditions in the transformer are reached. The decay time of the inrush current depends on the time constant resulting from the variable inductance of the transformer core as well as the resistive component of the system impedance and the copper loss of the transformer [2].

The magnitude and duration of the inrush current depend upon the following four factors [5,11,12]:

-

The exact point on the voltage wave at the instant when the transformer circuit is energized (i.e., the switching angle);

-

The impedance of the circuit supplying the transformer;

-

The value and polarity of the residual flux linkage in the transformer core;

-

The nonlinear magnetic saturation curve of the transformer core.

In recent research [13], an experimental task was carried out consisting of modeling of the unloaded three-phase Dy transformer dynamics in the first period of the supply voltage. A methodology has been developed and simulation results are presented for a solution that takes into account the influence of previously mentioned strategic factors. Particular emphasis is laid on implementation of the following:

-

Residual magnetism in individual core columns;

-

The time instant of the voltage waveform at which voltage is switched on to the winding of the middle column of the transformer.

However, before the three-column three-phase Dy transformer selected for testing is subjected to measurement verification (relative to the developed methodology), it must undergo a complete inspection [8,14].

Particular attention has been paid to dynamic/transient waveforms occurring in the initial few periods after the switch-on instant, and the first period of supply voltage in particular. Waveforms recorded in a quasi-steady state of the unloaded transformer, obtained for a long-time interval after the switch-on time instant, have also been analyzed; this has been carried out especially in relation to the time constants of dynamic events. Their importance arises from difficulties in integrating stiff differential equations describing the unloaded transformer. The overall cost of calculating (integration) of the differential equations is very high. In [15], it was shown that the result could be achieved without integrating from zero initial conditions. The steady state can be achieved computationally by selecting the initial integration condition for the system of nonlinear differential equations from the limit of the solution cycle. In this way, the result (i.e., reaching the steady state) can be obtained within two periods.

Following the measurements, significant upgrades have been introduced to the mathematical model. The performed measurement experiments supported by theoretical research have shown many practical and useful features accompanying the processes governing electromagnetic transformation in no-load mode.

The most relevant research features which are highlighted in this paper are as follows:

-

The synchronized measurement of 12 waveforms (both in transient and steady state) to demonstrate the evolution of the inrush phenomenon and the strong imbalance accompanying the transient state;

-

Effective obtainment of flux linkage waveforms through numerical computation and its further usage in thorough examination of the model and the mechanism study of the inrush phenomenon;

-

Practical testing of the proposed original strategy with forced conditions/solutions, both original pre-magnetization and control switching;

-

The consideration of a solution to the issue of residual magnetism, demonstrating the existence of a certain symmetric magnetization of the core, whereby the induction in the central column of the core is twice as large as in the side columns and has the opposite sign; under these assumed initial conditions in the transformer’s state equations, there exists a voltage switching phase for the transformer where specified current impulses are obtained, independent of the phase sequence of the three-phase system;

-

A comprehensive analysis leading to the proposed model update and its mathematical and numerical implementation.

2. Analyzed Transformer and Its Mathematical Model

A three-phase three-limb core-type Dy transformer characterized by a nonlinear magnetization curve was subjected to analysis.

Delta/Wye connection is one of the various winding arrangements used for transforming three-phase voltages from one level to another. It contains the following two most common configurations:

-

Delta, in which the polarity end of one winding is connected to the non-polarity end of the next winding;

-

Wye (star), where the beginnings of all three windings are connected together.

The three-phase transformer is designed with three sets of windings wrapped around a common core. Both primary and secondary windings of each phase are on the same leg; one is wound over another. The simplified equivalent circuit diagram of the examined transformer is shown in Figure 1.

In addition to the physical quantities describing the parameters of the mathematical model of the transformer, this diagram shows the measured and recorded electrical quantities: voltages and currents.

A list of all the designations used in this work can be found in the Terminology section. We analyzed both transient and steady-state responses of the discussed transformer operating in a no-load state.

3. Transformer Winding Arrangements and Vector Groups

With three-phase transformers, the vector group plays a large role in determining the shape of the no-load current. When the primary and secondary windings are connected differently, the secondary voltage waveforms will differ from the corresponding primary voltage waveforms by a multiple of 30 electrical degrees. First, a phase shift (vector group) of the investigated transformer was verified by measurements.

Figure 2a,b show the corresponding line voltages (primary and secondary), as well as the line voltage on the primary side and the corresponding phase voltage on the secondary side. The analyzed phase shifts between voltages are illustrated in the vector diagrams. The secondary side voltages are multiplied by the turn ratio for better readability and to show the correspondence in shape (pairing) with the primary side voltages.

It may be observed that the secondary line voltage waveforms differ from the corresponding primary line voltage waveforms by 330 electrical degrees (with clockwise rotation), calculated from the vector for the HV (primary) winding (Figure 2a). As a result of the above, the line voltage from the delta-connected primary side is in phase with the phase voltage of the star-connected secondary side (Figure 2b).

For the exemplary pair of time instants indicated in the waveform in Figure 2a, we may calculate that (0.0605469–0.0588867) × 180/(10 × 10⁻³) = 29.884 = 30 deg (el.).

Because the primary winding voltage leads the secondary winding voltage by 30 degrees (for clockwise rotation, this means a 330 deg shift), the Dy11 vector group was determined for the investigated transformer.

4. Nonlinear Characteristics of the Ferromagnetic Core

Since the transformer (especially the power transformer) is almost always switched on in a no-load state, the inrush current depends on the magnetic properties of the core (magnetization curves and remanence flux) and the point on the sine wave (the initial phase ${\phi }_0$ of the supply voltage) at the switch-on time instant.

The inrush current of an unloaded transformer is characterized by current pulses. The impulse generated in the first period of the supply voltage appears to be particularly dangerous. Its appearance is closely related to the first entry into the core saturation area. This justifies the need to use an accurate approximation of the magnetization curve, especially in the saturation region [16].

For the discussed transformer, the magnetization curve H(B) for the ferromagnetic core material is known. Since the magnetization curve does not have its own equation, and for the purpose of including it in the original mathematical model, it was necessary to convert the $B-H$ relationship to a $I_0-\Psi$ relationship.

Different methods for approximating magnetization curves have been propounded in the publications on the subject. We have chosen the approximation using a polynomial expressed by odd powers.

$$H(B)={\displaystyle \sum _{k=1}^{11}}{a}_k{B}^{2k-1}$$

(1)

where H—magnetic field strength, B—magnetic flux density, and a—estimated coeffi-cients of the function H(B).

Figure 3 shows the magnetization curve provided by the manufacturer of the sheet core metal (Figure 3a) as well as the curve approximated by the selected method (Figure 3b).

5. Remanence

The magnitude of starting currents may also be influenced by residual magnetism [1,17]. Transformers often acquire a certain remanence, either from the disconnection process, or from some previous tests involving direct current (e.g., winding resistance measurement) [7]. The value and polarity of the residual magnetism present at the time when the transformer is turned on are arbitrary. Therefore, each time the transformer is turned on, the unidirectional current may reach different peak values, even if the transformer is always turned on at the same instantaneous value of the supply voltage.

Since the residual magnetism reflects the history of the core magnetization, it is enough to determine its value to calculate the values of the flux linkages ${\Psi }_1(t_0),{\Psi }_2(t_0),{\Psi }_3(t_0)$ at the turn-on time instant (initial conditions for state variables in the transformer dynamics equations) [13]. It is difficult to measure the value of residual magnetism necessary to set the initial conditions in simulation tasks [18,19]. Therefore, another course of action is proposed.

Instead of making the effort to estimate the remanence, a specific value of flux density is imposed by magnetizing the columns to a given level and with assumed symmetry [20]. A detailed discussion of the validity of the assumed symmetry can be found in [13]. Therefore, the present work includes only a diagram illustrating the expected result of the magnetization of the core of the tested transformer (Figure 4).

Figure 5 shows a diagram of the system for pre-magnetizing the columns of the tested transformer (the described symmetry is maintained). The diagram shows the electrical quantities measured and recorded as a function of time.

After configuring a series–parallel connection of the primary side windings and with the secondary side windings open, a constant voltage was applied (in the system as in Figure 5, the switch is in position ‘a’) [18,21].

According to Figure 5, $I_2=\frac{2}{3}{I}_0;I_1=I_3=\frac{I_0}{3};I_0=I_1+I_2$.

If the set direct current $I_0$ is greater than the amplitude of the transformer no-load current, then transformer columns will become saturated. For established circuit conditions, the switch position changes from position ‘a’ to position ‘b’. As a result of short-circuiting the connected primary windings, the currents flowing in them are extinguished to zero, while the magnetic fluxes in the individual columns of the transformer change from values resulting from DC magnetization to the values corresponding to residual magnetism.

Changes in these fluxes over time cause, in accordance with Faraday’s Law, the induction of voltages on the secondary side of the transformer ($N_2$ turns) [22]:

$$e_{k(s)}(t)=\frac{\mathrm{d}}{\mathrm{d}t}{\Psi }_k(t)$$

(2)

or

$$e_{k(s)}(t)=N_2\frac{\mathrm{d}}{\mathrm{d}t}{\Phi }_k(t)$$

(3)

where $k=1,2,3$.

The flux density in individual columns of the transformer can be written as

$$B_k(t)=\frac{{\Phi }_k(t)}{s_{\mathrm{Fe}}}$$

(4)

By performing the integration operation for the above equations and taking into account the initial conditions in the form $B_k(0)$, we obtain [17,23]

$${\Phi }_k(t)=B_k(0)s_{\mathrm{Fe}}+\frac{1}{N_2}{\displaystyle \underset{0}{\overset{t}{\int }}{e}_{k(s)}(\tau )}\mathrm{d}\tau$$

(5)

or in the form of

$$B_k(t)=B_k(0)+\frac{1}{N_2{s}_{\mathrm{Fe}}}{\displaystyle \underset{0}{\overset{t}{\int }}{e}_{k(s)}(\tau )}\mathrm{d}\tau$$

(6)

If the instantaneous values of voltages induced at the secondary side of the transformer $e_{k(s)}(\tau )$ are known, the result of integration according to the formula

$$B_k(0)=\frac{{\Psi }_k}{N_1{s}_{\mathrm{Fe}}}$$

(7)

gives the appropriate values of flux density $B_k(t)$ of the transformer columns.

In the limiting case, the flux density values correspond to the magnetic hysteresis of the transformer cores $B_k=\underset{t\to \infty }{\lim }{B}_k(t)$.

In the experiment, the waveforms of decaying currents in the primary windings $i_{1(p)}(t);i_{2(p)}(t);i_{3(p)}(t)$ have been recorded, as well as the voltages $e_{1(s)}(t);e_{2(s)}(t);e_{3(s)}(t)$ induced at the secondary side of the transformer during a current decay occurring after changing the switch position from ‘a’ to ‘b’ (Figure 5).

Figure 6 shows the result of the practical application of magnetizing the central column of the tested transformer while maintaining the assumed symmetry in the windings of the primary side; recorded $i_{k(p)}(t),e_{k(s)}(t)$ waveforms and the calculated ${\psi }_k(t)$ waveform are presented. If it is necessary to filter the measured waveforms (e.g., Figure 6a), the optimal SASS method [24] can be used for these transient signals with discontinuities.

The experiment of pre-magnetizing the transformer core columns was repeated for several different values of the magnetizing current $I_0$, always with positive polarization [25].

The conducted experimental tests have shown that regardless of the initial value of the initial magnetization, the limits $B_2=\underset{t\to \infty }{\lim }{B}_2(t)$ are approximately the same, and for the considered example, they amount to $B_2\cong 1.0 \mathrm{T}$. This is the value of magnetic hysteresis/residual magnetism for the transformer’s central column (Figure 7). For the outer columns, these values are approximately $B_1=B_3=-\frac{1}{2}{B}_2$.

Direct current, flowing in the system defined as in Figure 5, provides magnetization in individual columns of the transformer in the following proportions:

$$B_2=B_0;B_1=B_3=-\frac{B_0}{2}$$

(8)

If the direction of the current $I_0$ in the system as in Figure 5 is reversed, then identical magnetization symmetry will be maintained, but with the opposite direction

$$B_2=-B_0;B_1=B_3=\frac{B_0}{2}$$

(9)

This symmetry of the initial magnetization ((7) and (8)) causes the pulses, generated after switching on the transformer, to retain the same location in time regardless of the phase sequence of the supply voltage.

6. Impact of the Initial Phase of the Supply Voltage

The size of the current pulses emerging when the transformer is energized depends strongly on the time instant of the supply voltage waveform, at which this voltage is switched on to the transformer winding [12,26].

When the primary winding of a transformer is connected to an AC voltage source at the exact moment in time when the instantaneous voltage is at its positive peak, there is no inrush current (best scenario). In order for the transformer to create an opposing voltage drop to balance against this applied source voltage, a magnetic flux of rapidly increasing value must be generated. The result is that the winding current increases rapidly, but no more rapidly than under normal conditions. Both the core flux and coil current start from zero and build up to peak values identical to those occurring during continuous operation.

If the transformer’s connection to the AC voltage source occurs at the exact moment in time when the instantaneous voltage is zero, a current surge is very likely to occur. In the steady state, the highest value of magnetic flux occurs when the supply voltage applied to the winding passes through zero. According to this rule, at the start-up, the voltage value should correspond to the maximum value of the flux. However, it is impossible for the forced flux (sinusoidal component) to suddenly reach its maximum value. When the transformer is turned on, the flux in the core must be equal to zero (residual magnetism is neglected); therefore, there must be a unidirectional flux component (aperiodic component) in the system. This component, when added to the steady-state(periodic) waveform, increases the peak value of the resultant flux, even to the point of doubling its value [4].

In such circumstances, the core will almost certainly be saturated during this first half-cycle of voltage. If a significant increase in magnetic flux is to be generated, then a disproportionate amount of magnetomotive force (mmf) is required. This means that winding current, which generates the mmf causing flux in the core, will disproportionately rise to a value easily exceeding twice its normal peak; typically, the magnitude will exceed the rated full-load current by 3.5 to even 40 times. After a time equal to ½ of the supply voltage period, the free component of the flux decays exponentially with the time constant resulting from the transformer parameters in the no-load state, but it does not change its polarity.

In the simulation studies described in [13], the expected impact of the initial phase of the supply voltage on the magnitude of the generated pulses was presented.

At the moment t = 0, the instantaneous values of the supply voltages can be written in the form

$$e_{12(p)}=\left|E_{12\mathrm{m}}\right|\sin (\omega t_0);e_{23(p)}=\left|E_{23\mathrm{m}}\right|\sin (\omega t_0-\frac{2\pi }{3});e_{31(p)}=\left|E_{31\mathrm{m}}\right|\sin (\omega t_0+\frac{2\pi }{3})$$

(10)

Changing the $t_0$ parameter, it will be possible to perform calculations taking into account the influence of the time instant of switching on the voltage on the transformer, measured in relation to the time instant of the voltage $e_{23(p)}(t)$ waveform crossing zero.

7. Dynamic Measurements of the Dy11 Transformer Switched on in No-Load State

In our experiments, the analyzed transformer was switched on repeatably. Each start-up was preceded by the column pre-magnetization procedure. The magnetization was carried out in accordance with the described arrangement (Figure 4); the adopted symmetry (8) was maintained. For each switching-on case, instantaneous waveforms of the following quantities have been recorded discretely, in accordance with the symbols given in Figure 1: $u_{12(p)}(t),u_{23(p)}(t),u_{31(p)}(t)$—voltage waveforms at the primary winding terminals; $e_{1(s)}(t),e_{2(s)}(t),e_{3(s)}(t)$—waveforms of voltages induced in the secondary windings; $i_{12(p)}(t),i_{23(p)}(t),i_{31(p)}(t)$—current waveforms in the primary windings.

We recorded numerous switching scenarios. One case was selected for presentation in this article. It is characterized by the occurrence of a current pulse with practically the highest observed amplitude [27]. Since the initial phase of the supply voltage of the transformer center column winding is the reference system for the entire analyses, this value was initially revised.

Figure 8 shows the measured and recorded waveforms of all three delta currents of the transformer’s primary windings and the reference supply voltage $u_{23(p)}(t)$ applied to the transformer center column winding at the time instant when this voltage passed through zero. The exact time instant of the unloaded transformer launch indicates the initial phase of the $u_{23(p)}(t)$ voltage waveform to be c. −10 deg (Figure 8).

If the voltages $e_{1(s)}(t),e_{2(s)}(t),e_{3(s)}(t)$ induced in the $N_2$ turns of the unloaded secondary side are known, then it is possible to verify the voltages induced in the $N_1$ turns of the primary side. In the modeling, the voltages induced in the $N_2$ turns of the unloaded secondary side correspond to the time derivative of the flux linkages $\frac{\mathrm{d}}{\mathrm{d}t}{\Psi }_k(t)$ (Figure 1). Taking into account the numerical integration [23,28] of Equation (2) as well as the initial condition obtained from the given level of core magnetization (according to the assumed column symmetry), we can calculate the flux linkage waveforms.

$$\frac{\mathrm{d}}{\mathrm{d}t}{\Psi }_k(t)=r_N{e}_{k(s)}(t)$$

(11)

where $r_N=\frac{N_1}{N_2}$—transformer turns ratio, $N_1$—number of turns of the primary (high-voltage) side, and $N_2$—number of turns of the secondary (low-voltage) side.

The groups of the measured $u_{k,l(s)}(t),i_{k(p)}(t),e_{k(s)}(t)$ and calculated ${\psi }_k(t)$ dynamic waveforms obtained in transient state caused by switching on the unloaded transformer at the time instant at which the voltage $u_{23(p)}(t)$ passes through zero are shown in Figure 9. The waveforms are limited to the most significant first three periods; they are distinguished with respect to individual transformer columns of the primary side windings in order to picture their separate contributions to the phenomena.

Nonlinearity and hysteresis are the main properties that transformer cores exhibit when magnetized. Since hysteresis is related to changes in the magnetization of the core, the future evolution of $\psi -i$ pairs depending on past history might be presented and further investigated (e.g., [29], a ferromagnetic core coil model and research with different levels of core saturation). The evolution may be particularly interesting when long-term steady-state effects are taken into account. Details such as the size and direction of changes of subsequent hysteresis loops are no less interesting and important. Although the evolution of hysteresis loops has been presented for each winding (and therefore each column) separately, the collected minor loops maintain the features resulting from the existence of magnetic couplings between these windings. Detailed information on the scope and size of changes occurring in the core-winding pairs as well as all three pairs simultaneously in the first period of the transient state caused by switching on the transformer in no-load conditions is given in Figure 10. An incomplete hysteresis major loop obtained from the first period of $\psi -i$ waveforms is distinguished against the background of the collection of minor loops heading towards a steady state.

The discriminated hysteresis major loops (taken separately for each column) show interesting regularity and symmetry accompanying the core’s entrance into deep saturation in particular columns of the Dy transformer, which is switched on in an unloaded state following its pre-magnetization [30].

A similar approach was used for the waveforms in a quasi-steady state, obtained after 380 periods, i.e., c. ~7.6 s. Measured and calculated waveforms of the current and flux linkage were used to discriminate their exemplary single period. This procedure made it possible to develop the final (steady-state) hysteresis loops linked with specific columns.

Long-term quasi-steady-state waveforms of the current and flux linkage are depicted in Figure 11. Exemplary single periods of $\psi -i$ signals were used to develop the final (steady) hysteresis loops with respect to specific core columns. The obtained final hysteresis loops are presented against the background of a family of changing minor loops [22] during the formation of the steady state (decaying inrush transient state).

The hysteresis evolution, presented from the very instant (Figure 10) when the voltage is applied to the windings up to achieving the quasi-steady state (Figure 11), shows the size and directions of changes taking place in the transformer’s given column during the transient introduced by transformer start-up in specific conditions (pre-magnetization of the core and settled value of initial phase angle in reference voltage) [14,31].

8. Phase Imbalance

When a three-phase transformer is magnetized, there is always an imbalance between the phases. Either the sum of the phase currents is non-zero or the sum of the phase voltages is non-zero (Figure 9 and Figure 10). The imbalance is caused by the interaction of electrical and magnetic effects in the transformer [2,9,10]. The magnetization behavior of a core is governed by the magnetization curve of the steel in such a way that the magnetizing field is always given by the flux density. In addition to this imbalance caused by the hysteresis, a three-phase transformer is also imbalanced by the fact that the middle column is magnetically shorter than the outer columns.

The literature of the subject shows that in transient states, with large and rapid changes in saturation, the core is unable to accommodate such a high flux density; so, the magnetic path closes in the air or the transformer casing.

As we can see, the waveforms $u_{12(p)}(t),u_{23(p)}(t),u_{31(p)}(t)$ differ significantly from the waveforms $r_N{e}_{1(s)}(t),r_N{e}_{2(s)}(t),r_N{e}_{3(s)}(t)$ in the time intervals in which current pulses appear $i_{12(p)}(t),i_{23(p)}(t),i_{31(p)}(t)$.

Figure 12 shows the measured waveforms of the electromotive forces induced in the secondary winding. The calculated unbalanced electromotive force (emf) sum emerges as a decaying waveform. Its amplitude peaks coincide with the largest distortions in waveforms of the voltages induced in the secondary winding.

Taking into account the diagram (Figure 1), which is based on Kirchhoff’s second law, we can write for each phase separately

$$u_{k,l(p)}(t)=r_N{e}_{k(s)}(t)+R_{sk}{i}_{k,l(p)}(t)+L_{sk}\frac{\mathrm{d}}{\mathrm{d}t}{i}_{k,l(p)}(t)$$

(12)

where $R_{sk}$—winding resistance, $L_{sk}$—leakage inductance of the k-th primary winding, and $r_N{e}_{k(s)}(t)$—phase voltage of the secondary winding expressed in terms of the primary side; this is in fact the time derivative of the flux linkage.

The performed measurements and calculations show that

$$r_N\left(e_{1(s)}(t)+e_{2(s)}(t)+e_{3(s)}(t)\right)=\frac{\mathrm{d}}{\mathrm{d}t}{\Psi }_1(t)+\frac{\mathrm{d}}{\mathrm{d}t}{\Psi }_2(t)+\frac{\mathrm{d}}{\mathrm{d}t}{\Psi }_3(t)=\frac{\mathrm{d}}{\mathrm{d}t}\left({\Psi }_1(t)+{\Psi }_2(t)+{\Psi }_3(t)\right)\ne 0$$

(13)

which means that when the secondary winding voltages are summed, there is no balance compliance in the transient state. This unbalance is clearly visible in the first period of the induced voltage waveforms from the time instant of switching on the unloaded transformer (Figure 12 and Figure 13). Figure 13 also shows the symmetric pattern (as a result of imposed magnetization symmetry (8)) of the residual fluxes before the transformer switch-on time instant.

This means that when it comes to circuit modeling the transformer with intended transient-state issues in no-load conditions, Kirchhoff’s first law cannot be used for its magnetic circuit in the form ${\Psi }_1(t)+{\Psi }_2(t)+{\Psi }_3(t)=0$.

The calculated waveforms of the unbalanced sums of the electromotive forces induced in the secondary windings and flux linkages were thoroughly assessed in order to develop an update of the mathematical model. The analysis clearly indicates that unbalancing only covers the transient state caused by switching on an unloaded transformer and is obviously associated with the occurring peaks of inrush currents; it decays in accordance with the time constant resulting from the transformer parameters (Figure 14). It is also confirmed in Figure 14 that, in the steady state, Kirchhoff’s first law for emf and flux linkages is fulfilled.

A solution that takes this phenomenon into account in circuit modeling is the use of an air magnetic shunt for the transformer columns (air shunt magnetic resistance $R_{\mathrm{m}\mathsf{\delta }}$) (Figure 15).

The value of this shunt is selected in such a way as to obtain the maximum value

$$\underset{t\in [0,T]}{\max }\left\{r_N\left(e_{1(s)}(t)+e_{2(s)}(t)+e_{3(s)}(t)\right)\right\}$$

(14)

from values acquired from measurements.

In the experiment, initial magnetization of the transformer core columns needs to be introduced. Only then we can compare the calculation results with results obtained from measurements.

The deviation of the measurement voltage $r_N\left(e_{1(s)}(t)+e_{2(s)}(t)+e_{3(s)}(t)\right)$ from the numerical calculations $\frac{\mathrm{d}}{\mathrm{d}t}{\Psi }_1(t)+\frac{\mathrm{d}}{\mathrm{d}t}{\Psi }_2(t)+\frac{\mathrm{d}}{\mathrm{d}t}{\Psi }_3(t)$ (Figure 16) was adopted as the criterion for selecting the reluctance $R_{\mathrm{m}\mathsf{\delta }}$. In practical implementation, a comparison between measured and simulated waveforms can be made using a variety of criteria (for instance, as in [32], when sums of squares are used or through an integral error [33]).

The new and corrected transformer model differs from the model given in [13] based on the presence of a carefully selected air shunt $R_{\mathrm{m}\mathsf{\delta }}$. The changed model will use a different system of differential equations.

Using the diagram in Figure 15 and Kirchhoff’s second law for the magnetic circuits, we may write that

$$\begin{array}{l}{\Theta }_1(t)-U_{\mathrm{m},1}\left({\Psi }_1(t)\right)-R_{\mathrm{m}\mathsf{\delta }}\left({\Psi }_1(t)+{\Psi }_2(t)+{\Psi }_3(t)\right)=0\\ {\Theta }_2(t)-U_{\mathrm{m},2}\left({\Psi }_2(t)\right)-R_{\mathrm{m}\mathsf{\delta }}\left({\Psi }_1(t)+{\Psi }_2(t)+{\Psi }_3(t)\right)=0\\ {\Theta }_3(t)-U_{\mathrm{m},3}\left({\Psi }_3(t)\right)-R_{\mathrm{m}\mathsf{\delta }}\left({\Psi }_1(t)+{\Psi }_2(t)+{\Psi }_3(t)\right)=0\end{array}$$

(15)

where magnetic voltage drops [1,22] assume the following analytical form

$$U_{\mathrm{m},k}\left({\Psi }_k(t)\right)=h_{\mathrm{Fe},k}H\frac{{\Psi }_k(t)}{N_1{s}_{\mathrm{Fe}}}$$

(16)

while magnetomotive forces of individual transformer coils are provided by

$$\begin{array}{l}{\Theta }_1(t)=\left(i_{12(p)}(t)-i_{\mathrm{Fe},1}(t)\right)N_1\\ {\Theta }_2(t)=\left(i_{23(p)}(t)-i_{\mathrm{Fe},2}(t)\right)N_1\\ {\Theta }_3(t)=\left(i_{31(p)}(t)-i_{\mathrm{Fe},3}(t)\right)N_1\end{array}$$

(17)

By transferring the developed updates to the general model (Figure 1), the model’s differential equations take the following form:

$$\begin{array}{l}\frac{\mathrm{d}}{\mathrm{d}t}{\Psi }_1(t)=R_{\mathrm{Fe},1}{i}_{12(p)}(t)-\frac{R_{\mathrm{Fe},1}}{N_1}\left(U_{\mathrm{m},1}\left({\Psi }_1(t)\right)+R_{\mathrm{m}\mathsf{\delta }}\left({\Psi }_1(t)+{\Psi }_2(t)+{\Psi }_3(t)\right)\right)\\ \frac{\mathrm{d}}{\mathrm{d}t}{\Psi }_2(t)=R_{\mathrm{Fe},2}{i}_{23(p)}(t)-\frac{R_{\mathrm{Fe},2}}{N_1}\left(U_{\mathrm{m},2}\left({\Psi }_2(t)\right)+R_{\mathrm{m}\mathsf{\delta }}\left({\Psi }_1(t)+{\Psi }_2(t)+{\Psi }_3(t)\right)\right)\\ \frac{\mathrm{d}}{\mathrm{d}t}{\Psi }_3(t)=R_{\mathrm{Fe},3}{i}_{31(p)}(t)-\frac{R_{\mathrm{Fe},3}}{N_1}\left(U_{\mathrm{m},3}\left({\Psi }_3(t)\right)+R_{\mathrm{m}\mathsf{\delta }}\left({\Psi }_1(t)+{\Psi }_2(t)+{\Psi }_3(t)\right)\right)\end{array}$$

(18)

The derivative of the flux linkage $\frac{\mathrm{d}}{\mathrm{d}t}{\Psi }_k(t)$ is the electromotive force induced at the terminals of the secondary winding. Based on Figure 1, we can write that, for each column,

$$\frac{\mathrm{d}}{\mathrm{d}t}{\Psi }_k(t)=R_{\mathrm{Fe},k}{i}_{\mathrm{Fe},k}(t)$$

(19)

In the resulting updated equations, it is assumed that the resistances $R_{\mathrm{Fe},k}$ model the total power losses in the transformer core. They can be determined by measuring the active power of unloaded transformer.

This shows that the developed mathematical model of the transformer may be used to calculate the dynamics of its start-up in the no-load state; in addition, it can be successfully used to determine the steady state as the limit cycle of solving the appropriate differential equations. This has already been demonstrated in [15,16].

9. Steady-State Assessment

Measurements performed in the quasi-steady state show that all three no-load currents are different (Figure 17a). The no-load current in a specific phase is a combination of the magnetization current in that phase and the leakage current (i.e., current caused by the leakage flux) from the other phases [34]. The reasons for this phenomenon have already been explained; see Figure 10 and Figure 11. The different lengths of magnetic paths in particular transformer columns definitely lead to differences in their saturation processes. Although the final saturation level may be the same or similar in different columns, the anatomy of the hysteresis phenomenon in individual columns is completely different (Figure 17b).

The recorded no-load current waveforms differ not only in value and shape, but they are seriously deformed as well. Due to the nonlinear nature of the core, the behavior of the harmonics is not always easy to predict or explain. That is why a steady-state analysis of the unloaded transformer almost always complements transient-state analysis.

Today in particular, this subject cannot be dismissed because of the requirements in the field of power quality improvement and voltage distortion (harmonics) [2,35,36,37]. Measured and recorded waveforms and subsequently calculated harmonic components may be analyzed and compared to qualitative predictions [38]. The discrepancies are discussed. Figure 18 shows the results of detailed analyses of nonsinusoidal primary delta winding currents in no-load steady state.

The magnitude and phase of the complex Fourier series coefficients calculated for current waveforms indicated in Figure 18 are shown in

Table 1. Calculated Fourier series complex coefficients of the periodic current waveforms shown in Figure 18.

	Left Column		Center Column		Right Column
h	Magnitude	Phase	Magnitude	Phase	Magnitude	Phase
1	0.028924	−90	0.020146	−93	0.034740	−107
2	—	—	0.000481	110	—	—
3	0.011229	89	0.001438	−21	0.011411	−12
5	0.004271	−85	0.002568	−98	0.004733	81
7	0.000604	51	—	—	0.000779	−115
9	—	—	0.000431	36	—	—
11	0.000460	63	0.000383	23	0.000447	33

.

Qualitative analyses can also be freely extended to the selected fragments of the dynamic current and voltage waveforms where, e.g., the dominant harmonic component might be verified [12,39,40]. The exemplary results of analyses performed for the first period of the transient delta winding current of the center column are presented in Figure 19 and shown in

Table 2. Calculated Fourier series complex coefficients of the nonperiodic current waveform shown in Figure 19.

h	Magnitude	Phase
0	−0.0003258	0
1	2.9702900	−161
2	1.9593100	28
3	0.9618150	−131
4	0.1563650	89
5	0.1913520	33
6	0.1917900	−111
7	0.0129269	175
8	0.0769568	79
9	0.0945583	−74
10	0.0203067	−141
11	0.0346355	114
12	0.0468005	−49

; the participation of the second harmonic [6,26] may be observed here. It might be related to the fact that when the inrush current starts, it flows in only one direction (either positive or negative). It will be a half-wave waveform instead of a full wave.

10. Conclusions

Measurements of the three-phase, three-column Dy transformer in no-load conditions have been conducted. The simulation results and estimated transformer behavior related to the inrush phenomenon have been confronted with experimental results.

Measured and recorded waveforms of $u_{k,l(s)}(t),i_{k(p)}(t),e_{k(s)}(t)$ as well as calculated ${\psi }_k(t)$ have been subjected to analysis and compared to the simulative predictions, and the discrepancies have been discussed. The demonstrated inrush scenario is similar to the least favorable conditions for switching on the transformer, i.e., when the source voltage passes through zero at the moment of switching on the transformer. This has made it possible to implement a research project consisting of investigating the occurrence and value of current pulses depending on the level of residual magnetism.

A ready-made strategy for energizing a three-column three-phase transformer with a suitable pre-magnetization of its columns (described in [13]) has undergone a practical test. This has eliminated the problem that the value and polarity of the residual magnetism when the transformer is turned on are random. Therefore, each time the transformer is switched on, the unidirectional current may reach a different peak value, even if it is always turned on at the same instantaneous value of the supply voltage.

The applied solution has succeeded. The effectiveness of this solution is that instead of measuring or estimating the value of residual magnetism (which is difficult), this value is preset prior to the switch-on time instant. For the core prepared in this way (pre-magnetized to a known value), the transformer can be further switched on with a given (set) value of the initial phase in the supply voltage.

The measurements have shown that the maximum starting/inrush current practically does not depend on the value of the pre-magnetization current. For the magnetization currents in the range above the value corresponding to the no-load condition, the obtained maximum starting currents were almost the same. Since the residual magnetism flux also has a significant impact on the way in which the flux associated with the primary winding of a transformer switched on for no-load operation is established, the transient state has been examined very thoroughly.

The evolution of the inrush phenomenon up to the quasi-steady state (currents and voltages and the dynamics of changes taking place in the core hysteresis loops) has been documented. In the research of the transient state, the strong imbalance in all measured and examined quantities (voltages, currents and magnetic fluxes) has been detected and carefully demonstrated. It has been diagnosed that during starting/inrush current surges, the magnetic field generated by the primary windings of all three phases closes outside the ferromagnetic core. With a three-column transformer design, the flux linkages produced by the powered windings do not add up and are forced to leave the core. They close in air and in the transformer’s structural elements.

All acquired transient-state regular phenomena have been identified and applied as an update to the mathematical model of the Dy transformer for the next stage of this research: a measurement verification of the controlled switching of the supply voltage with the pre-magnetization of the transformer’s core.