Inrush Current Reduction Strategy for a Three-Phase Dy Transformer Based on Pre-Magnetization of the Columns and Controlled Switching

: The methodology and test results of a three-phase three-column transformer with a Dy connection group are presented in this paper. This study covers the dynamics of events that took place in the first period of the transient state caused by the energizing of the transformer under no-load conditions. The origin of inrush currents was analyzed. The influence of factors accompanying the switch-on and the impact of the model parameters on the distribution and maximum values of these currents was studied. In particular, the computational methods of taking into account the influence of residual magnetism in different columns of the transformer core, as well as the impact of the time instant determined in the voltage waveform at which the indicated voltage is supplied to a given transformer winding, were examined. The study was carried out using a nonlinear model constructed on the basis of classical modeling, in which hysteresis is not taken into account. Such a formulated model requires simplification, which is discussed in this paper. The model is described using a system of stiff nonlinear ordinary differential equations. In order to solve the stiff differential state equations set for the transient states of a three-phase transformer in a no-load condition, a Runge – Kutta method, namely the Radau IIA method, with ninth-order quadrature formulas was applied. All calculations were carried out using the authors’ own software, written in C#. A ready-made strategy for energizing a three-column three-phase transformer with a suitable pre-magneti-zation of its columns is given.


Introduction
In transmission and distribution systems, transformer energization is a fairly routine operation.However, with the increase in distributed generation sources with inherent high intermittency resulting in more switching events, the transformers in service are becoming more and more vulnerable to electrical transients.A transient state during energizing can cause significant inrush currents [1].
Inrush currents generate forces comparable to those of short circuits.The problem is that they occur more frequently than short-circuiting and last longer.Thus, at every occurrence of an inrush, there is some degradation of the conductor and its insulation.After many such inrushes, local hotspots may emerge in the winding.The transformer inrush transient is not only dangerous because of its large current amplitude but also due to its rapid rise rate [2,3].When a transformer is frequently exposed to transients, it will deteriorate due to severe mechanical and thermal stresses and may eventually fail [4][5][6].The high inrush current may disturb or damage the operation of adjacent equipment in the circuit resulting in, e.g., the maloperations of power electronic converters [7,8] and protection relays [9].Apart from affecting power quality in terms of temporary undervoltage (sagging) [10], the inrush currents contain many high frequency harmonics which can also lead to harmonic resonant over-voltage [1].
One important issue is the correct selection of the mathematical description of the model (i.e., the choice of differential equations that define its dynamics) and method necessary to solve it [3,[22][23][24][25][26].In fact, this is a research matter, which leads to understanding the different phenomena involved and making suitable assumptions.
The research into transformer dynamics is a part of the much broader issue-the simulations of electromagnetic transients in power systems, which are essential for the adequate design of equipment and its protection [1,2].Power transformers play important roles in power transmission and their performances have significant impact on power quality and the lifetime of power system apparatuses [9,11].The major reasons for the failure of transformers are the thermal, electrical and mechanical stresses of transformer winding insulation.Therefore, the reliable models and methods associated with the representation of the transient states of transformers are required for investigation.

State of the Art
There are numerous mitigation techniques for transformer energization transients.These include introducing a pre-insertion resistor, controlling the switching time using the point-on-wave voltage at energization, varying the impedance of the power supply, and controlling the residual flux inside the transformer core during transformer energization [12][13][14][15][16][17][18][19][20][21].
A survey of scholarly knowledge on the topic also provides information about a number of controllable factors that are relevant to the theory of inrush currents.
The list of factors affecting inrush currents that are found in publications is given below:

•
Starting/switching phase angle of voltage; This paper describes the simulation studies on the inrush current for the delta side energization of the transformer.This is a new research subject and seems essential in terms of practicality.It is also a continuation of research carried out by the authors [3,[27][28][29].

Model Concept
A research task of transient analysis was formulated for an unloaded three-phase transformer with primary side windings connected in delta.In order to carry it out, differential equations describing the adopted mathematical model were first formulated along with the assumptions.
The equivalent scheme of the transformer for the discussed model is shown in Figure 1, together with physical quantities that describe the model.A list of all the designations used in this work can be found in Nomenclature.For the planned calculation process, it was assumed that the switching on of the transformer at time instant 0 t had the effect of the emergence of three voltages with in- stantaneous values in the following form: By making the parameter 0 t variable, it was possible to make the calculations de- pendent on the time instant specified in the waveform of the selected phase-to-phase voltage for the process of switching on the transformer, i.e., the start of the calculations.
In the investigation of the transient states, an invariant reference system was assumed.It was assumed that the reference point was the time instant at which the voltage We assume that the non-linear curve () HB of the core material is known in the case of discussed transformer.In our experiment, this task was implemented using a polynomial described by odd powers (2).The correctness of this method was verified experimentally [3].column of the transformer (Figure 1).Using the circuit description in modeling the magnetic circuit of the transformer, we can state that: where Fe, () k it is a current of the coil under consideration.

It was assumed that the parameters
Fe,k R could be determined by measuring the active power of a transformer operating at steady state under idling conditions.
The mathematical model of the transformer formulated in this way also offers the possibility of determining the steady state in the form of a limit cycle of the solution of the relevant differential equations.Details on the experimentally verified methodology of this subject may be found in [3,27].

Residual Magnetism
The formulated transformer model intentionally did not consider magnetic hysteresis.This was not necessary, since it was not intended to track the history of core magnetization, that is, the points at which reversals in the directions of changes in magnetic currents and fluxes occurred.Instead, the model offered the possibility of taking into account the magnetization of the transformer core at the particular time instant, when the transformer was switched on.The state of magnetization was taken into account in the model in the form of residual magnetism.If we know the amount of residual magnetism, then it is possible to set initial conditions for the state variables As indicated by numerous literature sources, e.g., [11,14], the energizing of unloaded transformer is accompanied by the occurrence of current pulses.Their presence is limited to the first period determined by the supply voltage.The amplitudes of these currents, as a rule, exceed the rated values, and not infrequently reach values close to short-circuit values.
Basing on these findings, we argue that if the transient state analysis of the transformer is limited to a single period from the energizing instant, the proposed model will make it possible (if initial conditions are set) to investigate the impact of residual magnetism on the magnitudes of generated current pulses.
The analysis of the physics of the formation of current pulses has already established that they (the pulses) first appear as a result of locating the working point of the inductor ( 0 I  − solution) far in the saturation region of transformer's core magnetization curve [3,12].
Therefore, the inclusion of the core's residual magnetism in the model must be supported by an accurate approximation of the core's nonlinear curve in the saturation region.

Dynamic Equations of the Unloaded Transformer Dy
In the developed mathematical model of the transformer, the magnetomotive forces (mmfs) generated by the coil currents (g) ( )( 1, 2,3) were supplemented by the magnetomotive forces generated by certain equivalent currents Such a concept for modeling a three-phase transformer with three columns and starconnected primary side windings has been comprehensively analyzed before, and the results of these studies were published in [27].There, the flux linkages were chosen as the state variables.
In the considered case, the windings of the primary side of the three-phase transformer were delta-connected.The flux linkages Taking into account Kirchhoff's first and second laws for the magnetic circuit of a three-phase transformer with three columns (Figure 1) and Equation ( 3) modeling the total iron losses, the equations of state for the variables were formulated: where When the system of Equation ( 4) is solved for the derivatives, the results are: where Equations for three independent circulations appeared as a result of applying Kirchhoff's second law to the modeled equivalent circuits of the primary side of the transformer (Figure 1): where The resulting system of equations was ordered in matrix form:  (7) where

R=
When the system of Equation ( 7) is solved for the derivatives, taking into account solution (5), the results are: (8) where While the inverse matrix L −1 of matrix L assumes the following analytical form: ( )

DL
The relationships formulated as differential Equations ( 5) and ( 8) form a general system of equations of state for an unloaded transformer.
The model formulated in this way does not take into account magnetic hysteresis.However, its simplifications (Section 6) make it possible to use it in the analysis of the influence of residual magnetism (and the parameter Fe,k R of the model) on the maximum values of the inrush currents of the unloaded transformer appearing during the first period determined by the supply voltage.

Investigation
The transformer data adopted in the test calculations are included in Appendix A. Powercore ® H 105-30 electrical steel was used in the experiment, which is manufactured by the ThyssenKrupp company.Material characteristics: density 7.65 kg/dm 3 , maximum specific loss at 1.7 T is 1.05 W/kg.

Definition of Supply Voltages Together with the Method That Takes into Account the Time Instant Determined (Set) in the Supply Voltage Waveform
To define the symmetrical supply voltages of the discussed transformer, it was assumed that they are sinusoidally alternating waveforms with pulsation 2 f  = and ar- bitrarily set rms values of phase-to-phase voltages, ,, E E E .It was also assumed that the initial phase angle for one (arbitrarily chosen) voltage was known.This was sufficient because the phase angles of other voltages were defined by the geometry of the triangle (Figure 2).
The introduced parameter phsq k made it possible to take into account the effect of the phase sequence on the calculated waveforms of transformer inrush currents.
The defined rms complex values of sinusoidally variable phase-to-phase voltages ,, E E E constituted the basis for determining their instantaneous values in the form of the following functions: Functions (11), representing input phase-to-phase voltages at the terminals of the primary winding of an unloaded transformer, met the requirements of the construction of Equations of states ( 5) and (8).The parameter 0 t present in the formula made it possible to define the time instant specified in the voltage waveform at which the indicated voltage appeared at the given transformer winding.
Since the argument of the reference voltage considered in the experiment The investigation of the transient state of the unloaded transformer was possible via the analysis of the solutions of the systems of differential Equations ( 5) and (8).In these equations, the initial point of integration was assumed to be zero ( 0 0 t = ).This meant that the starting point of the integral of the system of Equations ( 5) and ( 8

Study of the Influence of Residual Magnetism on the Maximum Values of Inrush Currents of an Unloaded Transformer
The study of the influence of residual magnetism on the maximum values of the current pulses accompanying the energizing of an unloaded transformer was carried out by setting initial conditions in the form of state variables BB are the residual flux density of the transformer core columns, namely the center and outer columns (Figure 1).
For the purposes of the study, four scenarios (variants) were also established, with non-randomly selected sets of residual flux density values ,, B B B (Table 1) in succes- sive columns of the transformer core.This procedure was carried out because of future applications, including the need to select the transformer core column whose winding will be supplied in the tests.The remaining initial conditions of the system of Equations ( 5) and ( 8) were assumed to be zero in all calculation cycles The influence of the time instant determined in the waveform of the supply voltage applied to the transformer was investigated by changing the parameter 0 t in functions (11) applied then to the equations of states ( 5) and (8).

Implementation of Calculations-Calculation Algorithm
Figure 4 shows the flowchart of the algorithm applied in both tests.The algorithm was input into the authors' own C# software [30].The program allows the precise calculation and visualization of the distributions of the maximum values of the inrush currents of an unloaded transformer depending on the time instant 0 t characterizing the time in- stant in the waveform of the switched supply voltage and also the values and initial scenarios of the pre-magnetization of transformer's different columns.To integrate the stiff differential Equations ( 5) and ( 8) of the transformer model, the authors' own ninth-order Radau IIA method was used [31][32][33].

Simulation Results
Figure 5 shows the results of the analysis of the distribution of the maximum values of inrush currents of the unloaded transformer for four variants of the pre-magnetization of the core (Table 1) in a positive phase sequence system (Figure 3).
The conditions of the experiment in Figure 5 were used in its repeated implementation, in which the phase sequence was changed to the opposite sequence.The calculation results obtained under the conditions of such a modified power supply are shown in Figure 6.A comparison of the results of the calculations presented in Figures 5 and 6 showed that for the pre-magnetization scenario in the center column with a value twice that of the outermost columns (this refers to the absolute value), the distributions of maximum currents do not depend on the phase sequence.
For pre-magnetization scenarios 1 and 4 (Table 1), the calculations of inrush currents of the unloaded transformer led to results highly consistent with each other (similar shape and values of the obtained waveforms in Figures 5 and 6).
In a further stage of the study, it was assumed that the initial magnetization of the transformer core would be carried out according to variant 1 (Table 1).
Figure 7 shows the result of the experiment of energizing the unloaded transformer for the residual magnetism set in variant 1 and for the set time instant  The calculated maximum instantaneous value of the unloaded transformer inrush current during the first transient period was In the case of energizing the transformer after a time A (Figure 8).

Algorithm for Selecting the Time Instant Specified in the Waveform of the Switched Supply Voltage
The results of the performed experiments and the diagnosed regularities of the model were used to develop an algorithm for selecting the time instant With regard to the selection of the supply voltage and the terminals of the primary side of the transformer to which this voltage is to be applied, it was determined that the most optimal configuration was provided by the computationally verified voltage () 12 () s et and the terminals of the primary winding wound on the center column of the transformer (Figure 1), as seen in scenario 1.

Method of Implementation of Pre-Magnetization of the Core Columns of the Tested Transformer
Due to the difficulties associated with measuring the residual magnetism of threephase transformers [12,18], it was decided that the pre-magnetization technique of the core [27] instead of its measurement should be used.The schematic diagram of the solution involving the pre-magnetization of the core of an unloaded transformer with deltaconnected primary side windings is presented in Figure 9a.
The concept of the pre-magnetization of the core of the transformer under study envisages the generation of a magnetization of twice the value and opposite polarity in the center column than is the case in the outermost columns (variant 1) (Figure 9b).As shown in the conducted numerical experiments, such as the proportion and symmetry of the initial magnetization guarantees the distribution of the maximum values of the inrush current of the unloaded transformer as a function of the switching time calculated with respect to the transition of the voltage () 12 () s et through zero and independent of the phase sequence of the supply voltages (

Selection of the Currents Required for Core Pre-Magnetization
The next step was to estimate what values of current are necessary for pre-magnetization in the system shown in Figure 9.
The starting point in the calculations was the value of the transformer winding resistance g R .Since the value of this resistance is relatively small, the condition for select- ing the resistance 0 R in the circuit, as shown in Figure 9a, was formulated as [34]: It was possible to determine the DC currents flowing through the transformer windings without any problem via the following: The currents determined in this way for the different windings of the three-phase transformer allowed the formulation of a system of nonlinear equations based on magnetic circuit theory (Figure 9b).In the equations, magnetic voltage drops were taken into account with the omission of the dynamic term (i.e., the Formula [27]).m, Fe, The system of nonlinear equations determining the distribution of flux linkages took the following form: ) where The formulated system of equations was solved for the given values of current 0 I by relating the solutions of to the corresponding average values of magnetic flux density: B= zs (10) The solution of the system of Equation ( 18), as shown in Equation (19), for different values of current 0 I is presented in Figure 10.
The calculated characteristic unambiguously proves that the proposed method of the pre-magnetization of the transformer core results in flux density levels in the different columns of the core according to variant 1: where 0 B depends on the magnetization current 0 I .For the calculated magnetization configuration (20), the opposite magnetization can also be considered.To obtain this configuration, it is enough to change the direction of current flow 0 I in the circuit shown in Figure 9a.Turning off the flow of current 0 I in the pre-magnetization system will reduce the value of flux density in the different columns of the core.For transformer sheets (soft magnetic materials), the value of flux density 0 B will correspond to the value of rema- nence r B , which is significantly lower than saturation flux density (by as much as 50%).
In the test data adapted for the simulation, it was assumed that the transformer sheet has a saturation flux density of 1.9 T. Hence, the value of the flux density 0 B , representing the initial magnetization of the core in the calculations in variant (20), will vary up to a value of 1.2 T.
Results of the calculations of the maximum values of inrush currents of the unloaded transformer in the first period of the transient state from the energization, depending on the adopted value of 0 B , are presented in Figure 11.
The calculated inrush currents are functions dependent on time The results were obtained for the initial magnetization, as seen in ( 20), enforced according to the scheme in Figure 9 with a positive phase sequence of supply voltage.   12and on the pre-magnetization of the core in configuration (20) for a posi- tive phase sequence of supply voltage.Outside the indicated time interval, the opposite phenomenon was observed: the increase in 0 B was accompanied by a sharp increase in the maximum values of inrush cur- rents.For 0 0 t = , the currents reached the highest possible values (kiloamperes) (Figure 11).This means that in order to eliminate the large inrush currents accompanying the energizing of an unloaded three-phase transformer, its switching should be performed The obtained solution is general and can be applied to any transformer, where primary winding is delta-connected and where columns are pre-magnetized according to (20) e t e t e t can be arranged into a pos- itive or negative phase sequence (without affecting the outcome).
The collected data can be used to construct a special device that will execute energizing the transformer at specific time instants in the supply voltage waveforms (Table 2).
Table 2. Switching conditions of the unloaded transformer for which the minimum values of inrush currents are obtained, taking into account the pre-magnetization in configuration (20).

The Strategy of Energizing a Three-Phase, Three-Column Transformer Where Primary Winding Is Delta-Connected and Where the Columns Are Pre-Magnetized According to (20) at Specific Time Instants in the Supply Voltage Waveforms
The developed strategy is illustrated by the flowchart in Figure 14.

•
The initial magnetization state of the transformer core can be taken into account as residual magnetism in the form of (assignable) initial conditions for the state variables BB is the residual flux density in the different columns of the transformer core.

•
The integration of the equations with the initial conditions formulated in this way leads in the first instance to a solution ( 0 I  − ) located deep in the saturation region of the ferromagnetic core curve; the maximum inrush currents of the unloaded transformer determined from this solution are calculated accurately.

•
Taking any initial conditions (e.g., zero) for the systems of Equations ( 5) and ( 8) as the starting point leads to a steady-state solution, which is the limit cycle of the transient state solution; the trajectories of the transition to this limit cycle have no physical interpretation in this case, since the assumed mathematical model does not take into account magnetic hysteresis (it does not track the history of core magnetization).

•
The model is suitable for the study of steady-state conditions of the transformer when the resistances

Nomenclature
The list of symbols and notations used in this paper:

Figure 1 .
Figure 1.Equivalent scheme of the analyzed three-phase Dy transformer in no-load conditions.


losses in the ferromagnetic material of the core.The total losses were taken into account according to the classical concept of transformer modeling.In each column of the three-phase transformer, an additional electromotive force was introduced in the form of a flux linkage derivative at the terminals of a coil with g z turns.The coil with zero internal resistance was loaded with a resistance Fe,k R , which modeled the total iron power losses in the th k −

Fe
they corresponded to real power losses in the ferromagnetic core.
were chosen as the state variables in the magnetic circuit equations.

(
) was 0 t away in time from the reference point determined by the passage of the voltage () 12 () s et through zero (Figure 3).
t e t shown in Figure3was taken into account in further calculations as a coefficient with the valuephsq 1 k =− .The presented method made it possible to set the parameter 0 t in the study of the effect of the time instant determined (defined) in the supply voltage waveform on the maximum values of the inrush current of the unloaded transformer in the first period determined by the supply voltage.

Figure 3 .
Figure 3. Visualization of how the

Figure 4 .
Figure 4. Structure of the computational algorithm examining the effect of the level of residual magnetism in the core and the time instant set in the supply voltage waveform on the maximum values of inrush currents of the unloaded transformer.

Figure 5 .
Figure 5. Distribution of the values of the maximum inrush currents of the unloaded transformer for the four variants of pre-magnetization, depending on the time instant 0 t , characterizing the time instant of energizing in the course of the supply voltage for the positive phase sequence.

Figure 6 .
Figure 6.Distribution of the values of the maximum inrush currents of the unloaded transformer for four variants of pre-magnetization, depending on the moment

Figure 7 .
Figure 7.The waveforms of inrush currents of the unloaded transformer in the first period of the transient state; energizing occurred with residual magnetism present (variant 1) and at the time instant zero, the maximum instantaneous value of the inrush current of the unloaded transformer during the first transient period was reduced to only

Figure 8 .
Figure 8.The waveforms of the inrush currents of the unloaded transformer in the first period of the transient state; energizing occurred with residual magnetism present (variant 1) and at the time instant 0 10 t = ms, defined in the waveform of the reference supply voltage () 12 () s e t .

Figure 9 .
Figure 9. Schematic diagram of the method of the pre-magnetization of the core of an unloaded transformer with delta-connected primary side windings (a) and an equivalent diagram of the magnetic circuit exhibiting forced flux distribution (b).

Figure 10 .
Figure 10.Dependence of magnetic flux density in different columns on magnetization current

0t
(the time instant of switching on) calculated with respect to the transition of voltage

Figure 11 .
Figure 11.Distribution of maximum inrush currents of the unloaded transformer for seven variants of pre-magnetization 0 B , depending on the time instant 0 t , defined in the voltage waveform

Figure 12 . 0 B
Figure 12.Visualization of the time instant of voltage switching on the transformer measured relative to the time instant when the voltage () 12 () s et passes through zero, for a positive phase sequence

Figure 13 .
Figure 13.Distribution of the maximum inrush currents of the unloaded transformer for two values of pre-magnetization

t
= , calculated with respect to the reference point determined by the tran- sition of the reference voltage (here:

Figure 14 .
Figure 14.Inrush current reduction strategy when energizing a three-phase transformer with deltaconnected primary side windings requiring the pre-magnetization of the core columns.

Fe, k R
are determined based on the idling losses, in which eddy currents and magnetic hysteresis are taken into account.•Thecompleted tests have shown that there is no significant influence* of the parameter Fe,k R of the model on the maximum values of the inrush currents of the unloaded transformer in the first period of the transient state caused by energizing the transformer (*-lossiness values provided by the manufacturer of the transformer plates vary within wide limits).

Table 1 .
Test sets of values (variants) of pre-magnetization in each column of the transformer core.

Table A1 .
Assumed data of the modeled transformer.

Table A2 .
Function H(B) coefficients obtained from solving the estimation problem based on the manufacturer's data.