Topology-Correct Reversible Transformer Model

IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 27, NO. 4, OCTOBER 2012

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Topology-Correct Reversible Transformer Model S. E. Zirka, Y. I. Moroz, C. M. Arturi, Member, IEEE, N. Chiesa, and H. K. Høidalen, Member, IEEE

Abstract—A topology-correct transformer model, which covers core operation under heavy saturation conditions, is presented. A method of accounting for magnetic fluxes outside the core and windings is proposed. Representation of the magnetization curve at high flux densities is considered. The equivalent air gap in the core is taken into account. The model is capable of reproducing inrush currents accurately regardless of which transformer winding (LV or HV) is energized. The model is illustrated by calculating inrush currents produced by subsequent energizations of a single-phase transformer. Index Terms—Hysteresis, magnetic fluxes, magnetization curve, saturation, transformer model, transient analysis.

A

I. INTRODUCTION

DVANCED transient models of electrical networks require enhanced representation of all their components, in particular, their transformers. The most widespread model of a two-winding transformer is the well-known T-scheme [1], which has been used for many decades in studying low- and mid-frequency transients. However, no clear correspondence has been found between the T-scheme parameters (“leakage inductances of separate windings”) and the physical arrangement of the windings [1], [2]. Also, as a terminal transformer representation, the T-scheme cannot reproduce the fact that the flux density in the yoke reaches substantially lower levels than that in the limb when the core approaches saturation [3]–[5]. For these reasons,transformer designers [2] and system modelers [6], [7] have turned to topology-correct models in which each section of the core is represented separately, and there is no need to split the leakage inductance between the windings.While the T-scheme has been used successfully in transient analysis [8], this paper focuses on topology-correct transformer models. We note that topologically correct models [1], [6], [9] concentrate on the magnetic fluxes in the core and those flowing within the windings (leakage fluxes), but less attention is paid to the fluxes, which close through the nonmagnetic space outside the windings. An exception is the zero-sequence flux in three-phase three-limb transformers whose air paths are represented by one or three linear reluctances in their magnetic networks [1], [6]. Unlike three-limb transformers, the zero-sequence flux in Manuscript received October 11, 2011; revised February 17, 2012 and May 17, 2012; accepted June 17, 2012. Date of publication July 31, 2012; date of current version September 19, 2012. Paper no. TPWRD-00865-2011. S. E. Zirka and Y. I. Moroz are with the Department of Physics and Technology, Dnepropetrovsk National University, Dnepropetrovsk, Ukraine 49050 (e-mail: [email protected]). C. M. Arturi is with the Electrical Engineering Department, Politecnico di Milano, Milan 20133, Italy (e-mail: [email protected]). N. Chiesa is with the Department of Electric Power Systems, SINTEF Energy Research, Trondheim N-7491, Norway (e-mail: [email protected]). H. K. Høidalen is with the Department of Electric Power Engineering, NTNU, the Norwegian University of Science and Technology, Trondheim N-7491, Norway (e-mail: [email protected]). Digital Object Identifier 10.1109/TPWRD.2012.2205275

five-limb transformers closes through the lateral limbs, so the fluxes beyond the windings (from yoke to yoke) are frequently neglected [10], [11]. Despite the cautions in [12] and [13], the same dubious simplification is still adopted when analyzing single-phase transformers [7]. A typical model drawback is outlined in Section II where the difference between analytical and modeling techniques for calculating inrush currents is discussed. In Section III, we generalize the method [12] for evaluating linear reluctances representing the air fluxes. The generalization is aimed at determining the reluctances which enable the model to be reversible (i.e., to reproduce inrush currents accurately), irrespective of which winding is energized. In Section IV, the important question of extrapolating the magnetization curve to high flux densities is considered. Section V presents a noniterative algorithm for transient analysis. A method of accounting for the equivalent air gap in the core is described in Section VI. In Section VII, the capabilities of the proposed model are illustrated by simulating subsequent energizations of a single-phase shell-type transformer (303/15 kV, 222 MVA). Section VIII compares the proposed model with a modified T-scheme model. II. ANALYTICAL EVALUATION OF INRUSH CURRENT The analytical method of evaluating inrush current peak is based on the simplest two-segment piecewise linear approxi, mation of the magnetizing curve having a breakpoint above which the core is considered to be heavily saturated (characterized by the vacuum permeability ). The value of is considered negligibly small, and the saturation level is set equal to 2.0 T [4], or 2.03 T [14]. Neglecting the winding resistance (it is justified for large transformers) and assuming the worst condition of switching (i.e., a network with voltage and inductance switched to the winding ( turns) wound on the core with cross section and initial can be written as magnetic induction , the value of (1) in (1) can be calculated accurately by The inductance the numerical method [15] or by formulae given in handbooks (e.g., in [16]). The value of the reactance is determined by short-circuit apparent power of the network . Different forms of (1), which is also known as Schwartz’s formula [2], [4], are given in many literature sources. For many decades, it has been used by transformer designers as a benchmark (inrush current upper bound) in the case of a single-phase transformer. It should be noted that inductance is often evaluated by the approximate formula

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(2)

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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 27, NO. 4, OCTOBER 2012

Fig. 2. Structure and magnetic fluxes of the shell-type transformer. Fig. 1. Unloaded single-phase transformer with a thin energized winding.

where is the height of the core window [4], or excited winding [14], and is its equivalent diameter [4], [9]. It can be seen from (2) that is the inductance of a thin cylindrical winding encircling the saturated limb placed between two non-saturated yokes. Regardless of the coil geometry, the value of is always smaller than the one calculated with (2). Although the use of instead of in (1) is acceptable for engineering practice [4], the reluctances of the model developed in this paper are calculated assuming hard saturation of the core. This requires that air-core inductances of the windings be calculated more accurately. The values of these inductances calculated by method [15] for the transformer considered (see Table II) are 15% (for LV) and 19% (for HV) less than those calculated with (2). At this point, we can leave analytical estimations and continue with transformer models. To analyze a drawback typical for many topology models, we can refer to the transformer magnetic network in Fig. 1. The fluxes in the main limb and yoke (the latter consists of two horizontal branches and the end limb) are designated as and , respectively. Assuming a no-load operation, only a thin energized winding is shown in Fig. 1 with being the flux between the winding and the limb. The role of the shunt air flux is of particular interest here. During the steady state at rated excitation, the unsaturated core has a much higher permeability than air, and flux can be neglected. However, during high saturation, the reluctance of the yoke may reach higher values than the reluctance of the space with flux . When the core reaches saturation and flux is still not accounted for, then flux is forced to flow (in the model) along the yoke whose length is much greater than the spacing between the yokes. This makes the calculated inductance of the winding substantially less compared to that of the air-core coil . This leads to an overestimation of inrush current peak compared to that calculated with typically used analytical formulae where the core geometry is neglected, and the value of is calculated from the excited winding geometry and number of turns. III. TRANSFORMER MAGNETIC NETWORK To describe the single-phase shell-type transformer whose upper part is shown schematically in Fig. 2, it is necessary to define the iron and air branches and indicate the boundaries between them. Since the real transformer is represented in this paper by lumped parameters, the choice of the branches is arbitrary to a certain extent.

Fig. 3. Transformer magnetic network.

By virtue of the vertical symmetry, the core can be divided into two branches referred to hereinafter as limb and yoke. The former is identified with the main limb (with cross-sectional area , length , and magnetic flux ), while the latter (with length ) combines the four yokes (right and left, top, and bottom) and the two end limbs. The yoke equivalent area and flux are considered two times the real values and to account for the combining. All of the elements of the distributed node 1 (bold lines in Fig. 2) are characterized by the infinitely high permeability. The yoke combining from before is also illustrated by the magnetic network in Fig. 3, where and are the mmf of the LV and HV windings; shaded reluctances and designate nonlinear iron branches; whereas unshaded (linear) reluctances , and represent the paths of magnetic fluxes , and in air. Reluctances and change with and dB/dt, so the calculation of their instantaneous values requires complex techniques [7], [17]. In the present model, iron-core branches are described by differential equations [18], which are solved together with other equations of the transformer and external circuits during the course of time-stepping transient calculation. The value of is uniquely defined by the leakage inductance , which is one of the main transformer parameters. (The leakage reactance can be measured in the conventional short-circuit test or calculated quite accurately from design parameters.) If is referred to the LV winding ( turns), then . There is no strictly defined value for the reactance and, thus, for the reluctance , which characterizes the channel between the core and inner (LV) winding [9]. So we

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can consider as an unknown variable and estimate it simultaneously with the reluctance , which represents the path of flux flowing from yoke to yoke through the surrounding nonmagnetic medium (“air”). Since the space containing is not clearly defined, reluctance can only be found in order to ensure proper model behavior at hard saturation when the role of is the most pronounced. The presence of two free components and allows us to make the model reversible (i.e., valid independently of which of two windings (LV or HV) are excited). Using design parameters, inductances and of the LV and HV windings in the air can be calculated accurately. Then, to estimate and , we can assume a heavy saturated core when the core reluctances and become magnetically linear and are determined by the leg and yoke geometries and . We can now require that input reluctances and seen from the LV terminals 1–2 and from the HV terminals 2–3 in Fig. 3 are equal to reluctances and . As the no-load regime is considered, the values of and are calculated for short-circuited terminals 2–3 and 1–2, respectively. It is easy to see that (3) (4) where symbol designates a parallel connection. To find and , it is convenient to introduce intermediate variables (5) (6) and, accounting for (3) and (4), rewrite the equalities imposed and ) as two simultafrom before ( neous equations for and (7) (8) System (7) and (8) is reduced to the quadratic equation for

(9) is substituted in (8) to give whose positive root and are known, then from (5) and (6)

. Since

(10) and are found, the diameters of equivalent thin After windings can be calculated. Although these diameters are not

Fig. 4. Magnetization curve of GO steel shown in three different scales.

used in the modeling, we can note (see Table II) that equivalent thin windings are situated within the actual coils. The found value of (it was also characterized by the reactance ) can be linked with by the factor [19]. For the transformer considered, we have obtained 0.491, which is close to that 0.5) proposed in [19]. The value of can be evaluated by the ratio . The small value of the ratio obtained (0.01685) points out the strong shunting effect of under saturation conditions, which will be shown in Section VII. IV. APPROXIMATION OF THE MAGNETIZATION CURVE AT HIGH FLUX DENSITIES The accurate description of the - curve above its knee is of primary importance when modeling regimes with a highly saturated core. As can be seen by examining the magnetization curve in Fig. 4, the notions of saturation and knee are somewhat ambiguous. In the scale of Fig. 4(a), the knee of the curve seems located below point 1, which is the point where ascending and descending branches of the measured hysteresis loop [20] merge into a single-valued curve. At the same time, it follows from Fig. 4(c) that point 1 is situated in the very beginning of the “large knee,” which ends at point 6. The lower (hysteretic) part of the curve ends at point whose ordinate 1.8011 T) was the maximum flux density reached experimentally at controlled sinusoidal induction for a 0.27-mm-thick GO steel at 0.003 Hz [20]. Point 1 and quasistatic loops of the steel practically coincide with those given in NSC catalog [21] for steel 27Z130 whose - curve above 1.8 T is represented in Fig. 4 by catalog points 2 to 5. In accordance with [21, p. 8], the tabulated point 2 (800 A/m, 1.84 T) and point 4 (2500 A/m, 1.92 T) also characterize steel M4, so the curve in Fig. 4 can be considered typical for

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IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 27, NO. 4, OCTOBER 2012

Fig. 6. Electrical scheme of transformer and external networks. Fig. 5. Static hysteresis loops at high flux densities.

V. MODEL IMPLEMENTATION 0.27-mm-thick GO steels. This is also illustrated by the encircled point 7 (14209 A/m, 2.02 T) measured recently in the Epstein tests on steel M4 [22]. in Fig. 4(c) was not measured: it lies on the Point 6 flat segment of the two-slope piecewise linear curve, so where the field is a fitting parameter. Points 1 to 5 can be approximated by the function

(11)

which gives at . Coefficients are found from the following three equations: (at ), and (at ). Here, is is the a known slope of the hysteresis loop at its tip 1, and adjusting value. Introducing the designations and , we find

A method for avoiding iterative calculations during transient analysis is the use of the dynamic hysteresis model [18] (14) Here, is the magnetic field at the sheet surface, is the field calculated by means of a static hysteresis model, and , where and are the thickness and resistivity of the steel, 1. Function makes the calculated dynamic loop agree with the measured loop of the steel considered [18]. The implementation of (14) is simplified if each mesh of the magnetic network contains no more than one iron branch. The network in Fig. 3 can be described by one nodal and four mesh equations (15) (16)

(12)

(17) (18) (19)

(13) 20 kA/m (it is shown in Fig. 4 The curve built for as curve 1-2-3-4-5-6) was found acceptable to be used in the model. The coefficients calculated with (12) and (13) are 8488.5, 4.09329, . The linear extension of the curve beyond point 6 has the slope . representation of the magnetization curve The inverse required for the model can be easily found analytically or approximated by an spline. For flux densities below 1.7–1.8 T, hysteresis properties of the core material should be taken into account in transient sim1.7 T is usually the highest ulations. Since the loop at catalog loop, there is a possibility at this stage of constructing a 1.8 T. Fig. 5 shows that the loops at static loop for 1.7 T and 1.8 T differ substantially. We should recall here that both of these loops have been measured (not constructed), so we show them for reference.

where in accordance with (14)

(20) In these equations, the iron branches are characterized by the magnetic-field strengths , and flux densities . Equations (15)–(20) for the magnetic network (as is shown conditionally by coupled windings in Fig. 6) are combined and solved simultaneously with equations describing external electrical networks connected to the transformer terminals. In Fig. 6, such feeding networks include the voltage sources of the generator and power network ( and ), their equivalent inductances ( and ), and resistances ( and ). Capacitances and account for transformer and bushings. Voltages and are those induced in transformer windings with resistances and .

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The electrical equations of the circuit can be written as

(21)

(22) (23) (24) (25) where the windings flux linkages (26) It is convenient to choose , and as state variables. Then, the intermediate variables and time derivatives needed for any library integration procedure are found in the following order:

Fig. 7. (a) Dynamic and (b) quasistatic hysteresis loops calculated with and without the air gap in the core.

and (30) becomes

(27) (28)

(29) and are expressed through their Derivatives square roots, which are found by solving quadratic equations, following from (16), (17), and (20):

(30)

(31) The other time derivatives are computed in accordance with (21)–(25). The above ordinary differential equations have been integrated in Fortran using a Gear method. VI. MODELING THE CORE AIR GAP To reproduce the residual flux pattern after transformer de-energization, the air gaps in the core should be taken into account. The simple core considered can be characterized by a single equivalent air gap , which may be related either to the limb or to the yoke. In the former case, an additional (air-gap) linear reluctance is placed in series with the nonlinear reluctance in Fig. 3. The magnetic potential drop across is . So the mesh equation (16) is rewritten as (32)

Despite many years’ efforts on evaluating equivalent air gaps [23]–[25], their reliable calculation is still a challenge. This calculation would require a complete 3-D model of the entire transformer with the detailed structure of laminations taken into account. According to [26], such a solution “is still a utopia.” This is not only because of “the available computing power” [26], but mainly due to the lack of knowledge about physical processes involved and different properties of GO steel in different directions (these include hysteretic, classical, and excess losses). It is considered that for the large laminated cores with step-lap joints and 5 to 7 sheets per step, the value of the equivalent air gap is of the order of several hundreds of micrometers. Hysteresis loops of the no-load transformer calculated with the model proposed at 0, 0.1 mm, and 0.2 mm are shown in Fig. 7. As can be seen in Fig. 7(a), the influence of the gap seems not so large at 50 Hz. (The distance between the “remanent points” , and is comparatively small.) However, this distance becomes substantial for quasistatic loops in Fig. 7(b) calculated at 0.1 Hz. This is essential because the residual flux pattern is determined by static characteristics of the core. VII. NUMERICAL RESULTS In order to eliminate the smearing influence of external networks and concentrate on the transformer itself, we first assume a generator of infinite power 0) and open a secondary network . Capacitive effects are also neglected. (Their influence will be evaluated later.) The roles of the air gap in the core and reluctance are illustrated by the waveforms in Fig. 8, which show a complex

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TABLE I FOR FIRST CURRENT PEAKS CALCULATED WITH AND WITHOUT DIFFERENT EXTRAPOLATIONS OF THE MAGNETIZATION CURVE

Fig. 8. (a) Flux density in the limb and yoke winding with and without the air gap in the core

. (b) Current in the LV and reluctance .

transient produced by two subsequent transformer energizations. The first energization (at 0) is the application of a sinusoidal voltage to the LV winding. (The HV winding is assumed open-circuited during the numerical experiment.) At 30 ms, an open command is sent to a circuit breaker (CB), which disconnects the LV winding at the next current zero crossing (at 35.02 ms). The CB contact is assumed to reclose at 280 ms when the voltage applied to the LV winding is again zero crossing. Before the first excitation, the core is in a demagnetized state . It has been found with the model that during the no-current pause, a positive flux density 1.593 T) is retained by the core at 0. At the same time, markedly lower values of (0.970 T and 0.503 T) are established at the end of the pause at 0.1 mm and 0.2 mm. These values of are consistent with the ordinates of points , and in Fig. 7(b). Because of the different at 0 and 0, flux density peaks after the second transformer energization are also different. The aforementioned findings that were discussed corroborate the important role of the core air gap, so all subsequent calculations are carried out at 0.2 mm. The first fact to note is that the first current peak (designated by the horizontal dash-dotted line 1 in Fig. 8(b)) is independent of . The difference between current peaks at 290 ms (horizontals 2, 3, and 4) are explained by different residual flux densities before reclosing CB. In accordance with the purpose of

the modeling, current peaks designated by horizontals 1, 2, 3, and 4 practically coincide with those calculated with (1), where , and is equal to 0 T, 1.593 T, 0.970 T, or 0.503 T, respectively. To show the role of in the model considered, a calculation was also carried out for multiplied by 1000. (This corresponds to the absence of in the magnetic network of Fig. 3.) As can be seen in Fig. 8(b), inrush current peaks calculated without this reluctance are 1.83 times higher than those evaluated with (1) and designated by horizontals 1 and 4. The role of is also illustrated by time dependence of the yoke induction in Fig. 8(a). Whereas almost coincide with the limb induction for the nonsaturated core, both peaks in are much lower than peaks in upon the saturation. (This shunting effect of was observed experimentally in [3].) The first current peaks predicted by the model with and without (78.05 and 143 kA, respectively) are given in the last row of Table I. We can now use Table I in conjunction with Fig. 4 to recall that the extrapolation of the - (or ) curve to high flux densities with keeping its final slope obtained in the open-circuit test can lead to a severe underestimation of current peak for any excitation level above the last specified point [27]. In the example considered in Fig. 8, the induction maximum (2.921 T) reached by during the first energization is much higher than points 1 to 6 in Fig. 4. (Coordinates and relative permeabilities at these points are listed in Table I.) If the magnetization curve is linearly extrapolated starting from points 1 to 5, then the first current peaks would be much lower than that (78.05 kA) calculated using the curve (11). At the same time, the first current peak calculated without but with the overstated final slope of 3.43 (it corresponds to point 5) would give an acceptable value (69.11 kA), which is a consequence of the cancellation of two model errors. A similar situation occurs when modeling transformer energization from the HV side. The model reversibility is corroborated by consistency between the first current peak of 2.424 kA predicted by the model and that (2.430 kA) calculated with (1), where . At the same time, the inrush current calculated with the topology model in the absence of is characterized by much larger overestimation (up to 400%) than in the case of the LV energization. The explanation is that the excited outer winding surrounds the limb (with flux ) and both air channels (with fluxes and ). So in this case, reluctance is the only branch shunting the yoke, and its influence is more pronounced. The mentioned errors show that is a mandatory element in topology models, and its absence makes the model inaccurate.

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Fig. 10. Nontopology circuital model.

Fig. 9. (a) Flux densities in the limb. (b) Currents in the LV winding calculated and transformer capacitances. with inductance

Fig. 9 illustrates the aforementioned reclosure scenario, but with inductance taken into account. As a representative example, it is reasonable to take equal to the transformer leakage reactance . In this particular case, the first current peak in Fig. 9(b) is 1.74 times less than that in Fig. 8(b). An additional decrease in the residual flux takes place if transformer capacitances ( 2000 pF, 600 pF. 1200 pF) are taken into account. As seen in Fig. 9, they do not influence the inrush current at the first energization. However, their discharge on the de-energized transformer (ringdown transient [18], [28]) leads to a decrease of the residual flux, which determines transformer behavior at subsequent energization. As before, the current peaks 1, 2, and 3 in Fig. 9(b), calculated with the model, coincide with those found analytically. This time, in (1), and (when calculating current peaks 2 and 3) have to be found with the model. As could be expected, the smoothing influence of is somewhat less in the presence of . (The first current peak is 1.35 times lower instead of 1.83 times at 0.) However, at the HV excitation and open-circuited LV winding, the 400% difference between current peaks calculated with and without is kept even with the network inductance taken into account . VIII. COMPARISON WITH ELECTRICAL MODELS The model proposed in the paper was compared with a nontopology circuital model based on a standard modeling

Fig. 11. Original 1) and adjusted corresponding inrush currents.

2.64

–

curves and

approach (BCTRAN based with an externally attached core and fictitious winding) [6], [29]. This model is shown in Fig. 10, where is the leakage inductance, represents the air channel between the LV winding and the core, is the nonlinear inductor representing the whole core (limb and yoke), and are the winding resistances, and IT is the ideal transformer. Reluctances and are the same as those in the topology model. If to confine the comparison to first current peaks, then hysteresis, core losses, and capacitances can be neglected. The - points given in Table I were used to construct the piece-wise curve (solid line in Fig. 11) and then recalculate it into the (flux linkage versus current) characteristic of the inductance . The simulation of the inrush current with such characteristics gives a current peak of 128.8 kA shown in the inset of Fig. 11. This overestimated value is explained by the fact that the value of at saturation is 5 times less than the air-core inductance of the energized winding. A possible, though palliative correction [30], [31] is to adjust the final slope of the curve that leads to the corrected -

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TABLE II CORE AND WINDINGS DESIGN PARAMETERS

Fig. 12. Topology circuital model.

curve shown by the dashed line in Fig. 11. The final slope of this curve is now such that the inductance seen from the LV terminal at saturation is close to . The relative permeability along the final segment of the - curve was thus increased from 1 to 2.64. The first current peak calculated with the corrected magnetization curve is 79.7 kA, which is close to that obtained with the topology model proposed in this paper. When the transformer is energized from the HV side, the model in Fig. 11 gives an inrush current peak of 3 and 2.38 kA with the original and adjusted final slopes, respectively. The latter value is consistent with that (2.424 kA) predicted by the topology model proposed. It is worth remarking that the adjustment of the final slope of the magnetization curve is an artificial technique, and the model in Fig. 10 does not reflect the processes in the real transformer. (It is only valid at the transformer terminals.) To obtain the topology circuital model, we can apply the dual transformation to the magnetic network in Fig. 3. This gives us the model shown in Fig. 12. It can be seen that this model is a refinement of the model in [1, Fig. 4.6]. It is important to note that in the absence of and especially , this model loses its physical meaning at high flux densities. A new method of making inductances and lossy and hysteretic will be proposed elsewhere. IX. DISCUSSION AND CONCLUSION The concept of a topology-correct reversible transformer model that covers core operation under heavy saturation conditions has been proposed. Unlike many electromagnetic models (e.g., [7] and [32]), which are correct with respect to the core geometry, the proposed model accounts for the topology of magnetic fluxes in the windings. This allows us to avoid the uncertainty in the separation of the transformer leakage inductance between individual windings [32] and, thus, exclude the inaccuracies pointed out in [1] and [33] that are caused by incorrect separation. Separate attention has been paid to the model’s ability to evaluate inrush currents. The first-noted point is the need to know the transformer design parameters when predicting inrush current behavior. Whereas the analytical evaluation of the inrush current requires the air-core inductance of the excited winding, construction of a reliable reversible model presupposes the knowledge of the core geometry in addition to the air-core inductances of both windings. It should also be noted that the standard no-load and short-circuit tests are insufficient to predict the inrush current behavior. Furthermore,

uncritical usage of the test data can lead to errors of different types and magnitudes. The most severe is the underestimation of inrush current peaks because of wrong extrapolation of the magnetization curve to the range of high flux densities. Another error, leading to different inrush current overestimations, is to neglect the magnetic fluxes in air (from yoke to yoke). It is shown that the air gap in the core and transformer capacitances do not influence the inrush current directly, but have a crucial impact on the residual flux in the core after transformer de-energization and, thus, on the inrush current during subsequent energization. Comparison with the nontopology circuital model in Fig. 10 shows reasonable agreement for inrush current peaks if the final slope of the curve is adjusted according to the given aircore inductance. However, the inclusion of nonlinear core losses in Fig. 10 and in the topology circuital equivalent in Fig. 12 is not straightforward. Besides the inrush events, the proposed model can be applied to regimes where transformer windings are fed with a sinusoidal or nonsinusoidal voltage that includes a floating dc offset [34]. It can also be applied to the study of resonance phenomena and overvoltages on transformer terminals. In the latter case, the structure and parameters of the upstream network should be specified [35], including the circuit-breaker characteristics and additional stray and system capacitances. Despite its simplicity, the model is also appropriate for estimating short-circuit currents and radial forces in the windings. A detailed investigation of these phenomena is beyond the scope of this paper and is the subject of a separate study. The reversibility concept has been implemented in a model of a single-phase transformer, which is based on a three-component dynamic hysteresis model (DHM). The static component of the DHM and the air gap in the core reproduce residual fluxes in all segments of the core. The dynamic (excess loss) component of the DHM can be used in fitting the transformer model to data obtained from an open-circuit test. The way in which the DHM is employed allows one to avoid iterative calculations when integrating state equations of the entire transformer network. An extension of the proposed approach to three-phase transformers will be described in future papers.

ZIRKA et al.: TOPOLOGY-CORRECT REVERSIBLE TRANSFORMER MODEL

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S. E. Zirka received the Ph.D. and D.Sc. degrees in electrical engineering from the Institute of Electrodynamics, Kiev, Ukraine, in 1977 and 1992, respectively. Since 1972, he has been with the Dnepropetrovsk National University, Ukraine, where he has been a Professor since 1992. His research interests include the modeling of magnetization processes in electrical steels, forming and transforming high-energy pulses, and transients in transformers of different types.

Y. I. Moroz was born in Ukraine in 1961. He received the Ph.D. degree in theoretical electrical engineering from the Institute of Modeling Problems in Energetics, the Ukrainian Academy of Sciences, Kiev, in 1991. Currently, he is an Associate Professor with the Department of Physics and Technology, Dnepropetrovsk National University, Ukraine.

C. M. Arturi (M’87) received the Electrical Engineering degree from the Politecnico di Milano, Milan, Italy, in 1975. Currently, he is Full Professor of Electrical Engineering, School of Engineering, Politecnico di Milano. His main fields of research activity include the modeling of parametric energy conversion devices, power transformers, and the thermodynamic approach to the electromagnetic conversion devices.

N. Chiesa was born in Italy in 1980. He received the M.Sc. degree from Politecnico di Milano, Milan, Italy, in 2005, and the Ph.D. degree from the Norwegian University of Science and Technology (NTNU), Trondheim, in 2010. Currently, he is a Research Scientist at SINTEF Energy Research, Trondheim, with a special interest in power transformers, transient simulations, power electronics, and energy storage systems.

H. K. Høidalen (M’05) was born in Norway in 1967. He received the M.Sc. and Ph.D. degrees from the Norwegian University of Science and Technology, Trondheim, in 1990 and 1998, respectively. Currently, he is a Professor at the Norwegian University of Science and Technology, with a special interest in electrical stress calculations and modeling.