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Threephase distribution transformer connections modeling based on matrix operation method by phasecoordinates
EURASIP Journal on Wireless Communications and Networking volume 2021, Article number: 66 (2021)
Abstract
This paper proposes a matrix operation method for modeling the threephase transformer by phasecoordinates. Based on decoupling theory, the 12 × 12 dimension primitive admittance matrix is obtained at first employing the coupling configuration of the windings. Under the condition of asymmetric magnetic circuits, according to the boundary conditions for transformer connections, the transformers in different connections enable to be modeling by the matrix operation method from the primitive admittance matrix. Another purpose of this paper is to explain the differences of the phasecoordinates and the positive sequence parameters in the impedances of the transformers. The numerical testing results in IEEE4 system show that the proposed method is valid and efficient.
Introduction
The distribution systems are unbalanced naturally. With the rapid development of the distributed generators and the wide use of electric vehicles, the unbalanced condition of distribution systems are getting worse by those singlephase power supply and loads increasingly [1, 2]. In this case, it is great need to promote the research and analysis of the unbalanced distribution systems. But for the unbalanced distribution systems, this unbalanced nature makes it difficult to generate the decoupled (1–2–0) networks for analysis. Consequently, it is direct and convenient to employ phase (a–b–c) coordinates for the analysis and solution of the unbalanced distribution system [3]. There are many connections and different neutral point states for the threephase transformers. In modern distribution system analysis, models of the transformers play an important role in powerflow analysis and shortcircuit studies. Therefore, it is necessary to study a new approach modeling threephase transformer connections by phasecoordinates in unified matrix analysis.
There are several representative approaches for modeling the threephase distribution transformer connections in the admittance matrix form by phasecoordinates proposed in [4,5,6,7,8,9,10]. Reference [4] developed an approach from the KCL and KVL, which was able to generate the 6 × 6 matrix in different transformer connections according to singlephase transformer symmetrical lattice equivalent circuits as the units grouping up. In the paper, the authors described the transformer in model relationship between the phasecoordinates and component coordinates as well. However, the interphase coupling did not be considered in this approach. Later, the improved models proposed in [5,6,7] derived from this approach of assembling singlephase transformer equivalent circuits by different connections. Another representative approach generated a primitive matrix by the six coins equivalent circuit of the transformer in (YN, yn0) connection [8, 9].The main characteristic of this approach was that the models were obtained from product relations between the primitive matrix and the incidence matrix in different connections, while it was not accurate description that the left incidence matrix and the right one were same. Reference [10] accounted for a method of handling matrix singularity in the use of the transformer powerflow models. And the modified augmented nodal analysis (MANA) [11] was proposed, which enriched the model application of transformers.
There were several flaws in the previous approaches for modeling the threephase transformers by phasecoordinates. For one thing, the phase selfimpedances and mutualimpedances come from the transformer positive impedances directly, though the phase impedances are much closed to the positive impedances. For another, the most of models described the parameters of 6 coins by 6 × 6 dimension to 7 × 7 dimension matrix in the models for transformer, only consider the injection currents at a–b–c phase between the primary and the secondary sides, but not all the currents. In this case, the models did not enable to cover the use of both in powerflow and short circuit calculations under the asymmetric magnetic circuits.
The threephase AC transmission theory used to be generated from the singlephase models of the system equipment by the AC circuit theory based on the symmetrical characteristics of the threephase voltages and currents, which the threephase symmetry is the characteristic of traditional power system analysis. And the unavoidable asymmetry conditions of the transformers are even more serious than the lines. The purpose of this paper is to model the threephase transformers in different connections by phasecoordinates by matrix operation method. The main work for studying is show as follows:

Generated the primitive admittance matrix from the coupling windings by decoupling method.

Modeling the transformers in different connections by matrix operation method based on the asymmetric magnetic circuits.

Method for obtaining phasecoordinate modified parameters for the models of the transformers.
Methods/experimental
The aim of this paper is to solve the problem for winding connections modeling of threephase transformer by phasecoordinates based on matrix operation method.
Firstly, we introduced the steps to obtain transformer admittance matrix by matrix operation method. Then we analyzed the modeling for the connections of threephase distribution transformer, and also we analyzed the differences of the impedance parameters between phasecoordinates and sequencecoordinates. Finally, we verified the effectiveness of the modeling methods by simulation.
Modeling methodology
In this section, the modeling approach for a transformer is described by the matrix operation method from the 12 × 12 dimension primitive admittance matrix. The complex variables and the values of parameters are given in perunit system. Firstly, the coupling configuration used prefers to describe and analyze the coupling phenomenon by comparing with the two circuit topologies for a singlephase doublewinding transformer. And then the phasecoordinates construction methodology for the threephase model is described by the following steps.

Definition of the transformer primitive admittance matrix Y_{P}.

Definition of the transformer admittance matrix Y_{T} by the matrix operation method in different connections.
Figure 1 shows the main steps of the conceptual scheme of the construction methodology [12]. Firstly, the primitive matrix is generated from the coupling windings shown as the inner blue frame. And then, according to the boundary conditions of the connections at the windings, the transformers are modeled by the matrix operation method.
Nomenclature
Subscripts
 i or 1:

Bus i or Bus 1 (Transformer primary side)
 j or 2:

Bus j or Bus 2 (Transformer secondary side)
 A, B, C or ABC (ia, ib, ic or iabc):

Transformer primary phases
 a, b, c or abc (ja, jb, jc or jabc):

Transformer secondary phases
Variables
 V :

Voltage complex vector.
 I :

Current complex vector.
Matrices
 [Z], Z:

Impedance matrix, element of impedance matrix.
 [Y] or Y, y:

Admittance matrix, element of admittance matrix.
Primitive admittance matrix Y _{P}
The two kinds of singlephase equivalent circuits are shown in Fig. 2. Figure 2a is the Tconfiguration, and Fig. 2b is the coupled configuration. In Fig. 2, “A”, “X” are the two buses at the primary side, “a”, “x” are the two buses at the secondary side respectively. I_{1}, I_{2} (I_{A}, I_{a}) are the Injection currents, and V_{1}, V_{2} (V_{A}, V_{a}) stand for the node voltages at the windings. In Fig. 2a, Z_{1}, Z_{2} are the windings selfimpedances, and jwM is the mutualimpedance between windings. w is the angular acceleration. In Fig. 2b, R_{A}, R_{a}, L_{A}, L_{a} are the resistances and selfinductances at the windings. M is the mutualinductance. And g_{0} + jb_{0} is the noload admittance. The circuit relationship in Fig. 2a (in p.u system) can be given by
There is potential difference between Bus “X” and “x” in transformer testing experiment. The Tconfiguration fails to express the electrical characteristics of the transformer, although the calculation enables to equipotential potential. When running in symmetrical operation, the two buses (X, x) are the same at "zero" potential point, so they can be connected in the form of equipotential. Generally, the two buses are not the same at the zero potential point when they run in threephase asymmetric operation. In order to reflect the potential offset phenomenon of neutral point and consider the various connections of transformers, threephase transformers for modeling should adopt the form of Fig. 2b.
Consequently, there is no electrical connection between Bus X and Bus x in Fig. 2b, which can indicate the potential difference between the two buses. The floating phenomenon without buses grounding and the different connections of the transformer enable to explain as well. While the Tconfiguration equivalent circuit of transformer in Fig. 2a utilizes the equipotential characteristics at the symmetrical operation, which is not suitable for asymmetrical operation analysis.
The application of Fig. 2b at Fig. 1, the threephase transformer contains 6 coins and 12 buses (A–B–C buses and X–Y–Z buses at the primary side; a–b–c buses and x–y–z buses at the secondary side) shown as Fig. 3, and the generalized primitive model can be given by a 12 × 12 matrix using the decoupling methodology. The branch current equation can be given as
where \(\left[ {Z_{Prim} } \right]_{6 \times 6} = \left[ {\begin{array}{*{20}l} {Z_{AA} } \hfill & {Z_{AB} } \hfill & {Z_{AC} } \hfill & {Z_{Aa} } \hfill & {Z_{Ab} } \hfill & {Z_{Ac} } \hfill \\ {Z_{BA} } \hfill & {Z_{BB} } \hfill & {Z_{BC} } \hfill & {Z_{Ba} } \hfill & {Z_{Bb} } \hfill & {Z_{Bb} } \hfill \\ {Z_{CA} } \hfill & {Z_{CB} } \hfill & {Z_{CC} } \hfill & {Z_{Ca} } \hfill & {Z_{Cb} } \hfill & {Z_{Cc} } \hfill \\ {Z_{aA} } \hfill & {Z_{aB} } \hfill & {Z_{aC} } \hfill & {Z_{aa} } \hfill & {Z_{ab} } \hfill & {Z_{ac} } \hfill \\ {Z_{bA} } \hfill & {Z_{bB} } \hfill & {Z_{bC} } \hfill & {Z_{ba} } \hfill & {Z_{bb} } \hfill & {Z_{bc} } \hfill \\ {Z_{cA} } \hfill & {Z_{cB} } \hfill & {Z_{cC} } \hfill & {Z_{ca} } \hfill & {Z_{cb} } \hfill & {Z_{cc} } \hfill \\ \end{array} } \right]_{6 \times 6}\), which presents the impedance relationship of the transformer, and its admittance is \(y_{prim} = Z_{prim}^{  1}\).
The node voltage equation of the transformer can be shown as
where \(\left[ {\begin{array}{*{20}l} {y_{prim} } & {  y_{prim} } \\ {  y_{prim} } & {y_{prim} } \\ \end{array} } \right]_{12 \times 12} = Y_{P}\), which is the full primitive model of the transformer in Ybus form.
Transformer admittance matrix Y _{T}
According to different connections, we obtain the admittance models based on the matrix operation method from the derivation of the initial admittance matrix Y_{P}.
Figure 4 is used to describe the (YN, d11) connection of transformer. According to potential relations, we obtains
The Eq. (4) presents the boundary conditions of the transformer in YN,d11 connection. The admittance matrix can be calculated by the matrix operation method according to the connected relationship of the transformer in YN,d11 connection.
Steps to obtain transformer admittance matrix by matrix operation method can be shown as follows:

(1)
YN connection of primary side: Bus X, Y and Z are grounding, and the rows and columns of them are retained in the primitive admittance matrix Y_{P}.

(2)
d11 connection of secondary side:

(3)
Block 1: Unchanged (the changes of secondary winding connection do not affect the array of the primary side at Y_{P});

(4)
Block 2: For the ja column in Y_{T}, sum with the elements of the j.jy column, (i.e.\(j.ja = j.ja + j.jy\)) in Y_{P};

(5)
Block 3: For ja rows in Y_{T}, sum with the jy row element jy.j, (i.e. \(ja.j = ja.j + jy.j\)) in Y_{P};

(6)
Block 4
Nondiagonal elements in Y_{P}:
a–y: column element: Y_{P} * (jb,jy) → Y_{T}Δ(jb,ja),row element: Y_{P} * (jy,jb) → Y_{T}Δ(ja,jb);
b–z: column element: Y_{P} * ( jc,jz) → Y_{T}Δ(jc,jb),row element: Y_{P} * (jz,jc) → Y_{T}Δ(jb,jc);
c–x: column element: Y_{P} * (ja,jx) → Y_{T}Δ(ja,jc),row element: Y_{P} *(jx,ja) → Y_{T}Δ(jc,ja);
Diagonal elements:
(in Y_{T}): \(y_{ja,ja} =  \sum\nolimits_{k \ne ja} {Y_{ja,k} }\) (in Y_{P}) (Ring network without grounding branch);
Preserve threephase voltage variables, and delete Buses (X, Y, Z, x, y, z) by needed, which can be obtained a 6 × 6 standard matrix (shown as Fig. 5) to a 7 × 7 matrix by retention of neutral buses.
Based on matrix operation method instead of scanning the branch, the models of the transformer are derived from the relationship of the connections, which directly forms the nodal admittance matrix. The analysis method can be used to threewinding transformers as well.
In accordance with the above rules, we can obtain two incidence matrixes (C_{Y} and C_{d11}). The Ybus model relationship (containing neutral voltage variable) of the transformer in YN,d11 connection between (C_{Y},C_{d11}) and Y_{P} can be given by
where
The model in the Eq. (5) is the complete full transformer admittance model. Generally, we need to retain the related parameters of the threephase voltage variables. Repeat Step 3, we can enable to obtain a 6 × 6 matrix.
Modeling of threephase transformers
In this section, the generalized modeling methodology is applied to represent the threephase constructions of transformers. The aim is to demonstrate the matrices of the models in derivation process. All the threephase two winding transformers are defined in magnetic circuits of asymmetry configurations in this paper, and symmetric configuration is a special kind of asymmetric configurations.
There are the magnetic circuits connecting closely of the transformer in threephase threelimb core, besides the magnetic coupling of the primary and secondary windings of each phase, as well as the magnetic circuits coupling of the different phase windings as shown Fig. 6. The effects of the coupling of the interphase windings are obvious when the transformer runs asymmetrically.
Considering mutual inductance between the windings, the branch current equation for conveniently analyzing mutual inductance is
Bus X and x, Bus Y and y, Bus Z and z are checked in unequal potentials, and those buses do not connect together. The selfimpedance of each winding is nearly equal in noload test, and the relationship of the selfimpedance can be given as
According to the nodal voltage Eqs. (6) and (7), the nodal admittance matrix can be shown by 12 × 12 dimensions as
where \(y_{m}^{\prime }\) is the mutual admittance between the primary and secondary sides of windings on the same ironcore. \(\dot{y}_{m}\) is the mutual admittance among different phases primary (secondary) sides of windings on the different ironcores.\(\ddot{y}_{m}\) is the mutual admittance among different phases from the primary sides to secondary sides of windings on different ironcores. y_{s} is the selfadmittance on the primary sides or the secondary sides of windings.
The Bus X, Y, Z and Bus x, y, z of transformer can be connected according to the connection of transformer to express the corresponding voltages in neutral points, based on the method of Sect. 2.

(1)
YN,yn0 connection
The network topology of the transformer in YN,yn0 connection can be shown as Fig. 7. And its 6 × 6 Ybus matrix can be given by the matrix operation method as
Ignoring mutualinductances among the three phases (\(\dot{y} = 0,\ddot{y} = 0,y_{m}^{\prime } = y_{m}\)), the 6 × 6 admittance matrix can be given by
The network topology of the transformer in the Eq. (10) can be shown as Fig. 8.

(2)
YN,d11 connection
In Fig. 9, the connected topology of the transformer in YN,d11 connection is presented. For the transformer in the asymmetric magnetic circuits (such as the transformer in threephase threelimb cores), the node admittance matrix is full because of the coupling among the windings.
The boundary condition is given by the Eq. (4). And the Buses X, Y, Z are grounded. We enable to obtain the 6 × 6 Ybus matrix by the matrix operation method retaining the threephase variables, shown as
where
According to different magnetic circuits, the Ybus model can be given from analyzing the Eq. (11):

(1)
Ignoring mutualinductances among the three phases, the model is able to be seen as the connection by three singlephase transformers assembling. The 6 × 6 admittance matrix can be given as
(12)
The network topology of the transformer in the Eq. (12) can be shown as Fig. 10.

(2)
Same ironcore magnetic circuit: \(y_{m}^{\prime } = \ddot{y}_{m} = \dot{y}_{m} = y_{m}\); and the admittance matrix can be given by
(13) 
(3)
Plane magnetic circuit layout: \(\dot{y}_{mab} = \dot{y}_{mbc} = \frac{2}{3}y_{m}^{\prime } ,\dot{y}_{mac} = \frac{1}{3}y_{m}^{\prime }\), \(\ddot{y}_{mab} = \ddot{y}_{mbc} = \frac{2}{3}y_{m}^{\prime } ,\ddot{y}_{mac} = \frac{1}{3}y_{m}^{\prime }\); the admittance matrix can be given by
(14) 
(4)
Three dimensional split magnetic circuit: \(\dot{y}_{mab} = \dot{y}_{mbc} = \dot{y}_{mac} = 0.5y_{m}^{\prime }\), \(\ddot{y}_{mab} = \ddot{y}_{mbc} = \ddot{y}_{mac} = 0.5y_{m}^{\prime }\), and the admittance matrix can be given by
(15)
Method obtaining the admittances by phasecoordinates
References in this paper attempt to convert directly the parameters of symmetry test into phasecoordinate parameters. In principle, unsymmetrical static threephase equipment fails to form decoupled 1–2–0 sequence circuits. As a result, the phasecoordinate parameters converted by the decoupled 1–2–0 parameters are approximate values. Compensation method enables to reduce the errors, but it cannot be equivalent. In order to distinguish between nominal values in this section, variables and symbols representing perunit values are marked “*” in the subscript. In this section, the admittance matrix in the Eq. (6) needs to analyze and calculate.
Impedances obtained in the symmetry
Considering the symmetry of structural parameters for threephase Transformer, the relationship of the Eq. (6) in perunit system enables to be expressed by
where
\(Z_{P*}\) is the selfadmittance on the primary winding;
\(Z_{s*}\) is the selfadmittance on the secondary winding;
\(Z_{m*}\) is the mutualadmittance between the primary and the secondary windings at the same ironcore;
\(Z^{\prime}_{m*}\) is the mutualadmittance between the primary windings.
\(Z^{\prime\prime}_{m*}\) is the mutualadmittance between the primary and the secondary windings at the different ironcores.
\(Z^{\prime\prime\prime}_{m*}\) is the mutualadmittance between the secondary windings.

(1)
Noload test: The test enables to obtain the two groups of data, and they are noload current I_{0} and noload loss P_{0}.The relationships of parameters in noload state can be given by
where
y0* is the noload admittance; z0* is noload impedance;
g0* is the noload conductance;
b0* is the noload susceptance;
STN is the transformer’s rated capacity.
The noload impedance can be expressed by

(2)
Short circuit test: The test enables to obtain the two groups of data, they are the percentage of shortcircuit voltage V_{s}% and short circuit power loss P_{k}. The relationships of parameters at shortcircuit state can be given by
$$\left\{ {\begin{array}{*{20}l} {R_{T*} = P_{k*} } \\ {\left {z_{T*} } \right = V_{s*} } \\ {x_{T*} = \sqrt {\left {z_{T*} } \right^{2}  R_{T*}^{2} } } \\ \end{array} } \right.$$(19)where
R_{T*} is the transformer winding resistance; \(\left {z_{T*} } \right\) is the leakage reactance; x_{T*} is the winding reactance. And the impedance in shortcircuit state can be given by
The Eqs. (10)–(20) deduced under the condition of the symmetrical currents or voltages, the above parameters stand for the positive sequence parameters. The steps for extrapolating the phasecoordinate parameters can be shown as follows:
Considering the same values of leakage reactance on the primary side and secondary side, the relationship (in value system) can be shown by
where
\(Z_{\sigma 1}\),\(Z_{\sigma 2}\) are the leakage reactances on the primary and secondary sides as Fig. 2b shown. The basic parameters in the symmetry in no load test can be given by
The reluctances of transformer’s magnetic circuit are mainly from the air gap between iron cores. According to the principle of magnetic circuit of phase separation test method, there is the relationship shown as follows:
The leakage fluxes run in the air different from those run in the ironcores. The threephase symmetrical currents have the effect of magnetization. Similar to threephase transmission lines, they have the basic relationships of positive sequence impedance and zero sequence impedance. The relationship between positive sequence leakage reactance and phase separation leakage reactance of transformer obtained from short circuit test is as follows:
where
\(x_{\sigma s}\) is selfinductance leakage flux;
\(x_{\sigma m}\) is mutualinductance leakage flux.
The selfimpedance and mutual impedance of the leakage reactance from the short current test can be given by
By substituting the Eqs. (14)–(25), the relationship of the impedance matrix by phasecoordinates in YN,yn0 connection can be given by the modified equation in perunit system shown as
Figure 11 presents the relationships of the vectors in the parametric systems. The relationships for the parameters in References are shown in Fig. 11a. While the relationships for our modified parameters are shown in Fig. 11b, which enable to explain the work in this section intuitively.
Impedances and admittances obtained in the asymmetry
The magnetic circuits of the threephase threelimb core transformer can be shown in Fig. 12. The branch Eq. (6) enables to express the relationships of the transformer conveniently.
Considering the relations of the linear circuits without magnetic saturation, the impedance parameters in the Eq. (6) can be obtained by the open circuit test. Figure 13 shows the coupling relations between the primary side and the secondary side.

(1)
At wye side for the open circuit test, when the voltage \(V_{AX} = V_{N}\)(where V_{N} stand for the rated voltage) is applied at the winding “AX”, and the rest windings in opening. We enable to obtain test data for the Eq. (6) shown as
$$\left[ {\begin{array}{*{20}l} {V_{AX}^{0} = V_{N} } \\ {V_{BY}^{0} } \\ {V_{CZ}^{0} } \\ {V_{ax}^{0} } \\ {V_{by}^{0} } \\ {V_{cz}^{0} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}l} {z_{AA} } & {z_{AB} } & {z_{AC} } & {z_{Aa} } & {z_{Ab} } & {z_{Ac} } \\ {z_{BA} } & {z_{BB} } & {z_{BC} } & {z_{Ba} } & {z_{Bb} } & {z_{Bc} } \\ {z_{CA} } & {z_{CB} } & {z_{CC} } & {z_{Ca} } & {z_{Cb} } & {z_{Cc} } \\ {z_{aA} } & {z_{aB} } & {z_{aC} } & {z_{aa} } & {z_{ab} } & {z_{ac} } \\ {z_{bA} } & {z_{bB} } & {z_{bC} } & {z_{ba} } & {z_{bb} } & {z_{bc} } \\ {z_{cA} } & {z_{cB} } & {z_{cC} } & {z_{ca} } & {z_{cb} } & {z_{cc} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}l} {I_{A0} } \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ \end{array} } \right]$$(27)
where
\(V_{AX}^{0} ,V_{BY}^{0} ,V_{CZ}^{0} ,V_{ax}^{0} ,V_{by}^{0} ,V_{cz}^{0}\) are the testing voltages;
\(I_{A0}^{0}\) is the testing current.
We can obtain the impedances by perunit system from the Eq. (20) expressed as
The rest impedances can be obtained by repeating the test two more again.

(2)
At delta side for the open circuit test, when the voltage \(V_{ax} = V_{NII}\) (where V_{NII} stands for the rated voltage at the delta side) is applied at the winding “ax”, and the rest windings in opening. We enable to obtain test data for the Eq. (6) shown as
$$\left[ {\begin{array}{*{20}l} {V_{AX}^{d0} } \\ {V_{BY}^{d0} } \\ {V_{CZ}^{d0} } \\ {V_{ax}^{d0} = V_{NII} } \\ {V_{by}^{d0} } \\ {V_{cz}^{d0} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}l} {z_{AA} } & {z_{AB} } & {z_{AC} } & {z_{Aa} } & {z_{Ab} } & {z_{Ac} } \\ {z_{BA} } & {z_{BB} } & {z_{BC} } & {z_{Ba} } & {z_{Bb} } & {z_{Bc} } \\ {z_{CA} } & {z_{CB} } & {z_{CC} } & {z_{Ca} } & {z_{Cb} } & {z_{Cc} } \\ {z_{aA} } & {z_{aB} } & {z_{aC} } & {z_{aa} } & {z_{ab} } & {z_{ac} } \\ {z_{bA} } & {z_{bB} } & {z_{bC} } & {z_{ba} } & {z_{bb} } & {z_{bc} } \\ {z_{cA} } & {z_{cB} } & {z_{cC} } & {z_{ca} } & {z_{cb} } & {z_{cc} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}l} 0 \\ 0 \\ 0 \\ {I_{a0}^{d} } \\ 0 \\ 0 \\ \end{array} } \right]$$(29)
where
\(V_{AX}^{d0} ,V_{BY}^{d0} ,V_{CZ}^{d0} ,V_{ax}^{d0} ,V_{by}^{d0} ,V_{cz}^{d0}\) are the testing voltages;
\(I_{a0}^{d0}\) is the testing current. And the impedances in perunit system from the Eq. (22) can be expressed as
The rest impedances can be obtained by repeating the test two more again. Consequently, the impedance matrix by phasecoordinates (in perunit system) in the Fig. 13 can be given as
Results and discussion
Results

(1)
Test by experiment
There are the transformers given as (Type I: Capacity: 60 k VA, Ratio: 400 V/110 V; Type II: Capacity: 1000 k VA, Ratio: 6300 V/400 V). In the Table 1, the mutual inductance coefficient is inversely proportional to the length of magnetic circuit. Ignore measurement errors, the mutual inductance (reactance) coefficients can be approximated as: aphase [1 0.66 0.33]; bphase [0.5 1 0.5]; cphase [0.33 0.66 1] (shown as Fig. 12).

(2)
Simulation calculation
The IEEE 4 node feeder test system network is shown in the Fig. 14. The parameters of the test system are as follows: transformer ratio is 12.47(kV):24.9 (kV), and the parameters of transformer and line present in [13]. The tolerance for calculation is 10^{−5} for testing. The unbalanced loads in bus 4 of the test system are 1250 kW, 1800 kW, 2375 kW, and the power factors are − 0.85, − 0.9 and − 0.95 respectively (Complex power of load marked as S_{1}).
There are 6 types and parameters of the transformers for testing by power flow calculations. Table 2 presents the types and the parameters of the transformers. In Table 2, The admittance matrix in No. 1 and No. 2 groups of the transformers employ the method in [8] and [12]. And the rest of the transformers use the method in this paper. Table 3 lists the results of power flow calculation for the transformers in D,yn1 connection. And Table 4 presents the results of power flow calculation for the transformers in YN,yn0 connection. Table 5 lists the power flow results for the different power supply in the transformer in YN,yn0 connection.
Discussion
The calculated results of No. 1 groups at Tables 3 and 4 are the same as [13].Comparing with the results in No. 1 groups of Tables 3 and 4, The results of No. 2 groups are merely smaller, but less errors considering noload parameters. While both the admittance matrix parameters in the two groups of transformers are calculated by the method in [8] and [12], which the parameters essentially are the positive sequence ones. The method employed in Sect. 5 of this paper is used to calculate the threephase admittance matrix parameters, and the experiments also show the differences in the test. Comparing with the previously results, the results in No. 3 group and No. 4 group at Tables 3 and 4 are a little smaller and closer in the same condition. Analysis from the physical definition, the true phase admittances are different from the positive sequence ones due to the demagnetization by threephase symmetrical currents. And the results of different magnetic circuits are given in Tables 3 and 4 as well. Those results also are different from each other.
In Table 5, comparing with the No. 1 group, cphase voltages are increasing gradually by the changes of loads in No. 6 group, due to the increased unbalances of loads’ currents. Comparing with symmetric component parameters, the effect of node voltages is more different by the phasecoordinates parameters.
Consequently, the threephase symmetrical currents have the demagnetization, which make the phaseimpedances slightly smaller than the synthetic impedances (positive sequence impedance). In addition, the mutualimpedances effect significantly by asymmetry.
Conclusions
In this paper, the authors propose the phasecoordinate modeling method by a 12 × 12 primitive full admittance matrix to structure the 6 × 6 admittance matrix (7 × 7 matrix by retention of 1 neutral point) for threephase transformers in different connections from asymmetric magnetic circuits by the matrix operation method. The phasecoordinate impedances have been corrected by considering the magnetization of threephase symmetrical currents. There are the advantages shown as follows:

Modeling the transformers in different connections and asymmetric magnetic circuits;

Real phasecoordinates parameters, which enable calculations accurately.
Finally, the matrix operation method also enables to use for modeling threephase threewinding transformers usefully, besides the threephase doublewinding transformers. And the method applies to model the nonstandard transformers as well.
Availability of data and materials
Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.
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ZZ finished the algorithm. MM put forward the idea of this paper and English writing of the paper. CW finished the experiments. All authors read and approved the final manuscript.
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Zhang, Z., Mo, M. & Wu, C. Threephase distribution transformer connections modeling based on matrix operation method by phasecoordinates. J Wireless Com Network 2021, 66 (2021). https://doi.org/10.1186/s1363802101945z
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DOI: https://doi.org/10.1186/s1363802101945z
Keywords
 Threephase transformer
 Phasecoordinate model
 Matrix operation method
 Asymmetric magnetic circuit
 Coupling