# Leakage Inductance Characteristics of Power Transformer

## Transcript Of Leakage Inductance Characteristics of Power Transformer

Open Access Journal

Journal of Power Technologies 99 (3) (2019) 231–238

journal homepage:papers.itc.pw.edu.pl

Leakage Inductance Characteristics of Power Transformer Winding Fault Based on ANSOFT

Na Wanga, Ke Xub,∗, Jie Yangb, Junzhe Huc

aHenan Polytechnic Institute, Nanyang Henan 473000, China; bHenan Institute of Technology, Xinxiang Henan 453000, China; cKaiserslautern Technical University, Kaiserslautern Rhineland-Pfalz 67663, Germany

Abstract

The development momentum of the power industry is pushed by the increasing demands and continuous progress in all sections of the economy. The improvement of the voltage grade of transmission lines is imminent, and the capacity of a single transformer continuously increases, thereby resulting in the serious problem of magnetic leakage, which that can aggravate the stray loss of transformers and the degree of winding deformation. The simulation model of leakage inductance under the different fault states of transformer winding was proposed on the basis of ﬁnite element method to reduce the threat to the safe and stable operation of power grids caused by the winding insulation damage from magnetic leakage. ANSOFT software was used to establish the physical model of 800 kVA and 10 kV true distribution transformers, and the leakage inductance values of transformer winding with a three-phase grounding short circuit, a turn-to-turn fault, and an inter-turn fault were analyzed. Finally, the accuracy of the model was veriﬁed through the ﬁnite element simulation. Results indicate that the leakage inductance of transformer increases by 30% in a three-phase short circuit, the leakage inductance is negatively correlated with the number of fault turns in a turn-to-turn fault, and the leakage inductance has a nonlinear positive correlation with the number of fault turns in an inter-turn fault. The proposed method provides references for the detection and evaluation of transformer operation fault.

Keywords: transformer; winding fault; leakage inductance; electromagnetic ﬁeld; ﬁnite element method

1. Introduction

As an important power equipment in the transmission network, the safe and stable operation of transformers is related to the safety of the entire system; hence, the cause of its failure needs special research attention [1]. At present, the transformers in a power system have two fault classiﬁcations, which are based on the different locations of transformer faults[2, 3, 4, 5, 6]. Transformer faults can be divided into ontology and external faults. According to the statistical analysis of previous types of transformer fault, ontology faults, which are internal faults, account for a large proportion. When a transformer has an internal fault and the ontology is seriously damaged, electrical and non-electrical characteristics can be observed, and fault detection is relatively easy. Currently, numerous transformer internal fault detection studies mainly focus on this phenomenon. However, the research on hidden and potential fault detection in transformers is still in its infancy. Meanwhile, the “hidden danger” of

∗Corresponding author Email address: [email protected] (Ke Xu)

transformers and the slight fault of transformer winding have not been investigated [7, 8, 9]. Existing literature shows that transformer winding is deformed when the transformer suffers from external faults during operation, thereby developing into internal inter-turn fault, which causes transformer fault [10].

Although the “healthy state” of internal winding in the normal operation of a transformer is poor and still not evident, it has already entered an adverse operation state. This feature describes a hidden fault, which has the property of trigger and accumulation [11, 12]. At present, the detection of hidden fault mainly depends on the analysis of oil chromatography, which is difﬁcult to combine with mature protection theory and the engineering application of large transformers. The detection and quantiﬁcation of this hidden fault in large transformers through electrical quantity have been reported rarely. In the actual operation of a power system, different short-circuit faults, which cause large electromagnetic shock in the system due to the short-circuit current, are inevitable. This shock deforms the winding of the transformer, which can inﬂuence the normal operation of the transformer and even

Journal of Power Technologies 99 (3) (2019) 231–238

lead to the failure of its operation. Thus, this study used the ﬁnite element method to propose

the leakage inductance characteristics of the power transformer winding fault based on ANS OFT . The leakage inductance characteristics of 800 kVA and 10 kW three-phase oil-immersed distribution transformers, which can be more accurately predicted under the conditions of three-phase grounding short circuit, turn-to-turn fault, and inter-turn fault, were investigated by establishing a two-dimensional (2D) simulation model of transformer winding deformation degree and its characteristics. Furthermore, the results can provide reference for the detection and evaluation of transformer operation fault.

2. State of the art

The main inﬂuence of leakage magnetic ﬁeld has two aspects. One is to accelerate the thermal aging of the internal metal structure of the transformer and paper insulation. The other is to produce a large electromagnetic force for the interaction between the current that ﬂows through the winding and the leakage magnetic ﬂux. Therefore, experts all over the world have not only studied the numerical value of the transformer leakage magnetic ﬁeld in recent years, but also achieved corresponding results in the calculation of the short-circuit electromagnetic force of winding.

The numerical solution of the electromagnetic ﬁeld problem has been developed rapidly with the development of computer and programming technology. The solution can be extensively applied in solving the electromagnetic ﬁeld problem. Currently, the most commonly used calculation methods include the least square, ﬁnite element, integral equation, and ﬁnite difference methods. Tsukerman I.A., Konrad A., and Lavers J.D. proposed the use of least square means to calculate the leakage inductance of transformers and applied it to transformer protection [13]. Silvester and Chari used the methods of 2D high-order ﬁnite element and 2D ﬁnite element with vortex term to calculate the constant magnetic ﬁeld of a transformer. O.W. Anderson used the axisymmetric ﬁnite element method to manage the magnetic ﬁeld leakage of transformers [14, 15]. In Britain, Coulson used the ﬁnite element method to calculate and analyze the electromagnetic ﬁeld distribution in a transformer based on the three-dimensional (3D) simulation model of the transformer [16]. In the United States, Kumbhar G. B. effectively simulated the electromagnetic operation process of a transformer by simplifying the geometric model of a transformer and applying the magnetization curve of the core in the model [17]. Dr. Wang Shenghui of Shenyang University of Technology used the A − V − A method to calculate and analyze the 3D transient ﬁeld of a transformer [18]. Liu Xiaoli and Bi Yanjun of Harbin University of Technology analyzed the transient leakage magnetic ﬁeld of a cable transformer using the ﬁeld circuit coupling and mirror image methods. Chongyou Jing, an engineer of Tianwei Transformer Factory in Bao Ding, obtained the converter transformer model under the non-sinusoidal condition and the distribution of the

transient leakage magnetic ﬁeld around the winding using a ﬁnite element simulation tool [19]. Xiaopo Ou of Guangdong Power Grid Company used the ﬁnite element method to calculate the transformer short circuit impedance to obtain the distribution of the leakage magnetic ﬁeld and the leakage inductance parameters [20]. Jianmin Wang of Baoding Tianwei Group not only investigated the transient electromagnetic ﬁeld characteristics of the UHV DC converter transformer, but also calculated the transient electromagnetic ﬁeld and harmonic loss of the converter transformer [21]. The magnetic leakage ﬁeld of a high-voltage converter transformer was calculated and analyzed in Reference [22], and the short-circuit impedance and the leakage magnetic ﬁeld distribution were obtained.

The short-circuit electromagnetic force of the transformer winding in the short-circuit state can cause serious mechanical damage to the transformer, bend or damage the winding, and even cause the transformer to explode. In addition, the monitoring of transformer short-circuit condition requires the use of a special test instrument. The requirements for test tools increase with the transformer capacity. Therefore, establishing the transformer numerical simulation model for predicting the short-circuit electromagnetic force of winding is crucial. S.V. Kulkarni took the lead in solving the electromagnetic ﬁeld of a transformer with tap winding using the ﬁeld circuit coupling method [23]. Jawad Faiz and Tahere Noori of Iran used the 2D and 3D ﬁnite element methods to deal with the short-circuit electromagnetic force of the core transformer [24]. Given that the short circuit electromagnetic force is mainly based on the overall fault of the transformer’s primary and secondary windings, the electromagnetic force at a ﬁxed winding position cannot be analyzed. Then, Hyun-Mo Ahn of South Korea divided the high- and low-voltage windings of a 50 kVA dry transformer into several parts and calculated the vector magnetic bits, magnetic ﬂux density, and electromagnetic force of each partial winding based on the ﬁnite element method [25]. In the literature [26], the simulation model of the three-phase core transformer was established by using the electromagnetic ﬁeld simulation software ANS YS , and the electromagnetic force on the winding was analyzed on the basis of the ﬁnite element method [27]. Jingqiu Qiao and Yunmiu Tang of China used a three-phase transformer with a rated working condition of 17 MVA as the simulation model. The ﬁnite element method was used to calculate the frequency spectrum distribution of the axial and radial electromagnetic forces when the winding was short-circuited in different initial states. Their results indicate that the calculation formula of the traditional electric power can basically meet the engineering requirement, but the numerical calculation can calculate the axial electric power [28]. Shuang Liu of Shenyang University of Technology used the ﬁnite element method to calculate the short-circuit electromotive force, which considers the properties of insulating materials, winding displacement, mechanicalproperties and elasticity. Meanwhile, a software for calculating the 2D transient symmetry ﬁeld and the winding shortcircuit strength was developed, and the calculation speed

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was improved on the basis of Windows platform [29]. Luliang Wang of Harbin University of Technology established a 2D axisymmetric ﬁeld-circuit coupling model of a transformer by using the ANS YS software and obtained the radial and axial force distributions of transformer windings under the three-phase symmetric short circuit condition. Although the ampere-turn imbalance and the leakage magnetic ﬁeld distribution between winding coils were considered, the interaction of each phase winding was ignored, and the result contained a certain error [30, 31].

This study used the ﬁnite element method to establish the simulation calculation model of the transformer winding electromagnetic ﬁeld, the leakage inductance characteristics of power transformers under a three-phase grounding short circuit, a turn-to-turn fault and an inter-turn fault were simulated and examined. The accuracy of the model was veriﬁed by ﬁnite element simulation, which provides reference for the detection and evaluation of transformer operation fault. The remainder of this study is organized as follows: section 3 describes the principle of ﬁnite element simulation and the construction of the transformer mathematical model, the parameters of transformer leakage inductance, and the ﬁnite element simulation model; section 4 discusses the use of the ﬁnite element method to calculate the transformer leakage inductance parameters and the analysis of the characteristics of a three-phase short circuit, a turn-to-turn fault and an inter-turn, and ﬁnally, the leakage inductance characteristics under different working conditions are obtained; section 5 summarizes the conclusions.

3. Methodology

3.1. Principle of ﬁnite element simulation

The core idea of ﬁnite element is summarized as variation and discretization. Variation establishes the “weak form” of the ﬁnite element. For example, the second-order partial differential equation under given boundary conditions is transformed into solving the extreme value of the ﬁrst-order partial differential equation, thereby approximately simplifying the complex problem [32, 33, 34, 35]. Discretization divides the entire continuous solution region into many small subregions, thereby obtaining the results in each small subregion, and then adds the results of each small subregion to obtain the solution of the entire region. The ﬁnite element method is brieﬂy summarized as follows: (1) the problem of solving the partial differential equation is transformed into solving a conditional variation problem [36]; (2) the entire solution domain is divided by partition unit as a triangle or quadrangle while constructing the linear interpolation function; (3) by constructing linear algebra equations, the extreme value of the universal energy function is transformed into the extreme value of energy function; (4) the linear algebra equation is solved.

Maxwell of ANS OFT Company is a powerful 2D/3D electromagnetic ﬁeld analysis software based on the ﬁnite element method. It includes a user-friendly interface, an ad-

vanced adaptive grid segmentation algorithm, and a powerful data processing ability and is preferred by research developers over other ﬁnite element software. The modeling and simulation steps based on the ANS OFT Maxwell software are shown in Fig. 1.

Figure 1: Calculation process of ANSOFT

3.2. Equivalent model of transformer Transformer is an important electrical equipment in a

power grid. To study transformers, the physical model of transformers must be simpliﬁed into a mathematical module, and the complex problem of magnetic coupling must be transformed into a simple circuit problem. Therefore, an equivalent circuit transformer model exists. The common equivalent model of a double-winding transformer is the T type equivalent model as shown in Fig. 2.

Figure 2: T-type equivalent model of double-winding transformer

In the T -type equivalent model as shown in Fig. 2 , the parameters of each side of the transformer are calculated by the same side voltage level. r1 and L1 respectively represent the resistance of the original winding and the leakage inductance of the winding as shown in Fig. 2; r2 andL2 represent the resistance of the secondary winding and the leakage inductance of the winding calculated at the primary side;

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u1 and i1 respectively represent the voltage and current values of the original winding side winding while the transformer is in operation; u2 and i2 respectively represent the voltage and current values of the secondary side winding of the original side while the transformer is running. In the equivalent transformer circuit diagram, in addition to these parameters, two important parameters characterize the excitation branch, namely, instantaneous excitation inductance Lm and short circuit turn leakage inductance Ls. However, these two parameters are unavailable in some situations. In the analysis of a speciﬁc operation state, the resistance of the excitation winding can be ignored, and the excitation branch only has the instantaneous excitation inductance Lm. In the analysis of the internal inter-turn fault of the transformer, in addition to calculating instantaneous excitation inductance Lm, the value must be calculated after it is parallel with short circuit turn leakage inductance Ls.

According to the circuit principle, the formula is obtained according to the T-type equivalent model of the transformer as shown in Fig. 2:

u = r i + L di1 + LmLs · d(i1 + i2)

(1)

1 1 1 1 dt Lm + Ls

dt

Formula 2 can be obtained by dividing Formula 1.

u = r i + i + (L + LmLs ) · d(i1 + i2) − r i − L di2 (2)

1 11 2

1 Lm + Ls

dt

1 2 1 dt

The resistance and leakage inductance on the original side of the transformer are small and can be ignored. Thus, Formula 3 can be expressed as follows:

u = r i + i + (L + LmLs ) · d(i1 + i2)

(3)

1 11 2

1 Lm + Ls

dt

When the transformer is in normal operation, the short circuit turn leakage inductance can be ignored. Thus, Formula 4 is expressed as follows:

u1 = r1 i1 + i2 + (L1 + Lm) · d(i1d+t i2) (4)

3.3. Calculation of transformer leakage inductance parameters based on ﬁnite element analysis

If the transformer is in normal operation, according to Formula 4, Lk = L1+Lm can be the equivalent instantaneous inductance of the transformer. If the transformer fails internally, the equivalent instantaneous inductance of the transformer can be deﬁned as Lk = L1+ LLmm+LLss according to Formula 3. The equivalent instantaneous inductance consists of two parts. The ﬁrst part is the leakage inductance of the original side winding L1. The second part is the parallel value of short circuit turn leakage inductance Ls and instantaneous excitation inductance Lm.

When i1=i1 + i2, Formula 3and Formula 4 can be uniﬁed as follows:

u1 = r1 i + Lk · d i (5) dt

The sampling period is T . By discretizing the differential Equation 5, Formula 6 is obtained:

u1(k)

=

r1

i(k) + Lk

i(k+1)− i(k−1)

u1(k + 1) = r1 i(k + 1) + 2LTk i(k+22T)− i(k) (6)

The calculation formula of equivalent instantaneous inductance Lk of the transformer can be obtained by simultaneously eliminating the original side winding resistance r1 from Equation 6:

Lk = 2T

i2(k) −

u1(k) · i(k − 1) ·

i(k + 1) − u1(k + 1) · i(k) i(k + 1) − i(k) · i(k + 2) +

i2(k + 1) (7)

3.4. Establishment of ﬁnite element simulation model for transformer leakage inductance magnetic ﬁeld

The 800 kVA and 10 kV three-phase three-column oilimmersed distribution transformer are selected as the simulation models. The main parameters of the transformer are shown in Table 1.

In the simulation model, the effects of the transformer structure, the coil skin, the iron core clamp, the fuel tank, and the support beam on the leakage magnetic ﬁeld can be simpliﬁed and even ignored. The current is set to uniform distribution mode in the winding zone, and the traverse vortex is ignored. The high- and the low-voltage winding have the same height, the demagnetization effect of the coil conductor is not considered, and each switch is evenly distributed.

Figure 3: 2D model diagram of the transformer

The cross-section of the three-phase distribution oil immersed transformer is selected to establish the model as shown in Fig. 3. The dark blue part is the upper and lower yokes of the transformer, the light blue surface is the core column, and the three-phase winding of transformer ABS is marked red, green, and yellow from right to left. The characteristic parameters of the iron core, the winding, and the

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Name

Table 1: Parameters of the real-scale transformer

Dimension, mm

Name

Core diameter Width of iron window Fuel tank width High-voltage coil Low-voltage coil Number of high-voltage winding Parameter name Rated capacity Rated current Loss of short circuit Connection group number

210 160 540 Ø276-360 Ø220-271 955 Parameter value 800 kVA 46.19/1154.7(A) 8200 W Dyn11

Height of iron window Height of upper and lower iron yoke

Height of fuel tank Height of high-voltage winding Height of low-voltage winding Number of low-voltage winding

Parameter name Rated voltage

No-load current Impedance voltage

Core material

Dimension, mm

505 825 1120 468 465

21 Parameter value 10±2×2.5%/0.4(kV)

0.18% 4.5%

B23P090

insulating oil are shown in Table 2. Fig. 4 shows the magnetization curve of the core in semi-logarithmic coordinates.

three voltage control switches are closed after 30 ms, and the three-phase short circuit of the transformer is obtained.

Figure 4: Magnetizing curve of the transformer iron core

The excitation source adopts the external circuit, and the external circuit adopts the same DT n11 connection group as the transformer. The inﬁnity boundary is set as the balloon boundary condition, which ensures that the boundary can be inﬁnity without setting a large boundary while reducing the number of computational grids. Several units are set to the maximum number (3000) by using the triangulation unit and the automatic subdivision algorithm. The solution termination time is set to 100 ms, the period to 5, and the step size to 1 ms.

4. Result Analysis and Discussion

4.1. Three-phase grounding short circuit

When the transformer fails, the winding has a large transient current, which leads to the complete damage of the transformer and serious power grid outage. Therefore, the magnetic leakage ﬁeld in the case of transformer failure must be studied. The three-phase short circuit fault position is set to occur on the low voltage side of the transformer external circuit, as shown in Fig. 5. When the initial phase angle of a phase is set to 0° and the corresponding short circuit time is 30 ms, the three-phase short circuit is the worst, that is, the transformer is under the load condition from 0 ∼ 30 ms. The

Figure 5: Three-phase short circuit of the secondary side

The current of A-phase of the low-voltage side can be obtained by simulation (Fig. 6).

Figure 6: Low-voltage side current of A-phase when shorted

Fig. 6 shows that after the three-phase short circuit, the short circuit current on the low-voltage side increases sharply, accompanied by a certain transient process. When the short circuit time is 1/4 cycles, which is 40 ms, the maximum value of 16.7 kA is reached, the short circuit steady state process is started, and the current remains very large. A large current often produces a large electromagnetic force, which causes winding deformation and insulation damage.

The magnetic leakage ﬁeld distribution of the three-phase transformer at 40 ms is shown in Fig. 7 .

In Fig. 7, the maximum magnetic density of the transformer is 2.6 T, and the core is saturated. The leakage inductance of the transformer is 76.7 mH, which is approximately

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Table 2: Properties of the materials in the transformer

Transformer component

Material Conductivity Relative dielectric constant

Winding Iron core Transformer oil

Copper Iron

Insulation oil

58000000 4550000 0

1 Nonlinear

2.4

Figure 7: Magnetic density distribution during shorted fault

Figure 9: Phase voltage of primary side with turn-to-turn fault

30% higher than that of the normal operation. 4.2. Turn-to-turn fault

If the end of the winding has a turn-to-turn fault for the primary side winding of the transformer’s A-phase, then the black part of the winding in the 2D model represents grounding, the number of turns is 8% of the total turns, and inter-turn fault occurs at 30 ms. The A-phase high-voltage winding is divided into normal and fault winding, and a wire grounding is drawn from the middle (Fig. 8).

Figure 8: Turn-to-turn fault of A-phase primary side winding

The voltage and current of the primary side of the fault phase can be obtained by simulation (Fig. 9 and Fig. 10).

In Fig. 10, the currents of A-phase and B-phase enter the transient process, the primary side is grounding because the winding grounding, and the current increases sharply. The current of C-phase is consistent with that when the primary

Figure 10: Phase current of turn-to-turn fault

side uses the triangle connection method. The calculated leakage inductance of A-phase is 58.8 mH.

The leakage inductance results of the transformer with different grounding are shown in Fig. 11 by constantly changing the grounding position, that is, changing the number of turns of fault winding.

In Fig. 11, the leakage inductance has a nonlinear negative correlation with the number of fault turns. When the number of fault turns is small, the leakage inductance is close to that in normal operation.

4.3. Inter-turn fault Assuming that the primary side winding of the A-phaseof

the three-phase transformer has an inter-turn fault, the Aphase in the model is divided into 10, the number of turns in each group is 90 turns, and the black part represents the occurrence of inter-turn fault (Fig. 12).

The voltage and current of the fault phase of the primary side are obtained by simulation as shown in Fig. 13 and Fig. 14.

Similarly, by increasing the number of fault turns, the black (fault) area of the winding in the 2D model of the corresponding turn is increased, and the relationship between leakage inductance and the number of fault turns is shown in Fig. 15.

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Figure 11: Relationship between leakage inductance and the number of fault turns

Figure 13: Phase primary side voltage with inter-turn fault

Figure 14: Phase primary side current with inter-turn fault

Figure 12: Inter-turn fault of phase A primary side winding

Fig. 15 shows that a nonlinear positive correlation exists between leakage inductance and the number of turns of the inter-turn fault. The leakage inductance tends to increase with the number of turns of the inter-turn fault.

5. Conclusion

To reduce the winding insulation damage caused by transformer magnetic ﬂux leakage, which can pose a great threat to the safe and stable operation of power grids, 800 kVA and 10 kV three-phase oil-immersed distribution transformers were selected as physical models, and the leakage inductance characteristics of the power transformer under the conditions of three-phase grounding short circuit, turn-to-turn fault, and inter-turn fault were studied using an ANS OFT software. Finally, the following conclusions could be obtained:

(1) When three-phase short circuit occurs on the secondary side of the transformer, the secondary side current increases sharply, the iron core reaches saturation, the leakage magnetic ﬁeld increases correspondingly, and the leakage inductance of the transformer increases by 30% compared with that in normal operation.

(2) When the turn-to-turn fault of transformer occurs, the leakage inductance is negatively correlated with the number of fault turns, and the leakage inductance tends to decrease with the increase in the number of grounding fault turns.

(3) When the inter-turn fault of transformer occurs, the leakage inductance is positively correlated with the number of turns in inter-turn fault. The leakage inductance tends to increase with the number of turn-to-turn fault turns.

Starting from the causes of transformer winding insulation damage, this study conducted a research of the leakage inductance characteristics under power transformer winding fault based on ANS OFT . The accuracy of the model was veriﬁed by ﬁnite element simulation, which provides a refer-

Figure 15: Relationship between leakage perception and the number of fault turns with inter-turn fault

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ence for detection and evaluation of transformer operation fault. Owing to the lack of actual ﬁeld fault monitoring date, an on-line monitoring system for transformer winding deformation must be designed in future research. By doing this, the running state data can be collected in real time according to the transformer operation process to judge the real-time shape of transformer winding according to the parameters and determine the reasonable maintenance time of transformers after comprehensive analysis. This system can not only ensure the maximum operation life of transformers, but also prevent the interruption of operation because of failure.

Acknowledgments

This research was supported by the Key Scientiﬁc Research Projects of Henan University in China, and its foundation is 20B470003.

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journal homepage:papers.itc.pw.edu.pl

Leakage Inductance Characteristics of Power Transformer Winding Fault Based on ANSOFT

Na Wanga, Ke Xub,∗, Jie Yangb, Junzhe Huc

aHenan Polytechnic Institute, Nanyang Henan 473000, China; bHenan Institute of Technology, Xinxiang Henan 453000, China; cKaiserslautern Technical University, Kaiserslautern Rhineland-Pfalz 67663, Germany

Abstract

The development momentum of the power industry is pushed by the increasing demands and continuous progress in all sections of the economy. The improvement of the voltage grade of transmission lines is imminent, and the capacity of a single transformer continuously increases, thereby resulting in the serious problem of magnetic leakage, which that can aggravate the stray loss of transformers and the degree of winding deformation. The simulation model of leakage inductance under the different fault states of transformer winding was proposed on the basis of ﬁnite element method to reduce the threat to the safe and stable operation of power grids caused by the winding insulation damage from magnetic leakage. ANSOFT software was used to establish the physical model of 800 kVA and 10 kV true distribution transformers, and the leakage inductance values of transformer winding with a three-phase grounding short circuit, a turn-to-turn fault, and an inter-turn fault were analyzed. Finally, the accuracy of the model was veriﬁed through the ﬁnite element simulation. Results indicate that the leakage inductance of transformer increases by 30% in a three-phase short circuit, the leakage inductance is negatively correlated with the number of fault turns in a turn-to-turn fault, and the leakage inductance has a nonlinear positive correlation with the number of fault turns in an inter-turn fault. The proposed method provides references for the detection and evaluation of transformer operation fault.

Keywords: transformer; winding fault; leakage inductance; electromagnetic ﬁeld; ﬁnite element method

1. Introduction

As an important power equipment in the transmission network, the safe and stable operation of transformers is related to the safety of the entire system; hence, the cause of its failure needs special research attention [1]. At present, the transformers in a power system have two fault classiﬁcations, which are based on the different locations of transformer faults[2, 3, 4, 5, 6]. Transformer faults can be divided into ontology and external faults. According to the statistical analysis of previous types of transformer fault, ontology faults, which are internal faults, account for a large proportion. When a transformer has an internal fault and the ontology is seriously damaged, electrical and non-electrical characteristics can be observed, and fault detection is relatively easy. Currently, numerous transformer internal fault detection studies mainly focus on this phenomenon. However, the research on hidden and potential fault detection in transformers is still in its infancy. Meanwhile, the “hidden danger” of

∗Corresponding author Email address: [email protected] (Ke Xu)

transformers and the slight fault of transformer winding have not been investigated [7, 8, 9]. Existing literature shows that transformer winding is deformed when the transformer suffers from external faults during operation, thereby developing into internal inter-turn fault, which causes transformer fault [10].

Although the “healthy state” of internal winding in the normal operation of a transformer is poor and still not evident, it has already entered an adverse operation state. This feature describes a hidden fault, which has the property of trigger and accumulation [11, 12]. At present, the detection of hidden fault mainly depends on the analysis of oil chromatography, which is difﬁcult to combine with mature protection theory and the engineering application of large transformers. The detection and quantiﬁcation of this hidden fault in large transformers through electrical quantity have been reported rarely. In the actual operation of a power system, different short-circuit faults, which cause large electromagnetic shock in the system due to the short-circuit current, are inevitable. This shock deforms the winding of the transformer, which can inﬂuence the normal operation of the transformer and even

Journal of Power Technologies 99 (3) (2019) 231–238

lead to the failure of its operation. Thus, this study used the ﬁnite element method to propose

the leakage inductance characteristics of the power transformer winding fault based on ANS OFT . The leakage inductance characteristics of 800 kVA and 10 kW three-phase oil-immersed distribution transformers, which can be more accurately predicted under the conditions of three-phase grounding short circuit, turn-to-turn fault, and inter-turn fault, were investigated by establishing a two-dimensional (2D) simulation model of transformer winding deformation degree and its characteristics. Furthermore, the results can provide reference for the detection and evaluation of transformer operation fault.

2. State of the art

The main inﬂuence of leakage magnetic ﬁeld has two aspects. One is to accelerate the thermal aging of the internal metal structure of the transformer and paper insulation. The other is to produce a large electromagnetic force for the interaction between the current that ﬂows through the winding and the leakage magnetic ﬂux. Therefore, experts all over the world have not only studied the numerical value of the transformer leakage magnetic ﬁeld in recent years, but also achieved corresponding results in the calculation of the short-circuit electromagnetic force of winding.

The numerical solution of the electromagnetic ﬁeld problem has been developed rapidly with the development of computer and programming technology. The solution can be extensively applied in solving the electromagnetic ﬁeld problem. Currently, the most commonly used calculation methods include the least square, ﬁnite element, integral equation, and ﬁnite difference methods. Tsukerman I.A., Konrad A., and Lavers J.D. proposed the use of least square means to calculate the leakage inductance of transformers and applied it to transformer protection [13]. Silvester and Chari used the methods of 2D high-order ﬁnite element and 2D ﬁnite element with vortex term to calculate the constant magnetic ﬁeld of a transformer. O.W. Anderson used the axisymmetric ﬁnite element method to manage the magnetic ﬁeld leakage of transformers [14, 15]. In Britain, Coulson used the ﬁnite element method to calculate and analyze the electromagnetic ﬁeld distribution in a transformer based on the three-dimensional (3D) simulation model of the transformer [16]. In the United States, Kumbhar G. B. effectively simulated the electromagnetic operation process of a transformer by simplifying the geometric model of a transformer and applying the magnetization curve of the core in the model [17]. Dr. Wang Shenghui of Shenyang University of Technology used the A − V − A method to calculate and analyze the 3D transient ﬁeld of a transformer [18]. Liu Xiaoli and Bi Yanjun of Harbin University of Technology analyzed the transient leakage magnetic ﬁeld of a cable transformer using the ﬁeld circuit coupling and mirror image methods. Chongyou Jing, an engineer of Tianwei Transformer Factory in Bao Ding, obtained the converter transformer model under the non-sinusoidal condition and the distribution of the

transient leakage magnetic ﬁeld around the winding using a ﬁnite element simulation tool [19]. Xiaopo Ou of Guangdong Power Grid Company used the ﬁnite element method to calculate the transformer short circuit impedance to obtain the distribution of the leakage magnetic ﬁeld and the leakage inductance parameters [20]. Jianmin Wang of Baoding Tianwei Group not only investigated the transient electromagnetic ﬁeld characteristics of the UHV DC converter transformer, but also calculated the transient electromagnetic ﬁeld and harmonic loss of the converter transformer [21]. The magnetic leakage ﬁeld of a high-voltage converter transformer was calculated and analyzed in Reference [22], and the short-circuit impedance and the leakage magnetic ﬁeld distribution were obtained.

The short-circuit electromagnetic force of the transformer winding in the short-circuit state can cause serious mechanical damage to the transformer, bend or damage the winding, and even cause the transformer to explode. In addition, the monitoring of transformer short-circuit condition requires the use of a special test instrument. The requirements for test tools increase with the transformer capacity. Therefore, establishing the transformer numerical simulation model for predicting the short-circuit electromagnetic force of winding is crucial. S.V. Kulkarni took the lead in solving the electromagnetic ﬁeld of a transformer with tap winding using the ﬁeld circuit coupling method [23]. Jawad Faiz and Tahere Noori of Iran used the 2D and 3D ﬁnite element methods to deal with the short-circuit electromagnetic force of the core transformer [24]. Given that the short circuit electromagnetic force is mainly based on the overall fault of the transformer’s primary and secondary windings, the electromagnetic force at a ﬁxed winding position cannot be analyzed. Then, Hyun-Mo Ahn of South Korea divided the high- and low-voltage windings of a 50 kVA dry transformer into several parts and calculated the vector magnetic bits, magnetic ﬂux density, and electromagnetic force of each partial winding based on the ﬁnite element method [25]. In the literature [26], the simulation model of the three-phase core transformer was established by using the electromagnetic ﬁeld simulation software ANS YS , and the electromagnetic force on the winding was analyzed on the basis of the ﬁnite element method [27]. Jingqiu Qiao and Yunmiu Tang of China used a three-phase transformer with a rated working condition of 17 MVA as the simulation model. The ﬁnite element method was used to calculate the frequency spectrum distribution of the axial and radial electromagnetic forces when the winding was short-circuited in different initial states. Their results indicate that the calculation formula of the traditional electric power can basically meet the engineering requirement, but the numerical calculation can calculate the axial electric power [28]. Shuang Liu of Shenyang University of Technology used the ﬁnite element method to calculate the short-circuit electromotive force, which considers the properties of insulating materials, winding displacement, mechanicalproperties and elasticity. Meanwhile, a software for calculating the 2D transient symmetry ﬁeld and the winding shortcircuit strength was developed, and the calculation speed

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was improved on the basis of Windows platform [29]. Luliang Wang of Harbin University of Technology established a 2D axisymmetric ﬁeld-circuit coupling model of a transformer by using the ANS YS software and obtained the radial and axial force distributions of transformer windings under the three-phase symmetric short circuit condition. Although the ampere-turn imbalance and the leakage magnetic ﬁeld distribution between winding coils were considered, the interaction of each phase winding was ignored, and the result contained a certain error [30, 31].

This study used the ﬁnite element method to establish the simulation calculation model of the transformer winding electromagnetic ﬁeld, the leakage inductance characteristics of power transformers under a three-phase grounding short circuit, a turn-to-turn fault and an inter-turn fault were simulated and examined. The accuracy of the model was veriﬁed by ﬁnite element simulation, which provides reference for the detection and evaluation of transformer operation fault. The remainder of this study is organized as follows: section 3 describes the principle of ﬁnite element simulation and the construction of the transformer mathematical model, the parameters of transformer leakage inductance, and the ﬁnite element simulation model; section 4 discusses the use of the ﬁnite element method to calculate the transformer leakage inductance parameters and the analysis of the characteristics of a three-phase short circuit, a turn-to-turn fault and an inter-turn, and ﬁnally, the leakage inductance characteristics under different working conditions are obtained; section 5 summarizes the conclusions.

3. Methodology

3.1. Principle of ﬁnite element simulation

The core idea of ﬁnite element is summarized as variation and discretization. Variation establishes the “weak form” of the ﬁnite element. For example, the second-order partial differential equation under given boundary conditions is transformed into solving the extreme value of the ﬁrst-order partial differential equation, thereby approximately simplifying the complex problem [32, 33, 34, 35]. Discretization divides the entire continuous solution region into many small subregions, thereby obtaining the results in each small subregion, and then adds the results of each small subregion to obtain the solution of the entire region. The ﬁnite element method is brieﬂy summarized as follows: (1) the problem of solving the partial differential equation is transformed into solving a conditional variation problem [36]; (2) the entire solution domain is divided by partition unit as a triangle or quadrangle while constructing the linear interpolation function; (3) by constructing linear algebra equations, the extreme value of the universal energy function is transformed into the extreme value of energy function; (4) the linear algebra equation is solved.

Maxwell of ANS OFT Company is a powerful 2D/3D electromagnetic ﬁeld analysis software based on the ﬁnite element method. It includes a user-friendly interface, an ad-

vanced adaptive grid segmentation algorithm, and a powerful data processing ability and is preferred by research developers over other ﬁnite element software. The modeling and simulation steps based on the ANS OFT Maxwell software are shown in Fig. 1.

Figure 1: Calculation process of ANSOFT

3.2. Equivalent model of transformer Transformer is an important electrical equipment in a

power grid. To study transformers, the physical model of transformers must be simpliﬁed into a mathematical module, and the complex problem of magnetic coupling must be transformed into a simple circuit problem. Therefore, an equivalent circuit transformer model exists. The common equivalent model of a double-winding transformer is the T type equivalent model as shown in Fig. 2.

Figure 2: T-type equivalent model of double-winding transformer

In the T -type equivalent model as shown in Fig. 2 , the parameters of each side of the transformer are calculated by the same side voltage level. r1 and L1 respectively represent the resistance of the original winding and the leakage inductance of the winding as shown in Fig. 2; r2 andL2 represent the resistance of the secondary winding and the leakage inductance of the winding calculated at the primary side;

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u1 and i1 respectively represent the voltage and current values of the original winding side winding while the transformer is in operation; u2 and i2 respectively represent the voltage and current values of the secondary side winding of the original side while the transformer is running. In the equivalent transformer circuit diagram, in addition to these parameters, two important parameters characterize the excitation branch, namely, instantaneous excitation inductance Lm and short circuit turn leakage inductance Ls. However, these two parameters are unavailable in some situations. In the analysis of a speciﬁc operation state, the resistance of the excitation winding can be ignored, and the excitation branch only has the instantaneous excitation inductance Lm. In the analysis of the internal inter-turn fault of the transformer, in addition to calculating instantaneous excitation inductance Lm, the value must be calculated after it is parallel with short circuit turn leakage inductance Ls.

According to the circuit principle, the formula is obtained according to the T-type equivalent model of the transformer as shown in Fig. 2:

u = r i + L di1 + LmLs · d(i1 + i2)

(1)

1 1 1 1 dt Lm + Ls

dt

Formula 2 can be obtained by dividing Formula 1.

u = r i + i + (L + LmLs ) · d(i1 + i2) − r i − L di2 (2)

1 11 2

1 Lm + Ls

dt

1 2 1 dt

The resistance and leakage inductance on the original side of the transformer are small and can be ignored. Thus, Formula 3 can be expressed as follows:

u = r i + i + (L + LmLs ) · d(i1 + i2)

(3)

1 11 2

1 Lm + Ls

dt

When the transformer is in normal operation, the short circuit turn leakage inductance can be ignored. Thus, Formula 4 is expressed as follows:

u1 = r1 i1 + i2 + (L1 + Lm) · d(i1d+t i2) (4)

3.3. Calculation of transformer leakage inductance parameters based on ﬁnite element analysis

If the transformer is in normal operation, according to Formula 4, Lk = L1+Lm can be the equivalent instantaneous inductance of the transformer. If the transformer fails internally, the equivalent instantaneous inductance of the transformer can be deﬁned as Lk = L1+ LLmm+LLss according to Formula 3. The equivalent instantaneous inductance consists of two parts. The ﬁrst part is the leakage inductance of the original side winding L1. The second part is the parallel value of short circuit turn leakage inductance Ls and instantaneous excitation inductance Lm.

When i1=i1 + i2, Formula 3and Formula 4 can be uniﬁed as follows:

u1 = r1 i + Lk · d i (5) dt

The sampling period is T . By discretizing the differential Equation 5, Formula 6 is obtained:

u1(k)

=

r1

i(k) + Lk

i(k+1)− i(k−1)

u1(k + 1) = r1 i(k + 1) + 2LTk i(k+22T)− i(k) (6)

The calculation formula of equivalent instantaneous inductance Lk of the transformer can be obtained by simultaneously eliminating the original side winding resistance r1 from Equation 6:

Lk = 2T

i2(k) −

u1(k) · i(k − 1) ·

i(k + 1) − u1(k + 1) · i(k) i(k + 1) − i(k) · i(k + 2) +

i2(k + 1) (7)

3.4. Establishment of ﬁnite element simulation model for transformer leakage inductance magnetic ﬁeld

The 800 kVA and 10 kV three-phase three-column oilimmersed distribution transformer are selected as the simulation models. The main parameters of the transformer are shown in Table 1.

In the simulation model, the effects of the transformer structure, the coil skin, the iron core clamp, the fuel tank, and the support beam on the leakage magnetic ﬁeld can be simpliﬁed and even ignored. The current is set to uniform distribution mode in the winding zone, and the traverse vortex is ignored. The high- and the low-voltage winding have the same height, the demagnetization effect of the coil conductor is not considered, and each switch is evenly distributed.

Figure 3: 2D model diagram of the transformer

The cross-section of the three-phase distribution oil immersed transformer is selected to establish the model as shown in Fig. 3. The dark blue part is the upper and lower yokes of the transformer, the light blue surface is the core column, and the three-phase winding of transformer ABS is marked red, green, and yellow from right to left. The characteristic parameters of the iron core, the winding, and the

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Name

Table 1: Parameters of the real-scale transformer

Dimension, mm

Name

Core diameter Width of iron window Fuel tank width High-voltage coil Low-voltage coil Number of high-voltage winding Parameter name Rated capacity Rated current Loss of short circuit Connection group number

210 160 540 Ø276-360 Ø220-271 955 Parameter value 800 kVA 46.19/1154.7(A) 8200 W Dyn11

Height of iron window Height of upper and lower iron yoke

Height of fuel tank Height of high-voltage winding Height of low-voltage winding Number of low-voltage winding

Parameter name Rated voltage

No-load current Impedance voltage

Core material

Dimension, mm

505 825 1120 468 465

21 Parameter value 10±2×2.5%/0.4(kV)

0.18% 4.5%

B23P090

insulating oil are shown in Table 2. Fig. 4 shows the magnetization curve of the core in semi-logarithmic coordinates.

three voltage control switches are closed after 30 ms, and the three-phase short circuit of the transformer is obtained.

Figure 4: Magnetizing curve of the transformer iron core

The excitation source adopts the external circuit, and the external circuit adopts the same DT n11 connection group as the transformer. The inﬁnity boundary is set as the balloon boundary condition, which ensures that the boundary can be inﬁnity without setting a large boundary while reducing the number of computational grids. Several units are set to the maximum number (3000) by using the triangulation unit and the automatic subdivision algorithm. The solution termination time is set to 100 ms, the period to 5, and the step size to 1 ms.

4. Result Analysis and Discussion

4.1. Three-phase grounding short circuit

When the transformer fails, the winding has a large transient current, which leads to the complete damage of the transformer and serious power grid outage. Therefore, the magnetic leakage ﬁeld in the case of transformer failure must be studied. The three-phase short circuit fault position is set to occur on the low voltage side of the transformer external circuit, as shown in Fig. 5. When the initial phase angle of a phase is set to 0° and the corresponding short circuit time is 30 ms, the three-phase short circuit is the worst, that is, the transformer is under the load condition from 0 ∼ 30 ms. The

Figure 5: Three-phase short circuit of the secondary side

The current of A-phase of the low-voltage side can be obtained by simulation (Fig. 6).

Figure 6: Low-voltage side current of A-phase when shorted

Fig. 6 shows that after the three-phase short circuit, the short circuit current on the low-voltage side increases sharply, accompanied by a certain transient process. When the short circuit time is 1/4 cycles, which is 40 ms, the maximum value of 16.7 kA is reached, the short circuit steady state process is started, and the current remains very large. A large current often produces a large electromagnetic force, which causes winding deformation and insulation damage.

The magnetic leakage ﬁeld distribution of the three-phase transformer at 40 ms is shown in Fig. 7 .

In Fig. 7, the maximum magnetic density of the transformer is 2.6 T, and the core is saturated. The leakage inductance of the transformer is 76.7 mH, which is approximately

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Table 2: Properties of the materials in the transformer

Transformer component

Material Conductivity Relative dielectric constant

Winding Iron core Transformer oil

Copper Iron

Insulation oil

58000000 4550000 0

1 Nonlinear

2.4

Figure 7: Magnetic density distribution during shorted fault

Figure 9: Phase voltage of primary side with turn-to-turn fault

30% higher than that of the normal operation. 4.2. Turn-to-turn fault

If the end of the winding has a turn-to-turn fault for the primary side winding of the transformer’s A-phase, then the black part of the winding in the 2D model represents grounding, the number of turns is 8% of the total turns, and inter-turn fault occurs at 30 ms. The A-phase high-voltage winding is divided into normal and fault winding, and a wire grounding is drawn from the middle (Fig. 8).

Figure 8: Turn-to-turn fault of A-phase primary side winding

The voltage and current of the primary side of the fault phase can be obtained by simulation (Fig. 9 and Fig. 10).

In Fig. 10, the currents of A-phase and B-phase enter the transient process, the primary side is grounding because the winding grounding, and the current increases sharply. The current of C-phase is consistent with that when the primary

Figure 10: Phase current of turn-to-turn fault

side uses the triangle connection method. The calculated leakage inductance of A-phase is 58.8 mH.

The leakage inductance results of the transformer with different grounding are shown in Fig. 11 by constantly changing the grounding position, that is, changing the number of turns of fault winding.

In Fig. 11, the leakage inductance has a nonlinear negative correlation with the number of fault turns. When the number of fault turns is small, the leakage inductance is close to that in normal operation.

4.3. Inter-turn fault Assuming that the primary side winding of the A-phaseof

the three-phase transformer has an inter-turn fault, the Aphase in the model is divided into 10, the number of turns in each group is 90 turns, and the black part represents the occurrence of inter-turn fault (Fig. 12).

The voltage and current of the fault phase of the primary side are obtained by simulation as shown in Fig. 13 and Fig. 14.

Similarly, by increasing the number of fault turns, the black (fault) area of the winding in the 2D model of the corresponding turn is increased, and the relationship between leakage inductance and the number of fault turns is shown in Fig. 15.

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Figure 11: Relationship between leakage inductance and the number of fault turns

Figure 13: Phase primary side voltage with inter-turn fault

Figure 14: Phase primary side current with inter-turn fault

Figure 12: Inter-turn fault of phase A primary side winding

Fig. 15 shows that a nonlinear positive correlation exists between leakage inductance and the number of turns of the inter-turn fault. The leakage inductance tends to increase with the number of turns of the inter-turn fault.

5. Conclusion

To reduce the winding insulation damage caused by transformer magnetic ﬂux leakage, which can pose a great threat to the safe and stable operation of power grids, 800 kVA and 10 kV three-phase oil-immersed distribution transformers were selected as physical models, and the leakage inductance characteristics of the power transformer under the conditions of three-phase grounding short circuit, turn-to-turn fault, and inter-turn fault were studied using an ANS OFT software. Finally, the following conclusions could be obtained:

(1) When three-phase short circuit occurs on the secondary side of the transformer, the secondary side current increases sharply, the iron core reaches saturation, the leakage magnetic ﬁeld increases correspondingly, and the leakage inductance of the transformer increases by 30% compared with that in normal operation.

(2) When the turn-to-turn fault of transformer occurs, the leakage inductance is negatively correlated with the number of fault turns, and the leakage inductance tends to decrease with the increase in the number of grounding fault turns.

(3) When the inter-turn fault of transformer occurs, the leakage inductance is positively correlated with the number of turns in inter-turn fault. The leakage inductance tends to increase with the number of turn-to-turn fault turns.

Starting from the causes of transformer winding insulation damage, this study conducted a research of the leakage inductance characteristics under power transformer winding fault based on ANS OFT . The accuracy of the model was veriﬁed by ﬁnite element simulation, which provides a refer-

Figure 15: Relationship between leakage perception and the number of fault turns with inter-turn fault

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ence for detection and evaluation of transformer operation fault. Owing to the lack of actual ﬁeld fault monitoring date, an on-line monitoring system for transformer winding deformation must be designed in future research. By doing this, the running state data can be collected in real time according to the transformer operation process to judge the real-time shape of transformer winding according to the parameters and determine the reasonable maintenance time of transformers after comprehensive analysis. This system can not only ensure the maximum operation life of transformers, but also prevent the interruption of operation because of failure.

Acknowledgments

This research was supported by the Key Scientiﬁc Research Projects of Henan University in China, and its foundation is 20B470003.

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