High frequency electric circuit modeling for transformer frequency response analysis studies

Electrical Power and Energy Systems 111 (2019) 351–368

Contents lists available at ScienceDirect

Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

High frequency electric circuit modeling for transformer frequency response analysis studies

T

Xiaozhen Zhaoa, , Chenguo Yaoa, , Ahmed Abu-Siadab, Ruijin Liaoa ⁎

⁎

a

State Key Laboratory Power Transmission Equipment & System Security and New Technology, School of Electrical Engineering, Chongqing University, Chongqing 400044, China b Electrical and Computer Engineering, Curtin University, Perth, WA 6845, Australia

ARTICLE INFO

ABSTRACT

Keywords: Power transformers Frequency response analysis High frequency electric circuit model Fault diagnosis

Power transformers are subject to winding and core deformations due to short circuit faults and other environmental conditions. Majority of these faults are of progressive nature and should be rectified as soon as they emerge. Frequency response analysis (FRA) has been widely accepted as a reliable diagnostic tool to detect such faults. As reliable interpretation codes for FRA signatures have not been fully developed and accepted yet, researchers have put much effort to investigate the impact of various mechanical deformations on the transformer FRA signature. Because of the intrusive nature of such faults when staged on a real transformer, most of the studies in the literatures were conducted on a transformer high frequency electric circuit model. Simplifications assumed in these models by ignoring the turn-to-turn capacitance, not taking into account the detailed winding structure and not considering all mutual inductances between coils in various phases have reduced the accuracy of the obtained results. To establish reliable FRA interpretation codes, it is essential to develop electric equivalent circuit models that can yield FRA signature as close as possible to the FRA signature trend of the real transformer. This paper proposes a detailed transformer high frequency electric circuit model which considers the winding structure, inter-turn capacitance and all mutual inductances. Calculation details of the model’s parameters are presented. The accuracy of the proposed model is assessed by comparing its FRA signature with that of an equivalent transformer hardware model during healthy, short circuit disks, radial deformation and axial disks buckling fault.

1. Introduction Power transformer is one of the key assets that confirm the reliability of electrical transmission and distribution networks. According to an international survey conducted by CIGRE, mechanical winding deformations within power transformers represented 19.4% of the global transformer failures during the period 1996–2010 [1]. Short-circuit faults [2], earthquakes, improper transportation and tap-changer failure [3] are the main causes of transformer core deformation and winding movement. Once emerged, minor mechanical deformations progress rapidly and may lead to a transformer collapsing if not attended at early stages [4]. Several techniques have been developed to detect such faults. This includes vibration analysis [5,6], short-circuit impedance [7,8] and frequency response analysis (FRA) [4,9]. The latest has been widely accepted as the most reliable technique to detect power transformer core and winding deformations [10–16]. However, current FRA

⁎

practice is lack of reliable interpretation codes, which is the main focus of recent FRA studies [17,18]. Majority of these studies are based on simulation analysis using 3D finite element modelling (FEM) [19], mathematical models [20] and equivalent electric circuit models [21,22]. Equivalent electric circuits included lumped parameter circuit model [23–30], ladder network equivalent circuit model [31,32], multi-conductor transmission line model [33–37] and black box model [38–40]. Among these models, lumped parameters-based circuit model has been widely adopted in the literatures for FRA studies to investigate the impacts of various winding deformations on the transformer FRA signature [29]. Hence it is extremely crucial that the equivalent electric circuit model be accurately established to reflect the transformer internal physical structure. The accuracy of the electric circuit model can be assessed by comparing the transformer practical FRA signature with the one obtained through simulation analysis of the model. Impact of various faults on the transformer equivalent electrical parameters and FRA signature is reported

Corresponding authors. E-mail addresses: [email protected], [email protected] (X. Zhao), [email protected] (C. Yao).

https://doi.org/10.1016/j.ijepes.2019.04.010 Received 8 November 2018; Received in revised form 5 March 2019; Accepted 7 April 2019 0142-0615/ © 2019 Published by Elsevier Ltd.

Electrical Power and Energy Systems 111 (2019) 351–368

X. Zhao, et al.

CHG1 2

in [4,13,14]. Correlation between transformer winding faults and the change in the equivalent electrical parameters using simulation analysis which is validated through practical measurements is investigated in [41]. In most of the equivalent electric circuit models proposed in the literatures, inter-turn capacitance was neglected [22,23,25–28,42]. Moreover, mutual inductance was either neglected or limited to one value between two adjacent coils. Also, none of these studies has considered the detailed winding structure (WS) in the proposed equivalent electric circuit model. This resulted in inaccurate trend of the obtained FRA signature that can emulate the practical signature of a real transformer. This paper proposes an improved transformer high frequency electric circuit model by considering the detailed winding structure, interturn capacitance and mutual inductance between various turns. The proposed improved model is aimed at investigating the impact of various power transformer internal mechanical deformations on the FRA signature. Calculation of these parameters is presented and simulation results are validated through experimental measurements.

CHL1 2 RHs1

GHG1 2

GH 1 C H1

CHG1 2

MH M HL 21 L12

CHG 2 2 GHG 2 2

RHs 2

GH 2 C H2 M MHL H 1n 12

GHGn 2

The investigated transformer in this paper is Υ/Υ three phase, 50 Hz, 400 kVA, 10/0.4 kV as shown in Fig. 1, with design specifications listed in Table A1 in the Appendix. The equivalent high frequency lumped parameter circuit model shown in Fig. 2 is simulated using Personal Simulation Program with an Integrated Circuit Emphasis (PSpice). The model comprises series inductance (LHs, LLs), and series resistance (RHs, RLs) that represent the high voltage (HV) and low voltage (LV) windings. Windings insulation is modelled by series/self-capacitance (CH, CL) and conductance (GH, GL) elements. The mutual inductance (M) between all coils is taken into account as shown in Fig. 2. The insulation between the HV and LV windings is modelled by a parallel branch comprising capacitance (CHL) and dielectric conductance (GHL). Similarly the dielectric insulation between the HV/LV windings and the tank/core is represented by a capacitance (CHG, CLG), shunted by dielectric conductance (GHG, GLG), respectively. Unlike other models proposed in the literatures [3,43], all the mutual inductances between HV and LV windings are taken into consideration in the model shown in Fig. 2 to increase the accuracy of the obtained results. Also, the inter-turn capacitance along with the nature of winding structure are taken into consideration while calculating the parameters of the proposed model. To obtain the HV winding FRA signature, an excitation voltage (Uin) is applied to the terminal of the winding through input resistance Rin while the response voltage (Uout) is measured at the other terminal of the winding across an output resistance Rout in a wide frequency range. The FRA signature (H(f)) in decibel (dB) can be numerically calculated from [44]:

CHGn 2

L12

CHL 2 2

GHL 2 2 CHL 2 2 M HLL12 2n

LHsn

GHLn 2 M MHLnnn HLnn

CHLn 2

GHLn 2

GHGn 2

GL1

GLG1 2

CLG1 2 GLG1 2

M MHLL12 12

CLG 2 2

RLs 2

M MHLn11 HL 22

MMH HLn1 MH C L12 2 M HLn HLn 2 L12

RHsn

CL1

MH M L12 HL12 MMH HL1n

LLs 2

GLG 2 2

CL 2

M HL M L1n 12

MH

L12

GHn C Hn

LLs1

GHL 2 2

MMH H 2n

CHGn 2

2. Investigated transformer and equivalent electric circuit model

LHs 2

R Ls 1

GHL1 2

M MHLH 12 MH M 12 L12HLn1

CHG 2 2

2

CHL1 2

LHs1

GHG1 2

GHG 2 2

G

HL1 M MHLn11 HL11

CLG1 2

GL 2

CLG 2 2 GLG 2 2

M

MHL L2n

CLGn 2

12

RLsn LLs 2

GLGn 2

CLn

GLn

CLGn 2 GLGn 2

Fig. 2. Improved electric circuit model for single-phase double-winding transformer.

H (f ) = 20 log

Uout (f ) Uin (f )

(1)

According to the simulation results in [4], the impact of bushing model on the FRA signature can be neglected. So the bushing model is not considered in the model shown in Fig. 2. 3. Finite element analysis To guarantee adequate mechanical and electrical capacity, a power transformer is assembled using complex structures of various comzponents including iron core, windings, clamping structures, tank and insulation system. Hence the lumped parameters in the high frequency equivalent circuit shown in Fig. 2 cannot be calculated accurately using exact mathematical formulas. The three dimension (3D) finite element analysis (FEA) model shown in Fig. 3 is built with the design specifications of the investigated transformer listed in Table A1. In this model, the LV winding is of 7-

ε we

ε de

Fig. 1. The investigated three phase transformer (a) Schematic diagram, (b) Front view.

Fig. 3. Finite element model of the three phase transformer hardware shown in Fig. 1(a) Diametric orientation view, (b) Configuration of one phase. 352

Electrical Power and Energy Systems 111 (2019) 351–368

X. Zhao, et al.

Fig. 4. Partial structure diagram of the transformer active part (a) Diagrammatic cross-section, (b) Disk top view and spacer.

Fig. 5. Transformer winding disks and its capacitance network (a) Enwinding diagram of the continuous disk winding, (b) Enwinding diagram of the interleaved disk winding, (c) Cross-section view and equivalent capacitance network.

turn helical type, and the HV winding consists of 10 parallel conductors with composite structure including 10 disks interleaved winding on the top, 10 continuous winding disks in the middle and 10 disks interleaved winding in the bottom. In the investigated transformer model, 30 HV winding cells and 7 LV windings cells have been taken into consideration to establish the electric circuit model. Design specifications of the investigated transformer as shown in Fig. 4 and insulation properties listed in Table A1 in the Appendix are used to calculate the electrical parameters of the equivalent electric

circuit model as elaborated below. 4. Calculation of the transformer equivalent circuit parameters Most of the electric circuit models proposed in the literature employed FEM to calculate the electric parameters used in the model without paying much attention to the nature of the winding structure; continuous winding layers-type was assumed in most of these models [13,19,21,36]. Moreover, while FEM can calculate the disk-to-disk 353

Electrical Power and Energy Systems 111 (2019) 351–368

X. Zhao, et al.

x 10

9

8

7

6

5

4

3

2

1

11

12

13

14

15

16

17

18

19

20

U disk U disk 2N

U disk

U disk 2

x B U

x

x

2l

Fig. 6. Cross-section view of the voltage distribution along a pair of disks in the continuous disk winding.

x 15

5

14

4

13

3

12

2

11

1

6

16

7

17

8

18

9

19

10

20

U disk U disk 2N

U disk

U disk 2

x B

U

x

2l

x

Fig. 7. Cross-section view of the voltage distribution along a pair of disks in the interleaved disk winding.

354

Electrical Power and Energy Systems 111 (2019) 351–368

X. Zhao, et al.

capacitance, winding resistance and inductance, it cannot extract the turn-to-turn series/self-capacitance which relies on the winding structure. This resulted in reduced accuracy of the electric circuit models proposed for power transformer FRA studies. This paper is aimed at proposing more detailed model through considering transformer WS and the inter-turn capacitance. Calculation of the electrical circuit parameters including insulation dielectric constants between adjacent disks, between HV and LV windings of the same phase, and between the HV winding and the tank is presented below. It is worth noting that the core magnetization characteristic affects the FRA signature within the low frequency range up to 10 kHz [45]. This is attributed to the fact that magnetic flux penetration to the core is significant within this range. In the mid-frequency and high frequency range, the FRA signature is dominated by the bulk windings and multiple resonant points are generated.

Table 1 Main parameters of the transformer equivalent circuit. RHs1

LHs1

CH1

GH1

CHG1

GHG1

CHL1

GHL1

4.5 Ω RHs13 4.5 Ω RLs1 0.5 Ω

97.93 μH LHs13 92.43 μH LLs1 2.13 μH

3.30 nF CH13 0.22 nF CL1 1.73 nF

0.14 μS GH13 0.14 μS GL1 0.14 μS

4.33 pF CHG13 1.75 pF CLG1 147.50 pF

0.29 μS GHG13 0.29 μS GLG1 0.29 μS

30.22 pF

0.2 μS

neutral bushing HV busking

4.1. Dielectric constant between adjacent disks within transformer winding

Tanformer tank

The insulation between adjacent disks within the HV/LV winding consists of oil-paper insulation and horizontal oil-duct with spacers. The equivalent relative dielectric constant between adjacent disks is calculated based on the following steps: (a) The equivalent relative dielectric constant εoe of the parallel insulation structure of the spacer and the transformer oil is calculated from [46]:

High frequency coaxial-cable

PC

oe Soe

Fig. 8. Experimental hardware setup.

Magnitude / dB

oe

0

-60 -80

Magnitude / dB

0

200

400

600

Frequency / kHz (a)

800

1000

de

-20 -40 -80 0

200

400

600

Frequency / kHz (b)

800

Model without WS

Model without Ctt

0.3888 355.9589

−0.1740 731.6609

−0.0365 782.6594

da B

bBn) + da B

c bBn

(3)

=

ao + ap ao

+

ap

(4)

p

The insulation between the HV and LV windings contains insulating cylinder, vertical oil-duct and winding supporting plate along with the insulation (oil-paper) between turns. According to the calculating method of the series capacitors, the equivalent relative dielectric constant (εwe) between the HV and LV windings within the same phase can be obtained from [46]:

Table 2 Correlation between simulation results and experimental measurement using numerical indices. Detailed model

o(

4.2. Dielectric constant between HV and LV windings within the same phase

1000

Fig. 9. FRA signatures obtained through (a) practical measurement and (b) simulation analysis.

CC ED

(2)

c Sc

where ad is the distance between the adjacent bare conductors within the transformer winding in the vertical direction, ao is the height of the equivalent horizontal oil-duct, ap is the thickness of the oil-paper insulation and εp is the relative dielectric constant of the oil-paper.

-60 -100

=

oe

Detailed model Model without WS Model without Ctt

0

+

where B is the radial width of the transformer winding disk as shown in Fig. 4(b); da is the average diameter of the transformer winding disk (Rinner + Router); b is the width of the spacers in the peripheral direction and n is the number of spacers within one winding disk. (b) The equivalent relative dielectric constant (εde) between adjacent disks within transformer winding is calculated from [46]:

Measurement

-40

-100

o So

where εo, and εc represent the relative dielectric constants of the horizontal oil-duct and the spacer respectively, and Soe, So, Sc represent the area of the parallel insulation structure of the spacer and the transformer oil and the area of the horizontal oil-duct and the spacer; respectively. Hence εoe can be obtained as below:

TDT6U FRA analyzer

-20

=

we

= dw

(

aw ap p dp

+

a0 0 d0

+

apc pc dpc

)

+…

(5)

where aw is the overall insulation thickness between the HV and LV windings, apc is the thickness of the insulating cylinder, do is the average diameter of the vertical oil-duct, dp is the average diameter of the 355

Electrical Power and Energy Systems 111 (2019) 351–368

X. Zhao, et al.

Label 1 Label 2 Label 4 Label 6

Label 3 Label 5

Copper conductor

Fig. 10. Short circuit disks fault configuration. 0

-10

Healthy Top-SC Middle-SC Bottom-SC

-10

-30

Magnitude / dB

Magnitude / dB

-20 -30 -40 -50 -60

Healthy Top-SC Middle-SC Bottom-SC

-20

-20 -30

-40 -20

-50

-30 -40

-60

-50

-70

-40 -50

-60 -70

-80

-80

-60

-70 0 10

10

10

1

Frequency / kHz

10

2

10

10

-90 0 10

3

3

2

10

10

1

Frequency / kHz

3

10

2

10

3

Fig. 11. Effect of the location of short-circuit disks on the FRA signature obtained through experimental measurements.

Fig. 12. Effect of the location of short-circuit disks on the FRA signature obtained through simulation analysis on the proposed detailed model.

insulation between turns, dpc is the average diameter of the insulating cylinder, dw is the average diameter of the insulating system between HV and LV windings and εpc is the relative dielectric constant of the insulating cylinder.

computation allows the determination of the charge q carried by each conductor. The energy stored in the electric field associated with the capacitance between two conductors is given by [47,48]:

Wij =

4.3. Dielectric constant between HV winding and the tank

1 2

Di × Ej d

(6)

where Wij is the electric field energy associated with the electric field lines connecting charges on conductor i to those on conductor j, Di is the electric field density associated with conductor i and Ej is the electric field associated with conductor j. Therefore, the disk-to-disk capacitance Cdd-ij between conductors i and j is [14]:

The space between HV winding and the tank is mainly filled with insulating transformer oil. So the equivalent dielectric constant between HV winding and the tank equals that of the insulating oil. 4.4. Calculation of the shunt capacitance The values of the shunt capacitances (CHGi, CLGi, CHLi) in the proposed electric circuit model are calculated using the materials properties and physical configuration of the related turns. To calculate the capacitance matrix of the transformer model, Maxwell 3D performs a sequence of electrostatic field simulations (for n-conductor system, n field simulations are performed). Each conductor is defined as a voltage source set to 1 V, while all other conductors are set to 0 V. The

Cdd

ij

=

2Wij V2

=

1 2

Di × Ej d

(7)

In the obtained capacitance matrix from finite element analysis, the offdiagonal elements represent the capacitances of the equivalent circuit model. It is worth noting that series capacitances CHi and CLi are calculated based on the disk-to-disk and the turn-to-turn capacitances [49]. This is 356

Electrical Power and Energy Systems 111 (2019) 351–368

X. Zhao, et al.

Table 3 Variations of resonant points in the experimental FRA signature due to SC disks location. Resonance frequencies

57 kHz

69 kHz

100 kHz

189 kHz

217 kHz

253 kHz

702 kHz

Top

f /kHz A /dB

+1 +1.9597

0 +1.5170

+3 +2.5201

+5 −1.0931

−3 1.7829

+8 +3.4376

+108 +0.2091

Middle

f /kHz A /dB

+1 +2.1779

0 +1.6415

+2 +2.5186

+8 −3.707

−1 +0.8393

+39 +5.9142

/ /

Bottom

f /kHz A /kHz

+1 +2.5020

0 +1.7324

+2 +2.6279

+11 −6.1756

0 +0.1539

+94 +5.2889

+255 +5.8815

Table 4 Variations of resonant points in the FRA signature obtained through simulation analysis due to SC disks location. Resonance frequencies Top

f /kHz A /dB f /kHz A /dB f /kHz A /dB

Middle Bottom

88 kHz

120 kHz

191 kHz

242 kHz

259 kHz

6271 kHz

417 kHz

0 1.3366 +1 4.2699 +1 4.7433

+1 1.6790 0 1.1815 0 0.2945

+3 −0.7579 +2 −2.2940 +1 −2.0133

+8 −5.3472 +11 −9.3965 +12 −8.9103

+1 −1.1482 +1 0.0178 +2 3.6598

+27 7.6261 +46 7.8227 +31 9.7762

+26 −4.7199 +100 −2.8466 +33 −10.1078

100

100

0

measurement detailed model model without WS model without Ctt

-50 -100

ED%

CC%

50

Top

Middle

Bottom

50

0

measurement detailed model model without WS model without Ctt Top

Middle

Location

Bottom

Location

Fig. 13. The CC% and ED% of FRA signatures for different location of SC disks. -10

0

Healthy 7%-SC 14%-SC 21%-SC

-10

-30

Magnitude / dB

Magnitude / dB

-20 -30 -40 -50 -60

Healthy 7%-SC 14%-SC 21%-SC

-20

-20 -40

-70 -60

-40 -50 -60

-20 -30 -40

-70

-50

-80

-70

-60

-80 -80 -90 0 10

10

2.3

10

10

-90 0 10

2.9

1

Frequency / kHz

10

2

10

3

10

10

3

1

Frequency / kHz

10

2

10

3

Fig. 15. Effect of short-circuit disks fault level on the FRA signature obtained through simulation analysis on the detailed model proposed in this paper.

Fig. 14. Effect of SC disks fault level on the experimental FRA signature.

shown in details below.

The enwinding diagram of the continuous disk and interleaved windings are shown in Fig. 5(a) and (b). The continuous winding disk contains two disks with equivalent capacitive network as shown in Fig. 5(c). The shunt capacitance has two components: the total turn-toturn capacitances Ctt and the disk-to-disk capacitance Cdisk. Energy

4.5. Calculation of series capacitance (1) Continuous disk winding 357

Electrical Power and Energy Systems 111 (2019) 351–368

X. Zhao, et al.

100

100

0

measurement detailed model model without WS model without Ctt

-50

-100

ED%

CC%

50

7%

14%

50

0

21%

measurement detailed model model without WS model without Ctt 7%

14%

21%

Level

Level

Fig. 16. CC% and ED% of FRA signatures for different levels of SC disks faults.

calculating method of the plate capacitor as [51]:

Ctt =

t 0

2

da h t

=

t 0

(Rinner + Router ) h 2 t

(9)

where εt is the relative dielectric constant of the turn-to-turn insulation, h is the height of the conductors within the winding disks, δt is the thickness of the turn-to-turn insulation around the disk conductors. It is assumed that the voltage distribution agrees with that of Fig. 6. Starting from conductor number 1 or number 20 and moving toward the middle of the winding (conductor number 10 or 11), the voltage changes linearly and monotonically. The voltage difference between adjacent turns in one disk is Udisk/2N. Hence, the steady state voltage distribution for the conductors in the upper and lower disks are respectively given by [49]: Fig. 17. Radial deformation (left side) and Axial disk buckling (right side) faults configurations on the HV winding of the transformer hardware model.

Uupper (n) =

N 1 Ctt 2N 2

x ) Udisk xUdisk , Ulower (n) = 2B 2B

(10)

Hence the equivalent series capacitance (Cs) in the continuous disk winding is given by:

summation method is applied to calculate (CHi, CLi). According to this method, the sum of the energies stored in the capacitances associated with the two disks represent the total energy stored within these disks. Assuming the number of turns in each winding disk is N, and the voltage of one pair of winding disks (containing two winding disks) is Udisk, there will be N number of Ctt in each disk. The number of turn-to-turn capacitances will be 2(N − 1) for one pair of disks. The equivalent turn-to-turn capacitance (Cturn) in one pair of disks is given by [50]:

Cturn =

(2B

Cs =

N 1 4 Ctt + 3 Cdisk for detailed model N2 4 C for model without Ctt 3 disk N 1 4 Ctt + 3 Cdisk for model without WS N2

(11)

(2) Interleaved disk winding In an interleaved winding, the series capacitance is considerably greater than that of the continuous disk winding [49]. The enwinding diagram of the simplest configuration of the interleaved disk winding is shown in Fig. 5(b) while the voltage distribution along the interleaved

(8)

The turn-to-turn capacitance can be derived according to the

Fig. 18. The effect of radial deformation fault level on the capacitance parameters (a) Shunt capacitance, (b) Series capacitance.

358

Electrical Power and Energy Systems 111 (2019) 351–368

X. Zhao, et al.

Fig. 19. The effect of radial deformation fault level on inductance parameters (a) inductance between HV deformed (No. 11) and other HV disks (b) mutual inductance between HV deformed disk (No. 11) and LV winding turns. 0

The equivalent series capacitance in the interleaved continuous winding disks is given by:

healthy 5%-RD 7%-RD 10%-RD

-10 -20

N

Magnitude / dB

N 1 Ctt N2

-40 -50

-30

-60

-80

-40

-90

-45 600 100

200

300

700

400

500

800 600

Frequency / kHz

900 700

800

-10

1000 900

1000

healthy 5%-RD 7%-RD 10%-RD

-20 -30

The equivalent lumped parameters circuit model includes series/ self-inductances (LHsi, LLsi) describing the generated electromotive forces across the turns and mutual inductances (MHHij, MHLij, MLLij) characterizing the induced voltage across the j-th turn caused by the current flowing through the i-th turn. To calculate the inductance, the average magnetic energy WAV is calculated first as [48]:

Magnitude / dB

-50 -60

-42

WAV =

-44

-48 700 100

200

300

400

500

800 600

Frequency / kHz

900 700

800

900

1 4

V

B × HdV

(13)

where B is the magnetic field density, H is the magnetic field intensity and V is the volume of the conductor. Then inductance can be calculated form the average magnetic energy and peak winding current IPeak as below [48]:

-46

-90

(12)

4.7. Calculation of inductance

-40

-80

for model without WS

The series resistance (RHs, RLs) and the shunt conductance (GH, GL) represent the Joule losses, which is determined by the transformer physical structure and material properties [41] along with the turn-toturn dielectric losses. The HV winding turn-to-ground dielectric losses and the LV winding turn-to-core dielectric losses are denoted by GHG and GLG, respectively. The series resistance in the equivalent circuit model is calculated using eddy current solver in the ANSYS Maxwell finite element software. While the resistance value is dependent on the frequency, this variation can be neglected [52]. The conductance (GH, GL, GHG and GLG) elements are calculated using the DC conduction solver in the finite element software.

Fig. 20. Effect of radial deformation fault level on the experimental FRA signature.

-70

+

for model without Ctt 4 C 3 disk

4.6. Calculation of resistance and conductance

-35

-70

-100

4

· Ctt + 3 Cdisk for detailed model

4 C 3 disk

Cs =

-30

-100

1 4

1,000

2 L = 4WAV / IPeak

Fig. 21. Effect of radial deformation fault level on the FRA signature obtained through simulation analysis on the detailed model proposed in this paper.

(14)

The inductance matrix is calculated using the magnetostatic solver in ANSYS Maxwell software, in which the diagonal elements represent the self/series inductance of the i-th turn, while the off-diagonal elements represent the mutual inductances. Main parameters of the transformer equivalent circuit shown in Fig. 2 are listed in Table 1. Also, HV winding parameters and the detailed inductance coupling coefficient matrix are listed in Tables A3 and

disk winding is shown in Fig. 7. In this figure, the solid lines represent the voltages of the upper disks and the dashed lines represent the voltages of the lower disks. The voltage difference between adjacent turns in one disk is Udisk/ 2N. 359

Electrical Power and Energy Systems 111 (2019) 351–368

X. Zhao, et al.

Table 5 Variations of resonant points in the experimental FRA signature due to various RD fault levels. Resonance frequencies 5%

f /kHz A /dB f /kHz A /dB f /kHz A /kHz

7% 10%

69 kHz

193 kHz

220 kHz

259 kHz

299 kHz

313 kHz

568 kHz

704 kHz

839 kHz

0 +0.0850 0 +0.2145 0 +0.1365

0 +0.1225 0 +0.1309 −1 +0.2945

0 +0.2958 0 +0.2918 0 +0.4478

0 +0.3039 −1 +0.2097 2 −1.3323

+1 −0.3065 1 −0.4406 1 −1.2465

+1 +0.0290 1 −0.0318 1 −0.5082

0 −0.1297 −6 +0.2542 −9 −0.2790

+14 +1.4124 6 +1.3408 −2 −2.0473

+29 −0.2333 23 −1.0085 −10 −0.8113

Table 6 Variations of resonant points in the FRA signature obtained through simulation analysis due to various RD fault levels. Resonance frequencies 5% 7% 10%

f /kHz A /dB f /kHz A /dB f /kHz A /dB

88 kHz

191 kHz

271 kHz

419 kHz

494 kHz

650 kHz

744 kHz

876 kHz

924 kHz

0 +0.3698 0 +0.2027 0 −0.1456

0 −0.1224 0 +0.0237 −1 +0.0349

−1 −0.9618 −2 −1.4185 −2 −1.6628

−4 −3.7133 −4.3617 −1.4681 −5 −5.5564

−2 −0.6213 −3 −0.6497 −3 −0.6567

−4 +0.6389 −8 +1.3741 −9 +1.4924

−18 −0.0224 −20 −0.0791 −21 −0.0810

−18 +0.1419 −2 +0.4390 −2 +0.5004

+6 +1.0868 −5 +0.2479 −5 +0.3117

100

100

99

80

measurement detailed model model without WS model without Ctt

97 96 95

ED%

CC%

98

5%

7%

Level

measurement detailed model model without WS model without Ctt

60 40 5%

10%

7%

Level

10%

Fig. 22. The CC% and ED% of FRA signatures under different levels of RD fault.

Fig. 23. The effect of axial disk buckling fault on the capacitance parameters (a) Shunt capacitance, (b) Series capacitance.

A4 in the Appendix. As stated in Section 4, the HV windings of the investigated three phase transformer is of composite structure. Unlike other models published in the literature, the model presented in this paper considers the detailed winding structure while calculating transformer model parameters. Table A5 in the Appendix gives a comparison of the series capacitances of such windings using the three modeling methods stated in (11) and (12). Due to the limitation of paper length, only three series capacitances of phase-A HV winding on the top, middle, and bottom locations are listed in the table. It can be seen that the proposed detailed model in this paper has the same series capacitance value (CH15)

for continuous disk winding calculated considering inter-turn capacitance. On the other hand, the value of the series capacitances for other winding structure in the proposed detailed transformer winding model differs significantly from those calculated using the other two simplified models. It can also be observed that the proposed detailed model has greater series capacitance values than the models that are not considering Ctt, especially for interleaved winding structure. 5. Simulation and experiment verification To investigate the functionality and accuracy of the proposed 360

Electrical Power and Energy Systems 111 (2019) 351–368

X. Zhao, et al.

Fig. 24. The effect of axial disk buckling fault on the inductance parameters (a) inductance between HV deformed (No. 11) and other HV disks (b) mutual inductance between HV deformed disk (No. 11) and LV winding turns. 0

linear mode. The experiment setup is shown in Fig. 8. It is to be noted that, the frequency response analyzer injects an ac voltage of low amplitude (only 10 V) to the transformer windings. Hence, saturation will never be reached and a linear B-H relation can be considered while developing the equivalent electric circuit model in Fig. 2. The practical FRA signature is compared with the FRA signature obtained through simulation analysis on the hardware equivalent electric circuit model shown in Fig. 2. End-to-end open-circuit connection scheme, which is widely used to detect various deformations within power transformers is employed to obtain the FRA signatures shown in Fig. 9 through simulation analysis and experimental measurements [53]. Fig. 9 reveals that the simulation considering winding structure (WS) and the turn-to-turn capacitance (Ctt) can provide a signature of a good trend agreement with the experimental FRA signature when compared with the simulation results obtained from the electric circuit model without considering these details. It is worth mentioning that the magnitude and phase of the experimental FRA signature show more variations in the high frequency range (above 200 kHz) that is not reflected in the signature obtained through simulation analysis. The discrepancies between the practical and simulation FRA signatures are inevitable due to the discrepancy of the physical transformer size and the FEA model, inconsistent insulation properties of the FEA model and the actual transformer, test leads and bushing tails [54], deficiency of stray capacitances in the detailed model [55], lack of required design data and material types due to manufacturing confidentiality and impossibility to consider the aging and erosion of the transformer insulation system. Different numerical indices have been proposed and used for the FRA interpretation [56] of which the correlation coefficient (CC) [57] and the Euclidean distance (ED) [58] are widely employed to evaluate the correlation between various FRA signatures. CC value varies between −1 and 1 and has a reverse relationship with the amount of changes in the FRA signature. It can be calculated from [44,59]:

healthy ADB

-10 -20

Magnitude / dB

-30 -40 -50 -20 -60 -30

-70

-80 -40 -90 -100 0 10

10

2.4

10

10

2.9

1

Frequency / kHz

10

2

10

3

Fig. 25. Effect of axial disk buckling fault on the experimental FRA signature. -10

healthy ADB

-20

Magnitude / dB

-30 -40 -50 -60 -70 -80 -90 -100

-30 -40 -50 -60 -70

0

300

400

600

10

CC(x , y ) =

800 1000

Frequency / kHz

100

1000

N i=1 N i=1

(x i

x¯)(yi x¯)2

(x i

N i=1

y¯) (yi

y¯)2

(15)

where xi and yi are the i-th vector elements of the two FRA traces. ED is an indication of the distance between two FRA signatures. ED is calculated from [58,60]:

Fig. 26. Effect of axial disk buckling fault on the FRA signature obtained through simulation analysis on the detailed model proposed in this paper.

ED = X

modeling method, FRA experimental measurement has been performed on phase-A of the HV winding of the transformer hardware model shown in Fig. 1. All experiments were carried out using a commercial frequency response analyzer (TDT6U) with a frequency range 1 kHz to 1 MHz, and the number of the measurement points was set to 1000 in

Y =

(X

Y )T (X

Y)

(16)

where X = [x1, x2, …, xn] and Y = [y1, y2, …, yn] are the two FRA vectors with n elements, and T is the transpose of the vector. CC mainly shows the shape variation of the trace, while ED indicates the shift of the trace even if the shape does not change at all. Large CC 361

Electrical Power and Energy Systems 111 (2019) 351–368

X. Zhao, et al.

Table 7 Variations of resonant points in the experimental FRA signature due to ADB fault. Resonance frequencies

f /kHz A /dB

69 kHz)

193 kHz)

220 kHz

259 kHz

299 kHz

313 kHz

568 kHz

704 kHz

839 kHz

0 +0.0728

−1 +0.4826

0 +0.1559

−4 −0.7979

0 −1.6685

0 −1.1760

+1 −1.4017

−27 −0.8267

−67 +1.2445

Table 8 Variations of resonant points in the FRA signature obtained through simulation analysis due to ADB fault. Resonance frequencies

f /kHz A /dB

88 kHz

191 kHz

271 kHz

419 kHz

494 kHz

650 kHz

744 kHz

876 kHz

924 kHz

0 −4.6502

−2 −0.7017

−4 −6.7544

−66 −7.9014

−95 −1.1296

−127 +6.8257

−183 +0.1649

−252 −1.4791

−107 +3.9674

the bottom of the winding [16]. To assess the impact of such fault on the FRA signature numerically, relative CC and ED as in (15) and (16) are calculated and plotted in Fig. 13. The calculated CC% and ED% reveal the variation between the faulty and corresponding healthy signature. It can be seen from Fig. 13 that the proposed modeling method has more trend agreement with the experimental signature than the models without considering WS and Ctt.

Table 9 Correlation between ADB and healthy condition using numerical indices.

CC ED

Measurement

Detailed model

Model without WS

Model without Ctt

0.9051 94.2680

0.8936 185.5107

0.7238 254.4502

0.7307 194.9143

value and small ED value indicate a better agreement between two compared objects [44,58–60]. These indices are calculated to measure the correlation between the simulation results and the actual FRA signature as listed in Table 2. Table 2 reveals that the proposed detailed model considering WS and Ctt has the maximum CC and minimum ED values which indicates a better performance of the proposed model in emulating the trend of real transformer FRA signatures.

CC % = CC(x, y ) / Max {CC(x , y) }

(15)

ED% = EDj / Max {EDj }

(16)

To validate the functionality of the proposed modeling method in quantifying the levels of SC disks, the impacts of three SC disks levels (implemented on the hardware model at the top section of the winding) on the FRA signature are shown in Figs. 14 and 15 including the experimental measurements and the FRA signatures obtained through simulation analysis on the detailed model proposed in this paper. In the electric circuit model, this fault is simulated by short circuiting 7%, 14% and 21% of the HV winding series impedance at the top side. From Figs. 14 and 15, it can be observed that the variation in the FRA signatures due to SC disks lies in the middle and high frequency bands. Higher levels of SC disks have observable impact on the FRA signature especially in the high frequency range. From the perspective of numerical indices, as show in Fig. 16, the trend of FRA signatures obtained through the electric circuit model that is considering the turnto-turn capacitance, winding structure and all possible mutual inductances are of good trend agreement with the experimental signatures when compared with the signatures obtained through less accurate models. Case Study 2: RD Fault The transformer hardware model is customized to implement radial deformation faults at various levels as shown in Fig. 17. RD fault levels of 5%, 7%, and 10% are implemented on the middle disks (disks 11–20) of the HV winding. Same fault levels and locations were simulated on the equivalent electric circuit model based on the simulation technique described in [3]. The variation in the electrical circuit parameters at each fault level is calculated using FEA. Fig. 18 shows the percentage variation in the capacitance elements relevant to the deformed disks at each radial fault level. As the location of fault affects the inductance elements significantly, results in Fig. 19 show the percentage change in the series/ mutual inductance between one deformed disk (disk number 11) and the other 30 disks in the HV winding and the 7 helical turns in the LV winding. It can be seen from Fig. 18(a) that the shunt capacitances (CHL, CHG) change almost linearly along with the RD level. While the capacitance between the HV winding and the earthed tank is increasing with the increase in RD level, the capacitance between the HV and LV windings is decreasing. This is attributed to the fact that, radial buckling in the

6. Transformer winding faults analysis To investigate the sensitivity of the proposed modeling method to identify and quantify short-circuit (SC) disks [3,61], radial deformation (RD) [14,61–63] and axial disk buckling (ADB) [64–67], the following case studies are conducted. To facilitate the implementation of various winding deformations, the original heavy tank of the investigated transformer was replaced by a light earthed iron shell and the oil filled in the original tank was drained out. It is to be noted that insulation oil is a capacitive component that affects the FRA signature at the high frequency range. Hence the accuracy of the obtained results will not be significantly affected within the investigated frequency range. Case Study 1: SC Disks Fault The hardware transformer model is customized to implement short circuit faults at various locations with various levels as shown in Fig. 10. Connecting labels (3, 4), (1, 2), and (5, 6) simulates short-circuit disks on the middle, top, and bottom sections of the winding, respectively. Fig. 11 shows the effect of SC fault location on the experimental FRA signature while Fig. 12 shows the effect of the same fault on the FRA signature obtained through simulation analysis on the detailed transformer model proposed in this paper. Implementation of SC fault on the simulated model is conducted by short circuiting a portion of the winding turns i.e. reducing the series impedance. It can be observed that the trends of the FRA signatures obtained through simulation analysis and experimental measurement are of satisfactory agreement. Tables 3 and 4 list the variations in the resonant frequencies and amplitudes of the FRA signatures obtained through the experimental measurement and simulation analysis. As can be seen from these results, resonance frequencies are increasing due to SC disks fault with a slight increase in the magnitude within the middle and high frequency ranges. Short-circuit fault in the middle of the winding has more impact than a short circuit at the top or 362

Electrical Power and Energy Systems 111 (2019) 351–368

X. Zhao, et al.

HV disk as shown in Fig. 17(a) results in a decrease in the distance between deformed HV disks and the tank while increasing the distance to the LV winding. On the other hand, Fig. 18(b) shows that the series capacitance increases slightly along with the increase in the RD fault level from 5% to 10%. The effect of RD fault on the series capacitance (CsIW) of interleaved disk winding is obviously less than that of the continuous disk (CsCD) which is in turns slightly larger than the capacitances between various disks (Cdisk). Because the relative distances of adjacent turns within the same disk is not influenced, the turn-to-turn capacitance (Ctt) is not affected by the RD fault. Fig. 19 shows that the inductance between the HV deformed disk (number 11) and other HV disks increases linearly with the increase in RD fault level. It can be also observed that variations in the inductance parameters associated with the middle 10 disks (where the actual fault takes place) is relatively higher than those in the top and bottom sections of the HV winding. The mutual inductance variation between disk 11 of the HV winding and LV 7 helical turns in Fig. 19(b) reveals that the mutual inductance drops at turn 2 then increasing linearly. This is attributed to the structure design of the two windings that places the 11 and 12 HV disks at the same height of 2nd LV winding disk. When RD takes place in disks 11 and 12 of the HV winding, the mutual coupling between these disks and disk 2 of the LV winding decreases. It is worth noting that the inductance variation of all winding disks cannot be neglected in spite of the small change within the individual ones. These practical results agree well with the results published in [14]. Fig. 20 shows the effect of RD fault on the experimental FRA signature while Fig. 21 shows the effect of the same fault on the FRA signature obtained through simulation analysis on the detailed transformer model shown in Fig. 2. It can be observed that the trend of the FRA signatures obtained through simulation analysis and experimental measurement are of satisfactory agreement. Tables 5 and 6 list the variations in the resonant frequencies and amplitudes of the FRA signatures obtained through the experimental measurement and simulation analysis, respectively. As can be seen from the tables, the effect of RD starts at the frequency range (300–600 kHz) and is more observable in the frequency range (600–1000 kHz). Relative CC and ED are calculated and plotted in Fig. 22. The calculated CC% and ED% reveal the variation between the faulty and corresponding healthy signatures. It can be seen from Fig. 22 that the proposed modeling method results in an FRA signature of a better trend agreement with the experimental signature than the models without considering WS or Ctt. Case Study 3: ADB Fault To investigate the sensitivity of the proposed modeling method to identify and quantify axial disk buckling fault, the hardware transformer model is customized to implement ADB fault at the middle 10 winding disks as shown in Fig. 17. Impacts of this fault when takes place at the middle 10 disks of HV winding on the electric circuit parameters are shown in Figs. 23 and 24. It can be seen from Fig. 23(a) that the values of CHG between the deformed disks (11–20) of the HV winding and the tank is increasing for disks number 16 and 17, decreasing for disks 11 and 20 and exhibit insignificant variations for the remaining disks. This is attributed to the irregular physical buckling deformation as can be seen in Fig. 17(b). On the other hand, CHL oscillates around zero because of the relative position change between the HV and LV windings. Because the relative distances of adjacent turns within the same winding disk is not influenced by the ADB fault, the variation in the turn-to-turn capacitance is insignificant and can be neglected. The variation of series capacitance is show in Fig. 23(b). It can be seen that the disk-to-disk capacitance (Cdisk) of the HV winding (except disk number 16) increases slightly. This is because the distance between adjacent disks slightly decreases due to ADB while the distance between disks 16 and 17 increases and the distance between the other adjacent disks exhibits insignificant variation. Results also show that the series capacitances in the interleaved winding do not exhibit much variation as in the continuous disk winding. This is because the turn-to-

turn capacitance of the interleaved winding contributes a large proportion in series capacitance than the continuous disk winding. The effect of axial buckling fault on the series and mutual inductance parameters is shown in Fig. 24. The figure shows that the inductance between disk number 11 and the upper part disks in the HV winding increases, while the coupling between this disk and the lower disks becomes weaker due to increased distance caused by the ADB fault. Similar trend in the mutual inductances between HV and LV winding disks can be observed in Fig. 24(b). Fig. 25 shows the effect of such fault on the experimental FRA signature while Fig. 26 shows the effect of the same fault on the FRA signature obtained through simulation analysis on the detailed transformer model proposed in this paper. Detailed simulation of AD fault can be found in [3,13]. It can be observed that the trend of the FRA signatures due to such fault agrees between the simulation and experimental results. Tables 7 and 8 list the variations in the resonant frequencies and amplitudes of the FRA signatures obtained through the experimental measurement and simulation analysis. As can be seen from the obtained results, the effect of ADB starts to appear in the frequency range (300–600 kHz) and is more observable in the frequency range (600–1000 kHz). CC and ED are calculated and listed in Table 9. Again, results reveal that the trend of the FRA signature obtained from the proposed modeling method has better agreement with the experimental signature than the models without considering WS or Ctt. As can be seen in the above results, the FRA signature obtained through simulation analysis is not perfectly matching with the practical measured signature. However, both signatures have close trends in terms of resonant frequencies variation for healthy and faulty signatures. The FRA is a relative technique that does not rely on the absolute signature of one phase. Various comparison techniques are adopted in this regard (with a reference signature, with a sister transformer or phase-to-phase). Hence, the FRA signature trend is of more concern than the absolute signature [68]. Results in this paper show a very close trend between simulation and actual measurements. As stated above, the difference between simulation and experimental signatures is attributed to the lack of precise equivalent circuit parameters of the investigated transformer. FEA modeling was based on the parameters and insulation properties stated during the manufacturing stage. These parameters and properties are changing due to aging and hence perfect simulation is not expected. The proposed model in this paper can be further improved by considering frequency dependency feature of various parameters in the circuit [23,69]. 7. Conclusion There are several simplified lumped parameters-based high frequency electric circuit models proposed in the literatures for FRA studies. Majority of these models did not consider the inter-turn capacitance nor winding structure. Moreover, these models did not take into account all mutual inductances between relevant coils. This simplification resulted in inaccurate FRA signature for the investigated transformer equivalent circuit models. To overcome this shortcoming, an improved detailed transformer high frequency model that takes into account the inter-turn capacitance, all possible mutual inductances between coils along with the winding structure has been proposed in this paper. The accuracy of the proposed detailed model is assessed via comparing the trend of the FRA signatures obtained through simulation analysis and practical measurement. Numerical indices reveal the good agreement of the simulation and the practical FRA signatures’ trends. The proposed model is utilized to study the impact of various faults including SC disks, RD and ADB on the FRA signature which is found to be of good trend agreement with the results obtained through experimental measurements. This model can be employed to understand the precise impacts of various transformer internal faults on the FRA signature and hence a reliable FRA interpretation code can be established. 363

Electrical Power and Energy Systems 111 (2019) 351–368

X. Zhao, et al.

For reliable identification and quantification of transformer internal faults, further research is required to develop a reliable expert system based on the detailed transformer equivalent electric circuit model to automate the assessment of the transformer mechanical integrity.

Acknowledgment The authors would like to express their appreciation to the National Natural Science Foundation of China (Grant no. 51377175) for its financial support.

Appendix See Tables A1–A5. Table A1 Three-phase transformer model—design specifications. Nominal value Rated voltage (kV) Rated power (kVA) Rated current (A) Frequency (Hz) Number of phase Connection group symbol HV insulation cell (mm) LV insulation ring (mm) Silicon steel core

10/0.4 400 23/577 50 3 Ynyn0 61 217 1390 335 300 520 250 210.5 520 1880 × 900 × 1415 ONAN

Yoke length (mm) Yoke height (mm) Limb width (mm) Limb height (mm) Outer radius (mm) Inner radius (mm) Height (mm)

HV winding Tank (mm) Cooling system

Table A2 Properties of insulating materials within the transformer. Material

Relative dielectric constant

Material

Relative dielectric constant

Oiled paper Oil

4.7 2.2

Insulating cylinder/ring Cell

4.5 4.7

Table A3 Detailed parameters of HV winding of three models for healthy condition. Unit: pF

Detailed model Model without WS Model without Ctt Detailed model Model without WS Model without Ctt Detailed model Model without WS Model without Ctt Detailed model/Model without WS/Model without Ctt

CH1

CH2

CH3

CH4

CH5

CH6

CH7

CH8

CH9

CH10

3329.02 854.24 751.13 CH11 1208.84 1208.84 1105.72 CH21 3688.09 1213.32 1110.20

3693.11 1218.34 1115.22 CH12 222.26 222.26 119.15 CH22 3695.72 1220.95 1117.83

3695.37 1220.60 1117.48 CH13 1204.36 1204.36 1101.24 CH23 3698.24 1223.47 1120.35

3701.47 1226.70 1123.58 CH14 222.54 222.54 119.42 CH24 3700.22 1225.45 1122.33

3703.30 1228.53 1125.41 CH15 1215.28 1215.28 1112.16 CH25 3710.43 1235.66 1132.54

3702.13 1227.35 1124.24 CH16 222.57 222.57 119.46 CH26 3707.71 1232.93 1129.82

3703.43 1228.66 1125.54 CH17 1208.56 1208.56 1105.44 CH27 3695.93 1221.16 1118.04

3699.72 1224.94 1121.83 CH18 222.43 222.43 119.31 CH28 3684.04 1209.26 1106.15

3683.29 1208.52 1105.40 CH19 1204.21 1204.21 1101.09 CH29 3322.74 847.97 744.86

2697.14 222.36 119.25 CH20 222.46 222.46 119.35 CH30 2634.80 160.02 56.91

CHG1 4.3020 CHG11 1.7479 CHG21 1.7365

CHG2 1.0111 CHG12 1.7478 CHG22 0.9171

CHG3 0.9364 CHG13 1.7499 CHG23 0.9122

CHG4 0.9238 CHG14 1.7534 CHG24 0.9114

CHG5 0.9175 CHG15 1.7546 CHG25 0.9154

CHG6 0.9171 CHG16 1.7518 CHG26 0.9178

CHG7 0.9172 CHG17 1.7520 CHG27 0.9259

CHG8 0.9125 CHG18 1.7531 CHG28 0.9384

CHG9 0.9154 CHG19 1.7497 CHG29 1.0113

CHG10 1.7349 CHG20 1.7444 CHG30 4.3110

364

365

H_1 H_2 H_3 H_4 H_5

H_1 H_2 H_3 H_4 H_5 H_6 H_7 H_8 H_9 H_10 H_11 H_12 H_13 H_14 H_15 H_16 H_17 H_18 H_19 H_20 H_21 H_22 H_23 H_24 H_25 H_26 H_27 H_28 H_29 H_30 L_1 L_2 L_3 L_4 L_5 L_6 L_7

0.07 0.07 0.08 0.08 0.09

H_20

H_19

0.08 0.08 0.08 0.09 0.10

0.84 1.00 0.85 0.71 0.61 0.53 0.47 0.42 0.38 0.34 0.26 0.24 0.19 0.17 0.14 0.13 0.10 0.10 0.08 0.07 0.06 0.06 0.05 0.05 0.05 0.05 0.04 0.04 0.04 0.04 0.31 0.22 0.15 0.11 0.08 0.06 0.04

H_2

1.00 0.84 0.70 0.60 0.53 0.47 0.42 0.38 0.34 0.31 0.24 0.22 0.17 0.16 0.13 0.12 0.10 0.09 0.08 0.07 0.06 0.05 0.05 0.05 0.05 0.04 0.04 0.04 0.04 0.04 0.28 0.20 0.14 0.10 0.07 0.06 0.04

H_1

0.06 0.06 0.07 0.07 0.07

H_21

0.70 0.85 1.00 0.85 0.70 0.60 0.53 0.47 0.42 0.38 0.28 0.26 0.20 0.19 0.15 0.14 0.11 0.10 0.08 0.08 0.07 0.06 0.06 0.05 0.05 0.05 0.05 0.04 0.04 0.04 0.34 0.23 0.16 0.12 0.08 0.06 0.05

H_3

Table A4 Inductance coupling coefficient matrix.

0.05 0.06 0.06 0.07 0.07

H_22

0.60 0.71 0.85 1.00 0.85 0.70 0.60 0.52 0.46 0.42 0.31 0.28 0.22 0.20 0.16 0.15 0.12 0.11 0.09 0.08 0.07 0.07 0.06 0.06 0.05 0.05 0.05 0.05 0.04 0.04 0.36 0.25 0.18 0.13 0.09 0.07 0.05

H_4

0.05 0.05 0.06 0.06 0.07

H_23

0.53 0.61 0.70 0.85 1.00 0.85 0.70 0.60 0.52 0.46 0.34 0.31 0.24 0.22 0.17 0.16 0.13 0.12 0.10 0.09 0.07 0.07 0.07 0.06 0.06 0.05 0.05 0.05 0.05 0.04 0.39 0.28 0.19 0.14 0.10 0.07 0.05

H_5

0.05 0.05 0.05 0.06 0.06

H_24

0.47 0.53 0.60 0.70 0.85 1.00 0.85 0.70 0.60 0.52 0.37 0.34 0.26 0.23 0.18 0.17 0.14 0.13 0.10 0.10 0.08 0.07 0.07 0.06 0.06 0.06 0.05 0.05 0.05 0.05 0.42 0.30 0.21 0.15 0.10 0.08 0.06

H_6

0.05 0.05 0.05 0.05 0.06

H_25

0.42 0.47 0.53 0.60 0.70 0.85 1.00 0.84 0.70 0.60 0.41 0.37 0.28 0.26 0.20 0.18 0.15 0.14 0.11 0.10 0.08 0.08 0.07 0.07 0.06 0.06 0.06 0.05 0.05 0.05 0.44 0.33 0.23 0.16 0.11 0.08 0.06

H_7

0.04 0.05 0.05 0.05 0.05

H_26

0.38 0.42 0.47 0.52 0.60 0.70 0.84 1.00 0.84 0.70 0.46 0.41 0.31 0.28 0.22 0.20 0.16 0.15 0.12 0.11 0.09 0.08 0.08 0.07 0.07 0.06 0.06 0.06 0.05 0.05 0.45 0.36 0.25 0.17 0.12 0.09 0.07

H_8

0.04 0.04 0.05 0.05 0.05

H_27

0.34 0.38 0.42 0.46 0.52 0.60 0.70 0.84 1.00 0.85 0.53 0.46 0.34 0.31 0.23 0.21 0.17 0.16 0.13 0.12 0.10 0.09 0.08 0.08 0.07 0.07 0.07 0.06 0.06 0.05 0.46 0.38 0.27 0.19 0.13 0.10 0.07

H_9

0.04 0.04 0.04 0.05 0.05

H_28

0.31 0.34 0.38 0.42 0.46 0.52 0.60 0.70 0.85 1.00 0.60 0.52 0.37 0.34 0.25 0.23 0.18 0.17 0.13 0.13 0.10 0.10 0.09 0.08 0.08 0.07 0.07 0.07 0.06 0.06 0.45 0.41 0.29 0.20 0.14 0.10 0.08

H_10

0.04 0.04 0.04 0.04 0.05

H_29

0.24 0.26 0.28 0.31 0.34 0.37 0.41 0.46 0.53 0.60 1.00 0.84 0.52 0.46 0.33 0.30 0.23 0.21 0.17 0.16 0.13 0.12 0.11 0.10 0.10 0.09 0.08 0.08 0.07 0.07 0.38 0.45 0.38 0.26 0.18 0.13 0.09

H_11

0.04 0.04 0.04 0.04 0.04

H_30

0.22 0.24 0.26 0.28 0.31 0.34 0.37 0.41 0.46 0.52 0.84 1.00 0.60 0.52 0.37 0.33 0.25 0.23 0.18 0.17 0.14 0.13 0.12 0.11 0.10 0.10 0.09 0.08 0.08 0.08 0.35 0.45 0.40 0.29 0.20 0.14 0.10

H_12

0.28 0.31 0.34 0.36 0.39

L_1

0.17 0.19 0.20 0.22 0.24 0.26 0.28 0.31 0.34 0.37 0.52 0.60 1.00 0.84 0.52 0.46 0.33 0.30 0.23 0.21 0.17 0.16 0.15 0.14 0.13 0.12 0.11 0.10 0.10 0.09 0.27 0.38 0.45 0.37 0.26 0.18 0.13

H_13

0.20 0.22 0.23 0.25 0.28

L_2

0.16 0.17 0.19 0.20 0.22 0.23 0.26 0.28 0.31 0.34 0.46 0.52 0.84 1.00 0.60 0.52 0.37 0.33 0.25 0.23 0.18 0.17 0.16 0.15 0.14 0.13 0.12 0.11 0.10 0.10 0.25 0.36 0.45 0.40 0.28 0.19 0.14

H_14

0.14 0.15 0.16 0.18 0.19

L_3

0.13 0.14 0.15 0.16 0.17 0.18 0.20 0.22 0.23 0.25 0.33 0.37 0.52 0.60 1.00 0.84 0.52 0.46 0.33 0.30 0.23 0.21 0.20 0.18 0.17 0.16 0.15 0.14 0.13 0.12 0.19 0.28 0.39 0.45 0.36 0.25 0.18

H_15

0.10 0.11 0.12 0.13 0.14

L_4

0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.20 0.21 0.23 0.30 0.33 0.46 0.52 0.84 1.00 0.60 0.52 0.37 0.33 0.25 0.23 0.22 0.20 0.18 0.17 0.16 0.15 0.14 0.13 0.18 0.25 0.36 0.45 0.39 0.28 0.19

H_16

0.06 0.06 0.06 0.07 0.07

L_6

0.04 0.04 0.05 0.05 0.05

L_7

0.09 0.10 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.21 0.23 0.30 0.33 0.46 0.52 0.84 1.00 0.60 0.52 0.37 0.34 0.31 0.28 0.26 0.24 0.22 0.20 0.19 0.17 0.13 0.18 0.26 0.37 0.45 0.39 0.27

H_18

(continued on next page)

0.07 0.08 0.08 0.09 0.10

L_5

0.10 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.23 0.25 0.33 0.37 0.52 0.60 1.00 0.84 0.52 0.46 0.34 0.31 0.28 0.26 0.23 0.22 0.20 0.19 0.17 0.16 0.14 0.19 0.28 0.40 0.45 0.36 0.25

H_17

X. Zhao, et al.

Electrical Power and Energy Systems 111 (2019) 351–368

366

0.10 0.11 0.12 0.13 0.13 0.17 0.18 0.23 0.25 0.33 0.37 0.52 0.60 1.00 0.84 0.52 0.46 0.41 0.37 0.34 0.31 0.28 0.26 0.24 0.22 0.10 0.14 0.20 0.29 0.40 0.45 0.35

0.10 0.10 0.11 0.12 0.13 0.16 0.17 0.21 0.23 0.30 0.33 0.46 0.52 0.84 1.00 0.60 0.53 0.46 0.41 0.37 0.34 0.31 0.28 0.26 0.24 0.09 0.13 0.18 0.26 0.37 0.45 0.38

H_20

0.08 0.08 0.09 0.10 0.10 0.13 0.14 0.17 0.18 0.23 0.25 0.34 0.37 0.52 0.60 1.00 0.85 0.70 0.60 0.52 0.46 0.42 0.38 0.34 0.31 0.08 0.10 0.14 0.20 0.29 0.41 0.45

H_21

0.07 0.08 0.08 0.09 0.10 0.12 0.13 0.16 0.17 0.21 0.23 0.31 0.34 0.46 0.53 0.85 1.00 0.85 0.70 0.60 0.52 0.46 0.42 0.38 0.34 0.07 0.10 0.13 0.19 0.27 0.38 0.46

H_22 0.07 0.07 0.08 0.08 0.09 0.11 0.12 0.15 0.16 0.20 0.22 0.28 0.31 0.41 0.46 0.70 0.85 1.00 0.85 0.70 0.60 0.53 0.47 0.42 0.38 0.07 0.09 0.12 0.17 0.25 0.36 0.45

H_23 0.06 0.07 0.07 0.08 0.08 0.10 0.11 0.14 0.15 0.18 0.20 0.26 0.28 0.37 0.41 0.60 0.70 0.85 1.00 0.85 0.70 0.60 0.53 0.47 0.42 0.06 0.08 0.11 0.16 0.23 0.33 0.44

H_24

H_i (L_i) represents the i-th disk of HV(LV) winding within the transformer.

H_6 H_7 H_8 H_9 H_10 H_11 H_12 H_13 H_14 H_15 H_16 H_17 H_18 H_19 H_20 H_21 H_22 H_23 H_24 H_25 H_26 H_27 H_28 H_29 H_30 L_1 L_2 L_3 L_4 L_5 L_6 L_7

H_19

Table A4 (continued)

0.06 0.06 0.07 0.07 0.08 0.10 0.10 0.13 0.14 0.17 0.18 0.23 0.26 0.34 0.37 0.52 0.60 0.70 0.85 1.00 0.85 0.70 0.60 0.53 0.47 0.06 0.08 0.10 0.15 0.21 0.30 0.42

H_25 0.06 0.06 0.06 0.07 0.07 0.09 0.10 0.12 0.13 0.16 0.17 0.22 0.24 0.31 0.34 0.46 0.52 0.60 0.70 0.85 1.00 0.85 0.70 0.61 0.53 0.05 0.07 0.10 0.14 0.19 0.28 0.39

H_26 0.05 0.06 0.06 0.07 0.07 0.08 0.09 0.11 0.12 0.15 0.16 0.20 0.22 0.28 0.31 0.42 0.46 0.53 0.60 0.70 0.85 1.00 0.85 0.71 0.60 0.05 0.07 0.09 0.13 0.18 0.25 0.36

H_27 0.05 0.05 0.06 0.06 0.07 0.08 0.08 0.10 0.11 0.14 0.15 0.19 0.20 0.26 0.28 0.38 0.42 0.47 0.53 0.60 0.70 0.85 1.00 0.85 0.70 0.05 0.06 0.08 0.12 0.16 0.23 0.34

H_28 0.05 0.05 0.05 0.06 0.06 0.07 0.08 0.10 0.10 0.13 0.14 0.17 0.19 0.24 0.26 0.34 0.38 0.42 0.47 0.53 0.61 0.71 0.85 1.00 0.84 0.04 0.06 0.08 0.11 0.15 0.22 0.31

H_29 0.05 0.05 0.05 0.05 0.06 0.07 0.08 0.09 0.10 0.12 0.13 0.16 0.17 0.22 0.24 0.31 0.34 0.38 0.42 0.47 0.53 0.60 0.70 0.84 1.00 0.04 0.06 0.07 0.10 0.14 0.20 0.29

H_30 0.42 0.44 0.45 0.46 0.45 0.38 0.35 0.27 0.25 0.19 0.18 0.14 0.13 0.10 0.09 0.08 0.07 0.07 0.06 0.06 0.05 0.05 0.05 0.04 0.04 1.00 0.55 0.30 0.18 0.12 0.08 0.06

L_1 0.30 0.33 0.36 0.38 0.41 0.45 0.45 0.38 0.36 0.28 0.25 0.19 0.18 0.14 0.13 0.10 0.10 0.09 0.08 0.08 0.07 0.07 0.06 0.06 0.06 0.55 1.00 0.55 0.30 0.18 0.12 0.08

L_2 0.21 0.23 0.25 0.27 0.29 0.38 0.40 0.45 0.45 0.39 0.36 0.28 0.26 0.20 0.18 0.14 0.13 0.12 0.11 0.10 0.10 0.09 0.08 0.08 0.07 0.30 0.55 1.00 0.55 0.30 0.18 0.12

L_3 0.15 0.16 0.17 0.19 0.20 0.26 0.29 0.37 0.40 0.45 0.45 0.40 0.37 0.29 0.26 0.20 0.19 0.17 0.16 0.15 0.14 0.13 0.12 0.11 0.10 0.18 0.30 0.55 1.00 0.55 0.30 0.18

L_4 0.10 0.11 0.12 0.13 0.14 0.18 0.20 0.26 0.28 0.36 0.39 0.45 0.45 0.40 0.37 0.29 0.27 0.25 0.23 0.21 0.19 0.18 0.16 0.15 0.14 0.12 0.18 0.30 0.55 1.00 0.55 0.30

L_5 0.08 0.08 0.09 0.10 0.10 0.13 0.14 0.18 0.19 0.25 0.28 0.36 0.39 0.45 0.45 0.41 0.38 0.36 0.33 0.30 0.28 0.25 0.23 0.22 0.20 0.08 0.12 0.18 0.30 0.55 1.00 0.56

L_6

0.06 0.06 0.07 0.07 0.08 0.09 0.10 0.13 0.14 0.18 0.19 0.25 0.27 0.35 0.38 0.45 0.46 0.45 0.44 0.42 0.39 0.36 0.34 0.31 0.29 0.06 0.08 0.12 0.18 0.30 0.56 1.00

L_7

X. Zhao, et al.

Electrical Power and Energy Systems 111 (2019) 351–368

Electrical Power and Energy Systems 111 (2019) 351–368

X. Zhao, et al.

Table A5 Comparison of series capacitance values for various models. Winding condition

Model

Healthy

RD fault

5%

7%

10%

ADB fault

Value of series capacitance (pF) CH5

CH15

CH25

Model without Ctt Model without WS Proposed detailed model

1125.41 1228.53 3703.30

1112.16 1215.28 1215.28

1132.54 1235.66 3710.43

Model without Ctt Model without WS Proposed detailed model Model without Ctt Model without WS Proposed detailed model Model without Ctt Model without WS Proposed detailed model

1070.09 1173.20 3647.97

1126.76 1229.87 1229.87

1111.75 1214.86 3689.64

1070.09 1173.20 3647.97

1129.18 1232.30 1232.30

1111.75 1214.86 3689.64

1070.09 1173.20 3647.97

1140.83 1243.95 1243.95

1111.75 1214.86 3689.64

Model without Ctt Model without WS Proposed detailed model

1070.09 1173.20 3647.97

1187.27 1765.30 1765.30

1111.75 1214.86 3689.64

Appendix A. Supplementary material Supplementary data to this article can be found online at https://doi.org/10.1016/j.ijepes.2019.04.010.

References [16]

[1] Transformer reliability survey. CIGRE Working Group A237, Brochure 642, France; 2015. [2] Arispe JCG, Mombello EE. Detection of failures within transformers by FRA using multiresolution decomposition. IEEE Trans Power Delivery 2014;29:1127–37. [3] Abu-Siada A, Hashemnia N, Islam S, Masoum MAS. Understanding power transformer frequency response analysis signatures. IEEE Electr Insul Mag 2013;29:48–56. [4] Hashemnia N, Abu-Siada A, Islam S. Detection of power transformer bushing faults and oil degradation using frequency response analysis. IEEE Trans Dielectr Electr Insul 2016;23:222–9. [5] Hong KX, Huang H, Zhou JP. Winding condition assessment of power transformers based on vibration correlation. IEEE Trans Power Delivery 2015;30:1735–42. [6] Seo J, Ma H, Saha TK. A joint vibration and arcing measurement system for online condition monitoring of onload tap changer of the power transformer. IEEE Trans Power Delivery 2017;32:1031–8. [7] Venikar PA, Ballal MS, Umre BS, Suryawanshi HM. A novel offline to online approach to detect transformer interturn fault. IEEE Trans Power Delivery 2016;31:482–92. [8] Masoum AS, Hashemnia N, Abu-Siada A, Masoum MAS, Islam SM. Online transformer internal fault detection based on instantaneous voltage and current measurements considering impact of harmonics. IEEE Trans Power Delivery 2017;32:587–98. [9] Alsuhaibani S, Khan Y, Beroual A, Malik NH. A review of frequency response analysis methods for power transformer diagnostics. Energies 2016;9:879-. [10] Pham DAK, Pham TMT, Borsi H, Gockenbach E. A new diagnostic method to support standard frequency response analysis assessments for diagnostics of transformer winding mechanical failures. IEEE Electr Insul Mag 2014;30:34–41. [11] Yao C, Zhao Z, Chen Y, Zhao X, Li Z, Wang Y, et al. Transformer winding deformation diagnostic system using online high frequency signal injection by capacitive coupling. IEEE Trans Dielectr Electr Insul 2014;21:1486–92. [12] Liu Y, Ji S, Yang F, Cui Y, Zhu L, Rao Z, et al. A study of the sweep frequency impedance method and its application in the detection of internal winding short circuit faults in power transformers. IEEE Trans Dielectr Electr Insul 2015;22:2046–56. [13] Hashemnia N, Abu-Siada A, Islam S. Improved power transformer winding fault detection using FRA diagnostics – Part 1: axial displacement simulation. IEEE Trans Dielectr Electr Insul 2015;22:556–63. [14] Hashemnia N, Abu-Siada A, Islam S. Improved power transformer winding fault detection using FRA diagnostics – Part 2: radial deformation simulation. IEEE Trans Dielectr Electr Insul 2015;22:564–70. [15] Zhao X, Yao C, Zhao Z, Abu-Siada A. Performance evaluation of online transformer

[17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]

367

internal fault detection based on transient overvoltage signals. IEEE Trans Dielectr Electr Insul 2017;24:3906–15. Zhao X, Yao C, Zhang C, Abu-Siada A. Toward reliable interpretation of power transformer sweep frequency impedance signatures: experimental analysis. IEEE Electr Insul Mag 2018;34:40–51. Bigdeli M, Vakilian M, Rahimpour E. Transformer winding faults classification based on transfer function analysis by support vector machine. IET Electric Power Appl IET Electric Power Appl 2012;6:268–76. Gomez-Luna E, Mayor GA, Gonzalez-Garcia C, Guerra JP. Current status and future trends in frequency-response analysis with a transformer in service. IEEE Trans Power Delivery 2013;28:1024–31. Zhang ZW, Tang WH, Ji TY, Wu QH. Finite-element modeling for analysis of radial deformations within transformer windings. IEEE Trans Power Delivery 2014;29:2297–305. Abu-Siada A. High frequency transformer modelling using state space representation for FRA studies. In: 2017 IEEE 11th International Symposium on Diagnostics for Electrical Machines, Power Electronics and Drives (SDEMPED); 2017. p. 422–6. Zhang H, Wang S, Yuan D, Tao X. Double-ladder circuit model of transformer winding for frequency response analysis considering frequency-dependent losses. IEEE Trans Magn 2015;51:1–4. Pham DAK, Gockenbach E. Analysis of physical transformer circuits for frequency response interpretation and mechanical failure diagnosis. IEEE Trans Dielectr Electr Insul 2016;23:1491–9. Abeywickrama N, Serdyuk YV, Gubanski SM. High-frequency modeling of power transformers for use in frequency response analysis (FRA). IEEE Trans Power Del 2008;23:2042–9. Mitchell SD, Welsh JS. Modeling power transformers to support the interpretation of frequency-response analysis. IEEE Trans Power Del 2011;26:2705–17. Rashtchi V, Rahimpour E, Fotoohabadi H. Parameter identification of transformer detailed model based on chaos optimisation algorithm. IET Electr Power Appl IET Electric Power Appl 2011;5:238–46. Rashtchi V, Rahimpour E, Shahrouzi H. Model reduction of transformer detailed RC-L-M model using the imperialist competitive algorithm. IET Electric Power Appl IET Electr Pow Appl 2012;6:233–42. Mukherjee P, Satish L. Construction of equivalent circuit of a single and isolated transformer winding from FRA data using the ABC algorithm. IEEE Trans Power Del 2012;27:963–70. Chanane A, Bouchhida O, Houassine H. Investigation of the transformer winding high-frequency parameters identification using particle swarm optimisation method. IET Electr Power Appl IET Electr Power Appl 2016;10:923–31. Toudji M, Parent G, Duchesne S, Dular P. Determination of winding lumped parameter equivalent circuit by means of finite element method. IEEE Trans Magn 2017;53:1–4. Chaouche MS, Houassine H, Moulahoum S, Colak I. BA to construction of

Electrical Power and Energy Systems 111 (2019) 351–368

X. Zhao, et al.

[31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51]

equivalent circuit of a transformer winding from frequency response analysis measurement. IET Electr Pow Appl IET Electr Pow Appl 2018;12:728–36. Shabestary MM, Ghanizadeh AJ, Gharehpetian GB, Agha-Mirsalim M. Ladder network parameters determination considering nondominant resonances of the transformer winding. IEEE Trans Power Del 2014;29:108–17. Tonekaboni Bandpey S, Vahidi B, Alizadeh Shabestary MM, Hosseinian SH, Gharehpetian GB. Sensitivity analysis on ladder network equivalent circuit parameters of power transformer. Electr Power Compon Syst 2015;43:2168–77. Liang G, Sun H, Zhang X, Cui X. Modeling of transformer windings under very fast transient overvoltages. IEEE Trans Electromagn C. 2006;48:621–7. Shintemirov A, Tang WH, Wu QH. A hybrid winding model of disc-type power transformers for frequency response analysis. IEEE Trans Power Delivery 2009;24:730–9. Tang WH, Shintemirov A, Wu QH. Detection of minor winding deformation fault in high frequency range for power transformer. IEEE PES General Meeting 2010:1–6. López ZL, Gómez P, Espino-Cortés FP, Peña-Rivero R. Modeling of transformer windings for fast transient studies: experimental validation and performance comparison. IEEE Trans Power Delivery 2017;32:1852–60. Gustavsen B. A filtering approach for merging transformer high-frequency models with 50/60-Hz low-frequency models. IEEE Trans Power Deliv 2015;30:1420–8. Jurisic B, Uglesic I, Xemard A, Paladian F. Difficulties in high frequency transformer modeling. Electr Power Syst Res 2016;138:25–32. Jurisic B, Uglesic I, Xemard A, Paladian F. High frequency transformer model derived from limited information about the transformer geometry. Int J Elec Power 2018;94:300–10. Gustavsen B, Portillo A. A black-box approach to interfacing white-box transformer models with electromagnetic transients programs. 2014 IEEE PES General Meeting | Conference & Exposition; 2014. p. 1–5. Liu S, Liu Y, Li H, Lin F. Diagnosis of transformer winding faults based on FEM simulation and on-site experiments. IEEE Trans Dielectr Electr Insul 2016;23:3752–60. Sharma U, Chatterjee S, Bhuyan K. Development of reference SFRA plot of transformer at design stage using high frequency modelling. 2012 1st International Conference on Power and Energy in NERIST (ICPEN); 2012. p. 1–4. Hashemnia N, Abu-Siada A, Islam S. Impact of axial displacement on power transformer FRA signature. IEEE Power and Energy Society General Meeting PESGM; 2013. p. 1–4. IEEE guide for the application and interpretation of frequency response analysis for oil-immersed transformers. IEEE Std C57149-2012; 2013. p. 1–72. Samimi MH, Tenbohlen S, Akmal AAS, Mohseni H. Effect of different connection schemes, terminating resistors and measurement impedances on the sensitivity of the FRA method. IEEE Trans Power Del 2017;32:1713–20. Scaife BKP. Principles of dielectrics: Oxford, United Kingdom; Oxford University Press; 1998. Ahn HM, Lee JY, Kim JK, Oh YH, Jung SY, Hahn SC. Finite-element analysis of short-circuit electromagnetic force in power transformer. IEEE Trans Ind Appl 2011;47:1267–72. Zhang H, Yang B, Xu W, Wang S, Wang G, Huangfu Y, et al. Dynamic deformation analysis of power transformer windings in short-circuit fault by FEM. IEEE Trans Appl Supercon 2014;24:1–4. Bagheri M, Phung BT, Naderi MS. Impulse voltage distribution and frequency response of intershield windings. IEEE Electr Insul Mag 2016;32:32–40. Karsai K, Kerenyi D, Kiss L. Large power transformers. United States: Elsevier Science Pub. Co. Inc., New York, NY; 1987. Yin L, Wu Z, Gui J. Diagnostics of transformer windings deformation based on

[52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68]

[69]

368

transfer function. In: Jia L, Liu Z, Qin Y, Zhao M, Diao L, editors. Lecture Notes in Electrical Engineering; 2014. p. 65–72. Aghmasheh R, Rashtchi V, Rahimpour E. Gray box modeling of power transformer windings based on design geometry and particle swarm optimization algorithm. IEEE Trans Power Delivery 2018;33:2384–93. Samimi MH, Shayegani Akmal AA, Mohseni H, Tenbohlen S. Detection of transformer mechanical deformations by comparing different FRA connections. Int J Elec Power 2017;86:53–60. Shintemirov A, Tang WH, Wu QH. Transformer winding condition assessment using frequency response analysis and evidential reasoning. IET Electr Pow Appl IET Electr Power Appl 2010;4:198–212. Nosratian Ahour J, Seyedtabaii S, Gharehpetian GB. Test system for detailed model of transformers for transient analysis using the electromagnetic transient program (EMTP). Electr Power Compon Syst 2018;46:511–20. Liu X, Yang Y, Yang F, jadoon A. Numerical research on the losses characteristic and hot-spot temperature of laminated core joints in transformer. Appl Therm Eng 2017;110:49–61. Khalili Senobari R, Sadeh J, Borsi H. Frequency response analysis (FRA) of transformers as a tool for fault detection and location: a review. Electr Power Syst Res 2018;155:172–83. Pourhossein K, Gharehpetian GB, Rahimpour E, Araabi BN. A probabilistic feature to determine type and extent of winding mechanical defects in power transformers. Electr Power Syst Res 2012;82:1–10. Behjat V, Mahvi M. Statistical approach for interpretation of power transformers frequency response analysis results. IET Sci Meas Technol 2015;9:367–75. Samimi MH, Tenbohlen S. FRA interpretation using numerical indices: state-of-theart. Int J Elec Power 2017;89:115–25. Picher P, Lapworth J, Noonan T, Christian J. Mechanical-condition assessment of transformer windings using frequency response analysis (FRA). CIGRÉ working group A 2008;2. Rahimpour E, Christian J, Feser K, Mohseni H. Transfer function method to diagnose axial displacement and radial deformation of transformer windings. IEEE Trans Power Deliv 2003;18:493–505. Ji TY, Tang WH, Wu QH. Detection of power transformer winding deformation and variation of measurement connections using a hybrid winding model. Electr Power Syst Res 2012;87:39–46. Jiang J, Zhou L, Gao S, Li W, Wang D. Frequency response features of axial displacement winding faults in autotransformers with split windings. IEEE Trans Power Deliv 2018;33:1699–706. Khanali M, Hayati-Soloot A, Høidalen HK, Jayaram S. Study on locating transformer internal faults using sweep frequency response analysis. Electr Power Syst Res 2017;145:55–62. Pandya AA, Parekh BR. Interpretation of sweep frequency response analysis (SFRA) traces for the open circuit and short circuit winding fault damages of the power transformer. Int J Elec Power 2014;62:890–6. Lei X, Li J, Wang Y, Mi S, Xiang C. Simulative and experimental investigation of transfer function of inter-turn faults in transformer windings. Electr Power Syst Res 2014;107:1–8. Hashemnia N, Abu-Siada A, Masoum MAS, Islam SM. Characterization of transformer FRA signature under various winding faults. In: Proceedings of 2012 IEEE International Conference on Condition Monitoring and Diagnosis (IEEE Cmd 2012). 2012:446–9. Wilcox DJ, Hurley WG, Conlon M. Calculation of self and mutual impedances between sections of transformer windings. IEE Proceed 1989;1365(5):308–14.