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Assessing mechanical deformations in two-winding transformer unit using reduced-order circuit model
Electrical Power and Energy Systems 79 (2016) 235–244
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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Assessing mechanical deformations in two-winding transformer unit using reduced-order circuit model Krupa Rajendra Shah ⇑, K. Ragavan Indian Institute of Technology Gandhinagar, Ahmedabad, India
a r t i c l e
i n f o
Article history: Received 25 August 2014 Received in revised form 8 December 2015 Accepted 9 December 2015
Keywords: Circuit synthesis Diagnostics Frequency response analysis Mechanical deformations Terminal measurement Transformer
a b s t r a c t Certain level of mechanical deformations in winding does not hamper the normal performance of power transformer. However, such incipient deformations if unattended could result in permanent failure of transformer. To this end, an approach is proposed to assess the severity of mechanical deformations in transformer winding. These deformations get reflected as changes in its high frequency behaviour. Hence, characterising the high frequency behaviour is essential. This requires building physically realisable ladder circuit corresponding to each winding. Thus, n-winding transformer is represented by n electrically and magnetically coupled ladder networks. In such scenario, the objective of fault diagnostics becomes very challenging. In this effort, realising the multi-winding unit by reduced-order ladder circuit model is explored. This approach essentially involves energising one winding at a time. Then, reducedorder ladder circuit of considered unit is synthesised from its measured driving-point impedance data. It is shown how these circuit models could be used for identifying mechanical deformations. To demonstrate capability of the method, two-winding model transformer is considered and deformations are introduced in its outer winding. Then, reduced-order circuit models are synthesised corresponding to healthy and faulty state of model transformer. The location of fault is identified by the changed parameter in the circuit. Further, the amount of change reveals the severity of introduced deformation. In all the cases, synthesised reduced-order circuit model agrees with that of model transformer with regard to driving-point impedance plot and results are found satisfactory. Ó 2016 Elsevier Ltd. All rights reserved.
Motivation Power transformer is a vital component of power system. Hence its uninterrupted functioning is of utmost importance. The mechanical integrity of a transformer is challenged by excessive electromagnetic forces generated in axial and radial direction due to short circuit current [1–3]. Apart from this, lightning and switching surges, transportation and rough handling of transformers are also responsible for reducing mechanical strength of the transformer. Cumulative effect of exposure to such abnormality would alter winding geometry. Such mechanical deformations are initially incipient faults. Eventually such fault-like condition grows and would lead to catastrophic failure. Therefore, identifying mechanical deformation and assessing its severity is paramount for smooth functioning of transformer and power system. Usually, changes in the mechanical structure of power transformer get reflected in the frequency range starting from few kHz to 1 MHz [4]. Hence, its high-frequency behaviour can be con⇑ Corresponding author. E-mail address: [email protected] (K.R. Shah). http://dx.doi.org/10.1016/j.ijepes.2015.12.035 0142-0615/Ó 2016 Elsevier Ltd. All rights reserved.
sidered. To this end, frequency response analysis (FRA) can be performed on transformer [5–14]. The acquired FRA data can be converted into network function by deploying curve fitting techniques [15,16]. A detailed analysis of such network function can be useful to detect deformation in transformer. Once, mechanical deformation is detected, the subsequent crucial task becomes identifying location of fault and its severity. To this end, it is necessary to look for some approach that is capable of addressing the goal of localisation. This becomes motivation for the research effort. Literature survey In order to locate deformation, it appeared worthwhile to have complete representation of the winding. This can be achieved if a physically realisable circuit model is built. The circuit should be synthesised such that it characterises accurately the highfrequency behaviour of transformer winding. Since most of the flux passes through air for high frequency excitation, linear circuit representation can be considered [17,18]. Such circuit models can be found in [10–12] [19–23]. Further, circuit model for two-winding transformer is shown in subsequent section (Fig. 1).
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1 C'’g1
2 C’'g2
R'’g1
R'’g3
C’'gM
2' r'’1
R’'g(M+1)
C’'sM M
3'
R’'s2 L’'22
C’'g(M+1)
R'’gM
C'’s2
R’'s1 L’'11
C’'g3
R'’g2
C'’s1 Line-end 1' Winding-2
M
r'’2
R’'sM L’'MM
r''M
Neutral-end Winding-2
L'12 L'2M L'1M Cw1
Cw2
Rw1
Rw2
Cw3
Rw3
CwN
RwN
Cw(N+1)
Rw(N+1)
L’1N L’2N
L’12 L’11 Line-end Winding-1 1
C’g1
L’22
r'1 R’s1
C’s1 R’g1 1
L’NN
r'2 R’s2
2
N
3
R’g2
Neutral-end Winding-1
C’sN
C’s2 C’g2
r'N R’sN
C’g3
R’g3
C’gN
C’g(N+1)
R’gN
R’g(N+1)
N
2
Fig. 1. High frequency circuit model of two-winding transformer.
The ladder circuits were used in [10–12,19,20] for the purpose of locating faults. In these publications, initially circuit corresponding to healthy transformer winding was synthesised. Later, some changes were introduced and then corresponding to faulty winding, one new circuit was synthesised. Then, these circuits were compared and faults were identified. Thus, the principle of identifying the changes remains the same. However, in [19,20], the ladder network was developed using design details. Whereas, in [10–12], terminal voltage and current data are utilised. Then, driving-point impedance (DPI) function was obtained. Utilising which physically realisable ladder network was synthesised. Thus, these two approaches are different. Further, there exists discrepancy with regard to test-arrangement as indicated below: In [10,12], single winding is considered. Then, discrete changes in capacitances were introduced in the winding and these changes were viewed as deformations. In [11], inductive changes were also introduced by shorting few discs along with the discrete capacitive changes. In [19,20] two-windings are considered. Then, axial displacement is introduced by moving one winding with respect to other. Further, radial deformations are introduced by deforming one of the windings. Thus, it is clear that the approach based on design details was validated considering ‘‘physical deformations” in the winding. Further, the test unit comprised of ‘‘two-windings”. On the other hand, the terminal measurement based approach was demonstrated considering ‘‘discrete changes” in ‘‘isolated winding”. Nonetheless, it is completely non-invasive.
From the above discussion, it is understood that there is a need to identify physical deformations through terminal measurement in two-winding unit. Hence, it becomes objective of the work. For this purpose, FRA is deployed.
Reduced-order circuit model The high frequency circuit model of two-winding transformer unit is comprised of two-ladder networks as in Fig. 1 with each ladder network corresponds to a winding. Further, the electric and magnetic coupling between these windings can be accounted by inter-winding capacitances and mutual-inductances respectively. The considered equivalent circuit has N and M sections corresponding to winding-1 and winding-2 respectively. The capacitances and inductances per section are represented as series capacitances ðC 0si ; C 00si Þ, shunt-capacitances ðC 0gi ; C 00gi Þ, resistances ðr 0i ; r00i Þ, self inductances ðL0ii ; L00ii Þ and inter-winding capacitances ðC wi Þ. Mutual inductances between any two sections ‘i’ and ‘j’ can be denoted by L0ij ; L00ij . The dielectric losses could be represented by large values of resistances ðR0gi ; R00gi ; R0si ; R00si ; Rwi Þ in parallel to capacitors. Thus, transformer with n-windings would be represented by nladder networks; coupled electrically and magnetically. Hence, it is clear that the number of variables (circuit parameters) to be estimated will increase with increase in number of windings. Thus, complexity increases during circuit synthesis. This problem could be simplified if the entire system is equivalently be represented by single ladder network. With this, it is possible to reduce number of variables significantly. Further, applicability of the diagnostic method would not be restricted with regard to number of windings in transformer.
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L1N L2N
L12
L11 Line-end
L22
r1 RS1
RS2
2
1 Rg1 1
Rg2
Neutral-end
N
CS2 Cg2
rN RSN
3
CS1 Cg1
LNN
r2
CSN Cg3
2
Rg3
CgN
RgN
Cg(N+1)
Rg(N+1)
N
Fig. 2. Reduced-order circuit model of multi-winding unit under considered terminal condition.
In order to check feasibility of representing transformer by single ladder network, several literature are explored. It is mentioned in [24] that the influence of all the neighbouring windings on the winding under investigation can be minimised if all the ‘‘neighbouring windings are shorted and grounded”. Further, it is reported that the transformer with said terminal condition can be considered equivalent to isolated winding and such system can be represented by single ladder network. To certain extent, findings in [24] can be supported by literature [17,18]. It is mentioned here that if neighbouring windings are shorted then the terminal behaviour of iron-core transformer would become similar to that of air-core transformer. Thus, influence of one winding on other can be ignored with regard to mutual inductance. However, it is not clear how inter-winding capacitances of multi-winding transformer are accounted by isolated winding? To this end, few more literature are explored. It is found, in [25] that the shorted neighbouring windings can be considered equivalent to shields. Since, neighbouring winding is grounded, equivalently the shield can also be considered grounded. Thus, multi-winding transformer is approximated by transformer with single winding and ground plane. To this end, inter-winding capacitances are now reflected between winding and ground plane. Thus, it can be inferred that ‘‘under the above mentioned terminal condition (shorted and grounded neighbouring windings), the iron-core transformer is equivalent to an air-core transformer with single winding and grounded shield(s)”. Hence, it is possible to represent such system by single ladder network. Its shunt capacitances would account for inter-winding capacitances. Since number of energy storing elements is less in single ladder network as compared to multi-ladder network, it can be termed as reducedorder circuit model. One such representation is shown in Fig. 2. It will be shown subsequently that such reduced circuit model is helpful in analysing mechanical condition of the considered unit. The number of reduced-order ladder networks would be same as number of windings in the considered unit. Proposed methodology This approach essentially involves measuring the driving-point impedance (DPI) of the considered unit. While acquiring DPI, one winding is energised by sinusoidal signal. Moreover, neighbouring windings are to be shorted and grounded. Later, the measured DPI will be used to acquire information about various characteristic features of the considered unit such as natural frequencies, dc resistance, effective shunt capacitance, equivalent inductance and capacitance. Using these characteristic features, the proposed method synthesises a mutually coupled, lumped parameter reduced-order ladder network.
For assessing the status of the system, these features are to be extracted at two different instances. Initially, healthy system is characterised by single ladder network and is termed as ‘‘Refer ence-circuit”. Then at the time of inspecting system, similar circuit is synthesised and this new ladder network is termed as ‘‘Testcircuit”. Comparing the parameters of Test-circuit with that of Reference-circuit, the condition of the considered unit can be understood. While synthesising circuit, convergence is deemed to have occurred, when differences between the characteristic features obtained from measurement and circuit are less than a preset tolerance. Let us say, the considered unit has n-windings. Then, similar procedure is to be repeated for remaining n 1 windings. Thus, for assessing status of n-winding transformer, n Reference- and Test-circuits can be synthesised. With this philosophy, the procedure for obtaining characteristic features of multi-winding transformer with the considered terminal condition (shorted and grounded neighbouring windings) is detailed below. Determining characteristic features Natural frequencies For determining natural frequencies of the system, FRA is performed. To this end, sinusoidal excitation is fed between the line- and grounded neutral-end of the winding under test. Then, magnitude ðjZjÞ and phase ðhÞ of the DPI are obtained. This exercise is done for wide range of frequencies and variation of magnitude of DPI with frequency is plotted. The peaks and troughs of magnitude plot correspond to open-circuit natural frequencies (f o , ocnf) and short-circuit natural frequencies (f s , scnf) respectively [26]. Occurrence of such peaks and troughs imply that the effect of both inductance and capacitances is present in the corresponding frequency range. Equivalent Inductance, Leq To obtain high-frequency circuit model, the value of winding inductance corresponding to high-frequency is needed. The obvious approach is to feed high-frequency excitation and determine the value of reactance using magnitude and phase of DPI. However, with high-frequency excitation, the capacitances get reflected and the reactance is due to both inductance and capacitance. Hence, it is not directly possible to obtain the value of high-frequency inductance. To this end, an alternative is to be identified. With the considered terminal condition, the effect of core is almost eliminated [17,18]. This gives a hint that the value of inductance of the considered unit is nearly equivalent to air-core inductance. Further, it is reported in [27] that the value of air-core inductance decreases with frequency and remains constant beyond
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certain frequency. To this end, variation in inductance with frequency is observed. The inductance can be obtained through Eq. (1).
L¼
jZjsin h 2pf
ð1Þ
Then, the frequency range is identified in which inductance remains nearly constant. Any of these frequencies can be considered as f 1 and corresponding inductance can be termed as Leq . Essentially, f 1 should be well below the first ocnf. If so, then the presence of capacitive effects can be ignored. Hence, it is possible to have a better estimate of air-core winding inductance. Equivalent Capacitance, C eq A ladder network of N sections with its neutral grounded and neighbouring windings shorted and grounded will have the following pole-zero form [10].
ZðsÞ ¼
Q b1 ðs sÞ N1 i¼1 ðs zi Þðs zi Þ QN i¼1 ðs pi Þðs pi Þ
ð2Þ
where b1 : scaling factor, s: real zero, zi ; zi : complex-conjugate zero pair, and pi ; pi : complex-conjugate pole pair. For high-frequency excitation, the input impedance is capacitive in nature. Hence,
b ¼ 1 C eq s s 1 b1 ¼ C eq 1
For low-frequency excitation, the input impedance is series connection of resistance ðRdc Þ and inductance ðLeq Þ. Hence, the real zero is,
s¼
Rdc Leq
ð3Þ
The peaks and troughs in the DPI plot correspond to the poles and zeros of DPI function. Also, as the absolute value of the imaginary parts of complex frequencies is significantly higher than their real parts (losses in the winding are low), pi ’s and zi ’s in Eq. (3) can be approximated to the corresponding imaginary parts. Hence Eq. (3) can be rewritten as,
Q 2 4p2 N f b1 Leq QN1i¼12 oi i¼1 f si
ð4Þ
Thus, determination of C eq is possible once natural frequencies and Leq are known. Effective shunt capacitance, C g;eff For this purpose, neutral-end of the winding under inspection is to be isolated from the ground. Further, line and neutral terminals are to be shorted. Then, sinusoidal excitation of frequencies (f) well below first ocnf is to be applied between line-end and ground. Let the corresponding impedance be Z\h. Next, Eq. (5) for calculating capacitance (C) can be used.
C¼
sin h 2pf jZj
Synthesising Reference-circuit Algorithm 1. Reference circuit 1: The resistance in series with inductor can be given by,
ri ¼
Rdc ; 8i ¼ 1; . . . ; N N
ð5Þ
Since end terminals of the winding are shorted and low frequency excitations are fed, the circuit in Fig. 2 can be represented by just parallel combination of resistances and shunt capacitances.
ð6Þ
whereRdc is dc resistance of winding. 2: The value of resistors in parallel with capacitors can be considered very high (in this work it is approximated as 100 kX). 3: To determine effective series capacitance ðC s;eff Þ of the winding, it is essential to know about C g;eff and initial voltage distribution constant ðaÞ.
C s;eff ¼
For DC excitation, that is, at s ¼ 0, the input impedance becomes resistive and is Rdc . Hence, the scaling factor b1 can be expressed as,
QN i¼1 ðpi Þðpi Þ b1 ¼ Leq QN1 i¼1 ðzi Þðzi Þ
Thus, it could be possible to relate C with C g;eff . However, for very low frequencies (in order of few Hz), the impedance would be very high and hence, current drawn from the source becomes feeble. This would lead to inaccurate current measurement. Hence estimated C may not indicate actual C g;eff of the winding. As frequency rises (in order of few kHz), the value of impedance will come down and hence accurate current measurement becomes possible. Thus, for those frequencies, it is possible to have C ¼ C g;eff . To this end, low frequency range (between few Hz to few kHz) is identified in which the estimated value of C is nearly constant. It can be inferred that this value does not suffer from measurement inaccuracy. Hence, for these frequencies C is same as winding C g;eff . The start of this frequency range is referred as f 2 . Once these characteristic features are known pertaining to system of healthy winding, the Reference-circuit can be synthesised using Algorithm 1.
C g;eff
ð7Þ
a2
However, the value of a varies with the type of winding and its estimation is to be done. For this purpose, the algorithm in [28] can be used. It involves varying a and N iteratively. For each value of a and N, equivalent capacitance ðC eq Þ is calculated by analytical formula.
C eq ¼
Cg 1 þ 2 C1s þ C þ1 1 g
ð8Þ
1 þ Cs
Then, the normalised error between equivalent capacitance determined by measurement and analytical formula can be expressed as,
C eq C eq 100 eða; NÞ ¼ C eq
ð9Þ
Thesum of errors corresponding to a particular value of a and different values of N is determined as,
EðaÞ ¼
N max X
eða; NÞ
ð10Þ
N¼Nmin
where N min is considered greater than number of observable ocnf and N max is 3Nmin . These limits are found suitable and hence considered. However, they can be altered. In the array EðaÞ, the global minimum is identified and the corresponding value of a is substituted in Eq. (7) for determining C s;eff . 4: Then eða; NÞ pertaining to considered a can be observed. In this array, global minimum is identified and the corresponding value of N is considered for synthesising circuit.
K.R. Shah, K. Ragavan / Electrical Power and Energy Systems 79 (2016) 235–244
5: Initially, the winding is free from faults. Hence, it is more appropriate to assume that it has symmetry and uniformity. As a result, its equivalent representation can be considered to have capacitances as mentioned below.
C si ¼ NC s;eff ; 8i ¼ 1; . . . ; N C gi ¼ C gi ¼
C g;eff 2N C g;eff N
;
8i ¼ 1; N þ 1
;
8i ¼ 2; . . . ; N
ð11Þ
6: From the known value of Leq and with the assumptions that (i) self inductances are identical, (ii) the mutual inductances between any two coils separated by same physical distance are equal and (iii) mutual inductances between the coils decrease with the increase in the physical separation, initial values and boundaries to L1j ; 8j ¼ 1; . . . ; N can be chosen. The initial value of self-inductance ðL11 Þ can be considered as small fraction of measured Leq . 7: The equivalent inductance ðLeq Þ can be estimated. N X N X Leq ¼ Lij
ð12Þ
i¼1 j¼1
L L 8: if eqLeq eq 100 < 2:5% then 9:
Construct the circuit model. Applying state-space analysis, determine the natural frequencies of the system under consideration. Then, determine ocnf and scnf ðf o ; f s Þ analytically [29]. 10: else 11: 12: 13:
14:
Go to step 15. if f offo o 100 < 2:5% and f s ffs s 100 < 2:5% then The corresponding circuit can be considered as the representation of the system under consideration with the proposed terminal condition. else
15:
Change the values of inductances iteratively and go to step 7. In this work, inductances are increased by 1% of L11 . 16: end if 17: end if
Synthesising Test-circuit The axial force is produced by interaction of radial component of leakage field with the current. Any minor misalignment between magnetic centres of the windings, asymmetry in ampere-turn due to tapping, unequal winding heights would generate enormous axial forces. This might result into displacement of the complete winding (or its turns) in axial direction. On the contrary, the radial forces are produced by interaction of axial leakage field with the current. This axial leakage field would produce hoop stresses on the outer winding and compressive stresses on the inner winding [1–3]. These events change geometry. That is, inductances and capacitances would get altered with regard to their reference values. To realise the system under inspection by a ladder circuit, once again characteristic features namely natural frequencies, effective shunt capacitance, equivalent inductance and capacitance are to be extracted as explained earlier. The procedure of synthesising the Test-circuit would involve few changes in Algorithm 1 and only those are indicated below.
239
Elements of the Reference-circuit are chosen as initial values. The number of sections are considered same as that of Reference-circuit. However, a is varied iteratively. Corresponding to each a; C s;eff is determined. Series- and shunt-capacitances are also iteratively estimated along with self- and mutual-inductances. Hence, the number of variables becomes,
C si ; 8i ¼ 1; . . . ; N C gi ; 8i ¼ 1; . . . ; N þ 1 Lij ;
8i; j ¼ 1; . . . ; N;
ð13Þ jPi
The effective series- and shunt-capacitance are also estimated along with equivalent-capacitance.
C g;eff ¼ C g1 þ þ C gðNþ1Þ 1 1 1 ¼ þ þ C s;eff C s1 C sN
ð14Þ
At convergence, following equations should be satisfied.
C g;eff C g;eff 100 < 2:5% C g;eff C s;eff C s;eff 100 < 2:5% C s;eff C eq C eq 100 < 2:5% C eq
ð15Þ
To assess the severity of the mechanical deformations, these two circuits are used. Results and discussion The objective is to examine capability of the method in interpreting high frequency behaviour of two-winding transformer unit. Further, it should be able to locate and assess the severity of mechanical deformation in it. To this end, two-winding unit is considered. The outer winding contains 15 discs with each disc has 10 turns. The conductor is of rectangular cross-section of 5 2 mm. The total height of the winding is about 180 mm. Inside disc winding, single layer winding of same height and diameter of 219 mm is kept centrally. As discussed earlier, behaviour of iron-core winding can be approximated to air-core winding under considered terminal condition. This equivalence avoids building iron-core unit for demonstration purpose. Once the windings are assembled together with iron-core, the overall unit becomes very rigid. It becomes difficult to introduce changes later, if needed. Considering this, it is decided to validate the proposed algorithm with air-core unit. The ground plane in actual transformer can be established by connecting core and tank to earth potential. Such equipotential surfaces can be established in the considered set-up by providing concentric non-magnetic shields [30]. To this end, two shields are used in the considered set-up and connected to earth potential. One of these shields has 210 mm diameter and is kept inside the inner winding. Another one has 332 mm diameter and is kept outside the outer winding. This entire set-up of two-windings and two shields is termed as model transformer. The test set-up comprising of outer-winding is shown in Fig. 3(a). The cut-section of model transformer is shown in Fig. 3(b). The schematic arrangement of model transformer in its healthy state is shown in Fig. 4(a). In Fig. 4(b) upward movement of disc winding is shown to represent axial displacement. The changes in radial direction can be achieved by removing some portion of shield as shown in Fig. 4(c). Such physical changes in geometry
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K.R. Shah, K. Ragavan / Electrical Power and Energy Systems 79 (2016) 235–244
Inner Shield
Tappings from Inner Winding Outer Winding
Outer Shield
(a)
(b)
Fig. 3. Test set-up for model transformer.
Axial Displacement Outer Shield
Outer Winding
Radial Deformation
Inner Shield
Inner Winding
(a)
(b)
(c)
Fig. 4. (a) Healthy state, (b) axial displacement and (c) radial deformation.
would change the area available for flux to flow. Further, it is obvious that few conductors may not be facing ground plane. As a result, magnetic and electric field would undergo changes. Hence, the winding inductances and capacitances would be different than that pertaining to healthy winding. Thus, with the considered setup, it is possible to introduce physical changes in the winding. In this work, outer disc winding is energised and terminal measurement is performed. While doing so, inner winding is shorted and grounded. Later, reduced-order circuit model of the twowinding model transformer is synthesised from the characteristic features and proposed algorithm. The corresponding circuit is termed as Reference-circuit. Then, physical deformations are introduced and once again reduced-order circuit models are built. Such models are termed as Test-circuit. It is shown that, by comparing Reference-circuit with Test-circuit, it is possible to reveal location and severity of introduced deformations in the considered unit. Determining characteristic features Initially, shields are connected to earth-terminal to established ground plane. Then, FRA is performed by energising the outer disc
winding. While doing so, its neutral terminal is connected to ground and terminal condition (shorted and grounded layer winding) is followed. Corresponding DPI Plot is shown in Fig. 5 with solid line. Its natural frequencies (f o ; f s ) are listed in Table 1. Using the acquired DPI data, inductance (L) is calculated for different frequencies below first ocnf. It is found that the inductance is nearly constant in the frequency range of 20–100 kHz. Any frequency in this range can be considered to identify winding inductance. In this work, f 1 is considered as 40 kHz and corresponding Leq is 1154 lH. Since f 1 is well below first ocnf, it can be assumed that capacitive effects are not pronounced at this frequency. Hence, Leq can be considered as good estimation of air-core inductance. The magnitude and phase of DPI at f 1 is shown in Table 2. The value of Rdc is 1.2 X. Next, the values pertaining to observable peaks and troughs are substituted in Eq. (4). In practice, the observable f oi ; f si might not be same as N, N 1 respectively [10]. Hence, N in the above equation will be replaced by the number of observable peaks. With this, b1 is obtained and corresponding C eq becomes 61.2 pF. For determining C g;eff , the neutral terminal of disc winding is isolated from earth. The line- and neutral-terminals are shorted
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K.R. Shah, K. Ragavan / Electrical Power and Energy Systems 79 (2016) 235–244
Winding Circuit
1
|Z| (k Ω)
10
1
2
3
4
5
6
7
8
L1j
57.9
24.9
18.8
6.6
3
2.7
2.5
0.2
Table 5 Inductance matrix.
−1
−2
10
0
0.5
1
1.5
2
2.5
Frequency (MHz) Fig. 5. DPI plot of model transformer with its healthy disc winding under test.
Table 1 Characteristic features – model transformer with its healthy disc winding under test. 400 403.9 871 868.4
(kHz) (kHz) (kHz) (kHz)
1129 1131 1610 1604
1760 1765 2290 2295
2420 2418
Max. error < 2:5%
Leq ¼ 1154 lH, C g;eff ¼ 320 pF, C eq ¼ 61:2 pF Leq ¼ 1154 lH, C g;eff ¼ 320 pF, C eq ¼ 61:5 pF
Max. error < 2:5%
Table 2 Measured terminal quantities – model transformer with its healthy disc winding under test. Excitation frequency (kHz)
Input impedance (X)
Calculated parameter
40 8
290\86.92 62000\-89
Leq ¼ 1154 lH C g;eff ¼ 320 pF
together. Then, sinusoidal excitation of various frequencies starting from 50 Hz is applied between line terminal of disc winding and ground. The capacitances are found nearly constant in the range of 8–150 kHz. Thus, the frequency f 2 is identified as 8 kHz. The corresponding magnitude and phase of DPI is presented in Table 2. Using these data C g;eff is determined as 320 pF. Later, input capacitances ðC eq Þ pertaining to various a and N are estimated iteratively and normalised error between C eq and C eq is obtained. Few of these values are presented in Table 3 for brevity. From Table, it is found that EðaÞ is least for a as 5.5. This value together with C g;eff as 320 pF can be used to estimate C s;eff . For this case it is 10.6 pF. Further, it can be observed from Table 3 that for
Table 3 Normalised error between C eq and C eq for different iterations – healthy disc winding.
a
N
lH) – model transformer with its healthy disc winding under test.
i
0
10
10
fo fo fs fs
Table 4 Inductance (in
4.5
5
5.5
6
6.5
5 6 7 8 9 11 12 13 14 15
27.38 24.06 22.01 20.67 19.74 18.57 18.19 17.89 17.66 17.46
16.90 13.28 11.03 9.55 8.52 7.23 6.81 6.48 6.22 6.01
8.49 4.57 2.14 0.52 0.60 2.01 2.47 2.83 3.12 3.35
1.63 2.57 5.19 6.93 8.14 9.67 10.17 10.56 10.88 11.13
4.06 8.52 11.31 13.17 14.47 16.12 16.66 17.08 17.42 17.69
EðaÞ
223.00
100.00
31.51
86.00
152.00
j
1
2
3
4
5
6
7
8
L1j L2j L3j L4j L5j L6j L7j L8j
57.9 24.9 18.8 6.6 3 2.7 2.5 0.2
24.9 57.9 24.9 18.8 6.6 3 2.7 2.5
18.8 24.9 57.9 24.9 18.8 6.6 3 2.7
6.6 18.8 24.9 57.9 24.9 18.8 6.6 3
3 6.6 18.8 24.9 57.9 24.9 18.8 6.6
2.7 3 6.6 18.8 24.9 57.9 24.9 18.8
2.5 2.7 3 6.6 18.8 24.9 57.9 24.9
0.2 2.5 2.7 3 6.6 18.8 24.9 57.9
the considered a, error is least with N as 8. Hence, it is decided to build a ladder network with N = 8. Synthesising Reference-circuit It is straightforward to calculate per section values of shuntand series-capacitances once C g;eff ; C s;eff ; a and N are known. The values of C si ; 8i ¼ 1; . . . ; 8 and C gi ; 8i ¼ 2; . . . ; 8 are found as 84.63 pF and 40 pF respectively. Further, C g1 and C g10 are considered half of the 40 pF. For estimating inductances, sequential iterations are performed. The inductances associated with first section are shown in Table 4. Assuming symmetry, the complete inductance matrix can be constructed as shown in Table 5. Then, the reduced order Referencecircuit can be built as in Fig. 6. The DPI plot obtained from Reference-circuit is shown in Fig. 5 with dotted line and characteristic features (with sign) are presented in Table 1. The normalised error between characteristic features is also determined. The maximum value of this deviation is found to be less than 2:5%. Thus, it can be said that the constructed Reference-circuit or the reduced-order model is good representation of model transformer with its disc winding under test and layer winding shorted and grounded. Synthesising Test-circuit Axial displacement This case study represents axial movement of the disc winding in upward direction by 7 mm (4% of winding height) with respect to layer winding and shields. Corresponding DPI plot is shown in Fig. 7 with solid line. To check the validity of algorithm corresponding to the faulty state of the winding, initially terminal measurement is performed and natural frequencies are identified. These values are shown in Table 6. Then frequencies f 1 and f 2 are identified as 40 kHz and 8 kHz respectively. Corresponding to these frequencies, various terminal quantities such as equivalent inductance and capacitance are obtained and shown in Table 6. As the characteristic features are different than that of the healthy winding, it can be assumed that the winding is faulty. To locate the faulty section and access the severity, once again sequential iterations are performed. As Reference-circuit has 8 sections, the Test-circuit is to be synthesised with N as 8. This reduced-order Test-circuit is shown in Fig. 8. It is found that the series capacitances remained unchanged. The changes are observed in C g1 and inductances pertaining to first
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57.9 Line-end
57.9
0.15
100
2
1 100 1
Neutral-end 9
84.63 40
100
40
0.15 100
8
3
84.63
84.63 20
57. 9
0.15
100
100
20
100
40
100
8
2
Fig. 6. Reference-circuit of model transformer with its healthy disc winding under test (resistances in series with inductances, resistances in parallel with capacitances, inductances and capacitances are in X, kX, lH and pF respectively).
1
10
|Z| (kΩ )
Table 7 Inductance (in
Winding Circuit
lH) – model transformer with axially displaced disc winding.
j
1
2
3
4
5
6
7
8
L1j
60
25
20
7
3.2
2.8
2.6
0.2
0
10
−1
Winding Circuit
10
1
10 0
0.5
1
1.5
2
|Z| (kΩ)
−2
10
2.5
Frequency (MHz) Fig. 7. DPI plot of model transformer with axially displaced disc winding.
−1
10
Table 6 Characteristic features – model transformer with axially displaced disc winding. fo fo fs fs
(kHz) (kHz) (kHz) (kHz)
410 413.4 872 865
1150 1149 1620 1604
1790 1779 2270 2295
−2
10
2410 2424
Max. error < 2:5%
fo fo fs fs
57 .9
0.15
100
2
1 84.63 40 1
1.5
(kHz) (kHz) (kHz) (kHz)
405 400.7 880 864.2
2
2.5
57. 9
0.15 8
3
2
1760 1778 2300 2316
2420 2439
Max. error < 2:5% Max. error < 2:5%
obtained as shown in Fig. 9 and characteristic features are obtained as listed in Table 8. The frequencies f 1 and f 2 are 40 kHz and 8 kHz respectively. Utilizing characteristic features and the proposed method, new Test-circuit is synthesised as shown in Fig. 10. Its capacitive variables are nearly same as that of reference case except C g7 ; C g8 and C g9 . These values are 37 pF, 37 pF and 17 pF respectively. Once again, it is found that series capacitances are
100 84 .63 100
1132 1132 1630 1616
Leq ¼ 1172 lH, C g;eff ¼ 310 pF, C eq ¼ 61:8 pF Leq ¼ 1168 lH, C g;eff ¼ 311 pF, C eq ¼ 61:5 pF
Radial deformation The change in radial direction is achieved by disturbing the geometry of outer shield near neutral-end. This exercise is done on shield rather than winding. This is just to avoid damage to the winding. With this emulated radial fault, once again DPI plot is
100
1
Table 8 Characteristic features – model transformer with radially deformed disc winding.
section. The value of C g1 is found as 13 pF. The inductive variable that has got deviated from its reference value (Table 5) is shown in Table 7. The remaining variables are found same as that of reference values and hence not shown in the Table. From the results, it is possible to say that fault is near line-end of the winding. This is same as that of actual fault location. The comparison between measured and estimated DPI plots is shown Fig. 7. It shows reasonably good agreement.
13
0.5
Fig. 9. DPI plot of model transformer with radially deformed disc winding.
Max. error < 2:5%
60
0
Frequency (MHz)
Leq ¼ 1160 lH, C g;eff ¼ 310 pF, C eq ¼ 54:3 pF Leq ¼ 1160:3 lH, C g;eff ¼ 313 pF, C eq ¼ 54:5 pF
Line-end
0
10
0.15 Neutral-end 9
100 84 .63
40
100
40
20
100
100
8
Fig. 8. Test-circuit of model transformer with axially displaced disc winding (resistances in series with inductances, resistances in parallel with capacitances, inductances and capacitances are in X, kX, lH and pF respectively).
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K.R. Shah, K. Ragavan / Electrical Power and Energy Systems 79 (2016) 235–244
57.9 Line-end
58.5
0.15
100
2
1
7
84.63 20
40
100
100
1
0.15 Neutral-end 9
100
8
84.63 100
37
58.5
0.15
100
84.63 100 8
37
7
17
100
Fig. 10. Test-circuit of model transformer with radially deformed disc winding (resistances in series with inductances, resistances in parallel with capacitances, inductances and capacitances are in X, kX, lH and pF respectively).
Table 9 Inductance (in
Table 11 Inductance (in
lH) – model transformer with radially deformed disc winding.
lH) – model transformer with disc-to-disc short.
j
1
2
3
4
5
6
7
8
j
1
2
3
4
5
6
7
8
L7j L8j
2.5 0.2
2.7 2.5
3 2.7
7 3
20 7
26.5 20
58.5 26.5
26.5 58.5
L1j L2j L3j L4j L5j L6j L7j L8j
50 24.9 18.8 6.6 3 2.7 0 0
24.9 50 24.9 18.8 6.6 3 0 0
18.8 24.9 50 24.9 18.8 6.6 1 0
6.6 18.8 24.9 50 24.9 18.8 2 1
3 6.6 18.8 24.9 50 24.9 9 2
2.7 3 6.6 18.8 24.9 50 11 9
0 0 1 2 9 11 30 11
0 0 0 1 2 9 11 15
Table 10 Characteristic features – model transformer with disc-to-disc short. fo fo fs fs
480 485.6 1060 1040.3
(kHz) (kHz) (kHz) (kHz)
1320 1306 1920 1894
2070 2062 2710 2694
2850 2824
Max. error < 2:5%
Leq ¼ 914 lH, C g;eff ¼ 320 pF, C eq ¼ 60:4 pF Leq ¼ 903:4 lH, C g;eff ¼ 320 pF, C eq ¼ 61:5 pF
tured correctly by reduced order model. Further, there is a good agreement between DPI pattern of the winding and circuit (Fig. 9).
Max. error < 2:5%
Disc to disc fault Next, disc to disc short is introduced near neutral-end. Corresponding DPI plot and characteristic features are presented in Fig. 11 and Table 10 respectively. Then, Test-circuit is synthesised as in Fig. 12. From the results, it is found that inductance matrix has changed dominantly (Table 11). Further, the major deviation is observed near neutral-end. The DPI plot in Fig. 11 shows reasonably good match between estimated and measured terminal behaviour. From the results, it emerges that the introduced faults are correctly identified. The severity of fault can be judged by comparing variables of Reference- and Test-circuit. In this section, outer disc winding is considered as test winding. Similarly, inner layer winding can also be considered and terminal measurement can be performed on it. While doing so, the outer winding is to be shorted and grounded. Correspondingly, reduced-order circuit models are to be synthesised. This exercise is very straightforward and hence the results are not presented. It is believed that the approach can also be extended to three phase transformer that has six windings. At a time, one winding is to be considered and the remaining five are to be shorted and grounded. Then, reduced-order circuit model is to be synthesised. This exercise is to be repeated six times and hence six Referencecircuits are to be synthesised. Later, mechanical deformations are
same as before. The inductive variable that has shown deviation form its reference value (Table 5) is presented in Table 9. From the results, it is clear that the shunt capacitances and inductances which are closer to neutral-end have undergone several changes. Thus, it can be concluded that the changes are cap-
Winding Circuit
1
|Z| (kΩ)
10
0
10
−1
10
−2
10
0
0.5
1
1.5
2
2.5
Frequency (MHz) Fig. 11. DPI plot of model transformer with disc-to-disc short.
50 Line-end
30
0 .1 5 100
2
1
7
84.63 20
100
40 1
100
40
15
0 .15 100
84.63 100 7
8 40
0 .1 5 Neutral-end 9
100 84.63 100 8
20
100
Fig. 12. Test-circuit of model transformer with disc-to-disc short (resistances in series with inductances, resistances in parallel with capacitances, inductances and capacitances are in X, kX, lH and pF respectively)
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to be introduced and six Test-circuits can be built. The comparison of Test-circuit with corresponding Reference-circuit would identify mechanical deformations. In this approach, DPI data are used to build circuit models. If the DPI plot does not exhibit peaks and troughs clearly then this approach cannot be deployed for fault diagnostics. Conclusion FRA-based method is proposed for assessing the mechanical integrity of model transformer. The axial fault is introduced by moving the outer winding vertically whereas, radial fault is emulated by disturbing the ground plane. Apart from this, disc to disc short is also considered. Such changes in the winding get reflected as changes in inductances and capacitances in the reduced-order circuit model. Comparing the values of circuit elements with that of its healthy counterpart reveals the location and extent of introduced changes. In all the case studies, there is a good agreement between the DPI plots of synthesised circuit and the model transformer. References [1] Waters M. The short-circuit strength of power transformers. London: McDonald & Co.; 1966. [2] Kulkarni SV, Khaparde SA. Transformer engineering – design and practice. New York: Marcel Dekker, Inc.; 2004. [3] Nirgude PM, Ashokraju D, Rajkumar AD, et al. Application of numerical evaluation techniques for interpreting frequency response measurements in power transformers. IET Sci Meas Technol 2008;2(5):275–85. [4] Secue JR, Mombello E. Sweep frequency response analysis (SFRA) for the assessment of winding displacements and deformation in power transformers. Electr Power Syst Res 2008;78(6):1119–28. [5] Dick EE, Erven CC. Transformer diagnostic testing via frequency response analysis. IEEE Trans PAS 1987;97(6):2144–53. [6] Ryder SA. Methods for comparing frequency response analysis measurements. In: Proc. IEE int. symp. electr. insul., Boston, MA USA; 2002. p. 187–90. [7] Saravanakumar A. Experimental investigation on terminal connection and system function pair during SFRA testing on three phase transformers. Int J Electr Power Energy Syst 2014;58:101–10. [8] 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 Electr Power Energy Syst 2014;62:890–6. [9] Herrera W, Aponte G, Pleite J, et al. A novel methodology for transformer lowfrequency model parameters identification. Int J Electr Power Energy Syst 2013;53:643–8.
[10] Ragavan K, Satish L. Localization of changes in a model winding based on terminal measurements: experimental study. IEEE Trans Power Deliv 2007;22 (3):1557–65. [11] Satish L, Sahoo SK. Locating faults in a transformer winding: an experimental study. Electr Power Syst Res 2009;79:89–97. [12] Pramanik S, Satish L. Localisation of discrete change in a transformer winding: a network-function-loci approach. IET Electr Power Appl 2011;5(6):540–8. [13] Hejazi MA, Gharehpetian GB, Moradi G, et al. Online monitoring of transformer winding axial displacement and its extent using scattering parameters and knearest neighbour method. IET Gener Transm Distrib 2011;5(8):824–32. [14] Mitchell SD, Welsh JS. Modeling power transformers to support the interpretation of frequency-response analysis. IEEE Trans Power Deliv 2011;26(4):2705–11. [15] Levy EC. Complex curve fitting. IRE Trans Autom Control 1959;AC-4:37–44. [16] De Tommasi L, Gustavsen B, Dhaene T. Roboust transfer function identification via an enhanced magnitude vector fitting algorithm. IET Control Theory Appl 2010;4(7):1169–78. [17] Soysal O. A method for wide frequency range modeling of power transformers and rotating machines. IEEE Trans Power Deliv 1993;8(4):1802–10. [18] Miki A, Hosoya T, Okuyama K. A calculation method for impulse voltage distribution and transferred voltage in transformer windings. IEEE Trans Power Ap Syst 1978;PAS-97(3):930–9. [19] Rahimpour E, Christian J, Feser K. Transfer function method to diagnose axial displacement and radial deformation of transformer windings. IEEE Trans Power Deliv 2003;18(2):493–505. [20] Karimifard P, Gharehpetian GB. A new algorithm for localization of radial deformation and determination of deformation extent in transformer windings. Electr Power Syst Res 2008;78(10):1701–11. [21] Bjerkan E, Hoidalen HK. High frequency FEM-based power transformer modeling: investigation of internal stresses due to network initiated overvoltages. Elect Power Syst Res 2007;77(11):1483–9. [22] Rashtchi V, Rahimpour E, Fotoohabadi H. Parameter identification of transformer detailed model based on chaos optimization algorithm. IET Electr Power Appl 2011;5(2):238–46. [23] Shintemirov A, Tang WJ, Tang WH, et al. Improved modelling of power transformer winding using bacterial swarming algorithm and frequency response analysis. Electr Power Syst Res 2010;80:1111–20. [24] Dent BM, Hartil ER, Miles JG. A method of analysis of transformer impulse voltage distribution using a digital computer. Proc IEE, A 1958;105 (23):445–59. [25] Hurter T, Ecklin G. The electromagnetic model for transformers. Brown Boveri Rev 1958;45(9):410–8. [26] Guillemin EA. Synthesis of passive networks. New York: John Wiley & Sons; 1957. [27] Mombello E. Impedances for the calculation of electromagnetic transients within transformers. IEEE Trans Power Deliv 2002;17(2):479–88. [28] Shah KR, Ragavan K. Estimation of transformer winding capacitances through frequency response analysis – an experimental investigation. Int J Emerg Electr Power Sys (IJEEPS) 2013;14(6):549–59. [29] Ragavan K, Satish L. An efficient method to compute transfer function of a transformer from its equivalent circuit. IEEE Trans Power Deliv 2005;20 (2):780–8. [30] Abetti PA, Maginniss FJ. Natural frequencies of coils and windings determined by equivalent circuit. AIEE Trans PAS 1953;72(2):495–504.
Contents lists available at ScienceDirect
Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Assessing mechanical deformations in two-winding transformer unit using reduced-order circuit model Krupa Rajendra Shah ⇑, K. Ragavan Indian Institute of Technology Gandhinagar, Ahmedabad, India
a r t i c l e
i n f o
Article history: Received 25 August 2014 Received in revised form 8 December 2015 Accepted 9 December 2015
Keywords: Circuit synthesis Diagnostics Frequency response analysis Mechanical deformations Terminal measurement Transformer
a b s t r a c t Certain level of mechanical deformations in winding does not hamper the normal performance of power transformer. However, such incipient deformations if unattended could result in permanent failure of transformer. To this end, an approach is proposed to assess the severity of mechanical deformations in transformer winding. These deformations get reflected as changes in its high frequency behaviour. Hence, characterising the high frequency behaviour is essential. This requires building physically realisable ladder circuit corresponding to each winding. Thus, n-winding transformer is represented by n electrically and magnetically coupled ladder networks. In such scenario, the objective of fault diagnostics becomes very challenging. In this effort, realising the multi-winding unit by reduced-order ladder circuit model is explored. This approach essentially involves energising one winding at a time. Then, reducedorder ladder circuit of considered unit is synthesised from its measured driving-point impedance data. It is shown how these circuit models could be used for identifying mechanical deformations. To demonstrate capability of the method, two-winding model transformer is considered and deformations are introduced in its outer winding. Then, reduced-order circuit models are synthesised corresponding to healthy and faulty state of model transformer. The location of fault is identified by the changed parameter in the circuit. Further, the amount of change reveals the severity of introduced deformation. In all the cases, synthesised reduced-order circuit model agrees with that of model transformer with regard to driving-point impedance plot and results are found satisfactory. Ó 2016 Elsevier Ltd. All rights reserved.
Motivation Power transformer is a vital component of power system. Hence its uninterrupted functioning is of utmost importance. The mechanical integrity of a transformer is challenged by excessive electromagnetic forces generated in axial and radial direction due to short circuit current [1–3]. Apart from this, lightning and switching surges, transportation and rough handling of transformers are also responsible for reducing mechanical strength of the transformer. Cumulative effect of exposure to such abnormality would alter winding geometry. Such mechanical deformations are initially incipient faults. Eventually such fault-like condition grows and would lead to catastrophic failure. Therefore, identifying mechanical deformation and assessing its severity is paramount for smooth functioning of transformer and power system. Usually, changes in the mechanical structure of power transformer get reflected in the frequency range starting from few kHz to 1 MHz [4]. Hence, its high-frequency behaviour can be con⇑ Corresponding author. E-mail address: [email protected] (K.R. Shah). http://dx.doi.org/10.1016/j.ijepes.2015.12.035 0142-0615/Ó 2016 Elsevier Ltd. All rights reserved.
sidered. To this end, frequency response analysis (FRA) can be performed on transformer [5–14]. The acquired FRA data can be converted into network function by deploying curve fitting techniques [15,16]. A detailed analysis of such network function can be useful to detect deformation in transformer. Once, mechanical deformation is detected, the subsequent crucial task becomes identifying location of fault and its severity. To this end, it is necessary to look for some approach that is capable of addressing the goal of localisation. This becomes motivation for the research effort. Literature survey In order to locate deformation, it appeared worthwhile to have complete representation of the winding. This can be achieved if a physically realisable circuit model is built. The circuit should be synthesised such that it characterises accurately the highfrequency behaviour of transformer winding. Since most of the flux passes through air for high frequency excitation, linear circuit representation can be considered [17,18]. Such circuit models can be found in [10–12] [19–23]. Further, circuit model for two-winding transformer is shown in subsequent section (Fig. 1).
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K.R. Shah, K. Ragavan / Electrical Power and Energy Systems 79 (2016) 235–244
1 C'’g1
2 C’'g2
R'’g1
R'’g3
C’'gM
2' r'’1
R’'g(M+1)
C’'sM M
3'
R’'s2 L’'22
C’'g(M+1)
R'’gM
C'’s2
R’'s1 L’'11
C’'g3
R'’g2
C'’s1 Line-end 1' Winding-2
M
r'’2
R’'sM L’'MM
r''M
Neutral-end Winding-2
L'12 L'2M L'1M Cw1
Cw2
Rw1
Rw2
Cw3
Rw3
CwN
RwN
Cw(N+1)
Rw(N+1)
L’1N L’2N
L’12 L’11 Line-end Winding-1 1
C’g1
L’22
r'1 R’s1
C’s1 R’g1 1
L’NN
r'2 R’s2
2
N
3
R’g2
Neutral-end Winding-1
C’sN
C’s2 C’g2
r'N R’sN
C’g3
R’g3
C’gN
C’g(N+1)
R’gN
R’g(N+1)
N
2
Fig. 1. High frequency circuit model of two-winding transformer.
The ladder circuits were used in [10–12,19,20] for the purpose of locating faults. In these publications, initially circuit corresponding to healthy transformer winding was synthesised. Later, some changes were introduced and then corresponding to faulty winding, one new circuit was synthesised. Then, these circuits were compared and faults were identified. Thus, the principle of identifying the changes remains the same. However, in [19,20], the ladder network was developed using design details. Whereas, in [10–12], terminal voltage and current data are utilised. Then, driving-point impedance (DPI) function was obtained. Utilising which physically realisable ladder network was synthesised. Thus, these two approaches are different. Further, there exists discrepancy with regard to test-arrangement as indicated below: In [10,12], single winding is considered. Then, discrete changes in capacitances were introduced in the winding and these changes were viewed as deformations. In [11], inductive changes were also introduced by shorting few discs along with the discrete capacitive changes. In [19,20] two-windings are considered. Then, axial displacement is introduced by moving one winding with respect to other. Further, radial deformations are introduced by deforming one of the windings. Thus, it is clear that the approach based on design details was validated considering ‘‘physical deformations” in the winding. Further, the test unit comprised of ‘‘two-windings”. On the other hand, the terminal measurement based approach was demonstrated considering ‘‘discrete changes” in ‘‘isolated winding”. Nonetheless, it is completely non-invasive.
From the above discussion, it is understood that there is a need to identify physical deformations through terminal measurement in two-winding unit. Hence, it becomes objective of the work. For this purpose, FRA is deployed.
Reduced-order circuit model The high frequency circuit model of two-winding transformer unit is comprised of two-ladder networks as in Fig. 1 with each ladder network corresponds to a winding. Further, the electric and magnetic coupling between these windings can be accounted by inter-winding capacitances and mutual-inductances respectively. The considered equivalent circuit has N and M sections corresponding to winding-1 and winding-2 respectively. The capacitances and inductances per section are represented as series capacitances ðC 0si ; C 00si Þ, shunt-capacitances ðC 0gi ; C 00gi Þ, resistances ðr 0i ; r00i Þ, self inductances ðL0ii ; L00ii Þ and inter-winding capacitances ðC wi Þ. Mutual inductances between any two sections ‘i’ and ‘j’ can be denoted by L0ij ; L00ij . The dielectric losses could be represented by large values of resistances ðR0gi ; R00gi ; R0si ; R00si ; Rwi Þ in parallel to capacitors. Thus, transformer with n-windings would be represented by nladder networks; coupled electrically and magnetically. Hence, it is clear that the number of variables (circuit parameters) to be estimated will increase with increase in number of windings. Thus, complexity increases during circuit synthesis. This problem could be simplified if the entire system is equivalently be represented by single ladder network. With this, it is possible to reduce number of variables significantly. Further, applicability of the diagnostic method would not be restricted with regard to number of windings in transformer.
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K.R. Shah, K. Ragavan / Electrical Power and Energy Systems 79 (2016) 235–244
L1N L2N
L12
L11 Line-end
L22
r1 RS1
RS2
2
1 Rg1 1
Rg2
Neutral-end
N
CS2 Cg2
rN RSN
3
CS1 Cg1
LNN
r2
CSN Cg3
2
Rg3
CgN
RgN
Cg(N+1)
Rg(N+1)
N
Fig. 2. Reduced-order circuit model of multi-winding unit under considered terminal condition.
In order to check feasibility of representing transformer by single ladder network, several literature are explored. It is mentioned in [24] that the influence of all the neighbouring windings on the winding under investigation can be minimised if all the ‘‘neighbouring windings are shorted and grounded”. Further, it is reported that the transformer with said terminal condition can be considered equivalent to isolated winding and such system can be represented by single ladder network. To certain extent, findings in [24] can be supported by literature [17,18]. It is mentioned here that if neighbouring windings are shorted then the terminal behaviour of iron-core transformer would become similar to that of air-core transformer. Thus, influence of one winding on other can be ignored with regard to mutual inductance. However, it is not clear how inter-winding capacitances of multi-winding transformer are accounted by isolated winding? To this end, few more literature are explored. It is found, in [25] that the shorted neighbouring windings can be considered equivalent to shields. Since, neighbouring winding is grounded, equivalently the shield can also be considered grounded. Thus, multi-winding transformer is approximated by transformer with single winding and ground plane. To this end, inter-winding capacitances are now reflected between winding and ground plane. Thus, it can be inferred that ‘‘under the above mentioned terminal condition (shorted and grounded neighbouring windings), the iron-core transformer is equivalent to an air-core transformer with single winding and grounded shield(s)”. Hence, it is possible to represent such system by single ladder network. Its shunt capacitances would account for inter-winding capacitances. Since number of energy storing elements is less in single ladder network as compared to multi-ladder network, it can be termed as reducedorder circuit model. One such representation is shown in Fig. 2. It will be shown subsequently that such reduced circuit model is helpful in analysing mechanical condition of the considered unit. The number of reduced-order ladder networks would be same as number of windings in the considered unit. Proposed methodology This approach essentially involves measuring the driving-point impedance (DPI) of the considered unit. While acquiring DPI, one winding is energised by sinusoidal signal. Moreover, neighbouring windings are to be shorted and grounded. Later, the measured DPI will be used to acquire information about various characteristic features of the considered unit such as natural frequencies, dc resistance, effective shunt capacitance, equivalent inductance and capacitance. Using these characteristic features, the proposed method synthesises a mutually coupled, lumped parameter reduced-order ladder network.
For assessing the status of the system, these features are to be extracted at two different instances. Initially, healthy system is characterised by single ladder network and is termed as ‘‘Refer ence-circuit”. Then at the time of inspecting system, similar circuit is synthesised and this new ladder network is termed as ‘‘Testcircuit”. Comparing the parameters of Test-circuit with that of Reference-circuit, the condition of the considered unit can be understood. While synthesising circuit, convergence is deemed to have occurred, when differences between the characteristic features obtained from measurement and circuit are less than a preset tolerance. Let us say, the considered unit has n-windings. Then, similar procedure is to be repeated for remaining n 1 windings. Thus, for assessing status of n-winding transformer, n Reference- and Test-circuits can be synthesised. With this philosophy, the procedure for obtaining characteristic features of multi-winding transformer with the considered terminal condition (shorted and grounded neighbouring windings) is detailed below. Determining characteristic features Natural frequencies For determining natural frequencies of the system, FRA is performed. To this end, sinusoidal excitation is fed between the line- and grounded neutral-end of the winding under test. Then, magnitude ðjZjÞ and phase ðhÞ of the DPI are obtained. This exercise is done for wide range of frequencies and variation of magnitude of DPI with frequency is plotted. The peaks and troughs of magnitude plot correspond to open-circuit natural frequencies (f o , ocnf) and short-circuit natural frequencies (f s , scnf) respectively [26]. Occurrence of such peaks and troughs imply that the effect of both inductance and capacitances is present in the corresponding frequency range. Equivalent Inductance, Leq To obtain high-frequency circuit model, the value of winding inductance corresponding to high-frequency is needed. The obvious approach is to feed high-frequency excitation and determine the value of reactance using magnitude and phase of DPI. However, with high-frequency excitation, the capacitances get reflected and the reactance is due to both inductance and capacitance. Hence, it is not directly possible to obtain the value of high-frequency inductance. To this end, an alternative is to be identified. With the considered terminal condition, the effect of core is almost eliminated [17,18]. This gives a hint that the value of inductance of the considered unit is nearly equivalent to air-core inductance. Further, it is reported in [27] that the value of air-core inductance decreases with frequency and remains constant beyond
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K.R. Shah, K. Ragavan / Electrical Power and Energy Systems 79 (2016) 235–244
certain frequency. To this end, variation in inductance with frequency is observed. The inductance can be obtained through Eq. (1).
L¼
jZjsin h 2pf
ð1Þ
Then, the frequency range is identified in which inductance remains nearly constant. Any of these frequencies can be considered as f 1 and corresponding inductance can be termed as Leq . Essentially, f 1 should be well below the first ocnf. If so, then the presence of capacitive effects can be ignored. Hence, it is possible to have a better estimate of air-core winding inductance. Equivalent Capacitance, C eq A ladder network of N sections with its neutral grounded and neighbouring windings shorted and grounded will have the following pole-zero form [10].
ZðsÞ ¼
Q b1 ðs sÞ N1 i¼1 ðs zi Þðs zi Þ QN i¼1 ðs pi Þðs pi Þ
ð2Þ
where b1 : scaling factor, s: real zero, zi ; zi : complex-conjugate zero pair, and pi ; pi : complex-conjugate pole pair. For high-frequency excitation, the input impedance is capacitive in nature. Hence,
b ¼ 1 C eq s s 1 b1 ¼ C eq 1
For low-frequency excitation, the input impedance is series connection of resistance ðRdc Þ and inductance ðLeq Þ. Hence, the real zero is,
s¼
Rdc Leq
ð3Þ
The peaks and troughs in the DPI plot correspond to the poles and zeros of DPI function. Also, as the absolute value of the imaginary parts of complex frequencies is significantly higher than their real parts (losses in the winding are low), pi ’s and zi ’s in Eq. (3) can be approximated to the corresponding imaginary parts. Hence Eq. (3) can be rewritten as,
Q 2 4p2 N f b1 Leq QN1i¼12 oi i¼1 f si
ð4Þ
Thus, determination of C eq is possible once natural frequencies and Leq are known. Effective shunt capacitance, C g;eff For this purpose, neutral-end of the winding under inspection is to be isolated from the ground. Further, line and neutral terminals are to be shorted. Then, sinusoidal excitation of frequencies (f) well below first ocnf is to be applied between line-end and ground. Let the corresponding impedance be Z\h. Next, Eq. (5) for calculating capacitance (C) can be used.
C¼
sin h 2pf jZj
Synthesising Reference-circuit Algorithm 1. Reference circuit 1: The resistance in series with inductor can be given by,
ri ¼
Rdc ; 8i ¼ 1; . . . ; N N
ð5Þ
Since end terminals of the winding are shorted and low frequency excitations are fed, the circuit in Fig. 2 can be represented by just parallel combination of resistances and shunt capacitances.
ð6Þ
whereRdc is dc resistance of winding. 2: The value of resistors in parallel with capacitors can be considered very high (in this work it is approximated as 100 kX). 3: To determine effective series capacitance ðC s;eff Þ of the winding, it is essential to know about C g;eff and initial voltage distribution constant ðaÞ.
C s;eff ¼
For DC excitation, that is, at s ¼ 0, the input impedance becomes resistive and is Rdc . Hence, the scaling factor b1 can be expressed as,
QN i¼1 ðpi Þðpi Þ b1 ¼ Leq QN1 i¼1 ðzi Þðzi Þ
Thus, it could be possible to relate C with C g;eff . However, for very low frequencies (in order of few Hz), the impedance would be very high and hence, current drawn from the source becomes feeble. This would lead to inaccurate current measurement. Hence estimated C may not indicate actual C g;eff of the winding. As frequency rises (in order of few kHz), the value of impedance will come down and hence accurate current measurement becomes possible. Thus, for those frequencies, it is possible to have C ¼ C g;eff . To this end, low frequency range (between few Hz to few kHz) is identified in which the estimated value of C is nearly constant. It can be inferred that this value does not suffer from measurement inaccuracy. Hence, for these frequencies C is same as winding C g;eff . The start of this frequency range is referred as f 2 . Once these characteristic features are known pertaining to system of healthy winding, the Reference-circuit can be synthesised using Algorithm 1.
C g;eff
ð7Þ
a2
However, the value of a varies with the type of winding and its estimation is to be done. For this purpose, the algorithm in [28] can be used. It involves varying a and N iteratively. For each value of a and N, equivalent capacitance ðC eq Þ is calculated by analytical formula.
C eq ¼
Cg 1 þ 2 C1s þ C þ1 1 g
ð8Þ
1 þ Cs
Then, the normalised error between equivalent capacitance determined by measurement and analytical formula can be expressed as,
C eq C eq 100 eða; NÞ ¼ C eq
ð9Þ
Thesum of errors corresponding to a particular value of a and different values of N is determined as,
EðaÞ ¼
N max X
eða; NÞ
ð10Þ
N¼Nmin
where N min is considered greater than number of observable ocnf and N max is 3Nmin . These limits are found suitable and hence considered. However, they can be altered. In the array EðaÞ, the global minimum is identified and the corresponding value of a is substituted in Eq. (7) for determining C s;eff . 4: Then eða; NÞ pertaining to considered a can be observed. In this array, global minimum is identified and the corresponding value of N is considered for synthesising circuit.
K.R. Shah, K. Ragavan / Electrical Power and Energy Systems 79 (2016) 235–244
5: Initially, the winding is free from faults. Hence, it is more appropriate to assume that it has symmetry and uniformity. As a result, its equivalent representation can be considered to have capacitances as mentioned below.
C si ¼ NC s;eff ; 8i ¼ 1; . . . ; N C gi ¼ C gi ¼
C g;eff 2N C g;eff N
;
8i ¼ 1; N þ 1
;
8i ¼ 2; . . . ; N
ð11Þ
6: From the known value of Leq and with the assumptions that (i) self inductances are identical, (ii) the mutual inductances between any two coils separated by same physical distance are equal and (iii) mutual inductances between the coils decrease with the increase in the physical separation, initial values and boundaries to L1j ; 8j ¼ 1; . . . ; N can be chosen. The initial value of self-inductance ðL11 Þ can be considered as small fraction of measured Leq . 7: The equivalent inductance ðLeq Þ can be estimated. N X N X Leq ¼ Lij
ð12Þ
i¼1 j¼1
L L 8: if eqLeq eq 100 < 2:5% then 9:
Construct the circuit model. Applying state-space analysis, determine the natural frequencies of the system under consideration. Then, determine ocnf and scnf ðf o ; f s Þ analytically [29]. 10: else 11: 12: 13:
14:
Go to step 15. if f offo o 100 < 2:5% and f s ffs s 100 < 2:5% then The corresponding circuit can be considered as the representation of the system under consideration with the proposed terminal condition. else
15:
Change the values of inductances iteratively and go to step 7. In this work, inductances are increased by 1% of L11 . 16: end if 17: end if
Synthesising Test-circuit The axial force is produced by interaction of radial component of leakage field with the current. Any minor misalignment between magnetic centres of the windings, asymmetry in ampere-turn due to tapping, unequal winding heights would generate enormous axial forces. This might result into displacement of the complete winding (or its turns) in axial direction. On the contrary, the radial forces are produced by interaction of axial leakage field with the current. This axial leakage field would produce hoop stresses on the outer winding and compressive stresses on the inner winding [1–3]. These events change geometry. That is, inductances and capacitances would get altered with regard to their reference values. To realise the system under inspection by a ladder circuit, once again characteristic features namely natural frequencies, effective shunt capacitance, equivalent inductance and capacitance are to be extracted as explained earlier. The procedure of synthesising the Test-circuit would involve few changes in Algorithm 1 and only those are indicated below.
239
Elements of the Reference-circuit are chosen as initial values. The number of sections are considered same as that of Reference-circuit. However, a is varied iteratively. Corresponding to each a; C s;eff is determined. Series- and shunt-capacitances are also iteratively estimated along with self- and mutual-inductances. Hence, the number of variables becomes,
C si ; 8i ¼ 1; . . . ; N C gi ; 8i ¼ 1; . . . ; N þ 1 Lij ;
8i; j ¼ 1; . . . ; N;
ð13Þ jPi
The effective series- and shunt-capacitance are also estimated along with equivalent-capacitance.
C g;eff ¼ C g1 þ þ C gðNþ1Þ 1 1 1 ¼ þ þ C s;eff C s1 C sN
ð14Þ
At convergence, following equations should be satisfied.
C g;eff C g;eff 100 < 2:5% C g;eff C s;eff C s;eff 100 < 2:5% C s;eff C eq C eq 100 < 2:5% C eq
ð15Þ
To assess the severity of the mechanical deformations, these two circuits are used. Results and discussion The objective is to examine capability of the method in interpreting high frequency behaviour of two-winding transformer unit. Further, it should be able to locate and assess the severity of mechanical deformation in it. To this end, two-winding unit is considered. The outer winding contains 15 discs with each disc has 10 turns. The conductor is of rectangular cross-section of 5 2 mm. The total height of the winding is about 180 mm. Inside disc winding, single layer winding of same height and diameter of 219 mm is kept centrally. As discussed earlier, behaviour of iron-core winding can be approximated to air-core winding under considered terminal condition. This equivalence avoids building iron-core unit for demonstration purpose. Once the windings are assembled together with iron-core, the overall unit becomes very rigid. It becomes difficult to introduce changes later, if needed. Considering this, it is decided to validate the proposed algorithm with air-core unit. The ground plane in actual transformer can be established by connecting core and tank to earth potential. Such equipotential surfaces can be established in the considered set-up by providing concentric non-magnetic shields [30]. To this end, two shields are used in the considered set-up and connected to earth potential. One of these shields has 210 mm diameter and is kept inside the inner winding. Another one has 332 mm diameter and is kept outside the outer winding. This entire set-up of two-windings and two shields is termed as model transformer. The test set-up comprising of outer-winding is shown in Fig. 3(a). The cut-section of model transformer is shown in Fig. 3(b). The schematic arrangement of model transformer in its healthy state is shown in Fig. 4(a). In Fig. 4(b) upward movement of disc winding is shown to represent axial displacement. The changes in radial direction can be achieved by removing some portion of shield as shown in Fig. 4(c). Such physical changes in geometry
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Inner Shield
Tappings from Inner Winding Outer Winding
Outer Shield
(a)
(b)
Fig. 3. Test set-up for model transformer.
Axial Displacement Outer Shield
Outer Winding
Radial Deformation
Inner Shield
Inner Winding
(a)
(b)
(c)
Fig. 4. (a) Healthy state, (b) axial displacement and (c) radial deformation.
would change the area available for flux to flow. Further, it is obvious that few conductors may not be facing ground plane. As a result, magnetic and electric field would undergo changes. Hence, the winding inductances and capacitances would be different than that pertaining to healthy winding. Thus, with the considered setup, it is possible to introduce physical changes in the winding. In this work, outer disc winding is energised and terminal measurement is performed. While doing so, inner winding is shorted and grounded. Later, reduced-order circuit model of the twowinding model transformer is synthesised from the characteristic features and proposed algorithm. The corresponding circuit is termed as Reference-circuit. Then, physical deformations are introduced and once again reduced-order circuit models are built. Such models are termed as Test-circuit. It is shown that, by comparing Reference-circuit with Test-circuit, it is possible to reveal location and severity of introduced deformations in the considered unit. Determining characteristic features Initially, shields are connected to earth-terminal to established ground plane. Then, FRA is performed by energising the outer disc
winding. While doing so, its neutral terminal is connected to ground and terminal condition (shorted and grounded layer winding) is followed. Corresponding DPI Plot is shown in Fig. 5 with solid line. Its natural frequencies (f o ; f s ) are listed in Table 1. Using the acquired DPI data, inductance (L) is calculated for different frequencies below first ocnf. It is found that the inductance is nearly constant in the frequency range of 20–100 kHz. Any frequency in this range can be considered to identify winding inductance. In this work, f 1 is considered as 40 kHz and corresponding Leq is 1154 lH. Since f 1 is well below first ocnf, it can be assumed that capacitive effects are not pronounced at this frequency. Hence, Leq can be considered as good estimation of air-core inductance. The magnitude and phase of DPI at f 1 is shown in Table 2. The value of Rdc is 1.2 X. Next, the values pertaining to observable peaks and troughs are substituted in Eq. (4). In practice, the observable f oi ; f si might not be same as N, N 1 respectively [10]. Hence, N in the above equation will be replaced by the number of observable peaks. With this, b1 is obtained and corresponding C eq becomes 61.2 pF. For determining C g;eff , the neutral terminal of disc winding is isolated from earth. The line- and neutral-terminals are shorted
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Winding Circuit
1
|Z| (k Ω)
10
1
2
3
4
5
6
7
8
L1j
57.9
24.9
18.8
6.6
3
2.7
2.5
0.2
Table 5 Inductance matrix.
−1
−2
10
0
0.5
1
1.5
2
2.5
Frequency (MHz) Fig. 5. DPI plot of model transformer with its healthy disc winding under test.
Table 1 Characteristic features – model transformer with its healthy disc winding under test. 400 403.9 871 868.4
(kHz) (kHz) (kHz) (kHz)
1129 1131 1610 1604
1760 1765 2290 2295
2420 2418
Max. error < 2:5%
Leq ¼ 1154 lH, C g;eff ¼ 320 pF, C eq ¼ 61:2 pF Leq ¼ 1154 lH, C g;eff ¼ 320 pF, C eq ¼ 61:5 pF
Max. error < 2:5%
Table 2 Measured terminal quantities – model transformer with its healthy disc winding under test. Excitation frequency (kHz)
Input impedance (X)
Calculated parameter
40 8
290\86.92 62000\-89
Leq ¼ 1154 lH C g;eff ¼ 320 pF
together. Then, sinusoidal excitation of various frequencies starting from 50 Hz is applied between line terminal of disc winding and ground. The capacitances are found nearly constant in the range of 8–150 kHz. Thus, the frequency f 2 is identified as 8 kHz. The corresponding magnitude and phase of DPI is presented in Table 2. Using these data C g;eff is determined as 320 pF. Later, input capacitances ðC eq Þ pertaining to various a and N are estimated iteratively and normalised error between C eq and C eq is obtained. Few of these values are presented in Table 3 for brevity. From Table, it is found that EðaÞ is least for a as 5.5. This value together with C g;eff as 320 pF can be used to estimate C s;eff . For this case it is 10.6 pF. Further, it can be observed from Table 3 that for
Table 3 Normalised error between C eq and C eq for different iterations – healthy disc winding.
a
N
lH) – model transformer with its healthy disc winding under test.
i
0
10
10
fo fo fs fs
Table 4 Inductance (in
4.5
5
5.5
6
6.5
5 6 7 8 9 11 12 13 14 15
27.38 24.06 22.01 20.67 19.74 18.57 18.19 17.89 17.66 17.46
16.90 13.28 11.03 9.55 8.52 7.23 6.81 6.48 6.22 6.01
8.49 4.57 2.14 0.52 0.60 2.01 2.47 2.83 3.12 3.35
1.63 2.57 5.19 6.93 8.14 9.67 10.17 10.56 10.88 11.13
4.06 8.52 11.31 13.17 14.47 16.12 16.66 17.08 17.42 17.69
EðaÞ
223.00
100.00
31.51
86.00
152.00
j
1
2
3
4
5
6
7
8
L1j L2j L3j L4j L5j L6j L7j L8j
57.9 24.9 18.8 6.6 3 2.7 2.5 0.2
24.9 57.9 24.9 18.8 6.6 3 2.7 2.5
18.8 24.9 57.9 24.9 18.8 6.6 3 2.7
6.6 18.8 24.9 57.9 24.9 18.8 6.6 3
3 6.6 18.8 24.9 57.9 24.9 18.8 6.6
2.7 3 6.6 18.8 24.9 57.9 24.9 18.8
2.5 2.7 3 6.6 18.8 24.9 57.9 24.9
0.2 2.5 2.7 3 6.6 18.8 24.9 57.9
the considered a, error is least with N as 8. Hence, it is decided to build a ladder network with N = 8. Synthesising Reference-circuit It is straightforward to calculate per section values of shuntand series-capacitances once C g;eff ; C s;eff ; a and N are known. The values of C si ; 8i ¼ 1; . . . ; 8 and C gi ; 8i ¼ 2; . . . ; 8 are found as 84.63 pF and 40 pF respectively. Further, C g1 and C g10 are considered half of the 40 pF. For estimating inductances, sequential iterations are performed. The inductances associated with first section are shown in Table 4. Assuming symmetry, the complete inductance matrix can be constructed as shown in Table 5. Then, the reduced order Referencecircuit can be built as in Fig. 6. The DPI plot obtained from Reference-circuit is shown in Fig. 5 with dotted line and characteristic features (with sign) are presented in Table 1. The normalised error between characteristic features is also determined. The maximum value of this deviation is found to be less than 2:5%. Thus, it can be said that the constructed Reference-circuit or the reduced-order model is good representation of model transformer with its disc winding under test and layer winding shorted and grounded. Synthesising Test-circuit Axial displacement This case study represents axial movement of the disc winding in upward direction by 7 mm (4% of winding height) with respect to layer winding and shields. Corresponding DPI plot is shown in Fig. 7 with solid line. To check the validity of algorithm corresponding to the faulty state of the winding, initially terminal measurement is performed and natural frequencies are identified. These values are shown in Table 6. Then frequencies f 1 and f 2 are identified as 40 kHz and 8 kHz respectively. Corresponding to these frequencies, various terminal quantities such as equivalent inductance and capacitance are obtained and shown in Table 6. As the characteristic features are different than that of the healthy winding, it can be assumed that the winding is faulty. To locate the faulty section and access the severity, once again sequential iterations are performed. As Reference-circuit has 8 sections, the Test-circuit is to be synthesised with N as 8. This reduced-order Test-circuit is shown in Fig. 8. It is found that the series capacitances remained unchanged. The changes are observed in C g1 and inductances pertaining to first
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57.9 Line-end
57.9
0.15
100
2
1 100 1
Neutral-end 9
84.63 40
100
40
0.15 100
8
3
84.63
84.63 20
57. 9
0.15
100
100
20
100
40
100
8
2
Fig. 6. Reference-circuit of model transformer with its healthy disc winding under test (resistances in series with inductances, resistances in parallel with capacitances, inductances and capacitances are in X, kX, lH and pF respectively).
1
10
|Z| (kΩ )
Table 7 Inductance (in
Winding Circuit
lH) – model transformer with axially displaced disc winding.
j
1
2
3
4
5
6
7
8
L1j
60
25
20
7
3.2
2.8
2.6
0.2
0
10
−1
Winding Circuit
10
1
10 0
0.5
1
1.5
2
|Z| (kΩ)
−2
10
2.5
Frequency (MHz) Fig. 7. DPI plot of model transformer with axially displaced disc winding.
−1
10
Table 6 Characteristic features – model transformer with axially displaced disc winding. fo fo fs fs
(kHz) (kHz) (kHz) (kHz)
410 413.4 872 865
1150 1149 1620 1604
1790 1779 2270 2295
−2
10
2410 2424
Max. error < 2:5%
fo fo fs fs
57 .9
0.15
100
2
1 84.63 40 1
1.5
(kHz) (kHz) (kHz) (kHz)
405 400.7 880 864.2
2
2.5
57. 9
0.15 8
3
2
1760 1778 2300 2316
2420 2439
Max. error < 2:5% Max. error < 2:5%
obtained as shown in Fig. 9 and characteristic features are obtained as listed in Table 8. The frequencies f 1 and f 2 are 40 kHz and 8 kHz respectively. Utilizing characteristic features and the proposed method, new Test-circuit is synthesised as shown in Fig. 10. Its capacitive variables are nearly same as that of reference case except C g7 ; C g8 and C g9 . These values are 37 pF, 37 pF and 17 pF respectively. Once again, it is found that series capacitances are
100 84 .63 100
1132 1132 1630 1616
Leq ¼ 1172 lH, C g;eff ¼ 310 pF, C eq ¼ 61:8 pF Leq ¼ 1168 lH, C g;eff ¼ 311 pF, C eq ¼ 61:5 pF
Radial deformation The change in radial direction is achieved by disturbing the geometry of outer shield near neutral-end. This exercise is done on shield rather than winding. This is just to avoid damage to the winding. With this emulated radial fault, once again DPI plot is
100
1
Table 8 Characteristic features – model transformer with radially deformed disc winding.
section. The value of C g1 is found as 13 pF. The inductive variable that has got deviated from its reference value (Table 5) is shown in Table 7. The remaining variables are found same as that of reference values and hence not shown in the Table. From the results, it is possible to say that fault is near line-end of the winding. This is same as that of actual fault location. The comparison between measured and estimated DPI plots is shown Fig. 7. It shows reasonably good agreement.
13
0.5
Fig. 9. DPI plot of model transformer with radially deformed disc winding.
Max. error < 2:5%
60
0
Frequency (MHz)
Leq ¼ 1160 lH, C g;eff ¼ 310 pF, C eq ¼ 54:3 pF Leq ¼ 1160:3 lH, C g;eff ¼ 313 pF, C eq ¼ 54:5 pF
Line-end
0
10
0.15 Neutral-end 9
100 84 .63
40
100
40
20
100
100
8
Fig. 8. Test-circuit of model transformer with axially displaced disc winding (resistances in series with inductances, resistances in parallel with capacitances, inductances and capacitances are in X, kX, lH and pF respectively).
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57.9 Line-end
58.5
0.15
100
2
1
7
84.63 20
40
100
100
1
0.15 Neutral-end 9
100
8
84.63 100
37
58.5
0.15
100
84.63 100 8
37
7
17
100
Fig. 10. Test-circuit of model transformer with radially deformed disc winding (resistances in series with inductances, resistances in parallel with capacitances, inductances and capacitances are in X, kX, lH and pF respectively).
Table 9 Inductance (in
Table 11 Inductance (in
lH) – model transformer with radially deformed disc winding.
lH) – model transformer with disc-to-disc short.
j
1
2
3
4
5
6
7
8
j
1
2
3
4
5
6
7
8
L7j L8j
2.5 0.2
2.7 2.5
3 2.7
7 3
20 7
26.5 20
58.5 26.5
26.5 58.5
L1j L2j L3j L4j L5j L6j L7j L8j
50 24.9 18.8 6.6 3 2.7 0 0
24.9 50 24.9 18.8 6.6 3 0 0
18.8 24.9 50 24.9 18.8 6.6 1 0
6.6 18.8 24.9 50 24.9 18.8 2 1
3 6.6 18.8 24.9 50 24.9 9 2
2.7 3 6.6 18.8 24.9 50 11 9
0 0 1 2 9 11 30 11
0 0 0 1 2 9 11 15
Table 10 Characteristic features – model transformer with disc-to-disc short. fo fo fs fs
480 485.6 1060 1040.3
(kHz) (kHz) (kHz) (kHz)
1320 1306 1920 1894
2070 2062 2710 2694
2850 2824
Max. error < 2:5%
Leq ¼ 914 lH, C g;eff ¼ 320 pF, C eq ¼ 60:4 pF Leq ¼ 903:4 lH, C g;eff ¼ 320 pF, C eq ¼ 61:5 pF
tured correctly by reduced order model. Further, there is a good agreement between DPI pattern of the winding and circuit (Fig. 9).
Max. error < 2:5%
Disc to disc fault Next, disc to disc short is introduced near neutral-end. Corresponding DPI plot and characteristic features are presented in Fig. 11 and Table 10 respectively. Then, Test-circuit is synthesised as in Fig. 12. From the results, it is found that inductance matrix has changed dominantly (Table 11). Further, the major deviation is observed near neutral-end. The DPI plot in Fig. 11 shows reasonably good match between estimated and measured terminal behaviour. From the results, it emerges that the introduced faults are correctly identified. The severity of fault can be judged by comparing variables of Reference- and Test-circuit. In this section, outer disc winding is considered as test winding. Similarly, inner layer winding can also be considered and terminal measurement can be performed on it. While doing so, the outer winding is to be shorted and grounded. Correspondingly, reduced-order circuit models are to be synthesised. This exercise is very straightforward and hence the results are not presented. It is believed that the approach can also be extended to three phase transformer that has six windings. At a time, one winding is to be considered and the remaining five are to be shorted and grounded. Then, reduced-order circuit model is to be synthesised. This exercise is to be repeated six times and hence six Referencecircuits are to be synthesised. Later, mechanical deformations are
same as before. The inductive variable that has shown deviation form its reference value (Table 5) is presented in Table 9. From the results, it is clear that the shunt capacitances and inductances which are closer to neutral-end have undergone several changes. Thus, it can be concluded that the changes are cap-
Winding Circuit
1
|Z| (kΩ)
10
0
10
−1
10
−2
10
0
0.5
1
1.5
2
2.5
Frequency (MHz) Fig. 11. DPI plot of model transformer with disc-to-disc short.
50 Line-end
30
0 .1 5 100
2
1
7
84.63 20
100
40 1
100
40
15
0 .15 100
84.63 100 7
8 40
0 .1 5 Neutral-end 9
100 84.63 100 8
20
100
Fig. 12. Test-circuit of model transformer with disc-to-disc short (resistances in series with inductances, resistances in parallel with capacitances, inductances and capacitances are in X, kX, lH and pF respectively)
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K.R. Shah, K. Ragavan / Electrical Power and Energy Systems 79 (2016) 235–244
to be introduced and six Test-circuits can be built. The comparison of Test-circuit with corresponding Reference-circuit would identify mechanical deformations. In this approach, DPI data are used to build circuit models. If the DPI plot does not exhibit peaks and troughs clearly then this approach cannot be deployed for fault diagnostics. Conclusion FRA-based method is proposed for assessing the mechanical integrity of model transformer. The axial fault is introduced by moving the outer winding vertically whereas, radial fault is emulated by disturbing the ground plane. Apart from this, disc to disc short is also considered. Such changes in the winding get reflected as changes in inductances and capacitances in the reduced-order circuit model. Comparing the values of circuit elements with that of its healthy counterpart reveals the location and extent of introduced changes. In all the case studies, there is a good agreement between the DPI plots of synthesised circuit and the model transformer. References [1] Waters M. The short-circuit strength of power transformers. London: McDonald & Co.; 1966. [2] Kulkarni SV, Khaparde SA. Transformer engineering – design and practice. New York: Marcel Dekker, Inc.; 2004. [3] Nirgude PM, Ashokraju D, Rajkumar AD, et al. Application of numerical evaluation techniques for interpreting frequency response measurements in power transformers. IET Sci Meas Technol 2008;2(5):275–85. [4] Secue JR, Mombello E. Sweep frequency response analysis (SFRA) for the assessment of winding displacements and deformation in power transformers. Electr Power Syst Res 2008;78(6):1119–28. [5] Dick EE, Erven CC. Transformer diagnostic testing via frequency response analysis. IEEE Trans PAS 1987;97(6):2144–53. [6] Ryder SA. Methods for comparing frequency response analysis measurements. In: Proc. IEE int. symp. electr. insul., Boston, MA USA; 2002. p. 187–90. [7] Saravanakumar A. Experimental investigation on terminal connection and system function pair during SFRA testing on three phase transformers. Int J Electr Power Energy Syst 2014;58:101–10. [8] 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 Electr Power Energy Syst 2014;62:890–6. [9] Herrera W, Aponte G, Pleite J, et al. A novel methodology for transformer lowfrequency model parameters identification. Int J Electr Power Energy Syst 2013;53:643–8.
[10] Ragavan K, Satish L. Localization of changes in a model winding based on terminal measurements: experimental study. IEEE Trans Power Deliv 2007;22 (3):1557–65. [11] Satish L, Sahoo SK. Locating faults in a transformer winding: an experimental study. Electr Power Syst Res 2009;79:89–97. [12] Pramanik S, Satish L. Localisation of discrete change in a transformer winding: a network-function-loci approach. IET Electr Power Appl 2011;5(6):540–8. [13] Hejazi MA, Gharehpetian GB, Moradi G, et al. Online monitoring of transformer winding axial displacement and its extent using scattering parameters and knearest neighbour method. IET Gener Transm Distrib 2011;5(8):824–32. [14] Mitchell SD, Welsh JS. Modeling power transformers to support the interpretation of frequency-response analysis. IEEE Trans Power Deliv 2011;26(4):2705–11. [15] Levy EC. Complex curve fitting. IRE Trans Autom Control 1959;AC-4:37–44. [16] De Tommasi L, Gustavsen B, Dhaene T. Roboust transfer function identification via an enhanced magnitude vector fitting algorithm. IET Control Theory Appl 2010;4(7):1169–78. [17] Soysal O. A method for wide frequency range modeling of power transformers and rotating machines. IEEE Trans Power Deliv 1993;8(4):1802–10. [18] Miki A, Hosoya T, Okuyama K. A calculation method for impulse voltage distribution and transferred voltage in transformer windings. IEEE Trans Power Ap Syst 1978;PAS-97(3):930–9. [19] Rahimpour E, Christian J, Feser K. Transfer function method to diagnose axial displacement and radial deformation of transformer windings. IEEE Trans Power Deliv 2003;18(2):493–505. [20] Karimifard P, Gharehpetian GB. A new algorithm for localization of radial deformation and determination of deformation extent in transformer windings. Electr Power Syst Res 2008;78(10):1701–11. [21] Bjerkan E, Hoidalen HK. High frequency FEM-based power transformer modeling: investigation of internal stresses due to network initiated overvoltages. Elect Power Syst Res 2007;77(11):1483–9. [22] Rashtchi V, Rahimpour E, Fotoohabadi H. Parameter identification of transformer detailed model based on chaos optimization algorithm. IET Electr Power Appl 2011;5(2):238–46. [23] Shintemirov A, Tang WJ, Tang WH, et al. Improved modelling of power transformer winding using bacterial swarming algorithm and frequency response analysis. Electr Power Syst Res 2010;80:1111–20. [24] Dent BM, Hartil ER, Miles JG. A method of analysis of transformer impulse voltage distribution using a digital computer. Proc IEE, A 1958;105 (23):445–59. [25] Hurter T, Ecklin G. The electromagnetic model for transformers. Brown Boveri Rev 1958;45(9):410–8. [26] Guillemin EA. Synthesis of passive networks. New York: John Wiley & Sons; 1957. [27] Mombello E. Impedances for the calculation of electromagnetic transients within transformers. IEEE Trans Power Deliv 2002;17(2):479–88. [28] Shah KR, Ragavan K. Estimation of transformer winding capacitances through frequency response analysis – an experimental investigation. Int J Emerg Electr Power Sys (IJEEPS) 2013;14(6):549–59. [29] Ragavan K, Satish L. An efficient method to compute transfer function of a transformer from its equivalent circuit. IEEE Trans Power Deliv 2005;20 (2):780–8. [30] Abetti PA, Maginniss FJ. Natural frequencies of coils and windings determined by equivalent circuit. AIEE Trans PAS 1953;72(2):495–504.