A novel methodology for transformer low-frequency model parameters identification

Electrical Power and Energy Systems 53 (2013) 643–648

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

A novel methodology for transformer low-frequency model parameters identification Wilder Herrera a,⇑, Guillermo Aponte a, Jorge Pleite b, Carlos Gonzalez-Garcia b a b

Universidad del Valle, Ciudad Universitaria Meléndez, Calle 13, No. 100-00, Cali, Colombia Universidad Carlos III de Madrid, Av. Universidad 30-28911 Leganés, Madrid, Spain

a r t i c l e

i n f o

Article history: Received 30 November 2012 Received in revised form 11 May 2013 Accepted 21 May 2013

Keywords: FRA Frequency response Magnetic circuit Magnetization inductance Magnetic core Modeling

a b s t r a c t This paper describes a novel methodology to estimate the parameters of a low-frequency model of a 3-phase transformer, by only using data from its frequency response. The described calculation procedure takes into account the magnetic coupling among different phases and allows their analysis separately, enabling the identification of a possible failure. This work describes the used core model, the procedure to obtain its parameters, and its application when interpreting the low frequency results of the Frequency Response Analysis (FRA) measurements. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction The Frequency Response Analysis (FRA) is a sensitive technique for evaluating the mechanical integrity of transformers by measuring either their electrical transfer function or impedance over a wide-frequency bandwidth [1]. Failure detection by using the FRA technique is mainly based on the graphical comparison of curves obtained when plotting the measured responses in different stages of the transformer life cycle. The frequency response of the transformer is obtained in a reference state, usually after manufacturing, when a healthy condition is supposed. Then the same transformer is measured in an evaluation state when a possible faulty condition is suspected. The variation between the reference and evaluation curves can point to a possible failure. However, quantification and localization of damage cannot be assessed with the degree of accuracy needed for diagnosis and decision making. To solve this task several researchers propose methodologies to interpret the FRA curves [2–5]. Alternatively there are proposals which suggest using a transformer model to interpret the measured frequency response. Additionally international organizations have recommended the use of models to face this difficulty [6–9]. The transformer can be modeled as an

⇑ Corresponding author. Tel.: +57 23321948. E-mail addresses: [email protected] (W. Herrera), [email protected] (G. Aponte), [email protected] (J. Pleite), [email protected] (C. Gonzalez-Garcia). 0142-0615/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijepes.2013.05.044

electrical network of R, L, and C parameters [10–13] whose values can be calculated to match the same frequency response that the transformer presents. Considering this similarity, any change in the transformer’s internal geometry, like winding displacement and/or deformation, will become a variation in the frequency response and hence, a change in the parameter’s value. Therefore a possible failure can be detected, quantified and located by an inspection of the R, L, and C parameter variations between reference and evaluation states. There are several works concerning transformer modeling for FRA interpretation distinguished by aspects like model structure and analyzed phenomena [14–17]. This contribution follows the methodology exposed in [18], and develops a transformer model that operates in the low-frequency bandwidth, obtained from the measured response curve data. The model allows the interpretation of response curve in the low frequency bandwidth which is usually 20–10 kHz. Section 2 describes the transformer’s magnetic core circuit. Section 3 describes the model structure. Sections 4 and 5 describe the mathematical procedure to calculate the parameters’ value. Finally, Section 6 contains the experimental results obtained by applying the methodology to a power transformer. 2. Magnetic circuit Transformers are made up of a magnetic circuit formed by laminated steel sheets on which the windings are wound. Power

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transformers are usually built core-type with three-limb. Each winding is wound on each magnetic core limb as shown in Fig. 1. The FRA method consists in measuring the transformer’s electrical equivalent impedance, over a wide-frequency bandwidth using a low voltage sinusoidal wave. The amplitude and phase of the measured impedance is plotted versus frequency. The transformer impedance can be measured in different conditions and through different accessible terminals, forming four types of test. In this paper only end-to-end open circuit test is treated. In the end-to-end open circuit test, the signal is applied to one end of winding (Vin) and the transmitted signal is measured at the other end (Vout), with the other windings terminals in open circuit as shown in Fig. 2, for a star connected winding. In this configuration the transformer behavior on the low frequency bandwidth is determined mainly for the magnetic core. This influence is negligible at frequencies above 10 kHz, since the magnetic field penetration depth decreases with increasing frequency. The equivalent core magnetic circuit is shown in Fig. 3, where UU, UV and UW yield the three magnetic fluxes of the respective U, V and W phases; RU , RV and RW represent the reluctance of three-phase magnetic circuits and NUIU, NVIV and NWIW yield the magneto-motive force present in every transformer winding. There are two common features in the core geometry of all three-limb core-type transformers:  The magnetic path longitude of the middle phase is lower than that of the lateral ones, making the middle leg reluctance (RV ) lower than that of the lateral ones (RU and RW ).  The magnetic paths of the lateral phases are equal due to symmetry as are the associated reluctances RU and RW .

These two effects can be synthesized by (1), where the k parameter depends on the transformer geometry.

R U ¼ RW ¼

RV k

where 0 < k 6 1

ð1Þ

The different values among lateral and central reluctances cause the curves representing the frequency responses to differ when the lateral and central windings are measured. When a lateral phase is excited by the voltage applied during the end-to-end open circuit measurement, the curve reproduces two resonance peaks corresponding to both the asymmetric magnetic paths (C1 and C2 in Fig. 4) of the magnetic flux. On the other hand, the curve obtained for the central phase reproduces a unique resonance peak, corresponding to two symmetric magnetic paths (C3 and C4 in Fig. 4) of the same reluctance. Three-limb core-type transformer presents an asymmetry between the magnetic paths corresponding to each transformer phase. In the case of transformer with other types of geometry such as shell-type or five-limb core-type transformers, the magnetic paths are practically symmetrical and therefore the frequency response from all phases tends to exhibit a curve

Fig. 2. End-to-end open circuit test in the U phase for a star connected winding.

Fig. 3. Transformer magnetic circuit.

characterized by only one resonance peak in the low frequency bandwidth. 3. Low-frequency model In this paper, the low-frequency bandwidth is defined as the range where the second resonance in the end-to-end open circuit measurement of a lateral phase (or the first in the central phase) is included. The electric equivalent core model is presented in Fig. 5 and is derived from the transformer magnetic circuit (Fig. 3) by applying the Duality Principle and adding the parasitic capacitances and resistances in each transformer phase. Parameters LU, LV and LW represent the magnetization characteristics of the respective phases U, V and W as expressed in (2) where RU , RV and RW are the magnetic reluctances shown in (1) and NU, NV and NW are the number of turns on each winding. Variations in LU, LV and LW parameters can indicate faults associated with the magnetic core, residual magnetization or turn-to-turn short circuit, among other.

LU ¼

LV ¼

N2V RV

LW ¼

N2W RW

ð2Þ

The active power losses of the magnetic core for each phase can be represented by adding the three resistive parameters RU , RV and RW that comply with the relationship illustrated in (3) where VU, VV and VW are the test voltages and P0U, P0V and P0W are the core power losses. Variations in RU, RV and RW parameters can indicate problems associated with the magnetic core.

RU ¼

Fig. 1. Three-phase three-limb core-type transformer.

N 2U RU

V 2U P0U

RV ¼

V 2V P0V

RW ¼

V 2W P0W

ð3Þ

Finally, the low-frequency model (Fig. 5) is completed by adding the C parameters representing the winding capacitance of each phase. Variations in CU, CV and CW parameters can indicate problems such as overall movement of windings. As soon as the model obtained represents the physical phenomena reproduced in the frequency response, it can be used to interpret the measurements and assess possible failures. For example, in a three-limb core-type transformers showing healthy condi-

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Fig. 4. Difference between the responses of the central and lateral phases.

Fig. 6. Transformer core model represented by three impedances ZU, ZV and ZW containing characteristics of each phase.

Fig. 5. Transformer core model.

tions, the inductance values of the lateral phases (LU and LW) are approximately equal, as the equivalent reluctances RU and RW . Also, the LV value must be greater than the inductances LU and LW, given the lower reluctance (RV ) of the magnetic circuit in the middle phase. The same conclusion can be applied to the resistive parameters. The lateral magnetic paths are larger than the central one, which leads to higher power losses and therefore, a lower value on the RU and RW parameters compared with the RV. For the same example under healthy conditions, and considering no manufacturing differences of the three phase windings, the capacitive parameters CU, CV and CW, representing the winding capacitance of each phase, must also be fairly equal as expressed by.

CU ¼ CV ¼ CW

and evaluation conditions, the damage will be allocated exclusively to the phase(s) whose model impedance(s) has changed. However, the effects of the ZU, ZV and ZW impedances are mixed due to magnetic coupling when an end-to-end open circuit measurement is obtained. For example, the measured impedance ZmU obtained between two circuit terminals (Fig. 6), is equal to the parallel connection of ZU and (ZV + ZW). In this case, a variation in the measured ZmU impedance can be derived from any ZU, ZV or ZW impedance, and the accurate location of the faulty phase is not straightforward. The same coupling effect and drawback is present for the ZmV and ZmW measured impedances. The solution consists in obtaining the ZU, ZV and ZW model impedance values independent of the ZmU, ZmV and ZmW measured impedances, through a mathematical algorithm. First, Eqs. (5)–(7) relate the measured and modeled impedances as:

ZmUðxi Þ ¼

Z U ðxi Þ  ðZ V ðxi Þ þ Z W ðxi ÞÞ Z U ðxi Þ þ Z V ðxi Þ þ Z W ðxi Þ

ð5Þ

ZmVðxi Þ ¼

Z V ðxi Þ  ðZ U ðxi Þ þ Z W ðxi ÞÞ Z U ðxi Þ þ Z V ðxi Þ þ Z W ðxi Þ

ð6Þ

ð4Þ

4. Calculation of the frequency characteristic impedance curve in each phase The resulting low-frequency transformer model can be constructed by three impedances (ZU, ZV and ZW) connected as shown in Fig. 6, where ZU is the impedance containing the physical characteristics of phase U, ZV of phase V and ZW of phase W. The coincidence of the physical effects of each phase into a single impedance, allows the simplest identification of possible failures. When comparing the parameters’ values between reference

ZmWðxi Þ ¼

Z W ðxi Þ  ðZ U ðxi Þ þ Z V ðxi ÞÞ Z U ðxi Þ þ Z V ðxi Þ þ Z W ðxi Þ

ð7Þ

Using these expressions, a three-equation system can be derived as:

ZmUðxi Þ  ðZ U ðxi Þ þ Z V ðxi Þ þ Z W ðxi ÞÞ  Z U ðxi Þ  ðZ V ðxi Þ þ Z W ðxi ÞÞ ¼ 0

ð8Þ

ZmVðxi Þ  ðZ U ðxi Þ þ Z V ðxi Þ þ Z W ðxi ÞÞ  Z V ðxi Þ  ðZ U ðxi Þ þ Z W ðxi ÞÞ ¼ 0

ð9Þ

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ZmWðxi Þ  ðZ U ðxi Þ þ Z V ðxi Þ þ Z W ðxi ÞÞ  Z W ðxi Þ  ðZ U ðxi Þ þ Z V ðxi ÞÞ ¼ 0

ð10Þ

Then the objective function is defined by (15). This function defines a relative error E between the measured impedance and the model simulated impedance to the measured frequency points m.

The solution for this system for the ZU, ZV and ZW model impedances is given by (11)–(13) respectively.

ZmU 2i þ ZmV 2i þ ZmW 2i Z U ð xi Þ ¼ 2  ðZmU i  ZmV i  ZmW i Þ ZmU i  ZmV i þ ZmU i  ZmW i þ ZmV i  ZmW i  ðZmU i  ZmV i  ZmW i Þ Z V ð xi Þ ¼

Z W ð xi Þ ¼

ZmU 2i þ ZmV 2i þ ZmW 2i 2  ðZmV i  ZmU i  ZmW i Þ ZmU i  ZmV i þ ZmU i  ZmW i þ ZmV i  ZmW i  ðZmV i  ZmU i  ZmW i Þ ZmU 2i

ZmV 2i

E¼

ð11Þ

ð12Þ

R¼

1 ¼ GV ðxi Þ þ j  BV ðxi Þ Z V ðxi Þ

ð14Þ

where GV(xi) and BV(xi) are the real part and imaginary part of the admittance curve at each frequency point (xi) in the low-frequency bandwidth.

Fig. 7. Circuit model of a transformer core branch.

1 jY V ðxi Þj2

ð16Þ GV ðxi Þ jY V ðxi Þj2

x4r  L¼

m X i¼1

1

x2i jY V ðxi Þj2

x2r 

 2x2r 

m X x B i¼1

C¼

After calculating the ZU, ZV and ZW model impedances from the measured data using Eqs. (11)–(13), the magnetic power losses and the electrical features of the U, V and W phases are represented respectively with the L, R and C parameters that confirm the impedance (Fig. 7). The resonance peak of the model impedance is simulated by a single parallel RLC branch in a given bandwidth. Considering the specific bandwidth A–B in Fig. 7, the resonance peak is simulated by a parallel RLC branch, where the positive slope is dominated by the inductive behavior of L, the negative slope is mainly dominated by the C capacitive behavior and the resonance peak is the value of the R parameter, as shown in Fig. 7. An optimization algorithm is applied to calculate the electrical parameter values that better fit the actual ZU, ZV and ZW and the frequency response. The fitting algorithm starts establishing the characteristic admittance of each phase, e.g. for phase V the admittance can be written as shown in the following equation:

i¼1 m X i¼1

ð13Þ

5. Estimating the low-frequency model parameters

ð15Þ

where xr, is the resonance frequency of the characteristic admittance of V phase. The parameters L, R and C of the branch (corresponding to the V phase in this example) are calculated as the values that ensure that (15) reaches the minimum value. As a result, the three Eqs. (16)–(18) can be used to directly calculate the L, R and C parameters. m X

Expressions (11)–(13) allow obtaining the low-frequency response of each phase independently, by only using the data from the frequency response of the three end-to-end open circuit tests, where the information of the three phases is mixed.

Y V ðxi Þ ¼

r

jY V ðxi Þj2

i¼1

ZmW 2i

þ þ 2  ðZmW i  ZmV i  ZmU i Þ ZmU i  ZmV i þ ZmU i  ZmW i þ ZmV i  ZmW i  ðZmW i  ZmV i  ZmU i Þ

    2   1 1 m Y V ðxi Þ  1  j  xi    X 2 R L x Lx i

i V ðx i Þ jY V ðxi Þj2

1 L  x2r

m X i¼1

 x4r 

1 jY V ðxi Þj2 m X i¼1

þ

m X i¼1

x2i

jY V ðxi Þj2

ð17Þ

BV ðxi Þ xi jY V ðxi Þj2

ð18Þ

Eqs. (16)–(18) are obtained by solving the system of equations resulting from the operation rE = 0 ? r2E > 0 which minimizes the objective function. The parameters L, R and C of the U and W phases are calculated by substituting the characteristic admittances YU(xi) and YW(xi) in (16)–(18). 6. Experimental results The methodology to obtain the low-frequency model explained in the previous paragraphs was applied to a 3-phase 10 MVA, 34.5/ 13.8 kV, Dy core-type transformer, in healthy condition. The ZmU, ZmV and ZmW measured impedances of the low-voltage side are plotted in the curves of Fig. 8, for the low-frequency bandwidth. The ZU, ZV and ZW model impedances are calculated by (11)–(13) using ZmU, ZmV and ZmW as input data. The plots of the three impedances obtained are shown in Fig. 9. Finally the L, R and C model parameter values for the U, V and W phases are calculated by (16)–(18), using the ZU, ZV and ZW model impedances as input data. The particular values for this example are shown in Table 1.

Fig. 8. Frequency response curves measured in the three-phase transformer.

W. Herrera et al. / Electrical Power and Energy Systems 53 (2013) 643–648

U

U

V

V

647

W W

Fig. 9. Characteristic impedance curves in frequency at each ZU, ZV and ZW phase.

Table 1 Parameter values per model. Parameter values per model Phase U RU [O] LU [H] CU [F]

36547.65 2.26 1.05e08

Phase V RV [O] LV [H] CV [F]

101583.12 6.666 9.95e09

Phase W RW [O] LW [H] CW [F]

35626.32 2.27 9.97e08

The results shown in Table 1 experimentally demonstrate the assumptions for a healthy transformer described in the previous paragraphs:  The LV inductance value associated with the central S phase is greater than the LU and LW inductances, indicating that the middle phase reluctance is lower than the lateral phase reluctance.  The LU and LW inductance values associated with the lateral phases are approximately equal (difference of 0.63%) due to the same reluctance of their magnetic paths.  The RV resistance value associated with the central phase has the greatest value, meaning that this phase has a lower loss because of the lower length of its magnetic circuit compared with the lateral one.  The RU and RW resistance values of the lateral phases are very close to a 2.52% difference, which shows that the magnetic circuit length of both phases is approximately equal.  The three phases’ capacitance values are approximately equal with a difference of 5.6% due to small differences in the manufacturing process of the windings.

Fig. 10. Actual end-to-end open circuit frequency responses measured in the transformer, and the same curves simulated by the low-frequency model.

The graphical comparison between measured and simulated frequency responses shown in Fig. 10 demonstrates that the model is able to fit the actual response accurately and hence, reproduce the transformer physical behavior. Small differences on the initial portion of measured and modeled curves are observed, these differences occur because the model does not include the winding’s series resistance that appears in lower frequencies. 7. Conclusions A novel methodology to obtain the parameters of a low-frequency transformer model based on the frequency response data is presented. The model takes into account the magnetic coupling present on a 3-phase transformer by applying the Duality Principle, and separating the magnetic losses and the electrical low-frequency effects of each phase by using the mixed information from end-to-end open circuit measurements. The inspection of the independent L, R and C model parameters eases the FRA curves’ interpretation, getting a physical significance of the phenomena occurring in the low-frequency bandwidth and leading to a most accurate diagnosis. Acknowledgements The authors thank to COLCIENCIAS and to the Spanish Government (Ministerio de Ciencia e Innovación, DPI2008-05890), for their support in the development of this work. References [1] Dick EP, Erven CC. Transformer diagnostic testing by frequency response analysis. IEEE Trans Power Apparatus Syst 1978;PAS-97:2144–53.

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