applied sciences

Article The Influence of Capacitance and Inductance Changes on Frequency Response of Transformer Windings

Szymon Banaszak * , Konstanty M. Gawrylczyk, Katarzyna Trela and Patryk Bohatyrewicz West Pomeranian University of Technology, 70-310 Szczecin, Poland; [email protected] (K.M.G.); [email protected] (K.T.); [email protected] (P.B.) * Correspondence: [email protected]; Tel.: +48-601-755-390



Received: 25 February 2019; Accepted: 8 March 2019; Published: 12 March 2019


Featured Application: The results presented in the paper are useful in the interpretation of Frequency Response Analysis measurements.

Abstract: Frequency Response Analysis (FRA) is a test method used for assessment of mechanical condition of transformer active parts. Its biggest problem is the interpretation of test results, namely the relationship between scale of differences between compared curves and the decision for further operation of the given transformer. Very often visible differences between two FRA curves do not mean that there is a deformation in the winding. The cause of the curve shift may come from other elements of the transformer that influence inductive or capacitive parameters. This paper takes under consideration the influence of both capacitance and inductance changes on transformer frequency response (FR). The analysis is performed with the computer model of a transformer and also some experimental results are presented, showing the influence of capacitance and inductance changes on the FR of real transformers. The results of research showed that this influence may lead to misleading effects on the shape of FR characteristics. The paper presents an analysis that can be used in the assessment of FRA measurement, especially in the case of uncertain data comparison results.

Keywords: transformer winding; FRA; Frequency Response Analysis; modeling

1. Introduction Transformers play a major role in power distribution systems. They are expected to be in operation for at least 30 years, sometimes even in very unfavorable conditions. One of the key values in power distribution is availability, as its increased value over longer timespans translates into less total distribution costs for operators and an increase in the volume of energy sales. When it comes to older units, owners often have to decide whether to further operate them with some technical restrictions, schedule major maintenance, or withdraw the from further operation and replace them with new units. During their lifetime, transformers undergo maintenance and measurements to assess their technical condition. The basic measurements do not require transformers to go offline for a long period of time, therefore they are commonly used. The advanced methods are used much less frequently (usually once in a couple of years), since they are costly and more time-consuming. Regular, comprehensive measurements, however, give users detailed information about the transformers actual technical state, and using this information the operator can decide if any actions should be taken in the future. A detailed survey [1] has been conducted to analyze the causes of transformer failure in high voltage systems. The main failure modes across all voltage classes are dielectrical (36.6%), mechanical (20.0%), electrical (16.5%), and thermal (10.9%). The methods, which have recently been used to comprehensively assess transformer technical condition, are described in [2]. Lately, approaches have

Appl. Sci. 2019, 9, 1024; doi:10.3390/app9051024 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 1024 2 of 14 been made to unify transformer measurement data into a single health index. Some of them [3,4] are already in use, but unfortunately they are not complex enough, as they mainly focus on dissolved gas analysis (DGA) and physical-chemical oil data. In a study including a mathematical Markovian model [5], authors attempt to find the optimum timespan between two specific inspections (DGA and FRA) in order to maximize availability and optimize maintenance costs. Frequency Response Analysis (FRA) is a test method used for assessment of mechanical condition of a transformer active parts. It is based on the measurement of winding response to a wide frequency sine signal. FRA is a comparative method, in which differences between two frequency response curves are analyzed, therefore it needs some “fingerprint” data measured in the transformer healthy state (the best option is right after installation). It is utilized as a diagnostic technique following such incidents as short circuits or lightning strikes. However, assessments using this method can provide critical information on a transformer winding state and may eventually lead to strategic decisions to remove the unit from further operation. At the current stage of FRA method development, it is possible to perform repetitive measurements. In 2012, an IEC standard on a measuring technology [6] was published, and most available commercial devices fulfill its requirements. The biggest problem of FRA method is interpretation of test results, namely the relationship between the scale of differences between curves and decision for further operation of a given transformer. Very often, visible differences between two FRA curves recorded for the same winding over time do not mean that there is a deformation in the winding. The cause of curve shift (resonance shift or damping change) may come from other elements of the transformer, e.g., different positions of the on-load tap changer (OLTC) or core magnetization caused by previous DC measurements, which influence inductance parameters. Similarly, some changes, like bushing replacement, may influence capacitive parameters [7–10]. Usually in industrial measurements, results are recorded in one test configuration: end-to-end open. In such cases, only the winding of the measured voltage side is taken into consideration, with some capacitive influence of other windings. At present, the FRA method is one of test methods commonly used in the group of advanced diagnostics of transformers, therefore any help in understanding the relationship between the results and the tested object leads to improvement of interpretation quality. Research on interpretation of FRA results has been conducted for years, and there are different experimental [11,12] or modeling techniques [13–15] used to help find correlations between visible changes in the frequency response curve and actual mechanical or electrical problems in transformers related to capacitance or inductance changes. This paper takes under consideration the influence these two parameters have on transformer frequency response (FR). The analysis is performed with a computer model of a transformer with introduced modifications resulting in capacitance and inductance changes. Results obtained from the model, in which changes of parameters are fully controlled, allow for good understanding of the relationship between curve shape changes and object properties. These results are free from additional interferences and effects, as is usually the case in industrial measurements, therefore providing direct relationships between introduced changes and obtained conclusions. In the next step there are some experimental results presented, showing the influence of capacitance and inductance changes on the FR of real transformers. It was achieved by measuring the frequency response for different on-load tap changer positions (inductance change) and by adding an additional capacitance to the bushing in the phase that was not under measurement (capacitance change). The main aim of the paper is to provide analysis on correlations between changes in capacitive and inductive parameters and their effect on the FR curve shape. With this knowledge it will be easier to identify measurement mistakes (like wrong tap position).

2. Parameters Influencing the Frequency Response of Transformer Windings FRA method, widely used in advanced transformer diagnostics for over a decade, has been standardized [6,16] and its main problem lies in interpreting detected changes in the shape of FR characteristics. Test results are usually presented as Bode plots, with amplitude values calculated as a Appl. Sci. 2019, 9, x FOR PEER REVIEW 3 of 15

FRA method, widely used in advanced transformer diagnostics for over a decade, has been standardized [6,16] and its main problem lies in interpreting detected changes in the shape of FR characteristics. Test results are usually presented as Bode plots, with amplitude values calculated as Appl. Sci. 2019 9 a signal damping, , 1024 (in dB) and phase angle shift in degrees. The amplitude is often marked as FRA3, of so 14 it can be written: signal damping (in dB) and phase angle shift in degrees. The amplitude is often marked as FRA, so it U2 can be written: FRA()20log dB = (1) UU FRA(dB) = 20 log 12 (1) − U1 ϕ =∠−∠tg1 () U U −1 21 (2) ϕ = tg (∠U2 − ∠U1) (2)

As was already mentioned, FR can be measured in various test configurations, however, according As was already mentioned, FR can be measured in various test configurations, however, to the standard [6], the main configuration is end-to-end open test setup, where the signal is applied to according to the standard [6], the main configuration is end-to-end open test setup, where the signal the beginning of the winding and measured at its end, with remaining windings left open. All results is applied to the beginning of the winding and measured at its end, with remaining windings left presented in this paper are based on this configuration, which is presented in Figure1. For delta open. All results presented in this paper are based on this configuration, which is presented in connected windings, the signal is given on the bushing connected to the beginning of a tested phase Figure 1. For delta connected windings, the signal is given on the bushing connected to the and measured at this point (Uin). The second measuring channel is connected to the bushing where beginning of a tested phase and measured at this point (Uin). The second measuring channel is the end of the winding in the tested phase is led out (Uout), shown in Figure1a. In the delta winding connected to the bushing where the end of the winding in the tested phase is led out (Uout), shown in configuration, part of the signal is also transferred through both remaining phases. In the case of star Figure 1a. In the delta winding configuration, part of the signal is also transferred through both connected windings, input voltage (Uin) is measured on bushings connected to the beginning of the remaining phases. In the case of star connected windings, input voltage (Uin) is measured on tested phase, while output voltage channel (Uout)) is connected to the neutral bushing. The signal bushings connected to the beginning of the tested phase, while output voltage channel (Uout)) is is transferred only through the tested winding, however obvious couplings to other windings also connected to the neutral bushing. The signal is transferred only through the tested winding, appear in FR. however obvious couplings to other windings also appear in FR.

UOUT ~ U IN H1 H2 H3 ~ H1 H2 H3 x1 x2 x3 n UIN x1 x2 x3 n UOUT

(a) (b)

Figure 1. End-to-end open test setup configuration on the example of a Dyn transformer: (a) delta Figurewinding, 1. End-to-end (b) star winding. open test setup configuration on the example of a Dyn transformer: (a) delta winding, (b) star winding. The frequency response depends also on the column on which the measurement is taken. When measurementThe frequency results response are compared depends between also on phases, the colu theremn on are which always the similar measurement visible differences is taken. When in the measurementlow frequency results range. are These compared changes between are the effect phases, of various there are magnetic always flux similar distributions, visible differences which for thein themiddle low frequency column is symmetricalrange. These (via changes two side are columns)the effect andof various for side magnetic columns flux asymmetrical distributions, (via middlewhich forone the and middle opposite column side columns).is symmetrical Additionally, (via two side some columns) smaller differences and for side in columns the middle asymmetrical frequency range (via middlecan be observed,one and ifopposite two phases side are columns). compared. Additionally, some smaller differences in the middle frequencyTheFRA range results can be are observed, presented if astwo the phases frequency are compared. function in logarithmic scale, allowing analysis of resultsThe in FRA each results frequency are presented range, but as occasionally the frequency linear func scaletion is in applied. logarithmic The FRscale, curve allowing shape isanalysis mainly ofdetermined results in each by winding frequency inductances range, but and occasionally capacitances. linear During scale assessment,is applied. The the FRFR rangecurve isshape usually is mainlydivided determined into three mainby winding subranges, inductances sometimes and including capacitances. additional During sub-bands. assessment, These the rangesFR range are is as usuallyfollows: divided low, medium, into three and main high subranges, frequencies. sometimes Their exact including borders additional depend onsub-bands. the construction These ranges of the aretransformer, as follows: mainly low, medium, on its geometrical and high frequencies. size, usually Their related exact to borders its power depend rating. on Therefore, the construction it is not ofpossible the transformer, to perform mainly analysis on inits predefined geometrical frequency size, usually ranges. related The to bigger its power the transformer, rating. Therefore, the higher it is notthe possible values of to inner perform capacitances, analysis thusin predefined the frequency frequency response ranges. is shifted The into bigger lower the frequencies transformer, [8]. the The higherbehavior the atvalues the low of inner frequency capacitances, (LF) range thus is usuallythe frequency considered response as the is shifted interaction into oflower the magnetizingfrequencies [8].inductance The behavior and bulk at capacitancethe low frequency to the ground (LF) range of the is winding usually taken considered under consideration. as the interaction In the of lowest the frequency range (starting at 20 Hz, according to [6]), winding inductances are dominant, resulting in a decreasing characteristic. When the first resonance is reached, the dominant influence of the winding capacitances appears (Figure2), reaching the end of the LF range. This range is strongly influenced by Appl. Sci. 2019, 9, x FOR PEER REVIEW 4 of 15 magnetizing inductance and bulk capacitance to the ground of the winding taken under consideration. In the lowest frequency range (starting at 20 Hz, according to [6]), winding inductances are dominant, resulting in a decreasing characteristic. When the first resonance is reached, the dominant influence of the winding capacitances appears (Figure 2), reaching the end of the LF range. This range is strongly influenced by short-circuits between the turns or coils, because the magnetic circuit still influences the response in such low frequencies [17–19]. The low-frequency rangeAppl. Sci. is2019 often, 9, 1024omitted in test result analysis, because it shows natural differences between the three4 of 14 phases, due to various magnetic flux distributions in the middle and side columns of the core, which results in a single or double minimum of the main parallel resonance. This frequency range can also beshort-circuits affected by between the previous the turns DC or coils,measurements, because the resulting magnetic circuitin core still magnetization influences the or response by core in constructionsuch low frequencies [17,20]. [The17–19 low]. The freq low-frequencyuency range rangefinishes is often at the omitted inflection in test point result of analysis, the increasing because capacitiveit shows natural slope differencesof the FR betweencurve until the threethe next phases, series due resonance, to various from magnetic which flux FR distributions analysis usually in the startsmiddle for and the side medium columns frequency of the core, range. which The results position in aof single the first or double resonanc minimume depends of the on mainthe winding parallel self-inductanceresonance. This and frequency total series range and can shunt also becapacitances. affected by the previous DC measurements, resulting in coreThe magnetization medium frequency or by core (MF) construction range is [17 sensitive,20]. The to low changes frequency caused range by finishes local deformations at the inflection or displacement,point of the increasing therefore capacitive it is usually slope used of thefor FRthe curveanalysis until of themeasured next series data. resonance, The high fromfrequency which HF FR rangeanalysis is influenced usually starts by fortest the setup medium itself, frequency connection range. quality The and position wave ofphenomena the first resonance in windings, depends so it on is usuallythe winding omitted self-inductance in the analysis. and total series and shunt capacitances.

-20 A-N -30 B-N C-N -40

-50

-60

FRA (dB) FRA -70

-80

-90 LF range MF range HF range

-100 100 1k 10k 100k 1M f (Hz)

Figure 2. Example of frequency response amplitude curves of a three-phase transformer, measured in Figure 2. Example of frequency response amplitude curves of a three-phase transformer, measured end-to-end open test setup, presented in a logarithmic scale. Three frequency ranges are marked. in end-to-end open test setup, presented in a logarithmic scale. Three frequency ranges are marked. The medium frequency (MF) range is sensitive to changes caused by local deformations or The changes of inductive or capacitive parameters of the transformer have strong influence on displacement, therefore it is usually used for the analysis of measured data. The high frequency HF LF range, especially on the position of the first resonance. Additionally, they also influence more range is influenced by test setup itself, connection quality and wave phenomena in windings, so it is complex phenomena resulting in visible changes in the medium frequency range. By observing usually omitted in the analysis. changes in both mentioned frequency ranges it is possible to determine whether visible changes are The changes of inductive or capacitive parameters of the transformer have strong influence on the results of deformation or if they come from some construction changes, e.g., the wrong tap LF range, especially on the position of the first resonance. Additionally, they also influence more position of OLTC during transformer measurement. FRA data are usually interpreted after complex phenomena resulting in visible changes in the medium frequency range. By observing performing measurements on-site, therefore it is difficult to check if there were no accidental changes in both mentioned frequency ranges it is possible to determine whether visible changes mistakes during tests (like the mentioned wrong tap position). The analysis presented in this paper are the results of deformation or if they come from some construction changes, e.g., the wrong tap allows to understand the influence of different parameters on FR curve. position of OLTC during transformer measurement. FRA data are usually interpreted after performing measurements on-site, therefore it is difficult to check if there were no accidental mistakes during tests 3. Model of Inductance and Capacitance Changes on FR Curve (like the mentioned wrong tap position). The analysis presented in this paper allows to understand the influenceThe computer of different model parameters has been on prepared FR curve. in order to verify the conception that capacitance from other windings and additional inductances (e.g., from the tap changer) could balance out. As a 3. Model of Inductance and Capacitance Changes on FR Curve The computer model has been prepared in order to verify the conception that capacitance from other windings and additional inductances (e.g., from the tap changer) could balance out. As a result, the first parallel resonance would remain in its original position, but some changes in the MF range could be expected. Appl. Sci. 2019, 9, x FOR PEER REVIEW 5 of 15

result, the first parallel resonance would remain in its original position, but some changes in the MF Appl. Sci. 2019, 9, 1024 5 of 14 range could be expected.

3.1.3.1. 2D2D ComputerComputer ModelModel ofof thethe WindingWinding inin FEMFEM SoftwareSoftware TheThe modelmodel isis basedbased onon thethe FiniteFinite ElementsElements MethodMethod (FEM)(FEM) inin two-dimensionaltwo-dimensional cylindricalcylindrical symmetry.symmetry. The geometry of the model shownshown inin FigureFigure3 3 corresponds corresponds to to the the dimensions dimensions of of the the typicaltypical LVLV windingwinding ofof thethe distributiondistribution transformertransformer (800(800 kVA,kVA, 15/0.4 15/0.4 kV,kV, Dyn5).Dyn5). TheThe complicatedcomplicated constructionconstruction of of the the actual actual transformer transformer makes makes the 3D the simulation 3D simulation computationally computationally costly and impractical,costly and thusimpractical, it was necessary thus it was to use necessary the simplified to use 2Dthe model. simplified During 2D themodel. analysis, During the solverthe analysis, assumed the that solver the 2Dassumed geometry that of the the 2D model geometry being studiedof the model sweeps be 360ing◦ studiedaround thesweeps z-axis 360° of aaround cylindrical the coordinatez-axis of a system.cylindrical This coordinate model, in system. particular, This does model, not takein particular, into account does the not existence take into of account the other the core existence columns, of butthe itother allows core for columns, the simulation but it allows to be performed,for the simu showinglation to thebe performed, influence of showingC and L theparameters influence with of C satisfactoryand L parameters accuracy. with satisfactory accuracy.

FigureFigure 3.3. 2D2D FEMFEM modelmodel ofof 450450 turnturn LVLV winding.winding.

SimulatedSimulated cross-overcross-over winding winding nominally nominally has has 450 450 turns. turns. Separate Separate models models were were made made to show to show the influencethe influence of changing of changing the position the position of the tap of changerthe tap +/-changer 10% in+/- tapping 10% in turn tapping range. turn These range. models These are similarmodels to are the similar basic oneto the shown basic in one Figure shown3, with in Figure the difference 3, with the being difference that they being contain that 405 they and contain 495 turns. 405 Oneand section495 turns. of theOne winding section hasof the been winding magnified has been to show magnified that all turnsto show with that inter-wire all turns insulationwith inter-wire were takeninsulation into accountwere taken during into analysis. account Onduring the topanalysis of the. On core the window, top of the additionalcore window, wire the provides additional the connectionwire provides to the the bushing connection capacitance. to the bushing This influence capacitance. of bushings This influence can be modeledof bushings as a can single be modeled element, becauseas a single during element, the measurement because during the winding the measur is seenement as a concentratedthe winding inductanceis seen as and a concentrated capacitance. Theinductance treatment and as additionalcapacitance. wire The also treatment allows modificationas additional of wire bushing also capacitancesallows modification [21,22]. On of externalbushing boundaries,capacitances the [21,22]. so-called On “ballooning”external boundaries, boundary the condition so-called has “ballooning” been stated. It boundary allows one condition to cut off thehas regionbeen stated. of analysis It allows near one to theto cut object. off the region of analysis near to the object. TheThe softwaresoftware used used for for the the simulations simulations was was ANSYS ANSY MaxwellS Maxwell v. 19.2, v. especially19.2, especially the 2D eddythe 2D current eddy solver,current which solver, computes which steady-state,computes steady-state, time-varying time-varying (AC) magnetic (AC) fields magnetic at a given fields frequency. at a given The followingfrequency. equations The following (3)–(10) equations were chosen (3)–(10) in the were Maxwell chosen calculation in the Maxwell package. calculation They allow package. calculation They ofallow necessary calculationR, L, andof necessaryC parameters. R, L, Theand eddyC parameters. current field The solver eddy calculates current field the electromagneticsolver calculates field the byelectromagnetic solving the magnetic field by vector solving potential the magnetic and the vector electric potential scalar potentialand the electric in the fieldscalar equation: potential in the field equation: (∇ × A) ∇ × = (γ + jωε)(−jωA − ∇φ) (3) µ Appl. Sci. 2019, 9, 1024 6 of 14 where A is magnetic vector potential, φ is electric scalar potential, µ is magnetic permeability, ω is angular frequency at which all quantities are oscillating, γ is conductivity, and ε is permittivity. The inductances of the coil, both self and mutual, have been computed by ANSYS Maxwell using the energy of the electromagnetic field delivered by FEM:

1 Z W = B·H∗dΩ (4) AV 8 8W Z = AV = · ∗ L 2 B H dΩ (5) I Peak where WAV is average energy of the magnetic field, L is inductance, B is magnetic flux density, H* is magnetic field intensity conjugate, and IPeak is peak current value. Wire resistance values were calculated from the power losses P due to current flow with density J:

1 Z P = J·J∗dΩ (6) 2γ

R ∗ 2P J·J dΩ = = R 2 2 (7) IPeak γIPeak In order to calculate self and mutual capacitances, the following electrostatic field equation has been solved: ∇·(εrε0∇φ(r, z)) = −ρ (8) where ρ is charge density, φ is scalar potential of the electric field induced with charge density ρ, ε0 is vacuum permeability, and εr is relative permeability of the environment. Similarly as in the case of winding inductance, the capacitance is obtained in Maxwell using the energy stored in the electric field associated with the capacitance between two conductors:

1 Z W = D ·E dΩ (9) ij 2 i j Ω

The capacitance between conductors i and j is therefore:

2Wij Z C = = D ·E dΩ (10) ij V2 i j Ω where Wij is energy of the electric field, Di is electric flux density, Ej is electric field intensity, and V is voltage. The computation domain is reduced due to the cylindrical symmetry of the model. The problem area was discretized with approximately 30,000 triangular elements of the second order. The parameters R, L, and C of the winding were determined as a function of frequency from the field model and used in further network calculations to evaluate the frequency response of the winding. The values of inductances and resistances were obtained in the form of impedance matrices for given frequencies. L and R values laying between the analyzed frequency points were obtained using linear interpolation. The capacitances were determined with the assumption that they do not depend on frequency. Obtained matrices served as input data for network analysis algorithm, calculating FRA- characteristics. This network, shortly presented in Figure4, was detailed in a previous study [23]. Appl. Sci. 2019, 9, x FOR PEER REVIEW 7 of 15

Obtained matrices served as input data for network analysis algorithm, calculating FRA-characteristics. This network, shortly presented in Figure 4, was detailed in a previous study Appl.[23]. Sci. 2019, 9, 1024 7 of 14

Figure 4. NetworkNetwork used used for for calculations calculations of of FRA FRA characteri characteristics.stics. The The circuit circuit in the in thebottom bottom relates relates to the to the bushing. bushing.

3.2. Core Modeling Proper representation ofof ferromagneticferromagnetic core core parameters parameters is is required required for for an an accurate accurate simulation simulation of offrequency frequency response response of the of winding the winding in the in wide the frequency wide frequency range. The range. ferromagnetic The ferromagnetic core has significant core has significantinfluence on influence the frequency on the response frequency of the response winding of below the winding approximately below 10approximately kHz, therefore 10 correct kHz, thereforesimulation correct of the simulation core material of th parameterse core material below parameters 10 kHz is especiallybelow 10 kHz important. is especially The behavior important. of Thethe corebehavior material of the is modeledcore material using is complex modeled permeability. using complex Equations permeability. (11)–(15) Equations were used (11)–(15) to evaluate were usedthe magnetic to evaluate permeability the magnetic given permeability for Maxwell give for eachn for frequency. Maxwell Eddyfor each currents frequency. flow inEddy the r-zcurrents plane, flowso it isin possiblethe r-z plane, to model so itit withis possible only one to componentmodel it with of magneticonly one vectorcomponent potential of magneticAφ. The modelvector potentialis based onAφ. an The 800 model kVA transformer,is based on an which 800 kVA is an transformer, old unit, and which its core is an is old made unit, from and cold-rolled, its core is madenon-grain-oriented from cold-rolled, steel, whichnon-grain-oriented was assumed st ineel, the which model. was Assuming assumed one-dimensional in the model. propagation Assuming one-dimensionalof the electromagnetic propagation wave into of thethe laminated electromagnetic ferromagnetic wave into core the with laminated 2D sheet thickness,ferromagnetic the wavecore withequation 2D sheet can be thickness, written asthe [24 wave]: equation can be written as [24]: cosh kz Hy(z) = Hy0 (11) = coshcoshkDkz Hzyy() H0 q (11) p (1+j) cosh kD2 where k = jωµγ = δ and δ is the skin-depth δ = ωµγ . Complex permeability()1 in+ laminatedj core is given by the equation [25]: where kj==ωμγ and δ is the skin-depth δ = 2 . δ D ωμγ 1 1 Z
sinhkD Complex permeabilityµ = in laminated·B = core is givenµHy bydz the= µ equation0µr· [25]: (12) Hy0 Hy0·2D kD· cosh kD 11−DD sinhkD μμμμ=⋅=BHdz() = ⋅ ⋅⋅ yr0 (12) H HD2cosh− kDkD where µ is complex permeability,yy00Hy0 is magneticD field intensity on the surface, B is average magnetic flux density, µ0 is vacuum permeability, and µr is relative permeability, which can be split into real and where μ is complex permeability, Hy0 is magnetic field intensity on the surface, B is average imaginary parts: magnetic flux density, μ0 is vacuum permeability, and μr is relative permeability, which can be split µ = µ µ0 + jµ00
(13) into real and imaginary parts: 0 Real and imaginary parts take the formμμμ=+ of: () μ 0 '"j (13)         Re µ sinh 2D + sin 2D Real and imaginary parts0 take the formµ rof:δ δ δ µ = =       (14) µ0 2D 2D + 2D cosh22DDδ cos δ sinh+ sin Re()μ μδδδ  r   μ ' ==        − μ 2D − 2D (14) Im 0µ 2D sinh22DDδ sin δ 00 µrδ cosh+ cos µ = =  δδ    (15) µ0 2D2D 2D cosh δ + cos δ Figure5 presents complex permeability as a function of frequency. The real part of the complex permeability represents the ability of the core material to conduct the magnetic flux, while the imaginary part represents the core losses caused by eddy currents circulating inside the laminations. Appl. Sci. 2019, 9, x FOR PEER REVIEW 8 of 15

22DD  sinh− sin − Im ()μ μδδδ  r   μ " == (15) μ 22DD 0 2D +  cosh cos  δδ 

Figure 5 presents complex permeability as a function of frequency. The real part of the complex Appl.permeability Sci. 2019, 9 ,represents 1024 the ability of the core material to conduct the magnetic flux, while8 ofthe 14 imaginary part represents the core losses caused by eddy currents circulating inside the laminations.

6 Figure 5.5. ComplexComplex permeabilitypermeability ofof thethe corecore materialmaterial ((µμrr == 560,560, γγ == 22·10·10 6 S/m,S/m, 2 DD = 0.23 mm) as a functionfunction ofof frequency.frequency.

The complexcomplex permeability permeability is calculated is calculated taking ta intoking account into theaccount maximum the equivalentmaximum permeability equivalent ofpermeability the ferromagnetic of the ferromagnetic material, equivalent material, anisotropic equivalent conductivity, anisotropic andconductivity, the thickness and of the the thickness steel sheet. of Inthe the steel computer sheet. In analysis, the computer it is assumed analysis, that it the is a permeabilityssumed that of the the permeability material varies of the with material the frequency varies ofwith the the real frequency part of the of complexthe real part permeability. of the comple Thex values permeability. of magnetic The values permeability of magnetic shown permeability in Figure5 wereshown introduced in Figure in5 were Maxwell introduced input data in Maxwell for each input frequency data separately.for each frequency separately.

3.3. Modeling Results 3.3. Modeling Results The FEM models have beenbeen preparedprepared forfor thethe threethree abovementionedabovementioned cases:cases: thethe neutralneutral positionposition isis 450 turns, turns, the the upper upper position position is is 495 495 turns, turns, and and th thee lower lower position position is 405 is 405 turns. turns. In the In thebeginning, beginning, the thecapacitance capacitance value value in the in the model model was was assumed assumed to to be be CC=1400= 1400 pF. pF. This This value value re refersfers to to the bushingbushing capacitance, connected parallel to thethe winding-tankwinding-tank (core)(core) capacitance.capacitance. Even though the bushingbushing capacitance is dominant, the otherother oneone waswas alsoalso evaluatedevaluated usingusing Maxwell.Maxwell. The analysisanalysis ofof bushingbushing influenceinfluence ofof FR FR curve curve can can be foundbe found in the in literature the literat [26ure], where [26], changeswhere changes are visible are in visible the high in frequencythe high range.frequency Authors’ range. experiments Authors’ experiments show that the show FR ofthat a transformer the FR of a changes transformer in the changes LF range in (up the to LF 10 range kHz) after(up to connecting 10 kHz) after additional connecting capacitance additional to thecapacitance bushing to in the otherbushing phase in the rather other than phase the rather tested than one, andthe tested in this wayone, modifyingand in this the way total modifying capacitance the of total the object capacitance [9], which of the will object be presented [9], which in Section will be4. Appl. Sci. 2019, 9, x FOR PEER REVIEW 9 of 15 Thepresented results in of Section the modeled 4. The frequencyresults of the response modeled are frequency given in Figure response6. are given in Figure 6.

0 405 turns, C=1400 pF -10 450 turns, C=1400 pF 495 turns, C=1400 pF

-20

-30 -55

FRA (dB) FRA -40 -60

-50

-65 -60

1,2k 1,3k 1,4k 1,5k 1,6k 1,7k 1,8k 1k 10k 100k f (Hz)

Figure 6. FR modeling for the three positions of the tap changer (405, 450, and 495 turns). Figure 6. FR modeling for the three positions of the tap changer (405, 450, and 495 turns).

Although all resulting parameters R, L, and C differ in the described three models, the inductance changes are dominant for the frequency response at low and middle frequencies. It can be seen that inductance changes influence the whole simulated frequency range, including the first parallel resonance (approximately 2 kHz). FR curve is shifted towards higher frequencies with decrease in number of turns, and toward lower frequencies with increase in the whole frequency range. This effect was expected. In the next step, there the change of bushing capacitance on opposite winding was simulated, which allowed the position of the first resonance resulting from inductance change to be balanced. The results are given in Figure 7.

0 405 turns, C=1400 pF -10 450 turns, C=1400 pF 405 turns, C=1585 pF

-20

-30 -55

FRA (dB) FRA -40 -60

-50

-65 -60

1,2k 1,3k 1,4k 1,5k 1,6k 1,7k 1,8k 1k 10k 100k f (Hz)

Figure 7. FR modeling for two positions of the tap changer for 405 and 450 turns, and for 405 turns with bushing capacitance change.

It can be seen that the first parallel resonance returned to the original position (nominal tap) after adding capacitance (at 2 kHz). This case may appear in field conditions where there is the wrong tap changer position and capacitance change (e.g., on the bushing insulator of the HV winding) [9]. The change of bushing capacitance can be the result of its earlier replacement or Appl. Sci. 2019, 9, x FOR PEER REVIEW 9 of 15

0 405 turns, C=1400 pF -10 450 turns, C=1400 pF 495 turns, C=1400 pF

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-30 -55

FRA (dB) FRA -40 -60

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-65 -60

1,2k 1,3k 1,4k 1,5k 1,6k 1,7k 1,8k 1k 10k 100k f (Hz)

Appl. Sci. 2019, 9, 1024 9 of 14 Figure 6. FR modeling for the three positions of the tap changer (405, 450, and 495 turns).

AlthoughAlthough all all resulting resulting parameters parametersR, L, R,and L,C anddiffer C indiffer the described in the described three models, three the models, inductance the changesinductance are changes dominant are for dominant the frequency for the responsefrequency at response low and at middle low and frequencies. middle frequencies. It can be It seen can thatbe seen inductance that inductance changes influencechanges influence the whole the simulated whole simulated frequency frequency range, including range, including the first parallelthe first resonanceparallel resonance (approximately (approximately 2 kHz). FR 2 curvekHz). isFR shifted curve towardsis shifted higher towards frequencies higher withfrequencies decrease with in numberdecrease of in turns, number and towardof turns, lower and frequenciestoward lower with frequencies increase in with the whole increase frequency in the range.whole Thisfrequency effect wasrange. expected. This effect In thewas next expected. step, thereIn the thenext change step, there of bushing the change capacitance of bushing on capacitance opposite winding on opposite was simulated,winding was which simulated, allowed which the position allowed of the the position first resonance of the first resulting resonance from resulting inductance from change inductance to be balanced.change to Thebe balanced. results are The given results in Figure are given7. in Figure 7.

0 405 turns, C=1400 pF -10 450 turns, C=1400 pF 405 turns, C=1585 pF

-20

-30 -55

FRA (dB) FRA -40 -60

-50

-65 -60

1,2k 1,3k 1,4k 1,5k 1,6k 1,7k 1,8k 1k 10k 100k f (Hz)

FigureFigure 7.7.FR FR modelingmodeling forfor twotwo positionspositions ofof thethe taptap changerchanger forfor 405405 andand 450450 turns,turns, andand forfor 405405 turnsturns withwith bushing bushing capacitance capacitance change. change. It can be seen that the first parallel resonance returned to the original position (nominal tap) after It can be seen that the first parallel resonance returned to the original position (nominal tap) adding capacitance (at 2 kHz). This case may appear in field conditions where there is the wrong tap after adding capacitance (at 2 kHz). This case may appear in field conditions where there is the changer position and capacitance change (e.g., on the bushing insulator of the HV winding) [9]. wrong tap changer position and capacitance change (e.g., on the bushing insulator of the HV The change of bushing capacitance can be the result of its earlier replacement or damage, but winding) [9]. The change of bushing capacitance can be the result of its earlier replacement or also by mistakes of testing personnel (leftover wires or tools on bushings of other phases and any other capacitive influence). The interesting behavior can be observed at higher frequencies: from approximately 5 kHz, the curve representing tap position 405, which is wired with additional capacitance, returns to the original position, before adding the capacitance. This means that in the case where the wrong tap position is covered by capacitance changes, this can be easily detected in higher frequencies. On the other hand, changes in FR visible in the MF range may be interpreted as a deformation (because LF range shows no differences), while they can be the result of mentioned LC phenomena. Similarly, this was simulated the contrary case, where turn number was increased, while bushing capacitance decreased (Figure8). Also in this case, the frequency shift of the first parallel resonance may be balanced to the original position by capacitance change. In MF range, the capacitance influence disappears and the curve returns to the original position. Appl. Sci. 2019, 9, x FOR PEER REVIEW 10 of 15

damage, but also by mistakes of testing personnel (leftover wires or tools on bushings of other phases and any other capacitive influence). The interesting behavior can be observed at higher frequencies: from approximately 5 kHz, the curve representing tap position 405, which is wired with additional capacitance, returns to the original position, before adding the capacitance. This means that in the case where the wrong tap position is covered by capacitance changes, this can be easily detected in higher frequencies. On the other hand, changes in FR visible in the MF range may be interpreted as a deformation (because LF range shows no differences), while they can be the result of mentioned LC phenomena. Similarly, this was simulated the contrary case, where turn number was increased, while Appl. Sci. 2019, 9, 1024 10 of 14 bushing capacitance decreased (Figure 8).

0 495 turns, C=1400 pF -10 450 turns, C=1400 pF 495 turns, C=1265 pF

-20

-30 -55

-40 FRA (dB) FRA -60

-50

-65 -60

1,2k 1,3k 1,4k 1,5k 1,6k 1,7k 1,8k 1k 10k 100k

f (Hz) FigureFigure 8. FR8. FR modeling modeling for for two two positions positions of of thethe taptap changerchanger for 405 and and 495 495 turns, turns, and and for for 495 495 turns turns withwith bushing bushing capacitance capacitance change. change.

4. ExamplesAlso ofin Industrialthis case, the Measurements frequency shift of the first parallel resonance may be balanced to the originalThe phenomena position by simulatedcapacitance in change. FEM software In MF range, can bethe foundcapacitance in industrial influence measurements. disappears and Somethe curve returns to the original position. examples of such results are given below. They were all measured by authors in the field conditions in the end-to-end open test setup, which was already discussed and presented in Figure1. The test 4. Examples of Industrial Measurements equipment was a commercial device Omicron FRAnalyzer. All details of the measurement technique are requirementsThe phenomena of the simulated IEC standard in FEM [6]. software In all examples,can be found only in oneindustrial phase measurements. is presented, toSome show differencesexamples coming of such directlyresults are from given introduced below. They changes were all to measured the transformer. by authors The in first the one,field presentedconditions in Figurein the9, showsend-to-end the influence open test of setup, capacitance which was on FR alre ofady a 250 discussed MVA, 400/110/30 and presented kV autotransformer.in Figure 1. The test The equipment was a commercial device Omicron FRAnalyzer. All details of the measurement technique additional capacitors were installed on the bushing of phase B, not the measured phase (phase A). The are requirements of the IEC standard [6]. In all examples, only one phase is presented, to show results of the measurement of phase A show that additional capacitance shifts the first resonance in differences coming directly from introduced changes to the transformer. The first one, presented in the LF range (100–300 Hz), but from approximately 10 kHz, its influence disappears. These results Figure 9, shows the influence of capacitance on FR of a 250 MVA, 400/110/30 kV autotransformer. confirmAppl. Sci. observations 2019, 9, x FOR obtained PEER REVIEW from the model that capacitance modifications influence only LF range.11 of 15 The additional capacitors were installed on the bushing of phase B, not the measured phase (phase A). The results of the measurement of phase A show that additional capacitance shifts the first resonance in the LF range (100–300 Hz), but from approximately 10 kHz, its influence disappears. These results confirm-20 observations obtained from the model that capacitance modifications influence only LF range.

-40

-60 FRA (dB)

-80 AN reference AN+2.5 nF on phase B AN+6.8 nF on phase B

100 1k 10k 100k 1M

f (Hz) FigureFigure 9. FR9. ofFR HV of phase HV Aphase of a 250 A MVA,of a 400/110/30250 MVA, kV400/110/ autotransformer30 kV autotransformer with additional with capacitances additional installedcapacitances on HV installed phase B. on HV phase B.

A similar effect is obtained if measurement is conducted on the LV winding of phase A, with capacitors attached to the HV phase B. The capacitive influence of the active part (even though it is in the other phase and in the other voltage side) is reduced in the MF range. The first resonance shifts in a range of 100–300 Hz, while differences between curves disappear from approximately 8 kHz. It can be seen in Figure 10.

-20

-40

-60 FRA (dB)

-80 a1n LV reference a1n LV+2.5 nF on phase B HV a1n LV+6.8 nF on phase B HV

100 1k 10k 100k 1M

f (Hz) Figure 10. FR of LV phase a of 250 MVA, 400/110/30 kV autotransformer with additional capacitances installed on HV phase B.

The next case necessary to verify the modeling results is the behavior of FR when the tap changer position is shifted. The first example is measured with a 160 MVA, 220/110/15 kV autotransformer, for three positions of OLTP. The results are given in Figure 11. The FR for change values of inductance (tap position) influence not only LF range, but also MF range. Again, the industrial measurement confirmed the results obtained from the computer model. A similar effect Appl. Sci. 2019, 9, x FOR PEER REVIEW 11 of 15

-20

-40

-60 FRA (dB)

-80 AN reference AN+2.5 nF on phase B AN+6.8 nF on phase B

100 1k 10k 100k 1M

f (Hz) Figure 9. FR of HV phase A of a 250 MVA, 400/110/30 kV autotransformer with additional Appl. Sci. 2019, 9, 1024 11 of 14 capacitances installed on HV phase B.

A similarA similar effect effect is obtainedis obtained if measurementif measurement is conductedis conducted on on the th LVe LV winding winding of of phase phase A, A, with with capacitorscapacitors attached attached to theto the HV HV phase phase B. The B. The capacitive capacitive influence influence of the of activethe active part part (even (even though though it is init is thein other the other phase phase and inand the in other the other voltage voltage side) side) is reduced is reduced in the in MF the range. MF range. The firstThe resonancefirst resonance shifts shifts in a rangein a range of 100–300 of 100–300 Hz, while Hz, while differences differences between between curves curves disappear disappear fromapproximately from approximately 8 kHz. 8 It kHz. can It becan seen be in seen Figure in Figure10. 10.

-20

-40

-60 FRA (dB)

-80 a1n LV reference a1n LV+2.5 nF on phase B HV a1n LV+6.8 nF on phase B HV

100 1k 10k 100k 1M

f (Hz) FigureFigure 10. FR10. ofFR LV of phase LV aphase of 250 MVA,a of 400/110/30250 MVA, kV400/110/30 autotransformer kV autotransformer with additional with capacitances additional installedcapacitances on HV installed phase B. on HV phase B.

TheThe next next case case necessary necessary to verify to verify the modeling the modeling results results is the behavior is the behavior of FR when of FR the tapwhen changer the tap positionchanger is shifted.position The is firstshifted. example The isfirst measured example with is ameasured 160 MVA, with 220/110/15 a 160 kVMVA, autotransformer, 220/110/15 kV Appl. Sci. 2019, 9, x FOR PEER REVIEW 12 of 15 forautotransformer, three positions of for OLTP. three The positions results areof OLTP. given inThe Figure results 11 .are The given FR for in change Figure values11. The of FR inductance for change (tap position) influence not only LF range, but also MF range. Again, the industrial measurement valuescan be observedof inductance for another (tap position) unit, the influence31.5 MVA, not 110/30 only kV LF transformer, range, but alsopresented MF range. in Figure Again, 12, alsothe confirmed the results obtained from the computer model. A similar effect can be observed for another industrialfor three positions measurement of OLTC. confirmed the results obtained from the computer model. A similar effect unit, the 31.5 MVA, 110/30 kV transformer, presented in Figure 12, also for three positions of OLTC.

-10

-20 AN 1 AN 14 -30 AN 27

-40

-50

-60 FRA (dB)

-70

-80

-90

100 1k 10k 100k 1M f (Hz)

FigureFigure 11. 11.FR ofFR phase of phase A of A 160 of 160 MVA, MVA, 220/110/15 220/110/15 kV kV autotransformer autotransformer for for three three positions positions of OLTC.of OLTC.

-10 AN 1 -20 AN 14 AN 27 -30

-40

-50 FRA (dB) -60

-70

-80

-90 100 1k 10k 100k 1M f (Hz)

Figure 12. FR of phase A of 31.5 MVA, 110/30 kV transformer for three positions of OLTC.

The abovementioned results show that the influence of inductance changes can be observed in the wide frequency range, including LF. If this is the only cause of changed FR measurement, it should be easy to detect. However, if these results were to be combined with capacitance changes, such as for modeling results shown in Figures 7 and 8, the outcome would be difficult to properly interpret. Such a transformer would possibly be described as faulty, while visible changes may be the result of the wrong tap position combined with some capacitive changes on the bushing of different phases than those measured. Yet another problem is core magnetization, which influences only LF range. An example of such a measurement is given in Figure 13. Appl. Sci. 2019, 9, x FOR PEER REVIEW 12 of 15

can be observed for another unit, the 31.5 MVA, 110/30 kV transformer, presented in Figure 12, also for three positions of OLTC.

-10

-20 AN 1 AN 14 -30 AN 27

-40

-50

-60 FRA (dB)

-70

-80

-90

100 1k 10k 100k 1M f (Hz)

Appl. Sci. 2019, 9, 1024 12 of 14 Figure 11. FR of phase A of 160 MVA, 220/110/15 kV autotransformer for three positions of OLTC.

-10 AN 1 -20 AN 14 AN 27 -30

-40

-50 FRA (dB) -60

-70

-80

-90 100 1k 10k 100k 1M f (Hz)

Figure 12. FR of phase A of 31.5 MVA, 110/30 kV transformer for three positions of OLTC. Figure 12. FR of phase A of 31.5 MVA, 110/30 kV transformer for three positions of OLTC. The abovementioned results show that the influence of inductance changes can be observed in The abovementioned results show that the influence of inductance changes can be observed in the wide frequency range, including LF. If this is the only cause of changed FR measurement, it should the wide frequency range, including LF. If this is the only cause of changed FR measurement, it be easy to detect. However, if these results were to be combined with capacitance changes, such as should be easy to detect. However, if these results were to be combined with capacitance changes, for modeling results shown in Figures7 and8, the outcome would be difficult to properly interpret. such as for modeling results shown in Figures 7 and 8, the outcome would be difficult to properly Such a transformer would possibly be described as faulty, while visible changes may be the result of interpret. Such a transformer would possibly be described as faulty, while visible changes may be the wrong tap position combined with some capacitive changes on the bushing of different phases the result of the wrong tap position combined with some capacitive changes on the bushing of than those measured. Yet another problem is core magnetization, which influences only LF range. different phases than those measured. Yet another problem is core magnetization, which influences An example of such a measurement is given in Figure 13. onlyAppl. Sci.LF 2019range., 9, x An FOR example PEER REVIEW of such a measurement is given in Figure 13. 13 of 15

-20

-30 Measurement 1 Measurement 2 -40

-50

-60 FRA (dB) FRA

-70

-80

-90 100 1k 10k 100k 1M f (Hz)

FigureFigure 13. 13.FR FR of distributionof distribution transformer transformer with wi differentth different core core magnetization. magnetization.

If theseIf these changes, changes, visible visible up toup 1 kHz,to 1 kHz, were were to be combinedto be combined with capacitive with capacitive influence influence coming coming from differentfrom different phases, thephases, result the may result be verymay similarbe veryto similar the original to the original curve. curve.

5. Conclusions This paper provides the analysis of the influence of capacitive and inductive parameters on the frequency response of transformer windings. The FRA method is one of the important test methods of power transformers, being a part of advanced diagnostics. The interpretation of measured data is not always easy, especially in cases where two compared curves (i.e., measured in the same test setup in time intervals) differ in some frequency ranges. The question is if such a difference is related to deformation or electric fault in the windings, or maybe it has its source in some additional effects. The first aim of the paper was to prepare a computer model of a transformer that would allow for analysis of capacitance and inductance changes on the frequency response. The results of modeling showed that changes of inductance of the winding or changes in capacitive parameters have an influence on FR curve. It can be seen as the frequency shifts of FR. Such an influence was expected, however combination of changes in these parameters may lead to misleading effects on the shape of FR characteristics. It was shown in modeling results that capacitance has its influence in the LF range, while inductance changes influence FR curve in the wider range. The second aim of the paper was to provide results from industrial measurements that would verify conclusions obtained from the model. There are two groups of such results presented: measured for different positions of OLTC, and with an additional capacitor installed on bushings in phases other than those tested. It can be seen that some differences between two compared curves, measured for the same unit, which are visible in MF range and related to local deformations, may be the effect of inductance changes (e.g., different tap positions for both measurements). The verification of such a measurement mistake by observing the LF range may be misleading if additional capacitive effects cover the inductive effect, or if the core of a transformer was magnetized by earlier DC measurement. Practical measurement values of changes of inductance or capacitance, which influence the shape of FR, are unknown. In addition, each transformer construction has a different FR curve shape, in which the position and number of resonances are different, therefore the influence of C and L parameters can appear in a different way. This makes it difficult to give one universal method for detection of these changes (e.g., a simple formula), which were described in this paper, but it is easy to analyze characteristic behavior in given frequency ranges, which are related to L and C changes. For example, the capacitance change influence of FR ends above 5–10 kHz. This value depends on the geometrical size of the unit (related to the power rating and nominal voltages). The problem of Appl. Sci. 2019, 9, 1024 13 of 14

5. Conclusions This paper provides the analysis of the influence of capacitive and inductive parameters on the frequency response of transformer windings. The FRA method is one of the important test methods of power transformers, being a part of advanced diagnostics. The interpretation of measured data is not always easy, especially in cases where two compared curves (i.e., measured in the same test setup in time intervals) differ in some frequency ranges. The question is if such a difference is related to deformation or electric fault in the windings, or maybe it has its source in some additional effects. The first aim of the paper was to prepare a computer model of a transformer that would allow for analysis of capacitance and inductance changes on the frequency response. The results of modeling showed that changes of inductance of the winding or changes in capacitive parameters have an influence on FR curve. It can be seen as the frequency shifts of FR. Such an influence was expected, however combination of changes in these parameters may lead to misleading effects on the shape of FR characteristics. It was shown in modeling results that capacitance has its influence in the LF range, while inductance changes influence FR curve in the wider range. The second aim of the paper was to provide results from industrial measurements that would verify conclusions obtained from the model. There are two groups of such results presented: measured for different positions of OLTC, and with an additional capacitor installed on bushings in phases other than those tested. It can be seen that some differences between two compared curves, measured for the same unit, which are visible in MF range and related to local deformations, may be the effect of inductance changes (e.g., different tap positions for both measurements). The verification of such a measurement mistake by observing the LF range may be misleading if additional capacitive effects cover the inductive effect, or if the core of a transformer was magnetized by earlier DC measurement. Practical measurement values of changes of inductance or capacitance, which influence the shape of FR, are unknown. In addition, each transformer construction has a different FR curve shape, in which the position and number of resonances are different, therefore the influence of C and L parameters can appear in a different way. This makes it difficult to give one universal method for detection of these changes (e.g., a simple formula), which were described in this paper, but it is easy to analyze characteristic behavior in given frequency ranges, which are related to L and C changes. For example, the capacitance change influence of FR ends above 5–10 kHz. This value depends on the geometrical size of the unit (related to the power rating and nominal voltages). The problem of finding universal markers for detecting inductance or capacitance changes will be the aim of further research. The paper presents analysis that can be used in the assessment of FRA measurement, especially in the case of uncertain data comparison results.

Author Contributions: Conceptualization, S.B.; methodology, S.B., K.M.G., K.T., and P.B.; resources, S.B.; supervision, S.B.; visualization, S.B. and K.T.; writing—original draft, S.B., K.M.G., K.T., and P.B.; writing—review and editing, S.B. and K.M.G. Conflicts of Interest: The authors declare no conflict of interest.

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