Modeling and Characterization of Leakage Inductances for Transformer Winding Fault Studies

Transcription

1 Modeling and Caracterization of Leakage Inductances for ransformer Winding Fault Studies Luís Oliveira, A. Cardoso o cite tis version: Luís Oliveira, A. Cardoso. Modeling and Caracterization of Leakage Inductances for ransformer Winding Fault Studies. Luis M. Camarina-Matos; Slavisa omic; Paula Graça. 4t Doctoral Conference on Comuting, Electrical and Industrial Systems (DoCEIS), Ar 013, Costa de Caarica, Portugal. Sringer, IFIP Advances in Information and Communication ecnology, AIC-394, , 013, ecnological Innovation for te Internet of ings. < / _45>. <al > HAL Id: al tts://al.arcives-ouvertes.fr/al Submitted on 5 Jul 016 HAL is a multi-discilinary oen access arcive for te deosit and dissemination of scientific researc documents, weter tey are ublised or not. e documents may come from teacing and researc institutions in France or abroad, or from ublic or rivate researc centers. L arcive ouverte luridiscilinaire HAL, est destinée au déôt et à la diffusion de documents scientifiques de niveau recerce, ubliés ou non, émanant des établissements d enseignement et de recerce français ou étrangers, des laboratoires ublics ou rivés. Distributed under a Creative Commons Attribution 4.0 International License

2 Modeling and Caracterization of Leakage Inductances for ransformer Winding Fault Studies Luís M. R. Oliveira 1,3 and A. J. Marques Cardoso,3 1 Instituto Suerior de Engenaria, Universidade do Algarve, Faro, Portugal University of Beira Interior, Deartment of Electromecanical Engineering, Portugal 3 Instituto de elecomunicações, Deartment of Electrical and Comuter Engineering, University of Coimbra, Pole II, P Coimbra, Portugal lolivei@ualg.t, ajmcardoso@ieee.org Abstract. is aer resents an analytical metod to comute te leakage inductances of ower transformers wit a turn-to-turn winding fault. A leakage inductance model to reresent te transformer wit faulty is also roosed. e results obtained from te alication of te analytical metod are validated by using data obtained from finite-element analysis and exerimental tests. Keywords: ransformers, winding faults, leakage inductances, modeling. 1 Introduction Power transformers are key elements of te electric generation, transmission and distribution network and constitute one of te most caital-intensive investments made by ower system utilities. e unexected failure of a ower transformer can generate substantial costs for reair and financial loss due to unsceduled electrical outage. erefore, it is of crucial imortance to detect internal defects in teir inciient stage, so tat te faulted unit can be immediately disconnected, avoiding te rogression of te defective condition into a catastroic failure, minimizing te damages in te transformer and oter exensive neigboring equiment, and tus reducing downtime and total outage costs. e develoment of new tecniques for transformer condition monitoring and fault rotection requires a detailed caracterization of te transformer beavior during te occurrence of turn-to-turn winding sort-circuits. e exerimental study of tese inciient internal faults resents some difficulties, mainly due to te ig magnitudes of te faulty currents involved, wic can damage te test transformer. erefore, a detailed analysis of tese enomena can be better investigated by te use of a suitable digital simulation transformer model [1]. Several circuit-based transformer models were resented in te last few years for winding fault studies [1]-[4]. One of te major difficulties in tese aroaces is to model te leakage inductances wen te turn-to-turn sort-circuit is resent. e difficulty arises because te distribution of te magnetic flux is substantially modified wen suc a fault occurs [5]. In [] te metod to determine te leakage inductances

3 418 L. M. R. Oliveira and A. J. M. Cardoso relies on correction factors, in order to take into account te radial comonent of te leakage flux. In [3] an analytical formula for te leakage inductance of te faulty winding is develoed, but only for te simler case of transformer no-load conditions. Neverteless, te derivation of te models is not straigtforward and difficult to be imlemented. More recently a simlified metod was roosed [4], in wic te leakage inductances are comuted from te namelate sort-circuit reactance. Errors u to 68% are reorted (as comared wit exerimental results) and only a model based on finite elements metod would yield good results. is work resents an analytical metod to comute te leakage inductances of ower transformers wit a turn-to-turn winding fault. e influence of te fault osition and fault rogression is also caracterized. Additionally, a leakage inductance equivalent circuit to reresent te transformer wit faulty is roosed. e results obtained from te alication of te analytical metod are validated by using data obtained from finite element analysis and exerimental tests. Relationsi to Internet of ings e smart grid can be seen as one of te Internet of ings alication domains, in wic intelligent electronic devices and teir communication caabilities can be used to rovide unrecedented reliability levels in te ower network. However, for tis to occur it is necessary te develoment of new fault detection metods, wic must be integrated wit te smart grid tecnologies. is in turn requires detailed and extensive modeling and simulation of te ower system network. A ower transformer model for winding fault studies is essential for tese uroses. 3 Winding Fault Caracterization For te exerimental investigation a tree-ase, two winding, tree limb transformer, of 10.3 kva, 30/13 V, was used. e rimary and te secondary windings ave 15 and 90, resectively. In eac transformer winding tere are five additional taings connected to te coils, allowing for te introduction of sorted at several locations in te winding, as sown in Fig. 1(a), for one ase of te rimary and te secondary windings. Only te coils of te center limb were used to erform te sort-circuit single-ase tests, wic results in a sell-tye core design. Wen a fault occurs in te rimary-side, te sort-circuited act as an autotransformer load on te winding, as sown in Fig. 1(b). e initial effect of te inter-turn sort circuit is limited to a sligt increase in te rimary current. However, te insulation failure can lead to a ig circulating current in te sorted, even if only a small number of is affected. Fig. (a) resents te current waveforms in te transformer windings for te case of a load-test during te occurrence of a turn-to-turn fault in te rimary winding. o rotect te transformer from comlete failure wen te fault was introduced te current in te sorted is limited by an auxiliary resistor, wic reresents te fault contact resistance, R s. It can be seen tat te current in te sorted ( ) and

4 Modeling and Caracterization of Leakage Inductances 419 te current in te secondary winding ( ) are in ase oosition wit te rimary-side current (i.e., tey are oosing te rimary magnetomotive force (MMF)). e leakage inductance is usually obtained by erforming te sort-circuit test. It migt be tougt tat te effect of te inter-turn sort-circuit could be analyzed by erforming a sort-circuit test were bot N s and N b were individually sorted, Fig. 1(c). However, as exlained next, tis test is far from reresenting te true leakage flux distribution. e resultant current waveforms obtained for tis case are sown in Fig. (b). It can be seen tat te current resents a comletely different beavior tan te one for te load test: it as a very small magnitude and is oosing te secondary MMF, instead of oosing te rimary MMF. e main reason is tat te mutual flux tat links te tree coils (N a, N b and N s ) is significantly different in tese two tests, and, terefore, te induced currents do not follow te same attern. As a result te flux distribution obtained by tis sort-circuit test is very different from te leakage flux comonent in te load test and it is not reresentative of tis latter condition. A better solution to study tese enomena can be obtained by erforming te sort-circuit test of Fig. 1(d), were te sort is alied to N s and N b connected in series-addition. Wit tis aroac bot currents in tese coils are oosing te rimary magnetomotive force, Fig. (c), wic is muc more similar to te results of te load-test of Fig. (a). Alternatively, te leakage inductances can be obtained by erforming te N R load N a N b N s i x R s v N a N b N s i x v a N a N s N b v a v N a a N b N s N a N b v b N s N a N b N s Fig. 1. (a) Location of te taings of te windings; (b) equivalent circuit for a fault occurring in te rimary winding; (c), (d) and (e) sort-circuit test scematics = (A) 5 0 (A) 5 0 (A) (a) (s) (b) (s) (c) (s) Fig.. Current waveforms for te case of: (a) load-test of Fig.1(b); (b) sort-circuit test of Fig. 1(c); (c) sort-circuit test of Fig. 1(d). (N b =.)

5 40 L. M. R. Oliveira and A. J. M. Cardoso traditional sort-circuit test were one winding is sorted at a time, Fig. 1(e). Wit tis aroac te tree-winding transformer teory can be alied to obtain te leakage inductance transformer model, wic must be consistent wit te conditions of te non-standard sort-circuit test of Fig. 1(d). 4 Leakage Inductance Analytical Comutation Several formulas for te analytical comutation of te leakage inductances ave been roosed in te ast. For te case of concentric transformer windings, wit te same eigt, te leakage flux tat flows due to te load current is virtually arallel to te axis (excet in te ends of te windings). Under tis assumtion, Fig. 3(a) sows te er unit MMF distribution (m k ) obtained for te windings geometric structure of te test transformer. It is considered tat te MMF's of te two windings are equal and oosite, wic is valid for normal oerating conditions (no fault). e leakage inductance can be comuted by using (notation as er Fig. 3) [6]: n π N1 µ 0 σ( ax) = σ k k gk + k + k 1 + k k 1 k wk k= 1 ( ) 3 L K m g r m m m m w r were N 1 is te number of of te excited winding, µ 0 is te ermeability of free sace, n is te number of vertical layers (7 in tis case), r wk is te mean radio of te coil k, r gk is te mean radio of te ga between coils k and k+1, and m k is te er unit MMF acting on ga between coils k and k+1, see Fig. 3(a).e Rogowski correction factor, K σ, is used to take into account te flux fringing at te to and bottom of te windings and te effect of te iron core. For te case of a sell-tye design: πe πe ( 1 ) ( 1 π C C π E+ C+ C e 1 e 1) L L1 L1 Kσ = π π e e C () E e L L L (1) m 0 m k m 1 m m 3 m 4 m k m 0 m m m m 4 m 5 45 m m 7 m 5 m 6 w = Fig. 3. Geometric structure and er unit leakage MMF distribution for te case of: (a) axial configuration; (b) radial configuration.

6 Modeling and Caracterization of Leakage Inductances 41 being E te eigt of te winding, te wave lengt of te MMF wave, L te mean coil erimeter, L 1 tat art of te erimeter wic as iron on bot sides, L te rest of te coil erimeter, and C and C 1 te distances from iron to coil on te two sides [6]. Equation (1) can also be used, wit roer adatations, for te comutation of te leakage inductance of a disk-tye winding configuration. Fig. 3(b) resents a generic disk-tye winding arrangement. In tis case it can be assumed tat te leakage flux as only one comonent in te radial direction and te radial leakage inductance becomes: n π N1 µ 0r σ( rad ) = σ k k + k + k 1 + k k 1 k w k= 1 ( ) 3 L K m g m m m m Wen tere are irregularities in te concentric windings, suc as a fault, te leakage flux is no longer arallel to te axis, but as significant comonents of radial flux, deending on te amount of asymmetry [7]. e formulas given by (1) and (3) are no longer valid under tese asymmetrical conditions and a direct comutation of te leakage inductance for tese winding arrangements can be very comlicated and extremely laborious. A very ingenious and useful way of dealing wit tese situations was roosed by Steens [7] (and generalized in [6]), in wic te leakage inductance is divided into two comonents, one axial and te oter radial. Eac comonent can be comuted searately and ten added togeter to give, very nearly, te value of te total leakage inductance. Fig. 4 illustrates te basic rincile of te metod wit a simlified diagram, assuming a fault in te middle of te rimary winding. e axial comonent is obtained by dividing te affected winding into two coils, one referring to te ealty ortion of te winding and te oter to te faulty art. ese two arts are uniformly distributed along te axis, resulting in a concentric design configuration, wic can be comuted by (1). e rocedure to obtain te leakage inductance radial comonent is sown in Fig. 4(b). First, te secondary-side current is converted to te rimary-side. Next, te secondary winding is divided into tree segments, wit te same dimensions of tose in te rimary winding. e MMF of eac segment is ten comuted assuming an uniform distribution of te amere in te windings. Finally, te MMF's of te corresonding segments of eac winding are summed u and te radial comonent of te leakage inductance can be comuted by using (3). e total leakage inductance is ten obtained: σ σ( ax) σ( rad ) (3) L = L + L (4) A finite elements metod (FEM) based transformer model [8] is also used to investigate te adequacy of te analytical calculations. e energy metod was used to comute te leakage inductance from te FEM results [9]. Fig. 5(a) resents te results obtained wen te osition of te faulty (N b ) are moved along te vertical axis of coil 1, from to to bottom, using te series-addition sort-circuit test of Fig. 1(d). e leakage inductance takes greater values wen te N b are located at te coil ends, because te radial comonent is iger in tis situation. Obviously, te asymmetry increases wit te number of te faulty, and, as a consequence, tere is also an increase in te variation of te leakage

7 4 L. M. R. Oliveira and A. J. M. Cardoso inductance wen te N b are moved along te winding. Only one exerimental result can be obtained for tis secific test conditions, due to te fixed ta ositions in te coils. Fig. 5(b) resents te variation of te leakage inductance wen te fault rogresses vertically, involving additional, first from te to to bottom in coil 1, and ten affecting te neigboring vertical layer (coil ). e leakage inductance initially grows, due te increasing values of te radial comonent. It is interesting to note tat tis beavior is oosing te fault rogression, since it tends to limit te faulty current. e radial comonent reaces its igest level wen te fault as extended to just about te vertical center of te coil and ten decreases more or less symmetrically. is attern is reeated in te oter vertical layers (only sown for coils 1 and ). e axial comonent of te leakage inductance dros, since te number of rimary is effectively decreasing as te fault evolves. N Nb N a m k N I + ( N + N ) I = 0 a b s s N1 = Na = Na1 + Na N a1 N b N a a1 b a A Na A1 Na N = N = Na a1 Na1 = Na w ws m k wb N b N 1 N b N a N = a Na b N = b w = b Na w Na1I NbIs NaI Is N a1 s NaI Na1I NbNaI N b s NaI NaI N a s NaI ( + ) Nb N N 1 I = A1 I N ( N + N ) a b s ( + ) Nb N Na I N ( Nb + ) ( + ) Nb N N I = A I N ( N + N ) a b s Fig. 4. (a) Equivalent reresentation of a fault in terms of two comonents, axial and radial; (b) diagram illustrating te rocedure to obtain te radial comonent. =10 N b L σfem L σformula L σmeasured = N b L σ ( ax )Formula L σ ( rad )Formula = N b = 5 N b Fig. 5. Leakage inductance as a function of te: (a) number of N b and teir relative osition along te winding (coil 1); (b) fault rogression. (Sort-circuit test of Fig. 1(d).)

8 Modeling and Caracterization of Leakage Inductances 43 e results obtained by te analytical calculation, te FEM analysis, and te exerimental tests are in relatively good agreement. e aforementioned metod is also valid for comuting te leakage inductances between airs of windings for te case of te standard sort-circuit tests of Fig. 1(e). Fig. 6(a) resents te corresonding results as a function of te N b, wic are also in good agreement wit te FEM analysis and te measured values (L σ(ij) leakage inductance wen te N i are excited and te N j are sorted). 5 Leakage Inductance Equivalent Circuit Model e equivalent circuit for te leakage inductance of tree-winding transformers roosed in [10] is adated ere to reresent te transformer wit faulty. e equivalent circuit is sown in Fig. 7 and its arameters are comuted from te sort- -circuit inductances between te airs of windings obtained by te tests of Fig. 1(e): L L (5) σ1 = σ( ab) ( ) L L N N (6) σ = σ( bs) a b ( ) Mσ = L σ( as) Lσ( ab) Lσ( bs) Na N b. (7) e leakage inductance equivalent circuit model can be used to simulate te sort-circuit test of Fig. 1(d). By analyzing te circuit of Fig. 7, wit a sort alied to te series-connected N b and N s, te equivalent inductance becomes: Lσ(a, b+s) = Lσ1 + Lσ ( ) + Mσ ( + N b ) (8) Fig. 6(b) comares te results obtained by alying (8) and te ones reviously resented in Fig. 5(b). Globally, te roosed leakage inductance model yields good and consistent results. e FEM results are almost coincident. Minor differences can be detected in te analytical comuted results, mainly due to aroximations in te calculation of te mean radius and te Rogowski correction factors. e leakage inductance network of Fig. 7 can be integrated wit oter transformer models [1], [10], in order to take into account te core and loss comonents. 6 Conclusions is aer as resented a metod for te analytical determination of te leakage inductances of transformers wit winding interturn sort-circuits. An equivalent circuit for te reresentation of te leakage inductance of transformers wit winding faults is also roosed. e exerimental and FEM analysis results confirm te adequacy of te roosed analytical calculation metod.

9 44 L. M. R. Oliveira and A. J. M. Cardoso Work is currently in rogress to furter simlify te leakage inductances comutation metod, in order to allow teir determination from namelate data and core window dimensions. L σfem L σ ( ab ) L σ ( as ) L σ ( bs ) L σ(a, b+s)fem L σformula L σ(a, b+s)formula L σmeasured L σ(a, b+s)measured Fig. 6. (a) Leakage inductance as a function of te number of N b for te case of te sortcircuit tests of: (a) Fig. 1(e); (b) Fig. 1(d). M σ v a Lσ1 L σ Na v s Na Nb v b Fig. 7. Leakage inductance equivalent circuit. References 1. Oliveira, L.M.R., Cardoso, A.J.M.: A Permeance-Based ransformer Model and its Alication to Winding Interturn Arcing Fault Studies. IEEE rans. Power Delivery 5, (010). Bastard, P., Bertrand, P. and Meunier, M.: A ransformer Model for Winding Fault Studies", IEEE rans. Power Delivery, 9, (1994). 3. Jablonski, M., Naieralska-Juszczak, E.: Internal Faults in Power ransformers. IE Electric Power Alications 1, (007) 4. Avendaño, A., Mork, B.A., and Høidalen H.K.: ransformer Internal Fault Modeling in AP. In: Int. Conf. Power Systems ransients (011) 5. Billig, E.: Mecanical Stresses in ransformer Windings, Journal IEE, Part II, 93, 7-43 (1946) 6. Blume, L.F. (Editor): ransformer Engineering. nd Ed., Jon Wiley & Sons (1951) 7. Steens, H.O.: ransformer Reactance and Losses wit Nonuniform Windings. AIEE rans. 53, (1934) 8. Meeker D.C.: Finite Element Metod Magnetics, Version 4., User's Manual, (010) 9. Kulkarni, S.V. and Kaarde, S.A.: ransformer Engineering: Design and Practice. Marcel Dekker (004) 10. León, F. Martinez, J.A: Dual ree-winding ransformer Equivalent Circuit Matcing Leakage Measurements. IEEE rans. Power Delivery 4, (009)