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 M.R. Oliveira 1,3 and A.J. Marques Cardoso,3 1 Instituto Superior de Engenaria, Universidade do Algarve, Faro, Portugal University of Beira Interior, Department of Electromecanical Engineering, Portugal 3 Instituto de elecomunicações, Department of Electrical and Computer Engineering, University of Coimbra, Pole II, P Coimbra, Portugal lolivei@ualg.pt, ajmcardoso@ieee.org Abstract. is paper presents an analytical metod to compute te leakage inductances of power transformers wit a turn-to-turn winding fault. A leakage inductance model to represent te transformer wit faulty is also proposed. e results obtained from te application of te analytical metod are validated by using data obtained from finite-element analysis and experimental 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 capital-intensive investments made by power system utilities. e unexpected failure of a power transformer can generate substantial costs for repair and financial loss due to unsceduled electrical outage. erefore, it is of crucial importance to detect internal defects in teir incipient stage, so tat te faulted unit can be immediately disconnected, avoiding te progression of te defective condition into a catastropic failure, minimizing te damages in te transformer and oter expensive neigboring equipment, and tus reducing downtime and total outage costs. e development of new tecniques for transformer condition monitoring and fault protection requires a detailed caracterization of te transformer beavior during te occurrence of turn-to-turn winding sort-circuits. e experimental study of tese incipient internal faults presents 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 penomena can be better investigated by te use of a suitable digital simulation transformer model [1]. Several circuit-based transformer models were presented in te last few years for winding fault studies [1]-[4]. One of te major difficulties in tese approaces is to model te leakage inductances wen te turn-to-turn sort-circuit is present. e difficulty arises because te distribution of te magnetic flux is substantially modified L.M. Camarina-Matos, S. omic, and P. Graça (Eds.): DoCEIS 013, IFIP AIC 394, pp , 013. IFIP International Federation for Information Processing 013

2 44 L.M.R. Oliveira and A.J. Marques Cardoso wen suc a fault occurs [5]. In [] te metod to determine te leakage inductances relies on correction factors, in order to take into account te radial component of te leakage flux. In [3] an analytical formula for te leakage inductance of te faulty winding is developed, but only for te simpler case of transformer no-load conditions. Neverteless, te derivation of te models is not straigtforward and difficult to be implemented. More recently a simplified metod was proposed [4], in wic te leakage inductances are computed from te nameplate sort-circuit reactance. Errors up to 68% are reported (as compared wit experimental results) and only a model based on finite elements metod would yield good results. is work presents an analytical metod to compute te leakage inductances of power transformers wit a turn-to-turn winding fault. e influence of te fault position and fault progression is also caracterized. Additionally, a leakage inductance equivalent circuit to represent te transformer wit faulty is proposed. e results obtained from te application of te analytical metod are validated by using data obtained from finite element analysis and experimental tests. Relationsip to Internet of ings e smart grid can be seen as one of te Internet of ings application domains, in wic intelligent electronic devices and teir communication capabilities can be used to provide unprecedented reliability levels in te power network. However, for tis to occur it is necessary te development 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 power system network. A power transformer model for winding fault studies is essential for tese purposes. 3 Winding Fault Caracterization For te experimental investigation a tree-pase, two winding, tree limb transformer, of 10.3 kva, 30/13 V, was used. e primary and te secondary windings ave 15 and 90, respectively. In eac transformer winding tere are five additional tappings connected to te coils, allowing for te introduction of sorted at several locations in te winding, as sown in Fig. 1(a), for one pase of te primary and te secondary windings. Only te coils of te center limb were used to perform te sort-circuit single-pase tests, wic results in a sell-type core design. Wen a fault occurs in te primary-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 primary 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) presents 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 primary winding. o protect te transformer from complete failure wen te fault was introduced te current in te sorted is limited by an auxiliary resistor, wic represents te

3 Modeling and Caracterization of Leakage Inductances 45 fault contact resistance, R s. It can be seen tat te current in te sorted ( ) and te current in te secondary winding ( ) are in pase opposition wit te primary-side current (i.e., tey are opposing te primary magnetomotive force (MMF)). e leakage inductance is usually obtained by performing te sort-circuit test. It migt be tougt tat te effect of te inter-turn sort-circuit could be analyzed by performing a sort-circuit test were bot N s and N b were individually sorted, Fig. 1(c). However, as explained next, tis test is far from representing 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 presents a completely different beavior tan te one for te load test: it as a very small magnitude and is opposing te secondary MMF, instead of opposing te primary 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 pattern. As a result te flux distribution obtained by tis sort-circuit test is very different from te leakage flux component in te load test and it is not representative of tis latter condition. A better solution to study tese penomena can be obtained by performing te sort-circuit test of Fig. 1(d), were te sort is applied to N s and N b connected in series-addition. Wit tis approac bot currents in tese coils are opposing te primary magnetomotive force, Fig. (c), wic is muc more similar to te results of te load-test of Fig. (a) N p R load N a N b N s i x R s v p 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 tappings of te windings; (b) equivalent circuit for a fault occurring in te primary 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 =.)

4 46 L.M.R. Oliveira and A.J. Marques Cardoso Alternatively, te leakage inductances can be obtained by performing te traditional sort-circuit test were one winding is sorted at a time, Fig. 1(e). Wit tis approac te tree-winding transformer teory can be applied 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 Computation Several formulas for te analytical computation of te leakage inductances ave been proposed in te past. For te case of concentric transformer windings, wit te same eigt, te leakage flux tat flows due to te load current is virtually parallel to te axis (except in te ends of te windings). Under tis assumption, Fig. 3(a) sows te per 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 opposite, wic is valid for normal operating conditions (no fault). e leakage inductance can be computed by using (notation as per Fig. 3) [6]: n πn1μ0 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 permeability of free space, 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 gap between coils k and k+1, and m k is te per unit MMF acting on gap 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 top and bottom of te windings and te effect of te iron core. For te case of a sell-type design: (1) 1 1 Kσ = e e e π E π E π( C1 C) π ( E+ C+ C e e L 1) L1 L1 4πC πe L L L () 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 per unit leakage MMF distribution for te case of: (a) axial configuration; (b) radial configuration

5 Modeling and Caracterization of Leakage Inductances 47 being E te eigt of te winding, te wave lengt of te MMF wave, L te mean coil perimeter, L 1 tat part of te perimeter wic as iron on bot sides, L te rest of te coil perimeter, and C and C 1 te distances from iron to coil on te two sides [6]. Equation (1) can also be used, wit proper adaptations, for te computation of te leakage inductance of a disk-type winding configuration. Fig. 3(b) presents a generic disk-type winding arrangement. In tis case it can be assumed tat te leakage flux as only one component in te radial direction and te radial leakage inductance becomes: n πn1μ0r 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 parallel to te axis, but as significant components of radial flux, depending on te amount of asymmetry [7]. e formulas given by (1) and (3) are no longer valid under tese asymmetrical conditions and a direct computation of te leakage inductance for tese winding arrangements can be very complicated and extremely laborious. A very ingenious and useful way of dealing wit tese situations was proposed by Stepens [7] (and generalized in [6]), in wic te leakage inductance is divided into two components, one axial and te oter radial. Eac component can be computed separately and ten added togeter to give, very nearly, te value of te total leakage inductance. Fig. 4 illustrates te basic principle of te metod wit a simplified diagram, assuming a fault in te middle of te primary winding. e axial component is obtained by dividing te affected winding into two coils, one referring to te ealty portion of te winding and te oter to te faulty part. ese two parts are uniformly distributed along te axis, resulting in a concentric design configuration, wic can be computed by (1). e procedure to obtain te leakage inductance radial component is sown in Fig. 4(b). First, te secondary-side current is converted to te primary-side. Next, te secondary winding is divided into tree segments, wit te same dimensions of tose in te primary winding. e MMF of eac segment is ten computed assuming an uniform distribution of te ampere in te windings. Finally, te MMF's of te corresponding segments of eac winding are summed up and te radial component of te leakage inductance can be computed 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 compute te leakage inductance from te FEM results [9]. Fig. 5(a) presents te results obtained wen te position of te faulty (N b ) are moved along te vertical axis of coil 1, from top 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 component is iger in tis situation. Obviously, te asymmetry increases wit te number of te faulty, and, as a

6 48 L.M.R. Oliveira and A.J. Marques Cardoso consequence, tere is also an increase in te variation of te leakage inductance wen te N b are moved along te winding. Only one experimental result can be obtained for tis specific test conditions, due to te fixed tap positions in te coils. Fig. 5(b) presents te variation of te leakage inductance wen te fault progresses vertically, involving additional, first from te top 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 component. It is interesting to note tat tis beavior is opposing te fault progression, since it tends to limit te faulty current. e radial component 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 pattern is repeated in te oter vertical layers (only sown for coils 1 and ). e axial component of te leakage inductance drops, since te number of primary is effectively decreasing as te fault evolves. N p Nb N a m k N I + ( N + N ) I = 0 a p b s s N1= Na = Na1+ Na N a1 N b N a a1 b a A Na A1 Na N = Np = Na a1 Na1 = Na wp ws m k wb N b N 1 N b N a N = a Na b N = b w = b Na wp Na1Ip NbIs NaIp NI s s N a1 s NaIp Na1Ip NbNaIp N b s NaIp NaI p N a s NaIp ( + ) Nb Np N 1 I = AI 1 N ( N + N ) a p p p b s ( + ) Nb Np Na Ip Np( Nb + ) ( + ) Nb Np N I = AI N ( N + N ) a p p p b s Fig. 4. (a) Equivalent representation of a fault in terms of two components, axial and radial; (b) diagram illustrating te procedure to obtain te radial component =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 position along te winding (coil 1); (b) fault progression. (Sort-circuit test of Fig. 1(d).)

7 Modeling and Caracterization of Leakage Inductances 49 e results obtained by te analytical calculation, te FEM analysis, and te experimental tests are in relatively good agreement. e aforementioned metod is also valid for computing te leakage inductances between pairs of windings for te case of te standard sort-circuit tests of Fig. 1(e). Fig. 6(a) presents te corresponding 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 proposed in [10] is adapted ere to represent te transformer wit faulty. e equivalent circuit is sown in Fig. 7 and its parameters are computed from te sort- -circuit inductances between te pairs 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 Lσ( 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 applied to te seriesconnected N b and N s, te equivalent inductance becomes: ( ) ( ) Lσ(a, b+s) = Lσ1 + Lσ + Mσ + N b (8) Fig. 6(b) compares te results obtained by applying (8) and te ones previously presented in Fig. 5(b). Globally, te proposed leakage inductance model yields good and consistent results. e FEM results are almost coincident. Minor differences can be detected in te analytical computed results, mainly due to approximations 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 components. 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)

8 430 L.M.R. Oliveira and A.J. Marques Cardoso M σ v a Lσ1 L σ Na v s Na Nb v b Fig. 7. Leakage inductance equivalent circuit 6 Conclusions is paper as presented a metod for te analytical determination of te leakage inductances of transformers wit winding interturn sort-circuits. An equivalent circuit for te representation of te leakage inductance of transformers wit winding faults is also proposed. e experimental and FEM analysis results confirm te adequacy of te proposed analytical calculation metod. Work is currently in progress to furter simplify te leakage inductances computation metod, in order to allow teir determination from nameplate data and core window dimensions. References 1. Oliveira, L.M.R., Cardoso, A.J.M.: A Permeance-Based ransformer Model and its Application to Winding Interturn Arcing Fault Studies. IEEE rans. Power Delivery 5, (010). Bastard, P., Bertrand, P., Meunier, M.: A ransformer Model for Winding Fault Studies. IEEE rans. Power Delivery 9, (1994) 3. Jablonski, M., Napieralska-Juszczak, E.: Internal Faults in Power ransformers. IE Electric Power Applications 1, (007) 4. Avendaño, A., Mork, B.A., 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. (ed.): ransformer Engineering, nd edn. Jon Wiley & Sons (1951) 7. Stepens, 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., Kaparde, 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)