Non-Linear Thermal Model for Transformers Study

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1 Aida Bulucea, George Manolea, iliana Perescu-Popescu Non-inear Teral Model for Transforers Study MARIUS CONSTANTIN POPESCU NIKOS E. MASTORAKIS CORNEIA AIDA BUUCEA GHEORGHE MANOEA IIANA PERESCU-POPESCU Faculty of Electroecanical and Environental Engineering, University of Craiova ROMANIA Military Institutes of University Education, Hellenic Naval Acadey GREECE Carles ogier College Craiova ROMANIA Abstract: Tis article begins wit a study of transforer additional losses due to te presence of non-sinusoidal loads, presents existing odels to represent tese losses and proposes an original odel. Consequent increent in transforer loss of life is studied by eans of a sensitivity analysis of te odel to transforer rated power, load aplitude, teral inertia, profiles correlation and aronic spectru content. Key-Words: Transforer odel, Non-sinusoidal loads, Teral paraeters Introduction Te proliferation of non-linear loads causes a aronic distortion on public and industrial networks. Te transforers teral ageing, estiated for sinusoidal loads, ay increase if aronic currents are considered. Tis is te ain subject of tis capter. Transforer additional losses due to power supply distortion can reside bot on voltage and current aronics. On section, te variation of additional losses wit frequency is analysed. Additional losses of a set of conductors located on an equivalent slot are deterined and coparisons for several equivalent cross sections of conductors are presented. Te influence between currents of adjacent conductors plays a deterinant role on additional losses deterination since tey liit, in frequency, te reduction in losses acieved by sub-dividing large cross section conductors. Tese results are also illustrated wit graps of conductor s current density for different frequencies. Models obtained for losses generated on conductors placed on slots are generalised for transforer windings wit rectangular cross section conductors. Te odels of tese additional losses, presented on section, are difficult to paraeterise since tey require a detailed knowledge of transforer electric and agnetic circuits. In te specialised literature several siplified versions wic validity is, liited in frequency, can be found. A new odel of transforer additional losses due to current aronics is presented. Te proposed odel is not frequency liited and is easy to paraeterise since it relays only on te transforer rated power. Te proposed odel is copared wit existing ones to evidence te acieved iproveents. Transforer osses Driving Non-Sinusoidal oads Transforer total losses can be separated into noload losses, also referred as core losses related to te voltage, and load losses, also referred as winding losses related to te current. No-load losses coprise ysteresis and eddy current losses in te transforer agnetic core. Tese losses are related to pereability and conductivity of core agnetic aterial, te core laination and design and also to te agnetic induction field, wic in turn depends upon te applied voltage [],[]. As a consequence of voltage distortion on te transforer power supply, no load losses will be affected, relatively to losses generated under a purely sinusoidal voltage. International Standards [],[4], ipose tat for distribution transforers, aronic distortion on te power syste voltage ust be below 5%, apart excessive aronic loading or resonance conditions [5],[6]. Under tese conditions and aving te MV power syste a iger internal sort-circuit power tan te rated apparent power of te considered transforer, te frequency of te applied voltage is predoinantly te fundaental. Tus, te voltage distortion level will not induce significant additional no-load losses and no correction is needed on no-load losses due ISSN: Issue 6, Volue 8, June 009

2 Aida Bulucea, George Manolea, iliana Perescu-Popescu to voltage aronics [4]. According to [7] a 0% distortion on te applied voltage will lead to an increase on no-load losses of about % only, relatively to rated values. Under tese conditions, te ain influence of aronics on transforer losses will be reflected on load losses. Te ost significant load losses arise fro te flowing of load current troug te windings resistance but, since tie varying fields are involved, additional losses (stray losses arise fro eddy-currents due to stray electroagnetic flux in transforer electric conducting aterials (windings, core, core claps, agnetic sields, tank walls. Altoug all tese losses contribute to increase transforer teperature, stray losses occurring in windings of distribution transforers are considered critical, [8],[9]. Windings stray losses, altoug not so significant at rated sinusoidal load (around 0% of total load losses [9], can be deterinant under a non-sinusoidal environent, due to its significant increase wit frequency. Stray oss Variations wit frequency. osses Forulation Under rated conditions (tis rated regie being denoted by te sub-script transforer load losses on windings can be traduced by: PWIN = RACI, ( were: R AC transforer apparent resistance associated to DC current and AC current at fundaental frequency [F]; I rated load current (RMS at rated frequency [A]. Introducing an apparent resistance associated to eddy losses at fundaental frequency, R, expression ( can also be written as PWIN = R DC I, ( being, by definition: = R, ( were: transforer equivalent resistance associated to DC current and taking into account te real winding teperature [F]. Under a non-sinusoidal load current environent, te total current flowing on te windings is: I =, (4 I = wit: I, load current (RMS [A], I, load current (RMS at aronic order [A] and tus, generalisation of ( for a non-sinusoidal load current, will lead to: P WIN = R I = R I, (5 = AC DC = were te transforer apparent resistance RAC is given by: = R R AC (6 and R represent an apparent resistance associated to eddy losses at aronic order. As transforers are projected to drive sinusoidal currents at rated frequency (rated regie, it would be useful tat additional losses due to te nonsinusoidal current were expressed as a function of losses generated under te rated regie. Fro (5 it can be derived: I = PWIN = R DC I, (7 I were te second ter in te parentesis represents additional losses, due to te non-sinusoidal current, in. p.u. values of losses generated under te rated regie. Correct analytical deterination of (7 is coplex. Since tie varying fields are involved, additional losses arise fro non-unifor distribution of current inside conductors (skin effect and terefore, te concept of spatially distributed currents ust be introduced. Detailed knowledge of transforer electrical and agnetic circuits is required wic depends upon any transforer specific caracteristics suc as: rated power, electric and agnetic aterial caracteristics, windings and conductors geoetry and anufacturing structure.. Skin Effect in Slot Placed Conductors To take into account te skin effect, te siplification of considering windings as line reduced conductors ust be witdraw, since skin effect represents a non-unifor distribution of current density inside conductors. Maxwell's equations ust be considered, to analyse spatial current distributions. Figure represents te cross section of a set wit conductors, eac one electrically insulated fro te oters and placed in a slot of a ferroagnetic aterial. Te pereability, to siplify te discussion, is assued to be infinite copared wit tat of te air. Slot widt is denoted by w s conductor widt by w s and conductor eigt by e. If te conductors lengt is assued to be long enoug copared to cross section diensions, so tat end effects can be neglected, te tree-diensional circuit can be reduced to a two diensional circuit (x and z ISSN: Issue 6, Volue 8, June 009

3 Aida Bulucea, George Manolea, iliana Perescu-Popescu according to Figure. w s W c (a z K=M e p= p= p= Fig.: a conductors placed in a slot and b noenclature of conductor s position in te slot. Tie variations of electric and agnetic fields are assued to be sinusoidal wit angular frequency = (8 were: rated angular frequency [rad.s - ] and aronic order. Considering tat every conductor is carrying te sae total current, it can be sown [9] tat te resistance of eac conductor, denoted by R p, will depend upon its position p in te slot, Figure (b, according to: R p sin( sin( sin( sin( = p( p (9 p cos( cos( cos( cos( were: p represent te DC resistance of eac conductor; for a conductor of lengt l c, its DC resistance is given by: l c p =. (0 wc e w Te variable t, represents te reduced conductor widt [8],[9] and is given by: e wc =, ( ws being te penetration dept given by: =. ( w µ 0 Te pereability of conductor s aterial is assued to be te pereability of vacuu, denoted by µ 0. Te penetration dept is a easure of ow deeply eac current aronic is distributed in te conductor and, terefore, te spatial distribution of te agnetic field. If is uc greater tan conductor widt e, te current and agnetic field tend to a spatial unifor distribution (DC liit case. At ig frequencies decreases, Figure (a, and te ajority of te current is carried near te surfaces of eac conductor, Figure (b, wic can be traduced by a reduction in te conductor cross section tat effectively carries te current. (b Tis effect is referred on literature as skin effect. To te skin effect contributes not only te conductor current itself, but also te currents of adjacent conductors. Tis is traduced by te second ter of (9. Te contribution of adjacent currents to skin effect occurring in a conductor is often referred as proxiity effect. To reduce skin effect, large cross section conductors are sectionalised into parallel conductors conveniently twisted and/or transposed so tat eac sub-conductor will experience equally te eddy effect due to different p positions. If twisting or transposing was not carried out, asyetries between sub-conductors would arise and te paralleling of te at te end of te layer would allow te circulation of eddy currents and, terefore, annul te expected iproveent by subdividing te large cross section conductor. For te conductors represented on Figure being correctly transposed, eac portion l c / of te conductors total lengt, l c, ust occupy all possible p positions and, terefore, its resistance will not depend upon its position in te slot but only upon te nuber of total conductors/transpositions,. a b Fig. : a Penetration dept as a function of aronic order, on 50 Hz per unit base, for copper and aluiniu; b Penetration dept of 50 Hz referred to conductor's eigt. Copper conductor. ISSN: Issue 6, Volue 8, June 009

4 Aida Bulucea, George Manolea, iliana Perescu-Popescu w s p= l c / l c p= Fig. : Diagra representing te longitudinal section of one conductor wit transpositions. Under tese conditions, one can consider tat eac conductor total resistance, R pt, will be given by te serial resistance of widt portions l c / of te conductor, denoted by R portion, eac one placed at different p positions in te slot, Figure. It will be: R portion = ( p( p (, ( R DCportion sin( sin( were: ( = and cos( cos( sin( sin( ( = (4 cos( cos( and te DC resistance is given by: lc / portion =. (5 wc e w Te serial resistance of te portions will be given by: = = ( RpT Rportion portion (. (6 p= Since, fro (0 and (5, it is: portion = p, (7 expression (6 is equivalent to: = ( R pt R DCp (. (8 Te conductor total resistance, R pt, can also be obtained by considering it as an average of te resistances a conductor lengt l c presents at eac position p, R p, given by (9. = = ( R pt R p R DCp (. (9 p= To realise te iproveents and te liits acieved in stray loss reduction by subdividing conductors wit large cross section, a nuerical exaple is presented... Nuerical Exaple osses generated on a single conductor wit a large cross section given by Al = wc e, were e represents te slot total eigt, will be copared to losses generated on a parallel of conductors wit cross section given by A = w e (Figure 4. c p= y z W c A (b (a Fig. 4: Diagra representing (a one large cross section conductor (b conductors of reduced cross section. It is assued tat te conductors are correctly transposed; ence te resistance of eac one will be represented by (8. Moreover, under tese conditions, te total current flowing in eac conductor will be l/, being I te total current flowing in te large section conductor. osses, P l generated in te large section conductor, A, are given by: P = R DC ( I, (0 p= were: te subscript refers variables to te diensions of te large conductor. osses generated in te set on transposed and paralleled conductors, P, will be given by te su of losses generated in eac conductor: I P = R pt. ( p= Attending tat = ( is equivalent to: P P = ( ( I, ( were te subscript refers te variable to te diensions of eac conductor. Generically, losses can be represented by an addition of losses associated to te DC resistance of windings, I, and stray losses, P s : P = R DC I PS ( Terefore, stray losses for te large conductor will be given by: PPS = ( (4 R DC I and for te set of parallel conductors by: PS = ( (. (5 R I DC e.. Results and Analysis A grapical representation of te iproveent in stray loss reduction, acieved by conductor s subdivision, is presented on Figure 5. Stray losses obtained under tree possible conductor A A A e ISSN: Issue 6, Volue 8, June 009

5 Aida Bulucea, George Manolea, iliana Perescu-Popescu subdivisions (=, and 4 are represented in p.u. values of stray losses obtained under te reference situation, defined by one conductor (= wit e=0, P S /P S. Copper windings are assued. Figure 5 sows tat te reduction in stray losses acieved by large cross section conductors occurs only up to a certain frequency; for iger frequencies, conductor s subdivision becoes unfavourable, and tis is due to te proxiity effect of adjacent conductors. To exeplify tis fact, stray loss generated on a set of conductors, placed far enoug fro eac oter, eaning isolated fro te agnetic field induced fro adjacent conductor currents, were deterined. Denoting tese stray losses by PS. isol, one obtains: PS. isol = (. (6 R DC I Grapical representation of (6 is presented on Figure 6. osses generated on te large section conductor (= and e=0 are exactly te sae bot on Figure 5 and Figure 6. On te oter and, losses generated on te set of conductors are substantially reduced wen te influence of adjacent conductors is not taken into consideration, Figure 6. Tese results can be confired fro te study of current density distribution inside conductors, for te different analysed situations. Figure 7 represents te aplitude of first aronic (=l, f=50 Hz current density inside conductor cross section, for a total driven current of I=00 A. Tree situations are considered: (a a single large section conductor, (b tree equivalent conductors and (c tree equivalent conductors agnetically isolated fro eac oter. Te asyetry tat current density presents (skin effect on te large section conductor, Figure 7 (a, is considerably reduced wen te conductor is sub-divided, Figure 7 (b, and practically null if no proxiity effects are considered, Figure 7 (c. A different situation occurs, wen te effect of iger frequencies is analysed. Fig. 5: Eddy losses variation wit te nuber and widt of conductors. Te p.u. base is te eddy loss at rated frequency for one conductor wit e=0. a Fig. 6: Eddy loss variation wit nuber and widt of conductors, witout considering proxiity effect. Te p.u. base is eddy loss at rated frequency for a conductor wit e=0. b ISSN: Issue 6, Volue 8, June 009

6 Aida Bulucea, George Manolea, iliana Perescu-Popescu c Fig. 7: Current density aplitude for =l and I=00A (a on one conductor wit e=0, (b on subconductors, considering proxiity effect, (c on subconductors, witout considering proxiity effect. Figure 8 represents current density aplitude for te conductors described above, but represents aronic order =50 (f=.5 khz: tis value corresponds to a practical liit for significant detectable values in current aronic spectru. a c Fig. 8: Current density aplitude for =50 and I=00A (a on one conductor wit e=0, (b on conductors, considering proxiity effect, (c on conductors, witout considering proxiity effect. Skin effect is ore pronounced on te large section conductor, Figure 8 (a, but its sub-division does not lead to a ore unifor distribution of current density. Fro te analysis of current density pase, one realises tat, considering te influence of adjacent conductors, te current in te lower part of te upperost conductors is antipase wit tat in te iger part; proxiity effect give rise to reverse currents inside upper conductors, Figure 8 (b. Tis asyetry in current density leads to iger losses relatively to tose generated on te single conductor, as represented on Figure 6. On te oter and, if conductors are assued agnetically isolated, altoug skin effect occurs inside eac conductor, te current density distribution is, in average and attending to grap scales, ore unifor tan in te large conductor, Figure 8 (c. Te perfored analysis leads to te conclusion tat, sub-division of large cross section conductors, wit correct transposition of sub-conductors, does reduce stray losses but only up to a certain frequency. For frequencies above, losses generated on te set of sub-conductors are iger tan on a single conductor wit an equivalent cross section, due to te proxiity of adjacent currents.. Skin Effect in Transforer Windings Expression ( is coonly used for te evaluation of eddy-current loss in transforers driving sinusoidal currents and wit rectangular cross section conductors [],[0],[]. b ISSN: Issue 6, Volue 8, June 009

7 Aida Bulucea, George Manolea, iliana Perescu-Popescu R li = = and RAC li = = AC (0 ( Fig. 9: (a longitudinal section of transforer winding (one side sowing layers and turns per layer (b view AB; cross section of transforer winding turn wit layers (conductors (c view CD; winding turn crosssection wit layers (conductors Considering a geoetrical siilitude between te slot and te transforer window, slot widt, w s, is replaced by te transforer window eigt, w T, and conductors widt, w c, is replaced by te winding total eigt, N wi, were N represents te nuber of winding turns per layer and w i a diension of conductor cross section, according to Figure 9. Transforer winding diagra is represented on Figure 9. Te winding equivalent resistance traducing alternating current effects (skin and proxiity, RAC relatively to winding DC resistance,, will, ten, be given by: R AC = ( ( (7 were te reduced conductor widt is: e Nwi = (8 wt and, te penetration deep, is given by (. Practical application of (7 does require precise knowledge of winding geoetry and construction procedures, data usually known by transforer anufacturer only and, not presented on te transforer data seet. Moreover, calculations ust be perfored for eac transforer winding. Obtained values ust be reduced to a single side (HV or V of te transforer and ten te total losses of all windings coputed..4 Asyptotic Tendencies Altoug coplex, te asyptotic beaviour of (7 can be studied fro series expansion of yperbolic trigonoetric functions. For a clearer result, fro (8 and ( one can obtain [],[]: e Nwi w µ 0 = = (9 wt were represents te reduced conductor widt, at fundaental frequency (section.. Te asyptotic study of (7 leads to: Te asyptotic analysis allows te conclusion tat te increase in te transforer equivalent resistance follows a quadratic beaviour for reduced aronic orders but after a so-called transition frequency, tis beaviour becoes a square root. iits traduced by (0 and ( are represented on Figure 0 for six different values of te conductor widt, and for a constant nuber of parallel conductors (=0. On tis diagra, a straigt line wit 45 slope traduces a quadratic increase of eddy losses wit frequency (or aronic order. It is interesting to reark tat te transition frequency (te frequency at wic te quadratic effect verified for lower orders ceases is reduced as te conductors widt is increased. Fig. 0: Eddy loss increase wit frequency for different conductor s widt and in p.u. values of eddy losses at rated frequency on te set of conductors wit 6 widt. 4 Stray oss Models Network anagers do ave to deal wit aronic effects in distribution transforers; odel (7 is, owever, of reduced usefulness for tis application due to te nuber of its paraeter. To study losses due to non-sinusoidal load currents in a generic way, it would be useful to derive a general forulation, quite siple, but easily paraeterised and valid for a given range of rated power transforers. ISSN: Issue 6, Volue 8, June 009

8 Aida Bulucea, George Manolea, iliana Perescu-Popescu 4. Existing Models One can find in specialised literature soe epirical forulae to approxiate expression (7, [4],[5]. Most of te proposed forulae are valid only witin a liited range of frequencies. In fact tey overweigt losses due to iger, frequencies since tey assue only te quadratic increase of stray losses wit frequency. Tese siplified forulas are derived by approxiating functions ( and (, (4 to a liited nuber of ters of teir series developent ( and ( ( 45 By inserting ( into (7 one obtains: R AC 5 = 4 ( 45 wic, attending to (9, can be rewritten as: R AC 4 5 = (4 45 Te validity of approxiations ( is strictly dependent upon te conductor reduced widt,, wic in turn, is a function of geoetric diensions and frequency. To realise te agnitude of te perfored approxiation, two nuerical exaples are presented, Figure. Figure (a and (b represents expression (7, denoted as "Original", and its '''Approxiation" given by (4, for copper conductors wit widt 0 and, respectively. Figure is illustrative of te frequency doain reduction for te validity of approxiation (4, as te conductor widt increases. a b Fig. : Eddy loss evaluated wit expression (7 and (4 for one conductor wit wit (a 0 and (b. Siplified odels based on approxiation fro Taylor series developent, are te ost coon ones, [],[4],[9], and can be rewritten as: R R AC = ( sdc (5 R DC were sdc represents te ratio of windings stray losses at rated frequency to losses associated to te DC resistance or, in oter words, te additional losses associated to te DC resistance, due to effects of sinusoidal current at rated frequency. R s DC (6 = Models of te for, R AC ( ( sdc (7 will underestiate lower aronic orders. Oter proposed odels, [6], [7], take into consideration tat windings and non-windings stray losses increase differently wit aronic order and are, generically, of te for: q r R AC ( R DC ( sdc ( sdc (8 were exponents q and r are obtained by fitting data values fro particular situations. Suc a odel does not, as well, fulfil te asyptotic iger and lower order tendencies. Moreover, q and r-values are dependent fro eac analysed situation (aronic spectru contents, since a ig aronic spectru will lead to q values closer to and wit a lower one, te obtained q values will tend to ½ (asyptotic tendency [7]. ISSN: Issue 6, Volue 8, June 009

9 Aida Bulucea, George Manolea, iliana Perescu-Popescu 4. Proposed Model According to [5],[6],[7],[8] stray losses on windings are te ost iportant due to te corresponding increase in transforer teperature. Terefore, tese are te only aronic stray losses tis work will consider. To odel te effects of non-sinusoidal currents on MV/V (ediu voltage/low voltage distribution transforers, tis work proposes te following odel [6], wic verifies te skin effect asyptotic tendency: R AC = (9 wit,! ( S R (40 were! (S R represents a generic function of transforer rated power, S R [VA]. Fro (9 and (40, one can verify tat te asyptotic tendency is fulfilled since: li = (4 0 and li =! ( S R (4 To deterine function! (S R, expressions ( and (4 ust be copared to conclude tat it ust be:! ( " S R (4 Attending tat is function of te conductors widt, e, (9, two extree scenarios ust be analysed. A oogeneous transforer series (siilar anufacturer structure and different rated powers will be considered. Current density, used in te series, will be assued constant, wic is a coon occurrence in practice, since it corresponds to te axiisation of aterials perforance. Te ratio Nwi wt will be considered as constant, also. i On te first scenario te conductors widt will be considered constant and tus, te constant current density wit increasing transforer rated power, iply a corresponding increase in te nuber of parallel conductors. ii On te second scenario, te opposite situation occurs: te nuber of parallel conductors will be kept constant and te conductor s eigt will increase wit te transforer rated power, in suc a way, tat te transforer rated current density can also be kept constant. For te first scenario, if one assues tat te nuber of parallel conductors will increase proportionally to te transforer rated power, eaning ", expression (4 leads to: S R S R! ( S R " #! ( S R " S R (44 For te second scenario, if one assues tat, to te necessary increase in te conductors eigt corresponds a proportional increase in sort-circuit losses, eaning e " Pcc, siilitude relationsips, / 4 lead to e " S R at constant rated density current. Terefore, te (4 proportionality relationsip will be equivalent to: /! ( S R " " e #! ( S R " S R (45 Neiter of tese two extree situations is expected to represent te reality, since a cobination of bot often occurs in practice. For a given nuber of parallel conductors, teir widt is expected to increase up to a certain rated power. Above tat power, conductors are subdivided (increase in and ten again, increase in power will be acieved by an increase of te conductor s eigt. A possible generic function accoplising (44 and (45 can be defined as: k S R ( S R $ wit S R0 % < k < (46 Terefore, te proposed odel assues te following for: k S R S R S! ( $ wit < k < (47 R0 were: S R0 reference transforer rated power [kva], $ epirical proportionality coefficient [diensionless]. Te proposed odel takes te for of a general siilitude relationsip and te $ epirical coefficient value will validate te relationsip for a given range of transforers rated power. Main advantages introduced by te proposed odel are: to fulfil te asyptotic tendency bot for lower and iger aronic orders, avoiding te unrealistic explosion of iger aronic losses [7] verified on odels were te losses are proportional to (5, and avoiding te underestiation of lower aronic losses as do odels wit losses proportional to (7. Moreover, odel paraeterisation becoes easy, since it relies on te transforer rated power only, traducing transforers different sensitivities to stray losses wit its rated power. Te present work is focused, ainly, on distribution transforers. Terefore, and on te lack of oter experiental data, values referred on [5] were used to "tune" te proposed odel. Model proposed by [5] is: ISSN: Issue 6, Volue 8, June 009

10 Aida Bulucea, George Manolea, iliana Perescu-Popescu R R AC AC = s AC ( d (48 were RAC represents winding resistance (DC = and AC effects at rated frequency, d is an exponent wit a value about.6 for distribution transforers, according to an European anufacturer and sac (losses at aronic referred to losses at fundaental frequency is a coefficient, wit typical values given in te sae reference, for power distribution transforers rated fro 00 kva to 000 kva. Model (48 can not be directly copared to (9 odel. However, considering tat te loss coponent due to stray effect is around 0% of total load loss at rated frequency, it can be derived: 9 R DC RAC (49 0 = and tus expression (48 takes te desired for of: 0 R [ ( ] b AC R DC sac (50 9 Model tuning will be perfored for te lower aronic orders since, according to te asyptotic study; te odel proposed [5] overestiates iger orders. Te reference transforer rated power, R0 S, is 000 kva, to be alost in te iddle point of te considered power range, 00 kva to 000 kva. Te iniisation of errors between te proposed and eetonen odels, for te fift aronic, attending to te k restriction in (47 and considering also $ > 0, leads to te values of $ and k paraeters listed in Table : Table : Values of $ and K paraeters. Trans, rated power [kva] $ k / / / / / / Te proposed odel will, ten, be defined by (9 and (40, wit: / S R (5 S R0 Figure represents coparisons between te proposed and te reference [5] odels, were differences are clearly seen. By coparing Figure and Figure, siilitude beaviour between lines representing "Original expression, (7, and "Proposed odel, (5, as well as between "Approxiated" expression, (4, and "[5] odel (50, can be found. Te next section will present soe siulation results of transforer loss of life, using te proposed odel to represent te transforer losses under non-sinusoidal. Fig. : Coparison between proposed and etonen [5] odel, for a 500 kva and a 000 kva transforer. Conclusions To estiate correctly transforer loss of life (or, reciprocally, its power derating, it is necessary to take into account te aronic currents spectru, te electrical caracteristics (losses, te teral beaviour (teral odel and te realistic load and abient teperature profiles (variability. Transforer loss of life decreases wen driving distorting loads, since additional losses are generated tat lead to an increase in te transforer ot-spot teperature. Due to te losses increase wit frequency, additional losses ust be taken into consideration on loss of life estiations, ainly if transforer is driving igly distorting loads. Correct estiation of additional losses is coplex since it requires a precise knowledge of transforer anufacturing procedures and paraeters, suc as: winding structure, conductor diensions and agnetic circuit geoetry. Coplex odels are useful if transforer agnetic and electrical circuits are fully specified bot on its pysical and ISSN: Issue 6, Volue 8, June 009

11 Aida Bulucea, George Manolea, iliana Perescu-Popescu geoetrical caracteristics. Correct estiation of transforer additional losses requires eavy paraeterisation and ediu coplexity calculation tools. Siplified odels of easy paraeterisation are coonly used. However, tey do not fulfil te asyptotic beaviour introduced by considering te skin and proxiity effects. Soe odels underestiate low order aronics, oters overweigt ig order aronics. A new odel was proposed and paraeterised only wit transforer rated power. Tis new odel fulfils asyptotic tendencies bot for low and ig order aronics and takes into consideration te current aronic spectru and te transforer rated power sensitivity to distorting loads. Fro siulations, it as been sown tat, for realistic distorting loads, transforer teral loss of life can approxiately be deterined fro te total aronic distortion factor of te current, once te load linear coponent and te abient teperature profiles as been caracterised. For deterinistic tie varying profiles, caracteristics of realistic input profiles, suc as aplitude, frequency variability and te correlation degree between te, are aspects tat can not be neglected and, in practice, ay ave a greater influence tan additional distorting loads, for a correct estiation of transforer expected life. References [] Crepaz S., Electrical Insulation Deterioration Treated as a Ceical Rate Penoenon, AIEE Transaction, Vol. 67, pp.-, 948. [] Degeratu Pr., Popescu M.C., Boteanu N., Te optiization of te acting wic ust acieve a speed variation, taking into account te criterion of energy losses. Te quality of te obtained results. International Conference on Applied and Teoretical Electricity, pp. 47-4, Craiova, 000. [] IEC-76, Part, International Electrotecnical Coission, Power Transforers Teperature Rise, Second Edition, 99. [4] IEC-54, International Electrotecnical Coission, oading Guide for Oil-Iersed Power Transforers, Second Edition, 99. [5] etonen M., Haronic losses and te local carrying capacity of power systes coponents in industrial networks, Report CIRED 9, iege, pp.-6, 99. [6] Resende M.J., Pierrat., Santana J., Sensitivity Analysis as Accuracy Measure of Siplified Strongly Non-inear Teral Model Transforers Reliability Studies, nd International Syposiu on Sensitivity Analysis of Model Output, pp.7-0, Venice, 998. [7] Pierrat, Métode d Identification et Analyse d Incertitude pour les Paraètres d une Courbe d Ecauffeent Tronquée, 8 t International Congress of Metrology, Besacon, 997. [8] Bulucea C.A., Nicola D.A., Introduction in Electrotecnics and Electrical equipents, SITECH Publising House, pp.5-8, 004. [9] Karsai K., Kerenyi D., Kiss., arge Power Transforers, Elsevier Science Publisers, Asterda, Te Neterlands, 987. [0] Mastorakis N.E., Bulucea C.A., Manolea G., Popescu M.C., Perescu-Popescu., Model for Predictive Control of Teperature in Oil-filled Transforers, Proceedings of te t WSEAS International Conference on Autoatic Control, Modelling and Siulation, ISSN: , ISBN: , pp.57-65,istanbul, Turkey, May 0-June, 009. [] Popescu M.C., Mastorakis N.E., Bulucea C.A., Manolea G., Perescu-Popescu., Transforer Model Extension for Variation of Additional osses wit Frequency, Proceedings of te t WSEAS International Conference on Autoatic Control, Modelling and Siulation, ISSN: , ISBN: , pp.66-7,istanbul, Turkey, May 0-June, 009. [] Mastorakis N.E., On te Solution of Ill-Conditioned Systes of inear and Non-inear Equations via Genetic Algorits (GAs and Nelder-Mead Siplex Searc, WSEAS Transactions on Inforation Science and Applications, Issue 5, Volue, 005, pp [] Mastorakis N.E., Nuerical Solution of Non-inear Ordinary Differential Equations via Collocation Metod (Finite Eleents and Genetic Algorit, WSEAS Transactions on Inforation Science and Applications, Issue 5, Volue, 005, pp [4] Popescu M.C., Siularea nueric< a proceselor, Reprografia UniversitSTii C. BrancuUi, Targu Jiu, pp.45-67, 996. [5] Popescu M.C., Olaru O., Mastorakis N.E., Equilibriu Dynaic Systes Intelligence, WSEAS Transactions on Inforation Science and Applications, ISSN: , Issue 5, Volue 6, pp.75-75, May 009. [6] Popescu M.C., Balas V., Manolea G., Mastorakis N.E., Regional Null Controllability for Degenerate Heat Equations, Proceedings of te t WSEAS International Conference on Sustainability in Science Engineering, ISSN: , ISBN: , pp.-7, Tiisoara, Roania, May 7-9, 009. [7] Popescu M.C., Siularea frecven=ei de excita=ie periculoas< în cazul fenoenului de rezonan=<, Sesiunea VtiinTifica Ecipaente electroecanice, Facultatea de ElectroecanicS, Craiova, 994. [8] Nicola D.A., Bulucea C.A. Electrotecnics, Electrical Equipents and Macines, SITECH Publising House, pp. 5-5, 005. ISSN: Issue 6, Volue 8, June 009