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ANALYTICAL COMPUTATION METHOD FOR ELECTRODYNAMIC FORCES ACTING OVER ELECTRICAL TRANSFORMER WINDINGS

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U.P.B. Sci. Bull. Series C Vol. 7 Iss. 9 ISSN ANALYTICAL COMPUTATION METHOD FOR ELECTRODYNAMIC FORCES ACTIN OVER ELECTRICAL TRANSFORMER WINDINS Ion-Drgoş DEACONU Aurel-Ionuţ CHIRILĂ Constntin HIŢĂ
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U.P.B. Sci. Bull. Series C Vol. 7 Iss. 9 ISSN ANALYTICAL COMPUTATION METHOD FOR ELECTRODYNAMIC FORCES ACTIN OVER ELECTRICAL TRANSFORMER WINDINS Ion-Drgoş DEACONU Aurel-Ionuţ CHIRILĂ Constntin HIŢĂ 3 Vlentin NĂVRĂPESCU 4 infrnco CHICCO 5 În prezent eistă mi multe metoe e clcul forţelor electroinmice e scurtcircuit in înfăşurările trnsformtorelor electrice trifzte. Aceste metoe fie sunt forte precise e eemplu meto elementului finit r necesită un timp înelungt e clcul fie permit un clcul rpi e eemplu meto clsică e clcul însă furnizeză rezultte proimtive. Lucrre e fţă prezintă o metoă rpiă e clcul cestor forţe vntul cestei fiin că permite oţinere unor rezultte precise. Pentru emonstr eficienţ metoei fost reliztă o comprţie între rezulttele oţinute cu utorul cestei şi rezulttele oţinute cu utorul principlelor metoe e clcul utilizte în prezent. Different methos re use t present to compute the short-circuit electronmic forces eerte over the winings of three-phse electricl trnsformers. Methos giving quite ccurte results e.g. the finite element metho require significnt computing resources n computtion time. Clssicl nlticl methos re fst ut not so ccurte. A fst computtion metho with ccurte results is presente in this pper. Comprisons etween the results given this metho n other well-known methos pplie to three-phse trnsformer is performe in orer to show n iscuss the effectiveness of the metho. Kewors: electricl trnsformer short-circuit electronmic forces nltic metho finite element metho. Introuction The forces tht stress the winings uring short-circuit re the most common cuse for the estruction of power electricl trnsformers [ ]. For this reson the computtion of these forces is ecoming n importnt step in the Tech. Assis. Electricl Engineering Fcult. Universitte POLITEHNICA Bucureşti Romni Tech. Assis. Electricl Engineering Fcult. Universitte POLITEHNICA Bucureşti Romni 3 Prof. Electricl Engineering Fcult Universitte POLITEHNICA Bucureşti Romni 4 Prof. Electricl Engineering Fcult Universitte POLITEHNICA Bucureşti Romni 5 Prof. Deprtment of Electricl Engineering Politecnico i Torino Itl 44 I.-D. Deconu A.-I. Chirilă C. hiţă V. Năvrăpescu. Chicco trnsformer s winings esign incluing the fiing mechnicl prts of the winings [3 4]. At present the trnsformers esigners re using ifferent methos in orer to preetermine the short-circuit forces. The most common re the clssicl metho n numericl methos. There re vrious numericl methos tht compute the trnsformer s lekge mgnetic fiel e.g. finite ifferences finite elements vritionl methos. Solving mgnetic fiel prolems with such methos cn e foun in [5] [6] for D n [7] [8] for 3D. Bse on this fiel ifferent quntities re erive such s lekge rectnce [9 ] trnsformer s equivlent circuits [] core s sturtion n non-smmetricl los influences over the lekge fiel [] str losses in ifferent soli metllic prts of the trnsformer tnk clmping pltes frme [3] n of course short-circuit electronmic forces [3]. The clssicl metho to estimte the short-circuit forces [4] is often use the trnsformers esigners/mnufcturers ue to its clcultion simplicit. Unfortuntel this metho provies onl the glol pproimte vlue of the forces without n stress istriution long the winings. This pper presents n nlticl metho to compute the istriution long the winings of the short-circuit electronmic forces il n ril cting over the winings of three-phse power electricl trnsformer. The metho is se on the lekge mgnetic fiel insie the trnsformer s winow clcultion. More ectl the Poisson eqution for the lekge mgnetic fiel s vectorpotentil is integrte using the reen s function metho [5 8]. The lekge mgnetic fiel insie the winow hs een etermine for the first time E. Roth [9]. The two epressions foun Roth hve poor convergence ecuse the re epresse using oule Fourier-series Fourier- Bessel respectivel. The clcultions prove tht for il component of mgnetic flu ensit oth epressions give the sme result. In cse of ril component the results re similr onl in cses with winings of equl mgnetomotive forces. When the mgnetomotive forces re not lnce supposing plne-prllel prolem the vlues otine re greter thn the mesure ones []. The metho presente in the pper is se on nlticl epressions in orer to etermine the trnsformers lekge mgnetic fiel contining simple Fourier-series tht ehiit fst convergence.. Computtion of the lekge mgnetic fiel insie the winow of three-phse trnsformer The lekge mgnetic fiel insie the winow is compute supposing the following simplifing hpotheses: Anlticl computtion metho for electron. forces cting over el. trnsformer winings 45 - wining s curvture is neglecte mgnetic fiel is inepenent over the z coorinte thus plne-prllel prolem cn e consiere; - isplcement currents etween the coils turns n the e currents of the ferromgnetic core re neglecte; - the core s mgnetic permeilit is suppose to hve high vlue 6 ; If the vector-potentil of the lekge mgnetic fiel is etermine then the mgnetic flu ensit cn e otine using the following epression: B rota where A represents the vector-potentil. This potentil is compute integrting the following liner constnt coefficients prtil erivtives eqution over the winow s surfce Σ with Neumnn efine conitions on the ounr Γ of the surfce Σ : ΔA μ where represents the current ensit. The integrtion of is performe using the three-potentils formul giving the vector-potentil: A ; Σ A ; s A μ Γ n 3 where ; represents the reen s function corresponing to the Lplce opertor n Neumnn ounr conitions n A is n ritrr constnt for simplicit it is usull consiere to e null. The reen s function is efine the following reltions: Δ ; μ δ δ 4 ; n μ lγ 5 where δ δ is clle i-imensionl Dirc s function for crtesin coorintes n l Γ is the length of the curve Γ see Fig.. The resulting lekge mgnetic fiel insie the winow B cn e consiere to e generte the superimposing of two fiels. One is efine in 46 I.-D. Deconu A.-I. Chirilă C. hiţă V. Năvrăpescu. Chicco the hpothesis of equl mgnetomotive forces winings enote with B thus neglecting the mgnetizing mgnetomotive force. The other one is efine in the hpothesis of the solel eisting mgnetizing mgnetomotive force n it is * enote with B. In consequence: * B B B 6 The solution of 4 with ounr conitions 5 when lnce mgnetomotive forces re consiere it is forme two ifferent epressions []. Ech epression is vli for one of the two iviing su-omins of surfce Σ. The ivision is efine the line see Fig. n for the reen s function epression this ivision is reflecte onl s simple sum inste of oule one s it woul hve een in the cse where no ivision ws consiere. Fig.. Trnsformer s winow computtion omin split into su-omins. In orer to compute the lekge mgnetic fiel trnsformer with ifferent numer of coils is consiere for oth high voltge HV n low voltge LV winings. Surfce Σ is ivie into five su-omins s it is presente in Fig.. LV coils hve the sme ril imension n ifferent heights ; HV coils hve lso the sme ril imension n ifferent heights where is the orer of the coil. For simplicit it is consiere the sme numer of coils for oth winings enote with l. Anlticl computtion metho for electron. forces cting over el. trnsformer winings 47 In consequence the lekge mgnetic fiel vector-potentil will hve five ifferent epressions corresponing to ech of the five su-omins. The sme is for mgnetic flu ensit whose epressions re resulting from. Fig.. Coils of orer elonging to the two winings of the trnsformer. The cse of plne-prllel fiel se on n 3 iels tht for ech point elonging to the rectngulr ounr Γ of surfce Σ see Fig. the quotient etween the norml erivtive of the lekge mgnetic fiels vectorpotentil for tht point n the voi s mgnetic permeilit is equl to the tngentil mgnetic fiel i.e. tngentil component of the lekge mgnetic fiel to ounr Γ. Neglecting the mgnetizing mgnetomotive force cuses the cnceling of the tngentil mgnetic fiel to ounr Γ n so 3 ecomes: Σ ; A 3 Eqution 3 will hve five ifferent epressions corresponing to the five su-omins: l I A 48 I.-D. Deconu A.-I. Chirilă C. hiţă V. Năvrăpescu. Chicco l II A l III A 7 l IV A l V A In 7 n hve s rguments n ; respectivel. The current ensities specific to the coil numer re otine with following epressions: ] [ ] [ i w i w 8 where w n w represent the numer of turns for coils while i n i represent the instntneous vlues of the phse currents for LV n HV coils. The integrtion of 7 llows computing the vector-potentil for ech suomin. Becuse the prolem is plne-prllel there will e onl two components for ech su-omin for the mgnetic flu ensit over the is ril component n over the is il component respectivel: A B A B 9 Along the seprtion lines etween the regions the il n ril components of the mgnetic flu ensit must meet the following equlities: II I B B II I B B etc. Anlticl computtion metho for electron. forces cting over el. trnsformer winings 49 The solution of 4 with ounr conitions 5 when ust mgnetizing mgnetomotive force is consiere is forme onl one epression []. This time the computtion omin surfce Σ see Fig. hs not een ivie into su-omins n thus the reen s function contins one oule-sum n two single-sums. In orer to compute the lekge mgnetic fiel the sme trnsformer shown in Fig. is consiere ut insie the winow is plce onl one conuctor. The current flowing through this conuctor is equl to the mgnetizing mgnetomotive force w i w i w i. The tngentil mgnetic fiel effective vlue of the lekge mgnetic fiel tngentil component over ounr Γ - insie the winow is efine : * μ A n Ht Tking into ccount 3 n the lekge mgnetic fiel vectorpotentil given solel the mgnetizing mgnetomotive force cn e eucte with the following epression: * * A ; Γ Ht s 3 As consequence eqution 3 cn e integrte fter otining the tngentil component of the lekge mgnetic fiel over the ounr Γ. This component m e compute using D finite element metho. The integrtion of 3 llows computing the vector-potentil for ech of the five su-omins see Fig. n then ecuse the prolem is plne-prllel * the two components for ech su-omin of the mgnetic flu ensit B these components re otine with epressions similr to 9: Bse on the suppose hpotheses the effect superposition principle cn e pplie. Thus the resultnt lekge mgnetic fiel B insie the trnsformer s winow cn e otine s sum of two fiels: one is generte the winings n B from Fig. 3 B B n the other one is given the winings c n C from Fig. 3 B C. Hence: B BB BC The currents re flowing through the trnsformer s winings tht re plce insie the trnsformer s lekge mgnetic fiel. In consequence il n 5 I.-D. Deconu A.-I. Chirilă C. hiţă V. Năvrăpescu. Chicco ril forces re eerting over the coils. In cse of short-circuit per unit length ril forces F n il forces F tht re cting over the coil prt of the LV wining II region Fig. n HV wining IV region Fig. respectivel re given the following epressions: F F BII II BIV IV 3 where B II B IV n B II resultnt lekge mgnetic fiels B IV re the il n ril components of the B II n B IV tht re compute with. Fig. 3. The computtionl omin use to etermine the resultnt lekge mgnetic fiel insie three-phse power trnsformer winow imensions re epresse in mm. Doule-integrls 3 cn e evlute with MATLAB using trpezoil metho []. The current ensities specific to prts of the winings where the short-circuit forces re compute re eucte using 8 where the currents i n i respectivel represent the mimum possile vlues uring the short-circuit perio when the trnsformer is supplie t rte voltge [4]. Anlticl computtion metho for electron. forces cting over el. trnsformer winings 5 3. Stu cse for three-phse power trnsformer In the following comprison is presente etween three ifferent computtionl methos regring the short-circuit electronmic forces il n ril components: numericl methos D n 3D finite element metho clssicl metho n nlticl metho presente in Section. The consiere three-phse electricl trnsformer hs the following rte t: S n 63 kva f n 5 Hz U n kv U n.4 kv u k 5.58 % w 8 w 5 group Δ 5. The min geometricl imensions epresse in mm re shown in Fig. 4. Fig. 4. Longituinl cross-section of the trnsformer through the lrgest step of the okes n columns imensions re epresse in mm. The computer tht hs een use for computtions hs the following configurtion: Intel Core Qu processor clocke t.83 Hz n 4 B RAM. The D n 3D finite element metho hve een pplie using MgNet 6.6. esigne Infoltic compn [3] the clssicl metho n the presente nlticl metho hve een pplie using MATLAB esigne MthWorks compn [4]. Fig. 5 shows the vrition of resultnt lekge mgnetic fiel B components with respect to the trnsformer s winow height for the right-sie trnsformers winow for the D n 3D finite element metho n the nlticl metho pplie t the sme coorinte 5 - see Fig. 3. In the cse of 3D finite element metho ue to the smmetr of the prolem ust hlf of the vrition hs een represente. 5 I.-D. Deconu A.-I. Chirilă C. hiţă V. Năvrăpescu. Chicco It cn e seen tht the nlticl metho n the D/3D finite element methos iel similr results regring the lekge mgnetic fiel insie the trnsformer s winow. For this reson the forces re comprle. Similr results hve een chieve for ll the coorintes shown in Fig. 3. Tle presents the glol vlues otine for the electronmic shortcircuit forces using the three methos. Fig. 5. Vrition of resultnt lekge mgnetic fiel components long the trnsformer s wiow height for 5. Electronmic short-circuit forces glol vlues D/3D finite element metho Ril force for wining B from Fig kn Ril force for wining from Fig kn Ail force for the ottom hlf of the wining B from Fig kn Ail force for the ottom hlf of the wining from Fig kn Ril force for wining C from Fig kn Ril force for wining c from Fig kn Ail force for the ottom hlf of the wining C from Fig kn Ail force for the ottom hlf of the wining c from Fig kn Tle computtion time ~ 8 s Clssicl metho onl the winings plce on the mile column re consiere Ril force kn computtion Ail force for hlf wining 9.78 kn time ~ s Anlticl metho Ril force for wining B from Fig kn computtion Ril force for wining from Fig kn time Ail force for the ottom hlf of the wining B from Fig kn ~ 6 s Anlticl computtion metho for electron. forces cting over el. trnsformer winings 53 Ail force for the ottom hlf of the wining from Fig. 3. Ril force for wining C from Fig. 3. Ril force for wining c from Fig. 3. Ail force for the ottom hlf of the wining C from Fig. 3. Ail force for the ottom hlf of the wining c from Fig kn 3.37 kn 9.5 kn 8.7 kn 4.6 kn 6. Conclusions Using the presente metho it is possile for ech point insie threephse trnsformer s winow to compute the lekge mgnetic fiel n the electronmic forces istriution long its winings in fst n ccurte w. The simultions show tht the RMS vlue of the lekge mgnetic fiel * generte the mgnetizing mgnetomotive force B hs low vlues for common three-phse power trnsformers when the short-circuit regime is consiere thus this component cn e neglecte. The presente nlticl metho hs een initill pplie to singlephse trnsformer. This pper presents how the metho is evelope n m e pplie on three-phse trnsformer. At present the results re vlite through the finite element metho see Fig. 5 ut in the ner future it is lso intene to vlite the results provie this metho with eperimentl mesurements of electronmic forces. The computtions involve the evelope metho cn e implemente on computer without mor ifficulties using more fforle softwre n computtionl resources thn the ones neee finite element methos. Another vntge over the finite element metho is tht this nlticl metho oes not require mteril specifictions for the trnsformers components ut onl the instntneous vlues of the winings flowing currents n the geometricl imensions of the trnsformers winow. Also it oes not require comple preprocessing stge e.g. rwing scle moel setting the pproprite mesh n so forth. R E F E R E N C E S []. S. nuszewski H. Switek K. Zmmer Consequences of internl short-circuits in ver high powerconverters Inustril Electronics ISIE 96 vol. pp []. M. Fortin P. Riffon. erth R. P. McLughlin Short-Circuit Strength n Short Circuit Testing of Power n Distriution Trnsformers IEEE/PES Trnsformers Committee Chrlotte North Crolin Mrch 7 8. [3]. M. Areili K. Aszeh S. mli H.A. Tolit Wining rrngement effects on electromgnetic forces n short-circuit rectnce clcultion in power trnsformers vi numericl n nlticl methos th Biennil IEEE Conference on Electromgnetic Fiel Computtion CEFC 6 rt. no 54 I.-D. Deconu A.-I. Chirilă C. hiţă V. Năvrăpescu. Chicco [4]. Z. Meng Z. Wng The nlsis of mechnicl strength of HV wining using finite element metho prt I clcultion of electromgnetic forces 39th Interntionl Universities Power Engineering Conference UPEC 4 - Conference Proceeings pp [5]. C. Hernánez M. A. Aron Design of istriution trnsformers se on knowlegese sstem n D finite elements ELSEVIER ournl - Finite Elements in Anlsis n Design Volume 43 Iss. 8 pp M 7. [6]. A. M. Kshtin A. Vhei A. Hlvei Investigtion of wining tpe effect on lekge flu of single phse shell tpe trnsformer using FEM Proc. of the 8th Interntionl Conference on Electricl Mchines n Sstems 3 rt. no pp ICEMS 5. [7]. K. Komez S. Wik Electromgnetic fiel clcultion of lekge trnsformer mens of finite element metho with hermitin elements The Interntionl ournl for Computtion n Mthemtics in Electricl n Electronic Engineering vol. 3 iss. pp [8]. X. Li Q. Chen R. Li. Long Clcultion of lekge rectnce for trnsformer winings with uneven istriution of mgnetic force oin ishu 3 3 pp [9]. C. hiţă Clculul inuctivităţii e ispersie l trnsformtore Buletinul Institutului Politehnic Bucureşti 3 3 pp []. A. Nerin. Fiz. Mohseni Clcultion of istriution trnsformer lekge rectnce using energ technique IEEE AUPEC Austrlsin Melourne Austrli. []. T. Henriksen Trnsformer lekge flu moeling Interntionl Conference on Power Sstems Trnsients IPST Rio e neiro Brzil une 4-8. []. M. Dolinr D. Dolinr.Štumerger B.Polžer Comprison of ifferent iron core moels of three-phse core-tpe trnsformers The Interntionl ournl for Computtion n Mthemtics in Electricl n Electronic Engineering vol. 6 iss. 4 pp [3]. L. Susnic Z. Hznr Z. Vlkovic Str losses computtion in power trnsformer th Biennil IEEE Conference on Electromgnetic Fiel Computtion rt. no [4]. C. hiţă Convertore electromecnice Eitur ICPE Bucureşti 998. [5]. V. S. Vlimirov Urveneni mtemticeskoi fiziki I-vo Nuk Moscv 967. [6]. R. Răuleţ I. Ciric Solutions es prolèmes u limites e l écution e Poisson utilisnt eu epressions e l fonction e reen en eu sous-omines Rev. Roum. Sci. Techn. Electrotechn. et Energ [7]. C. hiţă Clculul câmpului mgnetic e ispersie l trnsformtore Stuii şi Cercetări e Energetică şi Electrotehnică Eit. Ac. R. S. R. Tom 4 pp [8]. C. hiţă Dispersion mgnetic fiel inuce the mgnetiztion current in electricl trnsformers Buletinul Institutului Politehnic 8 3 pp [9]. E. Roth Etue nltique u chmp e fuites es trnsformteurs et es efforts mécniques eercés sur les enroulements R..E. pp Mi 98. []. V. N. Neelcu Regimurile e funcţionre le mşinilor e curent lterntiv Eitur tehnică Bucureşti 968. []. I. Ciric reen functions of the Lplce opertor for internl Neumnn prolems Rev. Roum. Sci. Techn. Electrotechn. et Energ []. [3]. [4].
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