Nonlinear identification of synchronous generators using a local model approach

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1 Seyyed Salma AHMADI, Mehd KARRARI Amrkabr Uversty f echlgy lear detfcat f sychrus geeratrs usg a lcal mdel apprach Abstract. A ew teratve apprach s prpsed t mdel sychrus geeratrs. Dfferet lcal structures are used fr the peratg cdts f the sychrus geeratr wth dfferet cmplexty. Ulke mst f the exstg methds, whch crease the mdel accuracy based creasg the umber f lcal mdel, the prpsed methd, there are tw chces fr creasg mdel precs at each terat: () creasg the umber f lcal mdels e reg, r () creasg lcal mdel cmplexty the same reg. he prpsed methd has bee tested expermetal data cllected a 3 kva mcr-mache. Streszczee. Zaprpwa wą teracyją metdę mdelwaa geeratrów sychrczych. W prówau d stejących metd w zaprpwaej metdze steją dwa wybry zwększaa dkładścteracj: przez zwększae lczby lkalych mdel lub przez zwększae złżśc mdelu lkaleg. Metdę sprawdz a przykładze geeratra 3 kva. (elwe mdelwae geeratrów sychrczych prze wykrzystau mdelu lkaleg) Keywrds: lcal mdel, pwer system mdellg, sychrus geeratr. Słwa kluczwe: mdelwae lklae, geeratr sychrczy Itrduct Icrease the umber f geeratg uts ad the cmplexty f the multple dstrbut grds has creased the mprtace f pwer system stablty ad dyamc aalyss. System stablty s depedet the perfrmace f the sychrus geeratrs whse behavur s lear due t such effects as eddy currets, ad the magetc saturat f the rtr ad statr r. herefre, Sychrus geeratr mdellg s fte the crtcal step pwer system stablty aalyss, desg ad smulat. Sychrus geeratr mdellg ca be classfed t three types: whte bx []-[], grey bx ad black bx. I the frst type, a mathematcal mdel s btaed usg physcal laws fr sychrus geeratr. he physcal parameters f btaed structure are estmated usg specfc tests. Such tests are specfed IEEE Stadards 5 [3]. Such a structure may be btaed by fte elemet methds, have tw ma dffcultes: ) the cmputat tme fte elemet methd smulat ad ) the large umber f parameters f the electrcal mache [4]. Usually, the frst categry vlves dffcult ad tme-csumg tests. hese tests ca maly be carred ut whe the mache s t servce. I the grey bx mdellg [5]-[7], a kw structure fr the sychrus mache s assumed; the physcal parameters are estmated frm measuremets. I [7], a clsed lp subspace parameter detfcat techque s prpsed t estmate Heffr-Phllps mdel parameters. Such a techque s used stead f pe-lp detfcat t avd bas errrs the estmated parameters. I the thrd type (the black-bx mdellg) [8]-[], the structure f the mdel s t kw a prr. he ly ccer s t map the put data set t the utput data set. Such a mdel s used ether a predctve ctrl structure fr applcats such as -le pwer system stablzer desg, r used as a smulatr t test a ff-le desg. Varus -lear mdels such as; Vlterra [8], eural etwrk [9] ad wavelet etwrk [] have bee preseted the lterature. A dffculty f such mdels s the large umber f parameters requred. herefre, several methds such as, aalyss f varace [], vsual spect methd [], ad rthgal least square wth D-ptmalty desg methd [3] have bee develped fr selectg the sgfcat terms. Ather apprach fr mdellg f sychrus geeratr s lcal mdel apprach [4]-[5]. I such methds, smlar lcal lear structures are used fr all peratg cdts. I [5], such a apprach s used t mdel sychrus geeratr. Parttg f peratg space s btaed usg lear ptmzat. After each terat f lear ptmzat, all lcal lear mdels parameters are estmated by glbal learg. herefre, the mult-mdel lses the lcal terpretablty. I such structures usg glbal learg the behavur f lcal mdels may t chage smthly as a fuct f the peratg reg. he use f such smth mdels ca lead t urelable ctrl. he parameters f lcal mdels are usually estmated by glbal r lcal learg algrthm. he lcal learg s faster tha glbal learg. he lcal learg leads t lcal terpretablty whch meas that the lcal mdels reflect the prcess behavur at the crrespdg peratg reg, but such prperty s t satsfed by glbal learg. he umber f effectve parameters mdel s less wth lcal learg tha wth glbal learg (see [6] ad refereces there). I ths paper, a ew teratve detfcat methd s preseted. he prpsed methd uses the lcal lear structure. At each terat, the algrthm uses sub ptmal lcal lear mdels stead f usg the same lcal lear mdels. he lcal mdels cmplexty s, therefre, dfferet at dfferet peratg regs. he prpsed methd cludes several teratve steps: Frst, the wrst lcal mdel s defed accrdg t lcal weghted least squares. he all dvss f ths lcal mdel peratg space are cstructed. he mst sgfcat terms f tw lcal mdels are determed tgether ad ther parameters are estmated smultaeusly by a vel methd. If all determed terms crrespd t e f the tw lcal mdels, e lcal mdel s detfed stead f the wrst lcal mdel. Fally the best glbal mdel s selected. he parameters are estmated by lcal learg algrthm the prpsed methd. I sect, the frmulat f the lcal mdel apprach s descrbed. I sect 3 the prpsed algrthm s preseted. Expermetal setup ad data cllect a mcr-mache are dscussed sect 4. I sect 5 the applcat f the prpsed methd s carred ut the mcr-mache ad the expermetal data s cmpared wth the smulated lear mdel f the sychrus geeratr. Sect 6 ccludes the paper. 66 PRZEGLĄD ELEKROECHICZY (Electrcal Revew), ISS 33-97, R. 87 R 8/

2 he lcal mdel apprach I the lcal mdel apprach, each lcal mdel shws the behavur a reg f the sychrus geeratr s peratg space. I these methds, glbal utput ŷ(k) s equal t the weghted sum f M lcal mdels y (k)=φ (k)θ, =,,,M. I ther wrds, glbal utput s btaed by the average f the lcal mdel utputs y (k) weghted by lcal mdel valdty fuct, that s M () ŷ( k ) y ( k ) ( z( k )) where φ (.) ad Θ are the regress ad parameter vectrs, respectvely, ad z(k)=[z (k),,z q (k)] s the peratg space vectr. he φ (.) ad z(k) ca be chse depedetly. hese vectrs ca be cmpsed f prevus mdel utput ad put. he valdty r weghtg fuct Φ (z(k)) descrbes the ctrbut f the -th lcal mdel t the utput. hs fuct causes smth trast betwee lcal mdels. Usually, the valdty fuct Φ (z(k)) s chse as rmalzed Gaussa fuct, that s q ( z ( k ) c ) m,m exp m,m () ( z( k )) M q ( z ( k ) c ) m j,m exp jm j,m where c,m ad σ,m are, respectvely, the cetre ad the stadard devat f Gaussa fuct. rmalzat meas that acrss the peratg space the sum f valdty fuct ctrbuts s uty. herefre, the ctrbuts f all lcal mdels sum up t uty everywhere the peratg space. he parttg strategy f the peratg space determes the valdty fucts parameters. If a prr kwledge ca be utlzed fr the parttg f the peratg space, ether a grd partt r a data-drve methd has t be chse. A vervew f exstg parttg strateges s gve [6]. I ths paper, the parttg f peratg space s t assumed t be kw a prr. he prpsed methd determes the mst sgfcat lcal mdels terms. Als, the lcal mdels ad valdty fucts parameters are estmated by lear ptmzat. he prpsed methd Assumg a set f put utput data s avalable; frst a glbal mdel s ftted t the avalable put-utput dataset {y(k), u(k)} where y(k) ad u(k) are utput ad put k measuremets, respectvely. ext, the system peratg space s splt t tw halves alg all dmess f peratg space. Fr each dvs, a lcal lear mdel s csdered, that s, (3) y( k ) y ( k ) ( z( k )) y ( k ) ( z( k )) he parameters f the tw lcal lear mdels are estmated such that the tw mdels lcal weghted least squares are mmzed tgether. he lcal weghted least squares are as fllw (4) J ( z( k ))( y( k ) y ( k ( y y ) Q ( y y ), k )) where Q =dag(φ (z()), Φ (z()),, Φ (z()))., he lcal weghted least squares (.e. J ad J )are cmbed t e: J J J ( L y L y ) ( L y L y ) ( L y L y ) ( L y L y ) (5) L y L U ( ) ( y' U' ' ) L y L U y' U' ' I eq. (5), Q, =, s decmpsed t Q =L L where L s a upper tragular matrx wth pstve dagal elemets ad y =UΘ, the memry matrx U s cmpsed f caddate regressrs. w a rthgal least squares wth D-ptmalty algrthm s used rder t determe the mst sgfcat terms ad t estmate the lcal mdels parameters, y (k) ad y (k). See the Appedx fr mre detals f the Orthgal Least Squares wth D-ptmalty algrthm. If all chse terms crrespd ly t e f the tw lcal mdels (fr example, y (k)), the ther mdel alg wth the crrespdg membershp fuct ca be elmated. I such a case t s recmmeded the rthgal least squares wth D-ptmalty algrthm be mplemeted ce mre t determe the mst sgfcat terms f y(k) =y (k).e. J J ( y y ) Q( y y ) (6) ( L y LU ) ( L y LU ) where Q s the valdty fuct crrespdg t the wrst mdel. he matrx Q s decmpsed t Q=L L, where L s a upper tragular matrx wth pstve dagal elemets. Wth the abve-meted methd, ew lcal lear mdels are btaed fr all peratg dmes m=,,,q. Amg all partts, the partt wth the smallest utput errr s chse. he, the lcal lss fucts fr each lcal lear mdel (S j,j=,,l) are cmputed by weghtg the squared mdel errrs wth the crrespdg value f valdty fuct (7) S ( z( k ))( y( k ) ŷ( k )), j,..., l j j k where l detes the umber f lcal lear mdels. ext the lcal lear mdel wth the maxmum lcal lss fuct s chse. w the algrthm s repeated wth regard t the peratg reg f the chse lcal lear mdel. he prpsed algrthm s utled belw: - Idetfy a glbal mdel usg the rthgal least squares wth D-ptmalty algrthm. - Calculate the lcal lss fuct eq. (7) fr all lcal lear mdels. Chse the wrst lcal lear mdel wth the hghest lcal lss fuct. 3- Splt the rectagle f the wrst lcal lear mdel t tw halves wth a axs-rthgal splt. ry dvss all dmess. Carry ut the fllwg steps fr each dvs: 3-a-Cstruct membershp fucts fr rectagle. he ceters f these membershp fucts are the ceters f the rectagle. he stadard devat each dmes s calculated as σ m =kδ m, where Δ m s the wdth f the hypercube the dmes m=,,q ad k a factr whch s chse a prr. 3-b- Cstruct all valdty fucts. 3-c-Usg eq. (5) f the rthgal least squares wth D-ptmalty algrthm determe a umber f y (k) ad y (k) PRZEGLĄD ELEKROECHICZY (Electrcal Revew), ISS 33-97, R. 87 R 8/ 67

3 terms as mst sgfcat terms ad estmate ther parameters. 3-c-- If all the chse terms are ly crrespdg t e f the tw lcal mdels, mplemet the rthgal least squares wth D-ptmalty algrthm fr eq. (6). 3-d - Calculate the sum f square errrs fr the glbal mdel (glbal lss fucts). 4- Amg all dvss cstructed step 3, select the e wth the smallest glbal lss fuct. 5- Check the termat crter: f satsfed, algrthm s stpped. Otherwse g t step. I the prpsed algrthm, the sgfcat terms f bth lcal mdels at each dmes are estmated tgether. he lcal mdels cmplexty s dfferet at dfferet peratg regs. Cmplex (r smple) mdel s used fr the peratg cdt f the sychrus geeratr wth mre (r less) cmplex. I ths methd, t s decded whch f the tw selects s better t acheve a mre precse mdel: Icreasg the umber f lcal mdels (f the selected parameters belg t bth regs), r creasg the umber f terms (f the selected parameters belg t e f the regs). Expermet sychrus geeratr he system uder csderat s a 3 kva, 8 V, 3 phase mcr-mache, drve by a DC mtr. he mcr mache ca represet dyamc respse f much larger sychrus maches whe the parameters ad varables are csdered a rmalzed vers (per ut system [7]). he ma prblem wth a mcr-mache ca be the feld tme cstat, whch s much lwer tha that f the larger maches. hs prblem has bee vercme usg a tme cstat regulatr, whch s used t crease the effectve feld tme cstat t match that f the larger uts. he expermetal setup used fr the expermet s shw Fg.. he sychrus geeratr s drve by a DC mtr. he excter put sgal s appled t the sychrus mache thrugh a D/A cverter. he feld vltage, termal vltage (v t ) ad the electrcal pwer (P) are measured ad sampled by the data acqust system. he mache s cected t a cstat vltage bus by a duble crcut trasmss le mdelled by lumped elemets. Each crcut cssts f sx π sects ad smulates the perfrmace f a 3 km lg 5 kv trasmss le. he samplg tme was selected t be 5 ms. hs samplg tme prved t be fast eugh t capture the requred dyamcs. A pseud radm bary sequece (PRBS) sgal wth 5% f the mal value was appled. he tal peratg cdt was selected t be P=.6, Q=.53, v t =.. he feld vltage, termal vltage ad electrcal pwer measured are shw Fg.. Fg.. Expermetal setup fr mcr-mache Smulat Results w that the system uder study s explaed, the prpsed methd s appled the sychrus geeratr. he rder f lcal lear mdels s csdered t be equal seve. I ths study, the put s the sampled feld vltage ad the utputs are the sampled termal vltage ad electrcal pwer. Fr cveece f mplemetat, the put-utput data s subtracted by ther mea values, respectvely. he utputs f a detfed mdel ca be recvered t the rgal system peratg reg. he cverted put ad utput are deted by u(k) ad y(k), respectvely. he peratg space s chse as z(k)=[u(k-), y(k-)]. he prpsed algrthm wth α=.5 s appled fr electrcal pwer ad termal vltage. Parttg f peratg space (z) based the prpsed methd fr mdellg termal vltage ad electrcal pwer are shw Fg.3 ad 4, respectvely. 68 PRZEGLĄD ELEKROECHICZY (Electrcal Revew), ISS 33-97, R. 87 R 8/

4 Fg.. Expermetal data wth a PRBS sgal appled t the feld. Valdty fuct.5.5 y(k-) u(k-) Fg.3. peratg space parttg s geerated by the prpsed algrthm fr termal vltage. Valdty fuct.5.5 y(k-) u(k-) Fg.4. peratg space parttg s geerated by the prpsed algrthm fr electrcal pwer. he umber f partts s depedet behavur f sychrus geeratr each reg. he umber f partts regs wth cmplex behavur s mre tha regs wth smple behavur. Idetfcat results wth the detfed mdel ad measured utput are shw Fg (a)emal Vltge tme(s) Feld Vltage me(s).65 Electrcal Pwer me(s).3 ermal Vltage me(s). measured prpsed methd measured prpsed methd (b)electrcal Pwer tme(s) Fg. 5. Idetfcat results wth the detfed mdel ad the mcrmache utputs It ca be see frm fg. 5 that the prpsed methd ca predct the sychrus geeratr behavur well, despte the fact that the geeratr peratg cdts chage sgfcatly... Ccluss lear detfcat f a sychrus mache usg lcal structure s descrbed ths paper. I the prpsed methd, t bta a precse mdel fr sychrus geeratr s peratg space, the peratg space s dvded t several regs. he umber f partts regs wth smple behavur s fewer tha regs wth cmplex behavur. Als, the prpsed methd uses dfferet lcal lear mdels fr dfferet peratg regs. bta a mre precse mdel, at each terat, a chce s made betwee creasg the umber f ptmal lcal mdels r creasg the cmplexty f each lcal mdel. he prpsed methd has bee verfed by studes usg actual data btaed a physcal sychrus mache. I ths paper, termal vltage ad the actve pwer are csdered as the utputs f the system ad the feld vltage as the put f the system. he btaed mdel ca be used fr system aalyss ad ctrller desg, ad s plaed t be used fr desgg a pwer system stablzer a predctve -le ctrl structure. Appedx: he Orthgal Least Squares wth D- ptmalty algrthm A lear--parameters mdel ca be frmulated as (8) y( k ) ( k ) e( k ),k,,..., where φ (k), =,,, are all caddate mdel terms, e(k) s a ucrrelated mdel resdual sequece wth zer mea ad varace f σ ad θ,=,,, are the ukw parameters t be estmated. Eq. 8 ca be represeted matrx frm as (9) y U E where y=[y(),,y()], U=[φ,,φ ], φ =[φ (),,φ ()], θ=[θ,, θ ], E=[e(),,e()]. A rthgal decmpst f U s () U PA where A s a ut upper tragular matrx ad P s a matrx wth rthgal clums p such that () P P dagp p,,..., s that eq. 9 ca be expressed as () y PA E P E he rthgal least squares slut θ ca be estmated frm (3) ˆ ( P P ) P y r (4) p y ˆ,,,..., he rgal parameters θ ca be estmated frm (5) A he mea squares errr (J) s cmputed as J y y (6) ( p y ) y y PRZEGLĄD ELEKROECHICZY (Electrcal Revew), ISS 33-97, R. 87 R 8/ 69

5 ehace mdel rbustess, eq. 6 s cmbed wth D- ptmalty crter. Such a crter s defed as: (7) maxdet( P P ) ( ) Eq. 6 ad D-ptmalty are augmeted as ( p y ) J y y lg( ) (8) ( p y ) y y lg( ) where pstve umber α regulates the trade ff betwee mdel apprxmat ablty ad rbustess. te that the et ctrbut f each term p ca be cmputed depedetly as ( p y ) lg( ). Eq. 8 ca be expressed as ( p y ) ( ) ( ) (9) J J lg( ) At the -th terat, a caddate term s selected as -th term f t prduces the smallest J () [3]. he select ( ) ( ) J J prcedure s termated f. he detfed mdel s expressed as () y( k ) p ( k ) e( k ), k,.., he mdel utput s represeted by meas f the rthgal mdel terms () y( k ) ( k ) e( k ), k,.., where the parameters,,...,, γ ={,,,} ca be calculated frm eq. 5. {γ, γ,, γ } s the dex set f -zer cmpets f θ where θ γs the γ -th cmpet the parameter vectr θ. REFERECES [] Adrzej B b ń, Stefa Paszek, Sebasta B e r h a use, Estmat f turbgeeratr electrmagetc parameters based verfed by measuremets wavefrms cmputed by the fte elemet methd, Przegląd Elektrtechczy, 86 (), r 8, 6-. [] Jerzy Kudła, Mathematcal mdel f a sychrus mache takg t accut saturat effect, Przegląd Elektrtechczy, 8 (5), r, 6-. [3] IEEE stadard IEEE Gude: test prcedures fr sychrus maches.part acceptace ad perfrmace testg. Part II test prcedures ad parameter determat fr dyamc aalyss. [4] Abdallah Barakat, Slm a, Gérard Champes, Emle Mu, Aalyss f sychrus mache mdelg fr smulat ad dustral applcats, Smulat Mdellg Practce ad hery, 8(),. 9, [5] Melgza J, Jesus R, Heydt G, Keyha A, A algebrac apprach fr detfyg peratg pt depedet parameters f sychrus mache usg rthgal seres expass, IEEE ras Eergy Cvers., 6() r, [6] Melgza JJR, Heydt G, Keyha A, Agrawal BL, S e l D., Sychrus mache parameter estmat usg hartley seres, IEEE ras Eergy Cvers, 6() r., [7] M. S lma D. W e stwck O.P. M alk, Idetfcat f Heffr Phllps mdel parameters fr sychrus geeratrs peratg clsed lp, IE Geer. rasm. Dstrb., (8), r. 4, [8] R. Dalrry Fard, M. karrar, O.P.Malk, Sychrus Geeratr Mdel Idetfcat fr Ctrl Applcat Usg Vlterra Seres, IEEE rasact O Eergy Cvers, (5), r. 4, [9] S h a msllah P, Malk OP. O-le detfcat f sychrus geeratr usg eural etwrks. Prceedgs f the Caada cferece electrcal ad cmputer egeerg, CCECE 96, Part, 996, p [] M. k a rrar, O.P.Malk, Idetfcat f sychrus geeratr usg adaptve wavelet etwrks, Electrcal pwer ad Eergy system, 7(5) r., 3-. [] L d, I., Ljug,L. Regressr ad structure select ARX mdels usg a structured AOVA apprach. Autmatca 44 (8), r., [] B a, E.-W., e m p, R. Represetat ad detfcat f -parametrc lear systems f shrt term memry ad lw degree f teract. Autmatc, 46(), r., [3] Hg, X. ad Harrs, C. J., lear mdel structure desg ad cstruct usg rthgal least squares ad D- ptmalty desg, IEEE ras. eural etwrk, 3(), r. 5,45-5. [4] Murray-Smth R., Jhase,.A. Multple Mdel Appraches t Mdelg ad Ctrl. aylr ad Fracs, Ld 997. [5] M. D. Brw, D. Fly ad G. W. Irw, Multple mdel lear ctrl f sychrus geeratrs, rasacts f the Isttute f Measuremet ad Ctrl 4(), r. 3, 5-3. [6] e l l es, O. lear System Idetfcat: Frm Classcal Appraches t eural etwrks ad Fuzzy Mdels. Sprger Verlag,. [7] Kudur P. Pwer system stablty ad ctrl, McGraw-Hll Ic., 994. Authrs: S. S. Ahmad, Electrcal Egeerg Departmet, Amrkabr Uversty f echlgy, ehra, Ira, E-mal: ahmad_salma@aut.ac.r; Prf. M. Karrar, Electrcal Egeerg Departmet, Amrkabr Uversty f echlgy, ehra, Ira, E- mal: karrar@aut.ac.r. 7 PRZEGLĄD ELEKROECHICZY (Electrcal Revew), ISS 33-97, R. 87 R 8/