GENERATOR PARAMETER IDENTIFICATION USING AN EXTENDED PARTICLE SWARM OPTIMISATION METHOD

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1 Corol 004, Uversy of Bah, UK, Sepeber 004 ID- GNRAOR PARAMR IDNIFICAION USING AN XNDD PARICL SWARM OPIMISAION MHOD J. S. Hu, C. X. Guo, Y. J. Cao* (College of lecrcal geerg, Zheag Uversy, Hagzhou 3007, Cha) (*Correspog auhor; -al: Keywors: Sychroous geeraor, eee parcle swar opsao, paraeer efcao Absrac hs paper preses a eee eho of parcle swar opsao () o he geeraor paraeer efcao proble. Copare wh he orgal parcle swar opsao () eho, eho uses ore parcles forao o corol he uao operao. A ew aapve sraegy for choosg paraeers s also propose o assure covergece of eho. Coparso aog he hree ffere ehoologes,, a P, s presee he paper o show he poeal of applcaos of eho o paraeer efcao a syse oelg, especally he case of large ose. Irouco Accurae eerao of geeraor paraeers as operag coos chage s pora o boh power syse aalyss a corol syse esg. he observe sulus respose aa of geeraor are usually use o efy he syse paraeers a a error crero of he respose aa s use as a obecve fuco o be se, whch s ypcally a fuco of he square precve errors. Varous ehos of paraeer efcao have bee aope he efcao of geeraor paraeers [3-5]. All of hese ehos usg grae of he obecve fuco as search recos coul o eal wh he probles of local a a aa polluo whle he quarac appg bewee he precve errors a he efe syse paraeers s geerally a cople, olear a possbly o-cove fuco. hese algorhs e o verge whe he easurees of geeraor oupus are hghly coaae by ose. I he pas wo ecaes, ay global search ehos have bee propose. Aog hese ehos, Sulae voluo (S), has bee cosere as a vable a prosg reco of eplorao. he S s spre by aural pheoea erve fro sulag Darwa evoluoary heory a clues hree broaly slar aveues: Geec Algorhs (GAs), voluo Sraeges (S), a voluoary Prograg (P) [3, 6]. Recely, a ew parcle swar opsao () eho was propose by Keey a berhar [, ]. hs ype of algorhs s oele o processes of he socologcal behavor assocae wh br flockg. eho s oe of he evoluoary copuao echques esseally. Base o he orgal parcle swar opsao eho, hs paper preses a ew eee parcle swar opsao () eho whch ca be apple o ffcul search probles. Copare wh eho, eho uses ore parcles forao o corol he uao operao a ees he orgal forulas of eho. eho ca search he global opal soluo ore effecvely. I hs paper, eho s apple o efy he paraeers of a geeraor uer oral operag coos. he geeraor oel s base o he Park s woas represeao [3]. he paraeers o be efe have efe physcal eags, whch are cooly use for sably aalyss a corol of power syses. hs oel sulaes a praccal evroe for suy of geeraor paraeer efcao probles. he hree ehos,, a P [3], are use o eal wh he sae efcao proble. he coparso of sulao resuls shows ha eho s he os powerful eho o search he real values of paraeers eve wh he aa of geeraor oupus ha are hghly coaae by ose. Maheacal escrpo of geeraor paraeer efcao. Geeraor oel o coser he paraeer efcao of a sgle geeraor coece o a large-scale power syse, ca be assue ha he geeraor s coece o a eeral syse whch has a large capacy a ufe bus volage a frequecy. hs s llusrae Fgure. he equaos of he geeraor ca be escrbe as followg equaos: δ ω ω () b K D K C ω δ ω V () q M M M M

2 Corol 004, Uversy of Bah, UK, Sepeber 004 ID- K K 3 4 δ q q 0 0 C + V f 0 0 K K K K f δ 5 6 q f KC K V + 3 u he oupu easurees ca be obae fro he followg equaos: ω ω (5) 5 6 q 3 (3) (4) V K δ + K + C V (6) P K δ + K + C V (7) q he sybols of he above equaos ()~(7) are gve he Appe. he coeffces are escrbe eale referece [3]. he equaos ()~(4) eoe he efcao oel of sgle ache fe-bus power syses as show Fgure. Whou kowg forao of sa power syses, geeraor paraeers ca be efe hrough he efcao equaos usg oupu easuree V as ecouple value. If I s use sea of V, whch represes he fluece of he eeral syse, equaos ()~(4) ake he followg for: δ ω ω (8) b K C C D 6 ω δ ω M M (9) K C C 7 C C 5 I q M M K C C K C C δ q q 0 0 (0) CC 5 + I f 0 0 K ( K C C ) K ( K C C ) f δ q () KCC K 3 5 I + u f he easuree equaos s ow gve as follows: ω ω () V ( K C C ) δ (3) + ( K CC) + CC I q 3 5 P ( K C C ) δ 6 + ( K CC ) + CC I 7 q 5 (4) geeraor V + Q R + X P I G+ B Fg. he geeral coeco of he geeraor. Geeraor paraeer efcao V syse Iefcao equaos ()~(4) a oupu easuree equaos (5)~(7) ca be escrbe as he followg couous-e geeraor oel: X() A( θ) X() + B( θ) u() + B ( θ) V () + ω() (5) e y( ) C( θ) X( ) + C ( θ) V ( ) + ν ( ) (6) k k e k k where k s he kh saplg e; θ s he paraeer vecor o be efe; X () s he saus vecor; y ( k ) s he oupu vecor; u () s a reverse pseuo-rao bary sequece (PRBS) pu apple o he ecer; V ( ) s a era k easuree; ω () a v ( k ) are he ose of he syse a easurees, whch are whe ose. he paraeer vecors are escrbe as follows: θ X, X, X,, M, D q 0 () δ, ω,, q f X y ( ) ω, V, P [ ] k hs couous-e oel ca be use as paraeer efcao oel of sychroous geeraor. he prcpal agra of paraeer efcao s show as Fgure. he respose oupu aa of geeraor uer PRBS pu apple o he ecer are use o efy he syse paraeers a a error crero of he respose aa s use as a obecve fuco o be se, whch s ypcally a fuco of he square preco errors. A each erao sep, we oba he error equao: e( θ, ) y( ) ( C( θ) X( ) + C ( θ ) V ( )) (7) k k k e k a he we oba obecve fuco: pu geeraor oel oupu oupu Y Y error Mofy paraeers of geeraor f (, k ) θ Iefcao eho Fg. Paraeer efcao prcpal agra of sychroous geeraor

3 Corol 004, Uversy of Bah, UK, Sepeber 004 ID- f ( θ, ) { e( θ, ) Λe ( θ, )} (8) k k k where Λ s a u ar. So, geeraor paraeer efcao s equal o sao of he obecve fuco equao (8). 3 ee parcle swar opsao eho 3. Parcle swar opsao eho eho s a geeral heursc searchg echque sulag br flockg [-]. Ially, each parcle represes a poeal soluo o a opsao proble a velocy value raoly. A each erao sep, each parcle keeps rack of s cooraes he proble space whch are assocae wh he bes soluo (fess) has acheve so far. he fess value s also sore a he value s calle pbes. Aoher bes value ha s racke by he local verso of parcle swar opsao eho s he bes value calle lbes, aae wh a local opologcal eghborhoo of parcles. he lbes oel res o preve preaure covergece of eho by aag ulple opal parcles. he searchg space ca be ve o several pars by soe rules (parcles space sace or parcles sequece uber). he global verso of eho offers a faser rae of covergece a he epese of robusess. he gbes oel aas a sgle bes value, calle gbes, across all he parcles he swar. All parcles ry o ofy velocy a poso usg her pbes a lbes or gbes. he ofe velocy of each parcle ca be calculae usg he curre velocy a he sace fro pbes a lbes or gbes as show below equao: v cv + c r () ( p, ) + c + 3 r() ( p ) (9) where c, c a c are posve cosa coeffces a r 3 a r are uforly srbue rao ubers [0,], v s he curre velocy of parcle a erao, v + s he ofe velocy, p s he pbes,, s he lbes or gbes. Usg he above equao, a cera velocy ha graually ges close o pbes a lbes or gbes ca be calculae. he curre poso (searchg po he soluo space) ca be ofe by he followg equao: + + v+ (0) Base o he equaos (9), (0) of eho, each parcle ofes s poso usg forao oly fro wo bes values. 3. ee parcle swar opsao eho eho uses oly wo bes values pbes a lbes (or gbes) o ofy parcle s poso a velocy a each erao sep, whou coserg ay oher parcles values. hs paper preses a ew eho of base o eho usg ore parcles forao o ofy parcle s p poso a velocy a offers a schee of eployg coeffces o guaraee he covergece of eho. he equaos of eho ca be escrbe as followg equaos: v w v + φ ( p ) + η ( p ) () +,, + + v+ () where φ c r (), η c r (). ach parcle of,,,, eho ofes s poso a velocy usg pbes soluos parcle acheve a gbes of all parcles (or lbes of eghborhoo parcles). Because of applyg ore parcles forao erao process, eho possess sroger global covergece ha eho. he for of he recurrece relao of parcle s poso ca be erve as follows: subsug equao () o () a fro equao v, we have he followg equao, + ( + ω φ η) ω (3) + φ p + η p,, whch s a o-hoogeeous recurrece relao ha ca be solve usg saar recursve echques. hs recurrece relao ca be wre as a ar-vecor prouc, + + w φ η w φp+ ηp 0 0 (4) 0 0 where φ φ, η η, φp φ p, ηp, η p,. he characersc polyoal of he equao (4) s, ( λ)( ω λ( + ω φ η) + λ ) whch has a rval roo of λ.0, a wo oher soluos (5) a (6), + w φ η+ γ α (5) + w φ η γ β (6) where γ φ η ( + w ) 4w (7) Noe ha α a β are boh egevalues of he equao (4). he eplc for of he recurrece relao (3) s he gve by equao, k + k α + k β (8) 3 where k, k a k are cosas eere by he al 3 coos of he syse a each erao sep. A pora aspec of he behavor of a parcle cocers wheher s raecory (specfe by ) coverges or verges. he coos uer whch he sequece { } + 0 wll coverge s eere by he ague of he values α a

4 Corol 004, Uversy of Bah, UK, Sepeber 004 ID- β, as copue usg equaos (5) a (6). Fro equao (7) s clear ha here are wo cases: ) Case A: ( + w φ η) < 4w I hs case, γ wll be a cople uber wh a o-zero agary copoe. A cople γ resuls α a β beg cople ubers wh o-zero agary copoes as well. Coser he value of he l, hus equao (8) becoes, θ σ k + k α e + k β e (9) 3 e σ α e θ a β are he epoe epresso of he rval roos. Clearly, equao (9) eplas he sequece { } + 0 wll coverge whe a( α, β ) <, so, l k + k α + k β k 3 ) Case B: ( + w φ η) > 4w I hs case, γ, α a β wll be real ubers, fro equao (8), f a( α, β ) <, he he sequece { } + 0 coverges. Fro he aalyss of above wo cases, we oba he + s covergece coos of he sequece { } 0 a( α, β ) <. Oe popular choce of upag paraeers s ha c,.4968, c.4968 a, w []. O accou of c r () a η,,,, φ c r (), so φ (0,.4968), η (0,.4968) a φ + η (0,.994). Whe φ + η (0, 0.0], fro equao (7), wll ply a realvalue γ, whch correspos o case B, he γ 0 a.798 a( α, β ) φ η + γ <. Slarly, φ + η (0.0,.994) wll resul a cople γ value, whch correspos o case A a fro equaos (5), (6) we ca assure α β <. he above aalyss of paraeers shows ha he popular choce of paraeers [] sasfy he covergece coos a wll assure covergece of he +. sequece { } 0 eho ffers fro eho s ha φ s he su of he coeffces of pbes parcles a η s he su of he coeffces of gbes parcles ( or lbes parcles ). he egree of porace of each parcle s weghe hrough parcle s fess value. he beer he parcle s fess value s, he ore pora he parcle s fluece s. So hs paper propose he rule of upag paraeers chose as follows: f φ.4968, f f η.4968 f (30) where f s he pbes parcle s fess value a f s he lbes (or gbes) parcle s value. hs aapve sraegy of upag paraeers ca assure covergece of eho a ehace he global covergece capably of eho. 4 Sulao resuls eho, eho a P eho have bee evaluae sulao sues. he paraeer efcao was uerake four cases wh ffere levels of easuree ose. he sulao resuls are presee o show he effecveess of eho. A reverse PRBS wh a aplue of 0.0 pu.. s eploye o ece he syse. he rage of efe paraeers are assue o be ± 00% of he real paraeers. he easuree ose s chose o be N(0, σ ), where σ s he varace of he ose. he ose varaces he four cases are lse able respecvely, whch correspos o ffere level of ose. o evaluae ucera value cobaos of a of eho, hey have bee eecue 30 es o solve he above paraeer efcao proble uer varous value cobaos. he resuls show ha he bes soluo ca be obae by eho whe a 3. case case case3 case4 able Measuree ose varaces ω V P V able lss he correspog efe paraeers hese four cases, where coparso resuls, obae usg eho, eho a P eho respecvely, are presee. I ca be see ha he case of ose free a oher low level ose, he efe paraeers obae by P eho are as goo as hose obae usg eho. However, he cases of larger ose, he perforace of P eho eeroraes quckly, whle eho oes o. I case, case 3 a case 4, he sulao resuls able show ha he efe paraeers usg eho are uch closer o he real values ha hose obae usg P eho proely. eho possesses sroger syhec capably of paraeer efcao. Fgures 3 shows he respose of ω whch s coaae by he ose Case. I ca be see ha eve he case of poor easuree aa coaae by large ose, he

5 Corol 004, Uversy of Bah, UK, Sepeber 004 ID- eho a he P eho boh ca sll work, bu he eho ca work ore effecvely ha he P eho fro he sulao resuls of able. Iefe paraeers Real value P case case case3 case4 P P P w (p.u.) able he resuls of he efcao es X X Xq 0 M D hree a error(%) RAL P e(sec.) Fg.3. Mache spee respose o PRBS pu 5 Cocluso he paper preses a ew eho of eee parcle swar opsao o he geeraor paraeer efcao proble a ercoece power syse base o he easurable oupu aa. A ew aapve sraegy for choosg paraeers s propose o assure covergece of eho. eho s ore geeral ha eho a ca be use for cople olear opsao probles. he sulao resuls show ha eho s a powerful search eho whch ca be use for paraeer efcao of cople syses a real evroe where here ess large easuree ose. Refereces [] M. Clerc a J. Keey, he parcle swar eploso, sably, a covergece a ulesoal cople space, I ras. o voluoary Copuao, 6, pp , (00). [] R. berhar a J. Keey, Parcle swar opzao, I Proceegs of I Ieraoal Coferece o Neural Neworks, Perh(Ausrala), pp , (995). [3] J.. Ma a Q. H. Wu, Geeraor paraeer efcao usg evoluoary prograg, I. J. lecrcal Power & ergy Syses, 7, pp , (995). [4] M. Naba,. Nshwak, S. Yokokawa, K..Ohsuka, a Y. Uek. Iefcao of paraeers for power syse sably aalyss usg Kala fler, I ras. Power Apparaus & Syses, 00, pp , (98). [5] R.Wakeue, I.Kawa, X.Da-Do, a A.Keyha, Ieravely reweghe leas squares for au lkelhoo efcao of sychroous ache paraeers fro o-le ess, I ras. ergy Coverso, 4, pp , (999). [6] Q. H. Wu, Y. J. Cao a J. Y. We, Opal reacve power spach usg a aapve geec algorh, Ieraoal Joural of lecrcal Power & ergy Syses, 0, pp , (998). Appe Noeclaure: cree of he varable δ orque agle ω agular spee ω base agular frequecy b q quaraure-as volage beh rase reacace f fel volage V bus volage u PRBS pu sgal o he ecer M era e cosa D apg coeffce 0 rec-as ope-crcu rase e cosa K ga of he ecao syse e cosa of ecao syse X rec-as sychroous reacace X X q rec-as rase reacace quaraure-as sychroous reacace. Ackowlegees he auhors woul lke o hak he referees for her valuable coes a apprecae he facal suppor fro Naoal Naural Scece Fouao of Cha (NNSF Nos , )