Application of Extended Kalman Filter to Parameter Estimation of Doubly-Fed Induction Generators in Variable-Speed Wind Turbine Systems

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1 Applcaton of Extene Kalman Flter to Parameter Etmaton of Doubly-Fe Inucton Generator n Varable-Spee Wn Turbne Sytem Mohame Abelrahem Stuent Member, IEEE Inttute for Electrcal Drve Sytem an Power Electronc Technche Unvertät München (TUM) Munch, Germany Emal: mohame.abelrahem@tum.e Chrtoph Hackl Member, IEEE Munch School of Engneerng Reearch Group Control of Renewable Energy Sytem (CRES) Technche Unvertät München (TUM) Munch, Germany Emal: chrtoph.hackl@tum.e Ralph Kennel Senor Member, IEEE Inttute for Electrcal Drve Sytem an Power Electronc Technche Unvertät München (TUM) Munch, Germany Emal: ralph.kennel@tum.e Abtract Th paper propoe a parameter etmaton metho for oubly-fe nucton generator (DFIG) n varablepee wn turbne ytem (WTS). The propoe metho employ an extene Kalman flter (EKF) for etmaton of all electrcal parameter of the DFIG,.e., the tator an rotor retance, the leakage nuctance of tator an rotor, an the mutual nuctance. The nonlnear tate pace moel of the DFIG erve an the egn proceure of the EKF ecrbe. The obervablty matrx of the lnearze DFIG moel compute an the obervablty checke onlne for fferent operaton conton. The etmaton performance of the EKF llutrate by mulaton reult. The etmate parameter are plotte agant ther actual parameter. The etmaton performance of the EKF alo tete uner varaton of the DFIG parameter to nvetgate the etmaton accuracy for changng parameter. Keywor DFIG, parameter etmaton, extene Kalman flter, vector control, wn turbne ytem NOTATION N, R, C are the et of natural, real an complex number. x R or x C a real or complex calar. x R n (bol) a real value vector wth n N. x the tranpoe an x = x x the Euclean norm of x. n = (,..., ) the n-th menonal zero vector. X R n m (captal bol) a real value matrx wth n N row an m N column. O n m R n m the zero matrx. x y z R 2 a pace vector of a rotor (r) or tator () uantty,.e. z {r, }. The pace vector expree n ether phae -, tator fxe -, rotor fxe r-, or arbtrarly rotatng k-coornate ytem,.e. y {,, r, k}, an may repreent voltage u, flux lnkage ψ or current,.e. x {u, ψ, }. E{x} or E{X} the expectaton value of x or X, rep. I. INTRODUCTION The electrcal power generaton by renewable energy ytem (RES; uch a e.g. wn turbne ytem) ha ncreae urng the lat year an, o, RES gnfcantly contrbute to the reucton of carbon oxe emon an therefore to a lower envronmental polluton [1]. The ncreae of electrcal power generaton of RES wll contnue a countre are extenng ther renewable acton plan. Therefore an, nce wn power alreay toay economcally compettve, n partcular the hare of wn power lkely to ncreae further worlwe. Among the varou type of wn turbne generator, the DFIG the mot commonly ue generator n on-hore an off-hore applcaton, accountng for aroun 5% of the ntalle wn turbne nomnal capacty worlwe [1]. DFIG can upply actve an reactve power, operate wth only a partal-cale power converter (aroun 3% of the machne ratng), an acheve a certan re through capablty [2]. Operaton above an below ynchronou pee feable. Due to ther we ue n the WTS, the evelopment of avance an relable control technue for DFIG ha receve gnfcant attenton urng the lat year [2]. Example of thee control technue are e.g. vector control, rect torue control, rect power control, an moel prectve control [2]-[5]. Vector control ha o far proven to be the mot popular control technue for DFIG n varable-pee WTS [2]. Th metho allow for a ecouple control of the actve an reactve power of WTS va regulatng the uarature component of the rotor current vector nepenently. However, vector control rele on the accurate knowlege of the (electrcal) parameter of the DFIG. If the control parameter o not match the actual value, the DFIG mght not work properly: the cloeloop ytem may be eterorate or even become untable. The electrcal parameter of DFIG are entve to temperature change, magnetc aturaton an ey current [6]. Therefore, n mot cae, the ue of contant parameter oe not allow to play the real ynamcal behavor of the DFIG accurately. Onlne parameter etmaton neceary. Although there are everal parameter etmaton technue avalable for nucton machne [6], only few reult are publhe on the parameter etmaton of DFIG n varable-pee wn turbne ytem [7]-[11]. In [7], two onlne parameter entfcaton metho for DFIG are propoe bae on moel reference aaptve ytem (MRAS). The frt metho reure a tetng gnal to excte (all) ytem egenvalue, whch not feable/reaonable when the machne connecte to the gr. The econ metho oe not rely on exctaton. However, th metho neglect the tator an rotor retance an not capable of etmatng the rotor leakage nuctance. In [8], a earch-bae algorthm for parameter entfcaton

2 Drve PWM / Drve PWM / r m e Lookup table Q, ref u c, ref u Q f, ref r PLL m e r l Q u c f Q f / / r, ref r, ref f, ref f f f, ref r r u / ( L l r r ( L l r r L ) l m L ) L u u m f f L f f DC Lnk RSC C C c u c GSC P c f R f r flter L f P f & Q f DFIG Encoer Q P Tr. m Wn turbne t gear box Gr Fgure 1: DFIG topology an control tructure for the varable-pee wn turbne ytem. of DFIG n wn turbne ytem propoe. However, f the ntal value of the parameter are not properly electe (a mentone n [8]), the objectve functon mght converge olely to a local optmum ntea of a global optmum. Moreover, the electrcal parameter of the DFIG are not etmate eparately: Combnaton of the parameter uch a the rato of rotor retance an nuctance or the rato of mutual an rotor nuctance are etmate. In [9], an aaptve etmaton algorthm ue for etmatng the DFIG rotor retance. However, all remanng DFIG parameter are aume to be known. In [1], a Levenberg-Maruart-Fletcher metho for parameter etmaton ue. However, olely tator retance, tator nuctance an mutual nuctance are etmate. Rotor retance an nuctance are aume to be contant an known. In [11] an algorthm evelope whch allow for onlne etmaton of tator, rotor an mutual nuctance. However, tator an rotor retance are aume to be contant an known. Snce the late 196, the Kalman flter ha receve huge attenton from varou fel n nutry an acaema an playe a key role n many engneerng cplne for trajectory planng, tate an parameter etmaton, gnal proceng, etc. [12], [13]. In [14] the behavor of two etmaton technue bae on the Kalman flter analyze. The metho utlze an Extene Kalman Flter (EKF) an an Uncente Kalman Flter (UKF) for parameter etmaton of DFIG n wn turbne ytem. However, the obervablty of the (lnearze) DFIG moel not aree an the two Kalman flter are not tete uner parameter varaton. Moreover, the two flter were not optmally tune (parameter etmaton converge after 6 ) an the ue nonlnear tate pace moel epen on tate, nput an output (whch not amble from a theoretcal pont of vew). In th paper, an Extene Kalman flter (EKF) propoe for parameter etmaton of all electrcal parameter of DFIG n wn turbne ytem: The etmate parameter are the tator an rotor retance, the leakage nuctance of tator an rotor, an the mutual nuctance. II. MODELING AND CONTROL OF THE WTS WITH DFIG The block agram of the vector control problem of WTS wth DFIG hown n Fg. 1. It cont of a woun rotor nucton machne mechancally couple to the wn turbne va a haft an gear box wth rato g r 1 [1]. The tator wnng of the DFIG are rectly connecte to the gr va a tranformer, wherea the rotor wnng connecte va a back-to-back partal-cale voltage ource converter (VSC), a flter an a tranformer to the gr. The tranformer wll be neglecte n the upcomng moelng. The rotor e converter (RSC) an the gr e converter (GSC) hare a common DClnk wth capactance C c [A/V] an DC-lnk voltage u c [V]. Detale moel of thee component can be foun n [15]. The tator an rotor voltage euaton of the DFIG are gven by [16]: u (t) = R (t) + t ψ (t), ψ () = 3 (1) u r (t) = r (t) + t ψ r (t), ψr () = 3 (2) }{{} ntal value where (aumng lnear flux lnkage relaton) ψ (t) = L (t) + r (t) (3) r (t) = L r r (t) + (t). (4) ψ Here u = (u a, u b, u c ) [V], ur = (ur a, ur b, ur c ) [V], = ( a, b, c ) [A], r = (r a, r b, r c ) [A], ψ = (ψ a, ψ, b ψ) c [V], an ψr = (ψr a, ψr b, ψr c ) [V] are the

3 tator an rotor voltage, current an fluxe, repectvely, all n the -reference frame (three-phae ytem). Stator L [V/A] an rotor L r [V/A] nuctance can be expree by L = + L σ an L r = + L rσ (5) where L σ an L rσ are the tator an rotor leakage nuctance an the mutual nuctance. R [Ω] an [Ω] are tator an rotor wnng retance. Note that the DFIG rotor rotate wth mechancal angular freuency ω m [ra/]. Hence, for a machne wth pole par number n p [1], the electrcal angular freuency of the rotor gven by ω r = n p ω m an the rotor reference frame hfte by the rotor angle φ r (t) = t ω r (τ)τ + φ r, φ r R (6) wth repect to the tator reference frame (φ r the ntal electrcal rotor angle). A. Moel n tator (tatonary) reference frame The euaton (1) an (2) can be expree n the tatonary reference frame a follow x = (x α, x β ) = T C x by ung the Clarke an Park tranformaton (ee, e.g., [15]), repectvely, gven by (neglectng the zero euence) [ 1 x 1 =γ 2 1 ] [ ] x & x k co(φ) n(φ) = x n(φ) co(φ) 2 }{{}}{{} =:T C =:T P (φ) 1 (7) where γ = 2 3 for an ampltue-nvarant tranformaton (or γ = 2/3 for a power-nvarant tranformaton). Expreng the rotor voltage euaton (2) alo wth repect to the tatonary reference frame (.e. ur = T P (φ r ) 1 T C ur ), the voltage euaton (1) an (2) can be rewrtten a u (t) = R (t) + t ψ (t), ψ () = 2 u r (t) = r (t) + t ψ r (t) ω r (t)jψ r (t), ψ r () = 2 where [15] J := T P (π/2) = B. Moel n tator voltage orentaton [ ] 1. 1 An eental charactertc of the DFIG control trategy that the generate actve an reactve power hall be controlle nepenently. It common to ue an ar-gap flux orentaton [17] or a tator flux orentaton [18]-[2] for the vector control cheme. However, t ha been hown that the tator flux orentaton can caue ntablty uner certan operatng conton [21]. Therefore, followng the ea n [16], [22], n th paper, a tator (gr) voltage orentaton for the vector control cheme ue. The tator voltage orentaton acheve by algnng the -ax of the ynchronou (rotatng) reference frame wth the tator voltage vector u whch rotate wth the tator (gr) angular freuency ω (uner eal conton wth contant } (8) ra gr freuency f >, t hol that ω = 2πf contant). Applyng the (nvere) Park tranformaton wth T P (φ ) 1 a n (7) wth φ (t) = t ω (τ)τ + φ, φ R to the voltage euaton (8) yel the ecrpton n the rotatng reference frame (neglectng ntal value) u k (t) = R k (t) + t ψk (t) + ω Jψ k (t), ur k (t) = r k (t) + t ψk r (t) + (ω ω r (t))jψr k (t), }{{} (9) =:ω l (t) where u k = (u, u ), u k r = (u r, u r ), k = (, ), k r = ( r, r ), ψ k = (ψ, ψ ), ψ k r = (ψ r, ψ r ), are the tator an rotor voltage, current an fluxe n the rotatng reference frame (k-coornate ytem wth axe an ), repectvely. ω l := ω ω r the lp angular freuency. Snce, e.g., ψ k = T P (φ ) 1 ψ = T P (φ ) 1 T C ψ, the flux lnkage are gven by ψ k = L k + r k } ψr k = L r r k + k (1). Moreover, f σ := 1 L2 m (5) L rσ + L σ + L rσ L σ = L r L L 2, m + L rσ + L σ + L rσ L σ one may re-wrte the current a functon of the tator an rotor flux lnkage a follow [23, Sec. 5.4] } k 1 = σl ψ k Lm σl L r ψr k r k 1 = σl r ψr k Lm σl L r ψ k (11). C. Dynamc of the mechancal ytem For a tff haft an a tep-up gear wth rato g r 1, the ynamc of the mechancal ytem are gven by t ω m = 1 ( m e m ) t, ω m () = ωm R (12) Θ g }{{} r =:m m where [23, Sec. 5.4] m e (t) = 3 2 n p (t) Jψ (t) = 3 2 n p k (t) Jψ k (t) (11) = 3 2 n p ψr k (t) Jψ k (t). (13) σl L r the electro-magnetc machne torue (moment), m t [Nm] the turbne torue prouce by the wn (ee Sec. III) an m m = mt g r [Nm] the mechancal torue actng on the DFIG haft. Θ [kgm 2 ] the rotor nerta an n p [1] the pole par number.

4 D. Overall nonlnear moel of the DFIG For the egn of the EKF, the ervaton of a compact (nonlnear) tate pace moel of the DFIG of the form t x = g(x, u), x() = x R 1 an y = h(x), (14) reure. Therefore, ntrouce the tate vector x, the output (meaurement) vector y an the nput vector u a follow: ( ) ψ x =, ψ, ψr, ψr, ω r, R R,, L σ, L rσ, L 1, m y = ( ) r r ω r R 5 (15), u = ( ) u u ur ur R 4. Combnng the ubytem of the DFIG a n (9), (1), (12) an olvng for t ψk, t ψk r, an t ω r = n p t ω m yel the nonlnear moel (14) wth ( u ψ ) R σl Lm σl L r ψr + ω ψ ( u ψ ) R σl Lm σl L r ψr ω ψ ( u ψ ) r r σl r Lm σl L r ψ + (ω ω r )ψ r ( u ψ ) r r σl r Lm σl L r ψ (ω ω r )ψ r g(x, u) = n p ( 3n Θ p 2σL L r ψr k (t) Jψ k (t) m m (ω r ) ) (17) an h(x) a n (16). Note that, n h(x), the parameter R = x 6, = x 7, L σ = x 8, L rσ = x 9, an = x 1 are alo tate varable an m m (ω r ) wll be approxmate n Sec. III a a functon of ω r = n p ω m. E. Overall control ytem of the WTS The complete control block agram of the DFIG n tator voltage orentaton epcte n Fg. 1. For the rotor-e converter (RSC), the -ax current ue to control the DFIG tator actve power (.e., proportonal to the electro-magnetc torue) n orer to harvet the maxmally avalable wn power (.e., maxmum power pont trackng, ee Sec. III), wherea the -ax current ue to control the reactve power flow of the DFIG to the gr. For the gr-e converter (GSC), alo tator voltage orentaton ue [22], [15], whch allow for nepenent control of actve (-ax current) an reactve power (-ax current) flow between gr an GSC. The man control objectve of the GSC to aure an (almot) contant DC-lnk voltage regarle of the magntue an recton of the rotor power flow. DC-lnk voltage control a nontrval tak ue to the poble non-mnmum-phae behavor for a power flow from the gr to the DC-lnk [15], [4]. More etal on controller egn, phae-locke loop an pule-wth moulaton (PWM) are gven n, e.g., [22], [15]. III. MAXIMUM POWER POINT TRACKING (MPPT) Wn turbne convert wn energy nto mechancal energy an, va a generator, nto electrcal energy. The mechancal (turbne) power of a WTS gven by [16], [15], [24]: p t = c p (λ, β) 1 2 ρπr2 t v 3 w }{{} wn power (18) where ρ [kg/m 3 ] the ar enty, r t [m] the rau of the wn turbne rotor (πr 2 t the turbne wept area), c p [1] the power coeffcent, an v w [m/] the wn pee. The power coeffcent c p a meaure for the effcency of the WTS. It a nonlnear functon of the tp pee rato λ = ω mr t g r v w [1] (19) an the ptch angle β [ ] of the rotor blae. The Betz lmt c p,betz = 16/27.59 an upper (theoretcal) lmt of the power coeffcent,.e. c p (λ, β) c p,betz for all (λ, β) R R. For typcal WTS, the power coeffcent range from.4 to.48 [16], [24]. Many fferent (ata-ftte) approxmaton for c p have been reporte n the lterature. Th paper ue the power coeffcent c p from [24],.e. c p (λ, β) = λ := ( 116 λ.4β 5 ) e 21 λ +.68λ 1 λ +.8β.35 β (2) For wn pee below the nomnal wn pee of the WTS, maxmum power trackng the ere control objectve. Here, the ptch angle hel contant at β = an the WTS mut operate at t optmal tp pee rato λ (a gven contant) where the power coeffcent ha t maxmum c p := c p (λ, ) = max λ c p (λ, ). Only then, the WTS can extract the maxmally avalable turbne power p t = c p 1 2 ρπr2 t v 3 w can be extracte. Maxmum power pont trackng acheve by the nonlnear pee controller [15] m e = k pω 2 m m m, k p := ρπr5 t 2g r c p (λ ) 3 (21) whch aure that the generator angular freuency ω m ω ajute to the actual wn pee v w uch that mr t! g rv w = λ hol. Accorng to (21) the optmum torue m e can be calculate from the haft pee ω m = ω r /n p an then t compare wth the actual electro-magnetc torue m e, a hown n Fg. 1. Bae on the fference m e m e the unerlyng torue controller 1 generate the rotor reference current r,ref. Remark: For wn pee above the nomnal wn pee, the WTS change to nomnal operaton,.e. m e = m e,nom, where m e,nom the nomnal/rate generator torue. Spee control acheve by (nvual) ptch control uch that the rate power m e,nom ω m,nom of the WTS generate. IV. EXTENDED KALMAN FILTER AND OBSERVABILITY A. Extene Kalman Flter The EKF a nonlnear extenon of the Kalman flter for lnear ytem. It egn bae on a crete nonlnear ytem moel [25]. For cretzaton the (mple) forwar 1 The torue control loop tll reure a thorough tablty analy whch not conere n th paper.

5 ( + L rσ ) 1 (L h(x) = m + L rσ ) (L σ + L rσ ) + L σ L rσ ( + L σ ), O 4 6 x (16) ( + L σ ) Euler metho wth amplng tme T [] apple to the tmecontnuou moel (14) wth (15), (17) an (16). For uffcently mall T 1, the followng hol x[k] := x(kt ) x(t) an x[k+1] x[k] tx(t) = T for all t [kt, (k + 1)T ) an k N {}. Hence, the nonlnear crete moel of the DFIG can be wrtten a x[k + 1] = =:f(x[k],u[k]) {}}{ x[k] + T g(x[k], u[k]) +w[k], y[k] = h(x[k]) + v[k], x[] = x (22) where the ranom varable w[k] := (w 1 [k],..., w 1 [k]) R 1 an v[k] := (v 1 [k],..., v 5 [k]) R 5 are nclue to moel ytem uncertante an meaurement noe, repectvely. Both are aume to be nepenent (.e., E{w[k]v[j] } = O 1 5 for all k, j N), whle (.e., E{w[k]} = 1 an E{v[k]} = 5 for all k N) an wth( normal probablty ) trbuton (.e., p(α ) = 1 σ α 2π exp (α E{α }) 2 2σ wth σ 2 α 2 α := E{(α E{α }) 2 } an α {w, v }). For mplcty, t aume that the covarance matrce are contant,.e., for all k N: Q := E{w[k]w[k] } an R := E{v[k]v[k] } >. (23) Note that Q an R mut be choen potve em-efnte an potve efnte, rep. Snce ytem uncertante an meaurement noe are not known a pror, the EKF mplemente a follow ˆx[k + 1] = f(ˆx[k], u[k]) K[k] ( y[k] ŷ[k] ) }, (24) ŷ[k] = h(ˆx[k]) where K[k] the Kalman gan (to be pecfe below) an ˆx an ŷ are the etmate tate an output vector, repectvely. The recurve algorthm of the EKF mplementaton lte n Algorthm 1 [25]. The EKF acheve an optmal tate etmaton by mnmzng the covarance of the etmaton error for each tme ntant k 1. A crucal tep urng the egn of the EKF the choce of the matrce P, Q an R, whch affect the performance an the convergence of the EKF. The ntal error covarance matrx P repreent the covarance (or mean-uare error) bae on the ntal conton (often P choen to be a agonal matrx) an etermne the ntal ampltue of the tranent behavor of the etmaton proce, whle uraton of the tranent behavor an teay tate performance are not affecte. The matrx Q ecrbe the confence wth the ytem moel. Large value n Q ncate a low confence wth the ytem moel,.e. large parameter uncertante are to be expecte, an wll lkewe ncreae the Kalman gan to gve a better/fater meaurement upate. However, too large element of Q may be lea to ocllaton or even ntablty of the tate etmaton. On the other han, low value n Q ncate a hgh confence n the ytem moel an may therefore lea to weak (low) meaurement correcton. Algorthm 1: Extene Kalman flter Step I: Intalzaton for k = : ˆx[] = E{x }, P := P [] = E{(x ˆx[])(x ˆx[]) }, K := K[] = P []C[] ( C[]P []C[] + R ) 1 where, for k, C[k] := h(x) x ˆx [k] Step II: Tme upate ( a pror precton ) for k 1: (a) State precton ˆx [k] = f(ˆx[k 1], u[k 1]) (b) Error covarance matrx precton P [k] = A[k]P [k 1]A[k] + Q where A[k] := f(x,u) x ˆx [k] Step III: Verfcaton of (local) obervablty k 1: n o [k] := rank ( S o [k] ) wth S o [k] a n (26) Step IV: Computaton of Kalman gan for k 1 K[k] = P [k]c[k] ( C[k]P [k]c[k] + R ) 1 Step V: Meaurement upate ( correcton ) for k 1: (a) Etmaton upate wth meaurement ˆx[k] = ˆx [k] + K[k](y[k] h(ˆx [k])) (b) Error covarance matrx upate P [k] = P [k] K[k]C[k]P [k] Step V: Go back to Step II (ue C[k]). The matrx R relate to the meaurement noe charactertc. Increang the value of R ncate that meaure gnal are heavly affecte by noe an, therefore, are of lttle confence. Coneuently, the Kalman gan wll ecreae yelng a poorer (lower) tranent repone. In [26] general gue lne are gven how to elect the value of Q an R. Followng thee gue lne, for th paper the followng value have been electe Q = 1 6 ag{2, 2, 2, 2, 5,.4,.4,.2,.2,.5} R = ag{1, 1, 1, 1, 1} (25) P = 1 3 ag{5, 5, 5, 5, 3,.2,.2,.4,.4,.3} x = 1 3 {,,,,, 1.4,.4,.4,.4,.5}. B. Obervablty The obervablty of a lnear ytem can be verfe by computng the obervablty matrx an t rank. For nonlnear ytem, t poble to analyze the obervablty locally by analyzng the lnearze moel aroun an operatng pont [27]. The obervablty matrx of the lnearze moel of the

6 ] [ m / v w Table I: Steay tate etmaton error of the EKF. Etmate tate R L σ L rσ Normal conton.5%.4%.2%.2%.4% R an.9%.7%.8% 1% 1.1% ncreae by 1% ncreae by 1% 1.4% 1.6% 1.5% 1.5% 1.6% r[pu] tme [] Fgure 2: Overall mulaton cenaro: Wn pee profle v w an correponng rotor pee ω r. conere DFIG a n (22) gven by C[k] C[k]A[k] C[k]A[k] 2 S o [k] :=. R 5 1, (26).. C[k]A[k] 9 where A[k] an C[k] are compute onlne for each amplng ntant k (ee Algorthm 1). The par {A[k], C[k]} (.e., the lnearze moel of the DFIG) obervable f an only f the obervablty matrx S o [k] ha full rank,.e., rank ( S o [k] ) = 1 for the conere DFIG a n (22). To check local obervablty, the rank of the obervablty matrx S o [k] compute numercally for each amplng ntant k n Step III of Algorthm 1. V. SIMULATION RESULTS AND DISCUSSION A mulaton moel of a 2 MW WTS wth DFIG mplemente n Matlab/Smulnk. The ytem parameter are lte n the Appenx. The mplementaton a n Fg. 1. For more etal on the mplementaton of e.g. back-to-back converter, PWM, current controller egn, ee [15]. The mulaton reult are hown n Fgure 2-6. The etmaton performance of the EKF are llutrate for normal operaton conton an for parameter varaton n R, an. Fg. 2 an 3 how the overall mulaton cenaro over tme: Wn pee profle v w, correponng rotor pee ω r, three-phae rotor current r an rank of obervablty matrx S o [k] a n (26). The obervablty of the lnearze DFIG moel ha been tete onlne for everal operatng pont (e.g. at low, hgh an ynchronou pee, ee Fg. 2): The lnearze ytem obervable even f the DFIG operate cloe to ynchronou pee,.e., the rotor freuency almot zero (a hown n Fg. 3). The obervablty matrx ha full rank for all tme,.e. rank ( S o [k] ) = 1 for all k. Fg. 4 how the mulaton reult of the propoe EKF uner normal operatng conton. Depte large ntal error n all electrcal parameter, the parameter etmaton fat; n partcular, for the mutual nuctance. The teay tate etmaton error of the EKF are mall (ee Tab. I). Thee reult llutrate the capablty of the EKF n etmatng all electrcal parameter of the DFIG. In orer, to llutrate the capablty of the EKF to track the electrcal parameter of the DFIG uner parameter varaton, the value of tator R an rotor retance are ncreae by 1% (e.g. ue to warmng or agng). For th cenaro, Fg. 5 how the etmaton performance of the propoe EKF. The EKF tll acheve a hgh etmaton accuracy. Moreover, the nfluence on the etmaton accuracy of tator an rotor leakage nuctance an mutual nuctance mall (ee lat three ubplot n Fg. 5). The teay tate etmaton error tll mall (ee Tab. I). Fnally, the etmaton performance of the EKF uner a varaton of the mutual nuctance (e.g., ue to magnetc aturaton) nvetgate. Therefore, ncreae tep-lke by 1%. Fg. 6 how the mulaton reult of the EKF for th cenaro. Agan, the EKF able to track the change n an the etmaton of the other parameter almot not affecte (ee Tab. I an Fg. 6). VI. CONCLUSION Th paper propoe a metho for etmatng all electrcal parameter of oubly-fe nucton generator (DFIG) n wn turbne ytem (WTS). The metho utlze an Extene Kalman flter (EKF). For the mplementaton of the EKF, the nonlnear tate pace moel of the DFIG ha been erve to etmate all electrcal parameter of the DFIG,.e.. the tator an rotor retance, the leakage nuctance of tator an rotor, an the mutual nuctance. The EKF egn ha [ pu ] r Rank tme [] Fgure 3: Overall mulaton cenaro: Three phaerotor current r of the DFIG an rank of the obervablty matrx S o [k] a n (26).

7 [ m ] R Rˆ [ m ] R Rˆ R [ m ] R Rˆ r [ m ] Rˆ r L 1 L [ H ] L ] [ H L r Lˆ L r Lˆr L r [ H ] L [ H ] Lˆ L r Lˆr [mh ] Lˆm [mh ] 3. Lˆm tme [] tme [] Fgure 4: Etmaton performance of the propoe EKF: Etmaton of tator retance R, rotor retance, tator leakage nuctance L σ, rotor leakage nuctance L rσ, an mutual nuctance for large ntal parameter error. Fgure 5: Robutne reult of the propoe EKF: Etmaton of tator retance R, rotor retance, tator leakage nuctance L σ, rotor leakage nuctance L rσ, an mutual nuctance for an 1% ncreae n R an. been preente n etal an the obervablty of the lnearze (funamental) moel of the DFIG ha been tete onlne for fferent wn pee. The reult llutrate that the lnearze moel locally obervable for all conere operatng conton. The etmaton performance of the EKF ha been llutrate by mulaton reult an compare to the actual parameter value. The reult howe that the EKF track all electrcal parameter of the DFIG wth hgh accuracy. Moreover, the EKF alo capable to etmate the electrcal parameter uner parameter varaton ue to temperature change or magnetc aturaton. REFERENCES [1] M. Lerre, R. Carena, M. Molna, an J. Rorguez, Overvew of Mult-MW Wn Turbne an Wn Park, IEEE Tranacton on Inutral Electronc, Vol. 58, No. 4, pp , Aprl 211. [2] R. Carena, R. Pena, S. Alepuz, an G. Aher, Overvew of Control Sytem for the Operaton of DFIG n Wn Energy Applcaton, IEEE Tranacton on Inutral Electronc, Vol. 6, No. 7, pp , July 213. [3] Z. Zhang, H. Xu, M. Xue, Z. Chen, T. Sun, R. Kennel, an C. Hackl, Prectve control wth novel vrtual flux etmaton for back-to-back power converter, IEEE Tranacton on Inutral Electronc, vol. PP, no. 99, p. 1 (IEEE Early Acce Artcle), 215. [4] C. Drcherl, C. Hackl, an K. Schechner, Explct moel prectve control wth turbance oberver for gr-connecte voltage ource power converter, to be publhe n the Proceeng of the 215 IEEE Internatonal Conference on Inutral Technology, 215. [5] Z. Zhang, C. Hackl, F. Wang, Z. Chen, an R. Kennel, Encoerle moel prectve control of back-to-back converter rect-rve permanent-magnet ynchronou generator wn turbne ytem, n Proceeng of 15th European Conference on Power Electronc an Applcaton, 213, pp [6] H. Tolyat, E. Lev, an M. Rana, A revew of rfo nucton motor parameter etmaton technue, IEEE Tranacton on Energy Converon, Vol. 18, No. 2, pp , Jun. 23. [7] S. Thomen, K. Rothenhagen, an F. Fuch, Onlne parameter entfcaton metho for oubly fe nucton generator, n Proceeng of IEEE Power Electronc Specalt Conference (PESC), pp , Jun. 28. [8] Y. Zhang, Y. Yuan, X. Chen, K. Qan, an B. Wu, The optmzaton ntal value an mcro-varaton earch algorthm an t mulaton n parameter entfcaton of oubly fe nucton generator, n Proceeng of Internatonal Conference on Sutanable Power Generaton an Supply, pp. 1-4, Apr. 29. [9] D. L, X. Ln, S. Hu, an Y. Kang, An aaptve etmaton metho for

8 [ m ] R [ m ] L r [ H ] L [ H ] [mh ] R Rˆ Rˆ r L Lˆ L r Lˆr Lˆm tme [] Fgure 6: Robutne reult of the propoe EKF: Etmaton of tator retance R, rotor retance, tator leakage nuctance L σ, rotor leakage nuctance L rσ, an mutual nuctance for an 1% ncreae n. parameter of oubly fe nucton generator, n Proceeng of Aa Pacfc Power an Energy Engneerng Conference, pp.1-4, Mar. 21. [1] X. Wang, J. Xong, L. Geng, J. Zheng, S. Zhu, Parameter entfcaton of oubly fe nucton generator by the Levenberg Maruart Fletcher metho, n Proceeng of IEEE Power an Energy Socety General Meetng (PES), pp. 1-5, July 213. [11] A. Djou, H. Chekreb, S. Bacha, On-lne entfcaton of DFIG parameter wth rotor current reconttuton, n Proceeng of Nnth Internatonal Conference on Ecologcal Vehcle an Renewable Energe (EVER), pp. 1-6, March 214. [12] F. Auger, M. Hlaret, J. Guerrero, E. Monmaon, T. Orlowka- Kowalka, S. Katura, Inutral Applcaton of the Kalman Flter: A Revew, IEEE Tranacton on Inutral Electronc, Vol. 6, No. 12, pp , Dec [13] M. Abelrahem, C. Hackl, an R. Kennel, Senorle Control of Doubly-Fe Inucton Generator n Varable-Spee Wn Turbne Sytem, to be publhe n the Proceeng of the 5 th Internatonal Conference on Clean Electrcal Power, Taormna, Italy, 215. [14] S. P. Aza, J. E. Tate, Parameter etmaton of oubly fe nucton generator rven by wn turbne, n Proceeng of Power Sytem Conference an Expoton (PSCE), pp. 1-8, March 211. [15] C. Drcherl, C. Hackl, an K. Schechner, Moellerung un Regelung von moernen Wnkraftanlagen: Ene Enführung (avalable at the author upon reuet), Chapter 24 n Elektrche Antrebe Regelung von Antrebytemen, D. Schröer (E.), Sprnger-Verlag, 215. [16] B. Wu, Y. Lang, N. Zargar, an S. Kouro, Power converon an control of wn energy ytem, Wley-IEEE Pre, 211. [17] M. Yameamoto an o. Motoyoh, Actve an reactve power control for oubly-fe woun rotor nucton generator, IEEE Tranacton on Power Electronc, Vol. 6, No. 4, pp , October [18] R. Pena, J. Clare, an G. Aher, Doubly fe nucton generator ung back-to-back PWM converter an t applcaton to varable-pee wnenergy generaton, IEE Proceeng on Electrc Power Applcaton, Vol. 143, No. 3, pp , May [19] A. Hanen, P. Sorenen, F. Lov, an F. Blaabjerg, Control of varable pee wn turbne wth oubly-fe nucton generator, Wn Energy, Vol. 28, No. 4, pp , June 24. [2] R. Faaeneja, M. Moallem, an G. Mochopoulo, Smulaton of a wn turbne wth oubly fe nucton generator by FAST an Smulnk, IEEE Tranacton on Energy Converon, Vol. 23, No. 2, pp. 69-7, June 28. [21] A. Peteron, L. Harnefor, an T. Thrnger, Comparon between tator-flux an gr-flux-orente rotor current control of oubly-fe nucton generator, proceeng IEEE 35 th Annual Power Electronc Specalt Conference, Vol. 1, pp , 2-25 June 24. [22] S. Müller, M Decke, an R. De Doncker, Doubly Fe Inucton Generator Sytem for Wn Turbne, IEEE Inutral Applcaton Magazne, Vol. 8, No. 3, pp , May/June 22. [23] D. Schröer, Elektrche Antrebe Grunlagen (5. Auflage), Sprnger- Verlag, Berln, 213. [24] Segfre Heer, Gr Integraton of Wn Energy Converon Sytem, John Wley & Son Lt, [25] G. Bhop, an G. Welch, An ntroucton to the Kalman flter, Techncal report TR 95-41, Department of Computer Scence, Unverty of North Carolna at Chapel Hll, 26. [26] S. Bolognan, L. Tubana, an M. Zglotto, Extene Kalman flter tunng n enorle PMSM rve, IEEE Tranacton on Inutry Applcaton, Vol. 39, No. 6, pp , November 23. [27] C. De Wt, A. Youef, J. Barbot, P. Martn, F. Malrat, Obervablty conton of nucton motor at low freuence, Proceeng of the 39 th IEEE Conference on Decon an Control, Vol. 3, pp , 2. APPENDIX The mulaton parameter are gven n Tab. II. Note that the rotor parameter (retance an nuctance) are converte to the tator of the DFIG. Table II: DFIG parameter Name Nomenclature Value DFIG rate power (bae power) p nom 2 MW Stator voltage (bae voltage) u rm 69 V Rotor voltage (bae voltage) u rm r 27 V Gr freuency (bae freuency) f = ω 2π 5 Hz Number of par pole n p 2 Stator retance R 2.6 mω Rotor retance 2.9 mω Stator nuctance L mh Rotor nuctance L r mh Mutual nuctance 2.55 mh