Amrican Journal of Signal Procssing 05, 5(A): 6-5 DOI: 0.593/s.ajsp.0.03 Frquncy Estimation of Unbalancd Thr-Phas Powr Systms Using th Modifid Adaptiv Filtring Amir Rastgarnia,*, Azam Khalili, Vahid Vahidpour, Md Kafiul Islam Dpartmnt of Elctrical Enginring, Malayr Univrsity, Malayr, Iran Dpartmnt of Elctrical and Computr Enginring, National Univrsity of Singapor, Singapor Abstract In this papr, th problm of frquncy stimation using adaptiv filtrs is addrssd basd on th augmntd complx normalizd last man squars () tchniqu. In othr words, motivatd from tchniqu, a nw vrsion calld that outprforms th prvious tchniqus is proposd. Th proposd mthod maks us of two wight cofficints, lik, togthr with normalizd tim variant stp siz, thus its promising in incrasing and nhancing both convrgnc rat and accuracy. Thn, using drivd wight cofficint, th frquncy is stimatd at ach stp. Th prformanc and convrgnc analysis of th proposd mthod along with simulation rsults comparing with two xisting tchniqus, CLMS and, ar providd in critical cass such as various unbalancd conditions and prsnc of harmonic distortion. Simulation rsults indicat that th proposd tchniqu achivs a bttr prformanc in trms of convrgnc rat and frquncy stimation accuracy as compard with CLMS and tchniqus. Morovr, achivs a smallr rror varianc than th othr mntiond cass. Kywords Adaptiv filtr, Frquncy stimation, Powr systms, Complx signal. Introduction In powr ntworks, knowing th dynamic stats of a systm is ncssary to utiliz th systm. Sinc frquncy indicats balanc/imbalanc btwn load and gnration, it has crucial rol in a powr systm. Furthrmor, th fact that th amount of activ powr producd should b always qual to th activ powr consumd plus losss, is on of th major difficultis in powr systms. If th consumd powr is mor than th producd powr, thn th frquncy would fall and vic vrsa. Consquntly, variations from nominal frquncy valu indicat lack of balanc, such as unxpctd abnormal systm conditions, btwn activ powr consumd plus losss and activ powr producd. Evn small dviations from this valu would damag synchronous machins and othr applicants. Thrfor, making sur of th frquncy bing constant is of grat importanc to powr systm ntworks. Control and protction of powr systm rquir fast and accurat frquncy stimation and thus it is a challnging problm that has attractd a lot of attntion. Akk [] considrd a tst critrion that rflcts rlvant dmands to compar diffrnt frquncy stimation and dtction mthod. Ths dmands ar: spd * Corrsponding author: a_rastgar@i.org (Amir Rastgarnia) Publishd onlin at http://journal.sapub.org/ajsp Copyright 05 Scintific & Acadmic Publishing. All Rights Rsrvd of convrgnc, accuracy, and nois rjction. Svral frquncy stimation tchniqus hav bn dvlopd in litratur, for xampl, zro-crossing-basd tchniqus [], last squar rror-basd adaptiv filtrs (LES) [3], Nwton typ algorithm [4], discrt Fourir transform (DFT) [5], last man squars mthod [6], phas lockd loop [7], rcursiv stat stimation-basd tchniqus, and Kalman filtrs [8], [9]. Among thm rcursiv stat stimation-basd tchniqus achiv accurat frquncy stimation undr critical conditions (unbalancd voltag). Most of ths mthods, which ar basd on a singl-phas masurmnt of a systm, ar limitd in trms of systm frquncy charactrization. In a thr-phas systm, du to xistnc of six diffrnt singl-phas voltag signals, no singl-phas signal can adquatly charactriz th systm frquncy vn though lin-to-lin voltags ar considrd [0]. Thus, on practical solution is considration of all thr-phas voltags simultanously, nabls unifid systm frquncy stimation as a whol and provids improvd robustnss. To construct a complx-valud signal from all thr-phas voltags, Clark s αβ transformation is utilizd. Among th solutions that hav bn drivd basd on Clark transformation, adaptiv algorithms ar broadly usd du to thir computational fficincy, simplicity, and robust prformanc on frquncy stimation in th prsnc of harmonic distortions and nois. Nvrthlss, in critical cass undr unbalancd voltag conditions, such as whn a voltag sag in on or two phass is taking plac, standard
Imaginary Amrican Journal of Signal Procssing 05, 5(A): 6-5 7 phas angl calculation tchniqus utilizd in a linar adaptiv filtr hav bn provn as suboptimal. In this papr, a modifid tchniqu for frquncy stimation, which is basd on th widly linar complx-valud signal modling drivd from thr-phas voltags by Clark transformation undr unbalancd conditions, is proposd. It maks us of two wight cofficints, lik, along with normalizd tim variant stp siz to nhanc convrgnc rat and spd using th fact that undr unbalancd conditions th complx-valud signal is scond ordr noncircular. To modl noncircular signals, not only th standard linar stimation, which is basd on covarianc matrix of a complx-valud random vctor x, i.., C xx =E[xx H ], is adquat, but th psudo-covarianc matrix P xx =E[xx T ] should also b considrd to dscrib th complt scond-ordr bhaviour [], []. In practic, this is accomplishd du to widly linar modling [], [3], whr both x and its complx conjugat x* ar combind into augmntd input x a =[x T, x H ] T. Thus, in ordr to stimat onlin frquncy of noncircular signals, widly linar modling-basd adaptiv filtring mthod is utilizd [4]. Th rmaindr of this articl is organizd as follows. In sction, w prsnt an ovrviw of CLMS frquncy stimation tchniqu Anothr mthod calld [5-9] is also introducd in this sction. Wand prsnt our proposd algorithm. In sction 3, numrical simulations ar prsntd to illustrat th ffctivnss and advantags of.5 th proposd algorithms. Sction 4 summarizs th main conclusions of th papr.. Frquncy Estimation Basd on Adaptiv Filtring Th voltags in th powr systm in a nois fr nvironmnt can b rprsntd as follows: v ( k) V ( k)cos( kt ) a a vb( k) Vb( k)cos( kt ) 3 vc( k) Vc( k)cos( kt ) 3 whr V a (k), V b (k), and V c (k) ar maximum (pak) valu of vry voltag signal at tim instant k, ΔT is sampling intrval, is th phas of fundamntal componnt, and ω=πf is th angular frquncy of th voltag signal, with f bing th systm frquncy. Th tim-dpndnt thr-phas voltag is transformd by th orthogonal αβ0 transformation matrix [0] into zro-squnc v 0 and dirct and quadratur axis componnts, v α and v β, as Clark s transform Fig. v0( k) va ( k) v ( k) v ( ) 3 b k v( k) 0 3 3 vc ( k) () () 0.5 0-0.5 - -.5 -.5 - -0.5 0 0.5.5 Ral Figur. Gomtric viw of circularity and noncircularity in complx plan. Th circl dnotd by "+" rprsnts circular complx-valud signal v(k) in balancd condition and llips dnotd by "o" rprsnts noncircular complx-valud v(k) in unbalancd situation
8 Amir Rastgarnia t al.: Frquncy Estimation of Unbalancd Thr-Phas Powr Systms Using th Modifid Adaptiv Filtring whn V a (k), V b (k), and V c (k) ar idntical, w havv 0 (k)=0, v ( k) A cos( kt ) v ( k) A cos( kt / ) whr A is constant amplitud, and v α (k) and v β (k) ar th orthogonal coordinats of a point whos position is tim variant at a rat proportional to th systm frquncy. Only th v α and v β componnts ar gnrally utilizd in th modling in practic. Thrfor, th complx voltag signal v(k) of th systm is givn by v( k) v ( k) jv ( k) (3) And can b stimatd as v( k ) A( k ) j( ( k ) T ) jt j( kt ) (4) jt A v( k) Whr th instantanous systm frquncy f is dnotd by th phasor jωδt. In normal conditions, sampls of v(k) ar placd on a circl with a constant radius A in complx plan. In Figur, thss sampls ar rprsntd by "+". v and v jθ hav th idntical distribution for any ral θ. This mans that v(k) is scond ordr circular, and in this cas, frquncy stimation can b prformd proprly by standard linar adaptiv filtr. Th CLMS algorithm, which is utilizd for frquncy stimation of thr-phas voltag, can b summarizd as vˆ( k ) v( k) w( k) ( k) v( k ) vˆ ( k ) w( k ) w( k) ( k) v ( k) Whr w(k) is th wight cofficint at tim instant k, v(k ˆ ) is th stimat of dsird signal v(k+), (k) is stimation rror, and μ is th stp siz. Comparing (4) and th linar stimation modl in (5), th systm frquncy can b stimatd from ˆ sin f ( ( w ( k )) T whr th oprator (.) dnots th imaginary part of a complx-valud numbr... Frquncy Estimation basd on Tchniqu Whn thr-phas powr systm dviats from its normal condition, such as, whn th voltags of thr channls show various lvls of transints and dips, voltags V a (k), V b (k), and V c (k) ar not idntical, and sampls of v(k) ar not locatd on a constant radius circl, as dnotd by llips in Fig., lad to braking down of th v(k) itrativ modl in (4). In such instanc, v(k) has noncircular distribution. From standard thr-phas systm in () and (), v α (k) and v β (k) coordinats of complx voltag v α (k) + jv β (k) is accuratly xprssd as (5) (6) j( kt ) j( kt ) v( k) A( k) B( k) (7) In othr words, sinc V a (k), V b (k), and V c (k) ar not idntical, A(k) is not constant and B(k) 0. Using th fact that in unbalancd conditions, v(k) xhibits a crtain dgr of circularity, both v(k) and its complx conjugat v * (k) should b considrd in frquncy stimation in unbalancd cass, i.., vˆ( k ) v( k) h( k) v ( k) g( k) (8) Whr h(k) and g(k) ar filtr wight cofficints rlatd to th standard and conjugat updats at tim instant k, rspctivly, and th stimation rror (k) and th cost function J(k) ar dfind as J( k) ( k) ( k) ( k) and ( k) v( k ) vˆ ( k ) (9) Th updat of both th standard and th conjugat trm wight cofficints can b drivd by using th stpst dscnt gradint as h( k ) h( k) J( k) (0) g( k ) g( k) J( k) () Whr th gradints dfind as J( k) ( k) ( k) hj ( k) ( k) ( k) () h ( k) h ( k) h ( k) J( k) ( k) ( k) gj( k) ( k) ( k) (3) g ( k) g ( k) g ( k) Sinc ( k) v ( k ) v ( k) h ( k) v( k) g ( k) (4) and (k)/ h*(k)= (k)/ g*(k)=0, w obtain h J ( k ) ( k ) v ( k ) (5) g J ( k ) ( k ) v ( k ) (6) Giving th standard and conjugat wight cofficints updats h(k) and g(k) in th form h( k ) h( k) ( k) v ( k) (7) g( k ) g( k) ( k) v ( k) (8) Equations (7) and (8) dscrib th algorithm, which is dsignd for widly linar adaptiv filtrs. In ordr to introduc th basd frquncy stimation tchniqu for thr-phas unbalancd systm, by substituting can b giv by (7) into (8), th stimat vk ˆ( ) g h
Amrican Journal of Signal Procssing 05, 5(A): 6-5 9 vˆ( k ) A( k) h( k) B( k) h( k) j( kt ) j( kt ) j( kt ) A ( k) g( k) j( kt ) B ( k) g( k) ( A( k) h( k) B ( k) g( k)) j( kt ) j( k T ) ( A ( k) g( k) B( k) h( k)) whil from (7), th trm v(k+) can b rwrittn as v( k ) A( k ) jt j( kt ) B( k ) jt j( kt ) (9) (0) Thus, at stady stat, th first trm on th RHS of (0) can b stimatd approximatly by countrpart in (9); hnc, th trm jωδt containing th frquncy information can b stimatd as ˆ ( ) ( ) ( ) ( ) j T A k h k B k g k () Ak ( ) Comparing th scond trm on th RHS of (9) and (0), th volution of th trm - jωδt can b xprssd as ˆ ( ) ( ) ( ) ( ) j T A k g k B k h k Bk ( ) Upon taking th complx conjugat, w obtain ˆ ( ) ( ) ( ) ( ) j T A k g k B k h k B ( k ) () (3) Undr th assumptions A(k+) A(k) and B(k+)=B(k), th quation () and (3) can b simplifid as jˆ T B () k h( k) g( k) (4) Ak () j ˆ T Ak () h ( k) g ( k) (5) B () k Th cofficint A(k) and B(k) hav ral and complx valud rspctivly, and thrfor, (B*(k)/A(k))= (B(k)/A(k))*. Sinc quations (4) and (5) should b qual, using a(k)= (B(k)/A(k))*, w can find a(k) by solving th following quadratic quation: g( k) a ( k) ( h( k) h ( k)) a( k) g ( k) 0 (6) Th discriminant of this quation is givn by ( h( k) h ( k)) 4 g( k) ( h( k)) g( k) (7) whr th oprator (.) dnots th imaginary part of a complx-valud numbr. Sinc a(k)is complx valud, th discriminant is ngativ, and th two roots can b found as j( h( k)) j ( h( k)) g( k) a() k gk () j( h( k)) j ( h( k)) g( k) a() k gk () (8) From (4), th phasor j ˆ T is stimatd ithr by using h(k)+a (k)g(k) or h(k)+a (k)g(k). Sinc th systm frquncy is far smallr than th sampling frquncy, th j ˆ T imaginary part of is positiv, thus xcluding th solution basd on a (k). Th systm frquncy fk ˆ( ) is thrfor stimatd in th form ˆ( ) sin f k ( ( h ( k ) a ( ) ( ))) k g k (9) T Th aformntiond quation is a gnric widly linar xtnsion of th standard linar frquncy stimation mthod and can b implmntd by any typ of widly linar adaptiv filtr. In addition, whn th systm is balancd, g(k) = 0, and th stimator in (9) simplifis into th standard linar CLMS-basd stimation... Th Proposd Tchniqu for Frquncy Estimation Basd on Adaptiv Filtring In ordr for frquncy stimation of unbalancd thr-phas powr systm to b prformd in ral-tim, th tchniqu outprforms th CLMS algorithm. This tchniqu can adapt itslf in a succssful and satisfactory way in unbalancd conditions and illustrats low snsitivity to thr-phas voltag amplitud variations during th tim and vn in th xistnc of high-ordr harmonics. Th main disadvantag of tchniqu is th spd of convrgnc. To ovrcom this, w ar motivatd to propos a novl tchniqu. Considring th wight cofficints updats h(k) and g(k) of algorithm from (7) and (8). Th stp siz μ tak constant valu. Th choic of stp siz μ is crucial, if not chosn with car; it can dstroy th dsirabl bhavior. In many instancs, it may b dsirabl to vary th valu of th stp siz in ordr to obtain bttr control ovr th spd of convrgnc of th algorithm. Consquntly, th biggr th stp siz, th highr th convrgnc spd. Th updats of both th standard and th conjugat trm wight cofficints h(k) and g(k) of (proposd algorithm) can b xprssd as h( k ) h( k) ( k) v ( k) (30) vk ()
Frquncy(Hz) 0 Amir Rastgarnia t al.: Frquncy Estimation of Unbalancd Thr-Phas Powr Systms Using th Modifid Adaptiv Filtring g( k ) g( k) ( k) v ( k) (3) vk () Comparing updats with ons, w find that th corrction trm that is addd to h(k) and g(k) is normalizd with rspct to th squard-norm of th input voltag. Morovr, as can b sn from (30) and (3), mploys a tim-variant stp siz of th form ˆ( k) v( k), as opposd to th constant stp siz, μ, which is usd by. Th prformanc of is lss snsitiv in critical stat and unbalancd condition than. Aftr calculation of wight cofficints h(k) and g(k), th stimatd frquncy at ach stp is xprssd as ˆ( ) sin f k ( ( h ( k ) a ( ) ( ))) k g k T (3) whr th oprator (.) dnots th imaginary part of a complx-valud numbr, h(k) and g(k) ar th standard and conjugat wight cofficints updats rspctivly, and a (k) is th accptabl root of th following quation g( k) a ( k) ( h( k) h ( k)) a( k) g ( k) 0 (33) 3. Prformanc Evaluation This sction is dvotd for prformanc valuation of th stimator and thn xamins th tracking abilitis of th stimator with rspct to svral typical powr systm oprating conditions. Capability of proposd tchniqu in (3) in comparison to and standard CLMS algorithm in (9) and (6), rspctivly, is also prsntd. Simulations wr prformd in Matlab programming nvironmnt with a sampling frquncy of 5 khz and th stp siz, μ and wight cofficint, w wr initializd at 0.00 and 0.998+j0.0634, rspctivly. Unbalancd thr-phas voltags wr st according to V a (k)=., V b (k)=0.7, and V c (k)=0.3. 3.. Convrgnc Spd Comparison of Frquncy Estimation undr Unbalancd Condition In this cas study, at t=0.05 s, th unbalancd conditions wr imposd to th systm. Figur illustrats th tracking prformanc of th,, and CLMS tchniqus. Both and algorithms achivd accurat frquncy stimation, howvr, th convrgnc spd in tchniqu is fastr than on. 5 48 Comparison for Unbalacd cas CLMS 46 44 4 40 38 36 0 0.05 0. 0.5 0. 0.5 0.3 0.35 0.4 Tim(sc) Figur. Comparison of th proposd,, and CLMS convrgnc spd undr unbalancd conditions at t=0.05 s
Frquncy(Hz) Varianc Amrican Journal of Signal Procssing 05, 5(A): 6-5 3.. Dtail Varianc Examination of Error Estimation In this sction, simulations addrssd th varianc prformanc of and algorithms on frquncy stimation of unbalancd systm against diffrnt lvls of nois. Th rsult was obtaind by avraging 000 indpndnt trails. Th unbiasd proprty of both and stimator can b obsrvd in high SNR rgion. In Figur 3, th achivd smallr rror varianc than. 0.5 0. Varianc of Estimation vrsus SNR 0.5 0. 0.05 0 5 30 35 40 45 SNR(dB) Figur 3. Comparison of th proposd and at diffrnt SNRs, obtaind by 000 indpndnt trials 55 Comparison for 00-sag at t=0.05(s) cas CLMS 45 40 35 30 5 0 0.05 0. 0.5 0. 0.5 0.3 0.35 0.4 Tim(sc) Figur 4. Frquncy stimation by,, and CLMS for up to 0.4 s whn a 00% singl-phas voltag sag occurrd in phas channl c (V c =0) at t=0.05 s
Frquncy(Hz) Amir Rastgarnia t al.: Frquncy Estimation of Unbalancd Thr-Phas Powr Systms Using th Modifid Adaptiv Filtring 3.3. Th Effct of 00% Singl-phas Voltag Sag This sction shows a 00% singl-phas voltag sag suddnly occurrd at t=0.05 s in channl c (V c =0) of th unbalancd thr-phas systm. Th tracking prformanc in Figur 4 indicats that th CLMS algorithm losts its frquncy tracking capability, whras aftr convrgnc, both and algorithm wr abl to accuratly track th systm frquncy, with no oscillations. Onc again, th spd of th rspons of th proposd mthod is highr than othrs, such that, at around t=0.5 s, th proposd mthod convrgs; whras, th othr algorithms still do not convrg. 3.4. Th Impact of Oscillatory Variations of Magnitud on th Estimatd Frquncy This st of simulations addrssd th impact of oscillatory variations of magnitud on th stimatd frquncy. In this cas study, at t = 0.05 s, th magnituds of balancd thr-phas voltags changd from thir normal valus. Figur 5 shows that th tracking prformanc of th proposd algorithm xhibitd smallr oscillatory stady-stat rror than algorithm. Ths small oscillatory stady-stat rrors of th and stms from th assumptions that, at two succssiv tim instants, A(k + ) A(k) and B(k +) B(k). 3.5. Th Effct of Harmonics on th Prformanc of Frquncy Estimator In this sction, ffcts of harmonic on th prformanc of th frquncy stimator ar studid. Whn th input signal of th stimator is contaminatd with harmonics, th stimatd frquncy is subjct to an oscillatory stady-stat rror and it also dviats from th nominal valu by an offst rror. In Figur 6(a), a balancd 0% third harmonic and a 5% fifth harmonic of th fundamntal frquncy wr addd into th unbalancd thr-phas powr systm at t = 0.03 s. Th algorithm achivd bttr prformanc with a smallr oscillation rror at th stady stat as compard with th CLMS algorithm; this advantag was most pronouncd whn th magnituds of th third and fifth harmonics varid ovr a rang of (0.0 0.4) p.u., as shown in Figur 6(b)..6.5 Comparison for Ocillatory cas.4.3.. 0 0.0 0.04 0.06 0.08 0. 0. 0.4 0.6 0.8 0. Tim(sc) Figur 5. Th impact of oscillatory variations of magnitud on th frquncy stimation by and
Frquncy(Hz) Frquncy(Hz) Amrican Journal of Signal Procssing 05, 5(A): 6-5 3 5.5 3 rd and 5 th Harmonic Effct at t = 0.03 s 49.5 49 48.5 0 0.0 0.0 0.03 0.04 0.05 0.06 0.07 0.08 Tim(sc) 5.5 3 rd and 5 th Harmonic Effct at t = 0.03 s 49.5 49 48.5 0 0.0 0.0 0.03 0.04 0.05 0.06 0.07 0.08 Tim(sc) Figur 6. Frquncy stimation for phas voltags contaminatd with harmonics. (top) 0% p.u. third harmonic and 5% p.u. fifth harmonic wr addd into th unbalancd powr systm at t = 0.03 s. (bottom) Man prcntag rror of (surfac blow) and (surfac abov) algorithms ovr a rang of amplituds of both third harmonic and fifth harmonic at th stady stat
Frquncy(Hz) Frquncy(Hz) 4 Amir Rastgarnia t al.: Frquncy Estimation of Unbalancd Thr-Phas Powr Systms Using th Modifid Adaptiv Filtring 3.6. Th Prformanc of th Proposd and Algorithm for th Cas of Frquncy Variation This sction is dvotd for prformanc valuation of th proposd and algorithms for th cas of frquncy variation. 57 56 Frquncy suddn fall at t = 0.05 and t = 0.3 55 54 53 5 5 0 0.05 0. 0.5 0. 0.5 0.3 0.35 0.4 0.45 0.5 Tim(sc) 5 Comparison for Frquncy ris or fall.8.6.4. 49.8 49.6 49.4 49. 49 0 0.0 0.04 0.06 0.08 0. 0. 0.4 0.6 0.8 0. Tim(sc) Figur 7. Frquncy stimation undr frquncy variations. (top) Unbalancd and noncircular systm input suffrd from suddn frquncy changs. (bottom) Unbalancd and noncircular systm input suffrd from th frquncy ris and dcay at a rat of 5 Hz/s
Amrican Journal of Signal Procssing 05, 5(A): 6-5 5 In Figur 7 a balancd voltag signal was affctd by th -Hz stp chang in frquncy from to 5 Hz at 0.05 s. In this cas, both algorithms achivd accurat frquncy stimation; howvr, at t = 0.3 s whn th systm signal bcam unbalancd and th frquncy was simultanously changd back to Hz, both and algorithms could still track th systm frquncy with no stady-stat rror. In Figur 7(bottom), th -Hz fundamntal frquncy of th unbalancd signal aros and dcayd at a rat of 5 Hz/s. Th tracking prformancs of both algorithms ar quit similar. 4. Conclusions A nw mthod for stimation of frquncy undr unbalancd condition for thr-phas powr systm is introducd. Th mthod utilizs th nwly dvlopd concpt of which is originally basd on th widly linar modling of th complx-valud signal, drivd from thr-phas voltags by th Clark transformation. Th algorithm was simulatd undr som unbalancd conditions and xhibitd good prformanc. In contrast to th prvious stimator mntiond in litratur, th proposd stimator can provid fast and accurat frquncy stimat. This papr also compars th proposd tchniqu with standard CLMS and tchniqus. Basd on our simulation rsults, w can conclud that th proposd tchniqu provids robust prformanc with rspct to distortion of th input signal,.g., harmonics and nois distortions; and also variations of its intrnal paramtr. Morovr, it has th capability to provid fast and accurat frquncy stimation undr various unbalancd conditions. REFERENCES [] M. Akk, Frquncy stimation by dmodulation of two complx signals, IEEE Trans. Powr Dl., vol., no., pp. 57 63, Jan. 997. [] C. T. Nguyn and K. Srinivasan, A nw tchniqu for rapid tracking of frquncy dviations basd on lvl crossings, IEEE Trans. Powr App. Syst., vol. PAS-03, no. 8, pp. 30 36, Aug. 984. [3] M.S. Sachdv, FIEEE, M.M. Giray, a last rror squars tchniqu for dtrmining powr systm frquncy, IEEE Transactions on Powr Apparatus and Systms, Vol. PAS-04, No., Fbruary 985, pp.437-444. [4] V. V. Trzija, M. B. Djuric, and B. D. Kovacvic, Voltagphasor and local systm frquncy stimation using Nwton typ algorithm, IEEE Trans. Powr Dl., vol. 9, no. 3, pp. 368 374, Jul. 994. [5] Tadusz Lobos and Jack Rzmr, Ral-Tim Dtrmination of Powr Systm Frquncy, IEEE transactions on instrumntation and masurmnt, VOL. 46, NO. 4, AUGUST 997, pp.877-88. [6] A. K. Pradhan, Mmbr, IEEE, A. Routray, Mmbr, IEEE, and Abir Basak, Powr systm frquncy stimation using last man squar tchniqu, IEEE transactions on powr dlivry, vol. 0, no. 3, JULY 005,pp.8-86. [7] V. Kaura and V. Blasko, Opration of a phas lockd loop systm undr distortd utility conditions, IEEE Trans. Ind. Appl., vol. 33, no.,pp. 58 63, Jan./Fb. 997. [8] A. Routray, A. K. Pradhan, and K. P. Rao, A novl Kalman filtr for frquncy stimation of distortd signals in powr systms, IEEE Trans. Instrum. Mas., vol. 5, no. 3, pp. 469 479, Jun. 00. [9] A.Girgis and T. Danil Hwang, Optimal stimation of voltag phasors and frquncy dviation using linar and non-linar kalman filtring: Thory and limitations, IEEE Trans. Powr App. Syst., vol. PAS-03, no. 0, pp. 943 95, Oct. 984. [0] V. Eckhardt, P. Hipp, and G. Hosmann, Dynamic masuring of frquncy and frquncy oscillations in multiphas powr systms, IEEE Trans. Powr Dl., vol. 4, no., pp. 95 0, Jan. 989. [] B. Picinbono and P. Chvalir, Widly linar stimation with complx data, IEEE Trans. Signal Procss., vol. 43, no. 8, pp. 030 033, Aug. 995. [] P. J. Schrir and L. L. Scharf, Scond-ordr analysis of impropr complx random vctors and procss, IEEE Trans. Signal Procss., vol. 5, no. 3, pp. 74 75, Mar. 003. [3] D. P. Mandic and S. L. Goh, Complx Valud Nonlinar Adaptiv Filtrs: Noncircularity, Widly Linar and Nural Modls. Hobokn, NJ: Wily, 009. [4] Y. Xia and D. P. Mandic, Widly linar adaptiv frquncy stimation of unbalancd thr-phas powr systms, IEEE Trans. Instrum. Mas., vol. 6, no., pp. 74 83, Jan. 0. [5] A. Khalili, A. Rastgarnia, S. Sani, Quantizd Augmntd Complx Last-man Squar Algorithm: Drivation and Prformanc Analysis, submittd to Signal Procssing, in rvision. [6] A. Khalili, A. Rastgarnia, Tracking analysis of augmntd complx last man squar algorithm, Intrnational Journal of Adaptiv Control and Signal Procssing, to appar, doi: 0.00/acs.594. [7] Azam Khalili, Amir Rastgarnia, Said Sani, Robust frquncy stimation in thr-phas powr systms using corrntropy-basd adaptiv filtr, IET Scinc, Masurmnt & Tchnology, to appar, doi: 0.049/it-smt.05.008. [8] A. Khalili, A. Rastgarnia, and W. Bazzi, A collaborativ adaptiv algorithm for th filtring of noncircular complx signals, in Tlcommunications (IST), 04 7th Intrnational Symposium on, Spt 04, pp. 96 99. [9] A. Khalili, A. Rastgarnia, W. M. Bazzi, and Z. Yang, Drivation and analysis of incrmntal augmntd complx LMS algorithm, IET Signal Procssing, 9 (4), 3-39 [0] E. Clark, Circuit Analysis of A.C. Powr Systms. Nw York: Wily, 943.