Unscented Kalman Filter for Frequency and Amplitude Estimation

Transcription

1 Paper aepted for presentaton at the 211 IEEE rondhem Powereh Unsented Kalman Flter for Frequeny and Ampltude Estmaton Happy Novanda, Pawel Regulsk, Franso M. González-Longatt, Senor Member IEEE and Vladmr erzja, Senor Member, IEEE Abstrat hs paper ntrodues a new dgtal sgnal proessng algorthm for frequeny and ampltude estmaton based on Unsented Kalman Flter (UKF). he results of omputer smulated and realst synthet data tests are presented. he ntal parameters used durng the tests were hosen arefully usng an establshed parameter estmaton method, the Self unng Least Square (SLS). It s onluded that the proposed algorthm s smple, effent and has low omputatonal demands ompare to SLS whh makes the UKF a very promsng method n next generaton of power qualty montorng deves. Index erms Kalman flters, unsented transformaton, power qualty, frequeny estmaton, ampltude estmaton. I. INRODUCION HE development of numeral algorthms for estmatng power qualty (PQ) parameters has beome an mportant subjet n reent years. More relable methods are requred for power qualty montorng and estmaton. In general, power system voltage and urrent waveforms are dstorted by harmon and nterharmon omponents, partularly durng system dsturbane. Faults or other swthng transents may hange the magntude and phase angles of the waveforms. Moreover, voltage and urrent an also be dstorted by nonlnear loads, power eletron omponents and nherent nonlnear nature of the system elements [1]. he assessment of PQ an be done ether by alulatng, measurng or estmatng PQ ndes (frequeny, spetrum, harmon dstorton et.). he estmaton of PQ ndes s stll an mportant and yet hallengng researh area to be explored. In the past dfferent tehnques have been developed for PQ ndes estmaton. Fast Fourer ransformaton (FF) [2], Least Squares (LS) [3-4], Newton ype Algorthm (NA) [5-7] and Self unng Least Square (SLS) [8-1] are some of aknowledged sgnal proessng methods used n the frequeny and ampltude measurement. One of the most popular methods for solvng nonlnear parameter estmaton problems s Extended Kalman Flter (EKF) [11-13]. he EKF lnearzes nonlnear systems so that H. Novanda (rrhappy.novanda@postgrad.manhester.a.uk) and P. Regulsk (pawel.regulsk@postgrad.manhester.a.uk) are PhD students at the Shool of Eletral and Eletron Engneerng, he Unversty of Manhester, Manhester, UK. F. González-Longatt (e-mal: fglongatt@eee.org) s Postdotoral Researh Assoate at the same Unversty. V. erzja (e-mal: terzja@eee.org) s the EPSRC Char Professor at the same Unversty. Kalman Flter equatons an be appled. In prate the EKF has several dsadvantages suh as when the assumptons of loal lnearty are volated, lnearzaton an produe hghly unstable flters. hs mght lead to dvergene phenomena [11]. Another drawbak of the EKF s that the use of Jaoban matres to lnearze the nonlnear model equatons often leads to sgnfant mplementaton dffultes, partularly nreased proessng requrements and exeuton tme, makng the algorthm dffult for real-tme applatons [11]. More teratve methods based on Kalman flter have been proposed n [16, 17]. hs approah an overome auray problems of the EKF due to negletng non-lnear terms. However, the teraton proedure leads to hgher proessng tme. hs paper arres out a study of Unsented Kalman Flter (UKF). he UKF s a novel measurng tehnque based on Unsented ransformaton (U) theory. hs method was orgnally proposed n [11], [15]. he UKF method an elmnate the dsadvantages of the EKF beause t does not lnearze the non lnear systems. Instead, t uses a statstal dstrbuton of the state whh s propagated through the nonlnear equatons. hs approah provdes better estmates of the atual state and the posteror ovarane matrx [14, 15]. In ths paper, the omputer smulated and realst synthet data reords are utlzed to analyze the valdty and performane of the proposed UKF algorthm for estmaton of frequeny and ampltude of the proessed sgnal (arbtrary voltage, or urrent). II. MEHODOLOGY A. Unsented ransformaton he Unsented ransformaton (U) s a transformaton based on the nsght that t s easer to approxmate a Gaussan dstrbuton than a hghly nonlnear funton [15]. he dea of U s to determnstally hoose a set so alled sgma ponts wth mean x and ovarane P xx and fnd the mean y and ovarane P yy by propagatng sgma ponts through a nonlnear transformaton. In U, the +1 sgma ponts an be obtaned by formula below: = x (1) ( ) = x + (n +λ) P xx (2) ( ) +n = x (n +λ) P xx (3) /11/$

2 where ( (n + λ) P xx ) s the th olumn of matrx (n + λ) P xx and λ = α 2 (n + κ) n. Parameter α s suggested to be between 1-4 and 1. [15] and value of parameter κ s 3 n or. After arefully seleted, sgma ponts are then propagated through the followng funton: = f ( ), where =,1,..., (4) he next step s alulatng mean and ovarane of the propagated ponts gven by: m y = W (5) yy W γ γ P = [( y)( y) ] (6) m he weghts W and W are defned below: m λ W = n + λ (7) λ 2 W = + (1 α + β ) (n + λ) (8) m 1 W = W = 2(n + λ) (9) B. Unsented Kalman Flter UKF s an algorthm whh an solve nonlnear systems n the followng form: x = k 1 f( x + k) + q (1) k y = h( x ) + r (11) k+ 1 k+ 1 k+ 1 where x s a dsrete state vetor, y s dsrete measurement vetor, q and r are the system and measurement Gaussan noses wth zero mean and ovarane matres Q and R, respetvely. here are three stages n the UKF method [14, 15]: 1. Sgma ponts alulaton Frstly, we have to defne an ntal state vetor x, ntal ovarane P, proess nose ovarane and measurementnose ovarane Q and R, respetvely. hs have to be defned n advane based on a pror knowledge of the system. In ths paper, ntal parameters are hosen arefully based on SLS algorthm results. In ths stage, sets of +1 sgma ponts are reated based on prevous state wth followng formula: X = [ x x ] + n+ λ [ P P ] (12) ( ) k-1 k-1 k-1 k-1 k-1 2. Kalman flter state predton hen sgma ponts n the frst stage are propagated through funton below: * = f( ) (13) kk 1 k 1 he followng step s omputng the predted state mean vetor x kk 1, and predted ovarane matrx P 1: m * kk 1 = W, k k 1 * * kk 1 W kk, 1 kk 1 kk, 1 kk 1 x (14) P = [( x )( x ) ] + Q (15) 3. Kalman Flter state orreton Next, the sgma ponts related to the predted state mean vetor and ovarane matrx are alulated wth the followng formula: = [ xkk 1, xkk 1 ± (n + λ) P ] (16) kk 1 kk 1 hen, the sgma ponts are propagated through measurement-update funton: = h( ) (17) kk 1 kk 1 he followng step s alulatng propagated ponts: m y kk 1 = W, k k 1 (18) hen, we an obtan the measurement ovarane matrx P and ross-ovarane of the state and measurement P : yy yy W, k k 1 kk 1, k k 1 kk 1 xy = W, k k 1 kk 1, k k 1 k k 1 P = [( y )( y ) ] + R (19) P [( x )( y ) ] (2) Fnally, we an ompute Kalman gan, the state mean and ovarane below: 1 Kk = PxyP yy (21) x = x + K ( y y ) (22) k k k 1 k k k k 1 k = k k 1 k yy k P P K P K (23) III. ALGORIHM ESING he UKF algorthm has been desgned n suh a way that he followng nstantaneous parameter model of the proessed nput voltage (t an also be a urrent) was used: ( ) sn ( ω ϕ) ξ ( ) ut = U t+ + t (24) n whh u(t) s an nstantaneous voltage at tme t, U s the magntude of fundamental omponent, ω s fundamental angular veloty, equal to 2πf where f s frequeny, ϕ s the phase angle and ξ(t) s a zero mean random nose. In ths model, the hgher harmons are onsdered as a random nose and need to be properly fltered out. In the testng of the algorthm ths flterng has not been done, just to demonstrate the orgnal algorthm propertes wthout some extra measures, kk xy

3 whh mght only mprove the overall algorthm performane. he only unknown model parameters are the sgnal magntude, ts frequeny and phase. he above algorthm has been tested thoroughly usng omputer smulated data reords. hree types of test were arred out: stat and random nose tests, dynam tests, and synthet data test. A. Stat and Random Nose est Durng the stat test, the followng sgnal model was used: Freq err, U err [p.u.] Freq error U error () os( ω 3).3os( 3ω 9).2os( 5ω 15) ut = t+ + t+ + t+ (25) where ω s fundamental angular veloty whh s equal to 2πf where f = 5 Hz. he sgnal model above was then used to nvestgate the senstvty of the method to random nose and seleton of the ntal ovarane. In ths smulaton, the samplng frequeny used was f s = 1.6k Hz. Fg. 1 presents the algorthm senstvty to random addtve nose wthout hangng the ntal ovarane. It s shown that the algorthm an trak ampltude and frequeny wth low maxmum error under very nosy ondton. However, these maxmum errors an be redued by hangng the ntal ovarane as shown n Fg. 2. he Sgnal to Nose Rato (SNR) s defned below: S SNR = 2log (26) ( 2σ ) S where s the root mean square value of the sgnal (S RMS ) 2 and σ s standard devaton of the nose. he maxmum errors are alulated usng funton below. ( f f ) max ( U U ) max est real est real max ( error ) = or (27) f U real he maxmum errors versus the SNR are plotted below. B. Dynam est Dynam tests were also arred out to analyze the algorthm senstvty to sgnal and frequeny dstorton. he dstorted voltage and urrent sgnals used for dynam test were gven below. u() t = os( ωt+ 45) +.5os( 3ωt+ 12) + (28) +.3os( 5ωt+ 15) +.2os( 7ωt+ 28) () t =.9os( ωt+ 45) +.4os( 3ωt+ 6) + (29) +.2os 5ωt os 7ωt+ 13 real ( ) ( ) SNR [db] Fg. 1. Maxmum ampltude and frequeny errors vs. sgnal to nose rato wth onstant ntal ovarane. Freq err, U err [p.u.] Freq error U error SNR [db] Fg. 2. Maxmum ampltude and frequeny errors vs. sgnal to nose rato wth adjusted ntal ovarane. Durng the test, the sgnal parameters are dynamally hanged. For the perod t<.158s, the test sgnals only onsst of the fundamental omponent, whh s the frst term of equaton (28), (29). At t=.158s, the nput sgnals were dstorted wth hgher harmons aordng to (28), (29) and frequeny sgnal was hanged sgnfantly from f = 5 Hz to f=45 Hz and then lnearly rased wth the rate 1 Hz/s. In Fg. 3, nput sgnals were presented. U, I [p.u.] U I Fg. 3. Input sgnal for dynam test.

4 It s shown n Fg. 4 that the UKF suessfully traked the ampltude. he hghest error durng ths measurement exludng the onvergene perod s less than 1-2 %. In Fg. 5, the result of estmated frequeny s ompared to the real one. he algorthm onvergene propertes are determned by ntal ovarane. Faster onvergene perod an be obtaned by redung measurement nose ovarane and ve versa..85 Voltages (V) 6 x me (s) Fg. 1. Voltage waveforms data reords. I RMS, U RMS [p.u.] Van (V) 5 x Urms Irms Fg. 4. Estmated RMS usng UKF n a dynam smulaton me (s) Fg. 2. Detals of the voltage on phase a, transformer s energzed at t =.5 s. Frequeny [Hz] Real Estmated Mag (% Fund) HDv (%) 3 2th 2 3th 4th 1 5th 6th me (s) me (s) Fg. 3. Harmon ontent on phase a Fg. 5. Estmated frequeny usng UKF n a dynam smulaton. 4 x 15 C. Proessng of a Steady State Dstorted Sgnal Obtaned hrough Dynam Smulaton of a Power System Fg. 1 shows the voltage waveforms of a three-phase transformer whh s energzed on a 5 kv network at t =.5s. he transformer rated 45 MVA, 5 kv/23 kv/6 kv onssts of three wndngs onneted n Y/Y/Delta. he voltage on phase a ontans a hgh level of 4th harmon, a detals of the hgh ontans of harmon s shown n Fg. 2 and Fg. 8. In Fg. 9, the estmated three-phase RMS voltages are presented. Before the dsturbane, all three phase RMS voltages are around V. It s shown that the algorthm an trak ampltude presely under hgher harmons ontent. Note that by knowng estmated RMS voltage and urrent, power omponents an be alulated based on the defnton gven n the IEEE Standard [22]. Urms [V] Urms-A Urms-B Urms-C Fg. 9. Estmated three-phase RMS voltage Fg. 1 reveals the system frequeny before, durng and after dsturbane. he estmated result s ompared wth real frequeny. he exat same value an be dretly reognzed from Fg. 1. hs result verfes that UKF s apable to trak frequeny durng large dsturbane.

5 Frequeny [Hz] Estmated Real Fg. 1. Estmated frequeny IV. CONCLUSION In ths paper, a new parameter estmaton method for frequeny, ampltude and phase trakng, based on Unsented Kalman Flter s presented. Varous smulatons are arred out to analyze ts steady state and dynam performane. It s shown that UKF obtaned hgh estmaton auray both under normal and nosy ondtons. It s verfed that ths method s not senstve to frequeny hanges, thus t an be effetvely mplemented as a relable tool for PQ ndes estmaton. Fnally, the smplty of the UKF due to the absene of the model lnearzaton gves promsng trakng performanes and effeny for PQ ndes montorng n harmons polluted power systems. he authors are now extensvely usng the proposed algorthm for PQ ndes estmaton for data reords obtaned through measurements n real systems. V. ACKNOWLEDGMEN he authors wsh to thank Prof. Knud Ole Helgesen Pedersen from Semens Wnd Power Denmark, Prof. Jaob Østergaard and Dr. John El Nelsen from DU Denmark for the support to omplete ths paper. Also thanks to Shlumberger Foundaton for the support through Faulty for he Future program. VI. REFERENCES [1] R.C.Dugan, M.F. MGranaghan, S. Santoso and H.W. Beaty, Eletral Power Systems Qualty, d. ed. MGraw-Hll, 22. [2] P.K. Dash, S.K. Panda, B. Mshra and D.P. Swan, Fast estmaton of voltage and urrent phasors n power networks usng an adaptve neural network, IEEE rans. Power System, vol. 12, no. 4, pp , [3] M.D.A. Rahaman and K.B. Yu, otal least squares approah for frequeny estmaton usng lnear predton, IEEE rans. On Aousts, Speeh and Sgnal Proessng, vol. 35, no. 1, pp , [4] R.Chudaman, K.Vasudevan and C.S.Ramalngam, Real-me Estmaton of Power System Frequeny Usng Nonlnear Least Squares, IEEE rans. on Power Delvery, vol. 24, ssue: 3, pp , 29. [5] V. erzja, M. Djur and B. Kovaev, Voltage phasor and loal system frequeny estmaton usng newton type algorthm, IEEE rans. Power Delvery, vol. 9, no. 3, pp , [6] V. erzja, V. Stanojev, M. Popov and L. van der Slus, Dgtal meterng of power omponents aordng to IEEE standard usng the Newton-type algorthm, IEEE rans. Instrumentaton and Measurement, vol. 56, no. 6, 27. [7] V. erzja, V. Stanojev, M. Popov and L. van der Slus, Dgtal meterng of power omponents aordng to IEEE standard usng the Newton-type algorthm, IEEE rans. Instrumentaton and Measurement, vol. 56, no. 6, 27. [8] V. erzja, M. Djur and B. Kovaev, A new self-tunng algorthm for the frequeny estmaton of dstorted sgnals, IEEE rans. Power Delvery, vol. 1, no. 4, [9] V. erzja and V. Stanojev, SLS algorthm for power-qualty ndes estmaton, IEEE rans. Power Delvery, vol. 23, no. 2, 28. [1] H. Novanda, V. Stanojev and V. erzja, Power qualty ndes estmaton durng sudden generator dsonneton n Faroe Islands power system, presented at Internatonal Conferene on Advaned Power System Automaton and Proteton, Jeju, Korea, 29. [11] S. J. Juler and J. K. Uhlmann, A New Extenson of the Kalman Flter to Nonlnear Systems, n Pro. of the 4th Conferene on Sgnal Proessng, Sensor Fuson and arget Reognton, vol. 368, pp , [12] B.F.L. Sala and R.R. Btmead, Desgn of an extended Kalman flter frequeny traker, IEEE rans. On Sgnal Proessng, vol.44,pp , [13] P.K. Dash, R.K. Jena, G. Panda and A. Routray, An extended omplex Kalman flter for frequeny measurement of dstorted sgnals, IEEE rans. on Instrumentaton and Measurement, vol. 49, pp , 2. [14] G. Valverde and V. erzja, Unsented kalman flter for power system dynam state estmaton, IE Generaton, ransmsson & Dstrbuton, vol. 5, pp , 21. [15] S. J. Juler and J. K. Uhlmann, "Unsented flterng and nonlnear estmaton," n Proeedngs of the IEEE, vol. 92, pp , 24. [16] N. Bretas, An teratve dynam state estmaton and bad dataproessng, Int. Journal Eletr Power Energy System, vol. 11, no.1, pp. 7-74, [17] E. Blood, B. Krogh and M. Il, Eletr power system stat state estmaton through Kalman flterng and load foreastng, IEEE Power and Energy Soety General Meetng Converson and Delvery of Eletral Energy n the 21 st Century, pp. 1-6, 28. [18] E.A.Wan, R. Van Der Merwe, he unsented Kalman flter for nonlnear estmaton, n Adaptve Systems for Sgnal Proessng, Communatons, and Control Symposum 2, pp , 2. [19] Z. Jan, A. Swan, N. K. C. Nar, and J. J. Lu, "Estmaton of power qualty usng an unsented Kalman flter," ENCON IEEE Regon 1 Conferene, 27, pp [2] P.K. Dash, S. Hasan and B.K. Pangrah, Adaptve omplex unsented Kalman flter for frequeny estmaton of tme-varyng sgnals, IE Sene, Measurement and ehnology, vol. 4, no. 2, pp , 21. [21] J.B.V. Reddy, P.K. Dash, R. Samantaray and K. Moharana, Fast trakng of power qualty dsturbane sgnals usng an optmzed unsented flter, IEEE ransatons on Instrumentaton and Measurement, vol. 58, no. 12, pp , 29. [22] A.E.Emanuel, Summary of IEEE Standard 1459: Defntons for the Measurement of Eletr Power Quanttes Under Snusodal, Nonsnusodal, Balaned, or Unbalaned Condtons, IEEE ransatons On Industry Applatons, Vol. 4, No. 3, May/June 24, pp VII. BIOGRAPHIES Happy Novanda obtaned the B.S degree n Eletral Engneerng from the Unversty of Indonesa n 26 and the Master degree n ehnology Management from the Unversty of New South Wales n 27. Currently she s a PhD student at the Unversty of Manhester workng n the area of development of estmaton methods for estmaton of power qualty ndes and nvestgatng the nature of harmon dstortons at renewable energy resoures, partularly nverter onneted wnd farms.

6 Pawel Regulsk obtaned the MS degree n Eletral Engneerng from the Wrolaw Unversty of ehnology (Poland) n 28. Currently he s a PhD student at the Unversty of Manhester workng on load modelng and voltage stablty. Hs man researh nterests nlude development of new estmaton tehnques and mplementaton of artfal ntellgene tehnques n power system applatons. Franso M. González-Longatt (S 1, M 3, SM 29) was born n Cagua-Venezuela, on July 3, He graduated on Eletral Engneerng of Insttuto Unverstaro Polténo de la Fuerza Armada Naonal, Venezuela (1994), and Master of Busness Admnstraton of Unversdad Bentenara de Aragua, Venezuela (1999) and PhD of Unversdad Central de Venezuela (28). Hs man area of nterest s renewable energy resoures and power system dynam n smart grds. He s former assoate professor on Eletral engneerng Department of Unversdad Naonal Polténo de la Fuerza Armada Naonal, Venezuela. He s urrently a Post Dotoral Assoate Researh Shool of Eletral and Eletron Engneerng, he Unversty of Manhester. Vladmr erzja (M 95, SM 2) s the EPSRC Char Professor n Power System Engneerng n the Shool of Eletral and Eletron Engneerng, he Unversty of Manhester, where he has been sne 26. From 1997 to 1999, he was an Assstant Professor at the Unversty of Belgrade. In 1999, he was awarded a prestgous Humboldt Researh Fellowshp. From 2 to 26, he was wth ABB AG, Germany, workng as an expert for swthgear and dstrbuton automaton. Hs man researh nterests are applaton of ntellgent methods to power system montorng, ontrol, and proteton, swthgear and fast transent proesses, as well as DSP applatons n power systems.