Estimation of Sequence Components using Magnitude Information

Transcription

1 6th NATIONAL POWER SYSTEMS CONFERENCE, 5th-7th DECEMBER, 2 47 Estimtion of Sequene Components using Mgnitude Informtion P.S. Ngendr ro nd Ssikirn Veknuru Deprtment of Eletril Engineering Indin Institute of Siene, Bnglore, Indi 562, ngendr@ee.iis.ernet.in ssi.7@gmil.om Astrt In this pper simple sheme to otin the sequene omponents of the unlned voltges, with out using ny omplex rithmeti is proposed. The error mesurement of the proposed shemes is provided, y onsidering set of voltges. The effetiveness of the proposed sheme is provided y ompring with the existing stndrds. I. INTRODUCTION Voltge unlne ommonly ours in power system network, due to ny of the following resons []: single phse loding, unlned impednes of trnsmission lines nd trnsformers, non-uniform ompenstion of three-phse with pitor nks nd inomplete trnsposition of lines. The symmetril omponents method hs een known s the most ommonly-used tehnique for nlyzing unlned systems. In the onventionl method, positive nd negtive sequene omponents hve een estimted using two steps: the mgnitude nd phse-ngle of eh line/phse unlned voltges re first estimted nd the symmetril omponent trnsformtion is then pplied. Symmetril omponents re so widely used tht, it will e useful if they n e estimted y simple mens. This hs een well reognized, in ft oth the NEMA nd IEC stndrds provide suh shemes. The importne of this issue is lso refleted in the ontinuing investigtion in this pper. In [2], method to derive the symmetril omponents sed on the mgnitude of the three phse urrents in system hs een proposed. In [3], the sequene omponents of the system hve een estimted, first y trnsforming the unlne system into αβ, nd then pplying the onstnt trnsformtion mtrix. In [4], reursive method for mesurement of sttionry nd instntneous symmetril omponents hve een proposed. In [5-7], Klmn hs een used s powerful tehnique to estimte the sequene omponents. In [8-], vrious tehniques using disrete fourier trnsform nd fst fourier trnsform hve een proposed to estimte the sequene omponents. In [], stte oserver hs een used to estimte the urrent nd voltge symmetril omponents in three phse eletril network. In [2], dynmi tehnique hs een proposed to mesuring the sequene omponents of unsymmetril urrent nd voltge wveforms. In [3], n lgorithm sed on four-smple method to extrt the rms vlues of sequene omponents hs een proposed. In this pper sheme to estimte the sequene omponents, using simple rithmeti is proposed. The errors in the estimtes re provided y onsidering set voltges tht over the rnge normlly enounter in prtil system. The oservtion mde on the existing stndrds hs een used to find the effetiveness of the proposed sheme. II. SCHEME FOR ESTIMATING SEQUENCE COMPONENTS Consider three phse three wire system where the zero sequene omponents re sent. nd eh line/phse quntity is the sum of the orresponding positive nd negtive sequene omponents. Consider the produt of the three line voltges V, V nd V. ( V V V ) = ( V + V 2 )( V + V 2 )( V + V 2 ) () ( V V V ) = (( V + V 2 )(α V +α 2 V2 )(α 2 V +α V 2 )) (2) where the susripts nd 2 represent the positive nd negtive sequene omponents nd α = e j2 Expnding nd simplifying the right hnd side of (2) it is esy to show tht ( V V V ) = ( V 3 + V 3 2) (3) Tking the ue root on oth sides of (3) gives ( V V V ) 3 = V { + ( V 2 V ) 3 } 3 (4) By treting x = ( V 2 V ) 3, the right hnd side of (4) n e expnded using the inomil theorem s ( V V V ) 3 = V ( + 3 x 9 x x3...) (5) In ny prtil system the negtive sequene voltge will e generlly very smll frtion of the positive sequene. Hene, the right hnd side n e pproximted to V,

2 6th NATIONAL POWER SYSTEMS CONFERENCE, 5th-7th DECEMBER, 2 47 V Sign of Qudrnt of θ Denomintor Numertor + + st - - 3rd + - 4th - + 2nd V V 2 V TABLE I METHOD OF CHOOSING THE QUADRANT OF θ WHILE USING (2) V 2 V V V 2 V V 2 V 2- θ Using () in (8) to (9) nd sutrting (9) from (8) nd simplifying we get V V = 3 V2 sin θ () From (7) nd (), we get θ = tn V V ( 3( V V ) ) (2) V 2 θ Fig.. VBB V Vetor digrm V V V 2+θ VB2B sine the lowest power of x the rtio of V 2 to V in the expnsion is 3. As mtter of ft, even when the negtive sequene is of the order of.5 of the positive sequene, the error in this pproximtion will e only in the rnge.5% to 4%, depending on the ngle etween the positive nd negtive sequene omponents. Therefore, we hve V ( V V V ) 3 (6) Now onsider the vetor digrm shown in Fig. The ngle etween V nd V 2 is θ. Ignoring the ngle etween the V nd V, whih will e invrily smll for smll mgnitudes of V 2, we n write V V + V2 os θ (7) Similrly negleting the ngle etween V nd V s well s V nd V, we n write the following two equtions. V V + V2 os(2 θ) (8) V V + V2 os(2 + θ) (9) As is known, V = V = V nd V2 = V2 = V2 () Sine θ is the ngle etween two vetors, it ould e in ny of the qudrnts. Whenever the rtio ( V V ( V V ) ) is positive θ will e in either the first or the third qudrnts nd if the rtio is negtive θ will e in either the seond or the fourth qudrnts. The prtiulr qudrnt is seleted sed on the signs of the numertor nd the denomintor quntities in (2) s indited in the TABLE I. Hene, using the θ found in (2) the negtive sequene voltge n e omputed using V2 V V = os θ (3) III. ERROR IN THE ESTIMATES OF THE PROPOSED SCHEME In order to ssess the ury of the proposed estimtion sheme, we hve omputed the error in the estimtes for rnge of unlned voltges tht is generlly enountered in prtie. A rnge of.85p.u to. p.u hs een onsidered for eh of the line/phse voltges in steps of.5p.u. Choosing one of these six vlues for eh of the three phse voltges (.85,.9,.95,.,.5,.), we generte 26 (6 3 ) three line voltge omintions. The error for ll these 26 omintions is omputed. This hoie overs positive sequene voltge rnge of.85p.u to.p.u nd negtive sequene omponent rnge of to.8p.u. The omintion of line voltges tht orrespond to eh of these 26 ses is shown in Fig 2. In Fig 3() the estimted positive sequene voltges for ll these 26 ses re plotted. The orresponding true vlues re plotted in Fig 3(). The perentge error in the estimte for these ses re plotted in Fig 3(). From the Fig 3() we see tht the perentge error in our estimte is greter thn.% only in 5 ses. In most of the ses the error is less thn.5%. Similr results for the negtive sequene quntities re presented in Fig 4. In Fig 4() the estimted negtive sequene voltges for the 26 ses nd in Fig 4() the tul negtive sequene voltges re shown. The error in the estimted vlues re plotted in Fig 4(). It is seen from Fig 4() tht in most of

3 6th NATIONAL POWER SYSTEMS CONFERENCE, 5th-7th DECEMBER, V V ve seq voltge tul -ve seq voltge estimted V The 26 ses onsidered Error Fig. 2. The 26 sets of Line voltges onsidered for study Fig. 4. Estimted nd tul negtive sequene voltges nd the orresponding errors %error Estimted +ve seq volt Atul +ve seq volt Error(rd) Angle diff estimted(rd) Atul Angle diff(rd) Fig. 3. error Estimted nd tul positive sequene voltges nd the perentge Fig. 5. Estimted nd tul ngles etween sequene omponents s well s the errors in estimtion these ses solute error is less thn.5p.u. For out 6 ses the error is etween -.5p.u nd -.5p.u, for nother 6 ses error is etween.5p.u nd.p.u. The estimted ngle etween the positive nd negtive sequene quntities, their tul ngle vlue nd the error in their estimtion for these ses is plotted in Fig 5. It n e seen from Fig 5() tht the solute error in the estimted ngle etween the positive nd negtive sequene voltge is lwys less thn.5 o. IV. EXISTING STANDARDS TO CHARACTERIZE THE UNBALANCE IN THE SYSTEM There re two generl definitions in use [4] to hrterize the voltge unlne in the system. The first one is NEMA (Ntionl Eletril Mnufturer Assoition Motor nd Genertor Stndrd), the line voltge unlne ftor (LVUF) ording to NEMA stndrd is defined s = mx[ V V lvg, V V lvg, V V lvg ] V lvg (4) where, V lvg = V + V + V 3 (5) Essentilly, NEMA requires the mgnitudes of line voltges to evlute the voltge unlne ftor. The seond definition for voltge unlne ftor (VUF) is given y Interntionl Eletromehnil Commission (IEC) s follows, V UF = V 2 V (6)

4 6th NATIONAL POWER SYSTEMS CONFERENCE, 5th-7th DECEMBER, where, V 2 is the negtive sequene omponent of the line voltges nd V is positive sequene omponent of the line voltges. V nd V 2 re otined y pplying symmetril trnsformtion on the line voltges. Therefore generlly, [5] industries nd utilities dopt LVUF to hrterize the unlne. VUF gives preise unlne ftor sine it uses tul positive nd negtive sequene omponents. But omputtion of this ftor involves omplex rithmeti. From [4], it implies tht verge of the line voltges is the pproximtion to the positive sequene omponent of the line voltges nd mximum devition of line voltges from the verge of the line voltges is the pproximtion to the negtive sequene omponent of the line voltges. This is used to hrterize the effetiveness of the proposed sheme. V. EFFECTIVENESS OF THE PROPOSED SCHEME In order to ompre the proposed estimtion sheme with the NEMA sheme, we hve estimted the positive nd negtive sequene quntities using the NEMA pproh for ll the ses onsidered. Results re summrized in Fig 6. For the positive sequene quntities ompring Fig 2() with Fig 6(), we see tht the perentge errors in the estimtes y the proposed method re signifintly lower in most of the ses. The verge of perentge error(solute) for ll these ses y the proposed sheme is.32 nd it is.2652 for the NEMA sheme. In the NEMA se only 63 of 26 ses hve n error whih is less thn.%. The errors re summrized in TABLE II. Error in -ve seq voltge % error in estimtion of +seq voltge Fig. 6. Error in the positive nd negtive seq omponents, estimted from the NEMA sheme It is lso seen tht the mximum error in the estimted positive sequene omponents for this rnge of voltge oservtions is % in NEMA sheme where s it is less thn.24% in the proposed sheme. Another feture tht my e noted is tht the error in the estimted vlues of the proposed sheme ould e either positive or negtive, ut in the NEMA sheme the error is lwys positive. Proposed method NEMA Positive seq Negtive seq Positive seq Negtive seq Mx error.24%.2p.u %.6p.u Aver error.32%.23p.u.2652%.49p.u TABLE II MAXIMUM(ABS) ERROR AND AVERAGE(ABS) ERROR FROM THE PROPOSED AND NEMA METHOD The errors in the negtive sequene omponents estimted from the NEMA sheme for the 26 ses is given in Fig 6(). It is seen from figure tht for 24 ses the error is etween -.p.u nd -.6p.u. The verge of solute error from proposed sheme is.23p.u nd for NEMA sheme it is.49p.u. The mximum error(solute) seen for the proposed sheme is.2p.u nd tht for the NEMA sheme is.6p.u. It is seen from the ove disussion tht the proposed method gives signifintly etter estimtion s ompred to the NEMA estimtion, with the dded dvntge of simpliity. VI. CONCLUSIONS A simple sheme to otin the sequene omponents of the line voltges nd the ngle etween them is proposed. It seen the tht the proposed sheme is estimting the sequene omponents with firly good ury. The effetiveness of the proposed sheme is presented y ompring with the existing stndrds. REFERENCES [] P. Gninski, Derting of n indution mhine under voltge unlne omined with over or undervoltges, Energy Conversion nd Mngement 5, 29, -7. [2] J. Brndolino, nd R. D. Findly, Prtil Mesurement of Symmetril Component Currents in Indution Motors, Cndin Conferene on Eletril nd Computer Engineering, IEEE Ctlog 94TH823, Vol. I, pp , 994. [3] R. A. Almmri, S.A.Solimn, M.A.Mostf, M.. A. El-Hwry, Two Digitl Filtering Algorithms for Fst Estimtion of Symmetril Components in Power System: A Stti Estimtion Approh, Eletri Power Systems Reserh, Volume 66, Issue 2, August 23, Pges [4] Miodrg D. Kuljevi, Symmetril Components Estimtion Through Mximum Likelihood Algorithm nd Adptive Filtering, ieee trnstions on instrumenttion nd mesurement, vol. 56, no. 6, deemer 27, [5] S. A. Solimn nd M. E. El-Hwry, Applition of Klmn filtering for online estimtion of symmetril omponents for power system protetion, Elet. Power Syst. Res., vol. 38,997, pp [6] A. A. Girgis, W. Chng, nd E. B. Mkrm, Anlysis of highimpedne fult generted signls using Klmn filtering pproh, IEEE Trns. Power Delivery, vol. 5, 99, pp [7] Rfel A. Flores, Irene Y.H. Gus nd Mth H.J. Bollen, Positive nd Negtive Sequene Estimtion for Unlned Voltge Dips, IEEE Generl Meeting, Toronto, ON, Cnd, 23. [8] E. Orn Brighm, The Fst Fourier Trnsformer And Its Applitions, Prentie Hll Interntionl, 988. [9] T. Loos, Fst estimtion of symmetril omponents in rel time, in Pro. Inst. Elet. Eng. C, vol. 39, 992, pp [] A. Cmpos, G. Joos, P. D. Ziogs, nd J. F. Lindsy, A dsp-sed reltime digitl filter for symmetril omponents, in Pro. Athens Power Teh. Joint Int. Power Conf., vol., 993, pp [] E. Rosolowski, nd M. Mihlik, Fst Estimtion of Symmetril Components y Use of Stte Oserver, IEE Pro.- Genertion, Trnsmission nd Distriution, Vol. 4, No.6, 994, pp

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