Active and Reactive Power Metering in Non-Sinusoidal Conditions Using Newton Type Algorithm

Transcription

1 Acte and Reacte Power eterng n Non-Snusodal Condtons Usng Newton ype Algorthm Vladmrerza, Vladmr Stanoec, Zoran Lazarec, aran Popo 3, ABB Calor Emag ttelspannung GmbH, Ratngen, Germany, e-mal: Vladmr.erza@de.abb.com Faculty of Electrcal Engneerng, Belgrade, Ser&o, e-mal: lazarec@lop.etf.bg.ac.yu; stanoec@web.de 3 Delft Unersty of echnology, Power Systems Laboratory, Delft, he Netherlands, e-mal:.popo@eee.org Abstract: hs paper presents a new two stage Newton ype numercal algorthm for acte and reacte power measurement n electrc power systems wth snusodal, non-snusodal lnear and non-snusodal nonlnear crcuts. In order to estmate ther spectra, n the frst stage of the algorthm the current and oltage sgnals are separately processed, whereas n the second stage the power components are calculated. As the man adantage, ths technque prodes measurements not senste to frequency deatons. he algorthm performance s tested usng computer smulated tests. Keywords: power systems, power measurement, nonlnear estmaton, computer smulaton, laboratory testng.. Introducton Harmonc polluton s of rasng presence n the contemporary power systems. he man cause of ths are large non-lnear elements such as power electroncs used for the system operaton (e.g. thrstor controlled compensers or reactors), non-lnear loads such as the arc furnaces, or hgh power conerters. here s a great number of papers concerned wth deelopment of measurement technques and algorthms for electrc power systems wth non-snusodal lnear and non-lnear crcuts. Howeer, many of those numercal algorthms are senste to frequency aratons, for example, DF (Dscrete Fourer ransformaton) or LSE algorthm (Least Squares Error). he frequency deaton occurs after a large bloc of load or a large generator unt s connected to or dsconnected from the power system. Frequency aratons are much more lely to occur for the loads suppled by a generator solated from the utlty system. Numercal algorthm presented n ths paper consders the system frequency as an unnown parameter of the model to be estmated and n ths way soles the problem of senstty to frequency aratons n wde range. Wth the ntroducton of the power frequency n the ector of the unnown model parameters, the model tself becomes nonlnear and the strateges of nonlnear estmaton must be used. For the purpose of the oltage and current spectra estmaton, the Newton ype Algorthm (NA) s deeloped and successfully mplemented n the acte and reacte power measurement. he NA can be also appled n determnng some other quanttes, for example, those defnng electrc power qualty or n the procedure of dentfyng harmonc sources n the power system.. NA Algorthm Deelopment Let us assume the followng obseraton model of the nput oltage (or current) sgnal, dgtzed at the measurement dece locaton: ( t) = h(, t) + ξ ( t) x () 4 RE&PJ, Vol., No.3, arch 5

2 n whch (t) s an nstantaneous oltage at tme t, ξ ( t) a zero mean random nose, x a sutable parameter ector and h nonlnear functon expressed as: () h( x, t) = V + V sn( ω t + ϕ ) () = For the generc model (), a sutable ector of unnown parameters s gen by: [ V ω, V, K,, ϕ,, ϕ ] x = (3), V K where V s the magntude of the decayng DC component at t =, s the hghest order of the harmoncs presented n the sgnal, ω s the fundamental angular elocty, equal to πf, f beng frequency, V s the magntude of the -th harmonc and ϕ s the phase angle of the -th harmoncs ( =,...,). he equalent sgnal model can be assumed for the current sgnal. he number of unnowns,.e. the model order, s n = +. he nput sgnals are sampled durng a fnte perod of tme, called data wndow. ang N samples to a data wndow, a set of N ( N n = + ) nonlnear equatons, gen by () and (), n n unnowns are obtaned. he problem s to sole the oerdetermned system of nonlnear equatons,.e. to determne the unnown model parameters. he ey relaton of the NA algorthm s gen by: ( J J ) J h( ˆ t) ( ) = xˆ + x, xˆ + (4) where s an teraton ndex, xˆ s estmated # ector, J ( J J ) J = s referred to as a left pseudonerse of Jacoban J, s an ( N ) measurement ector, h ( ˆx,t) s an ( N ) ector of nonlnear functons determned by the assumed mathematcal model of the nput sgnal and N s the number of samples belongng to the data wndow. Jacoban matrx J s an ( n) N matrx, hang as ts elements the partal derates of the sgnal model (). If s an arbtrary row of the Jacoban, then [,, ] = L (5), 3, + = = V = = Vt ω cos = ( ωt + ϕ ) (6) (7) + = = sn V where =, K,. ( ω t + ϕ ) + + = = V cos ϕ ( ωt + ϕ ) Equaton (4) s dered frst by lnearzaton,.e. by aylor seres expanson of () and neglectng hgher order terms, and then by applyng Least Square Error estmaton method. 3. wo Stage Numercal Algorthm Bloc dagram of the two stage numercal algorthm s presented n Fgure. NA (current) NA (oltage) I stage x x u II stage P, Fgure : Bloc dagram of the two stage NA algorthm. In Fgure, and are current and oltage samples, respectely. he nputs to the second stage of the algorthm are the oltage and current sgnals spectra, contaned n the ectors x and x u, estmated n the frst stage of the algorthm. In the second stage of the algorthm, acte and reacte powers are calculated. he aerage power s expressed as: Pa = V I + V I cos( ϕ ψ ) () = where ϕ and ψ are oltage and current -th harmoncs phase angles. here s no unque defnton for the reacte power under non-snusodal condtons. In ths paper Budeanu s reacte power defnton [] s used: Bud = V I sn = ( ϕ ψ ) (8) (9) () 4 RE&PJ, Vol., No.3, arch 5

3 4. Algorthm estng he proposed numercal measurement algorthm s tested through computer smulated tests, as follows:. dynamc test, and. test usng dynamc smulaton of power system. A. Dynamc test Dynamc test conssts of processng the followng computer generated dstorted oltage and current sgnals: () t = cos( ω t + 45) + 5cos( 3ω t + ) + 3 cos( 5 t + 5) + cos( 7ω t + 8) ω [V] () () t = cos( ω t) + 4cos( 3ω t + 6) + cos( 5 t + 3) + cos( 7ω t + 3) ω [A] (3) At frst, from to.3s, pure snusodal waes of oltage and current were generated, wth magntude of V (A) and 5Hz. At t=.3s the magntude step change of % occurred. o approe the NA algorthm nsenstty to frequency deatons, the frequency was also step changed at.3s from 5Hz to 49Hz, and then snusodal changed wth tme: f ( t) = 49 + sn(π.t) (4) wth frequency of.hz (see Fgure 4) whch s approxmately the natural frequency of synchronous generators swngng n a power system. Note that such a large frequency deatons are not lely to occur n a real power system due to nerta of generators rotors. In ths paper such large frequency aratons are ntentonally mposed to the oltage and current sgnals n order to nestgate the robustness of the algorthm. Generated oltage and current are presented n Fgures and 3. Real and estmated frequency s presented n Fgure 4, whle estmated power components are shown n Fgure 5. u (V) Fgure : Frst fe perods of computer generated oltage. (A) Fgure 3: Frst fe perods of computer generated current. f (Hz) P ( W ), ( VAr ) orgnal estmate Fgure 4: Orgnal and estmated oltage frequency Fgure 5: Power components estmated by NA. From Fgure 5 t could be notced that after a short perod of conergence, the proper results for acte and reacte power are obtaned by the NA algorthm: 37W and 439 VAr. On the contrary, the results obtaned by usng the Fast Fourer echnque are ncorrect due to the leaage effect of the FF n case when the assumed frequency (5Hz) dffers from the actual one. hese results are shown n Fgure 6. P 43 RE&PJ, Vol., No.3, arch 5

4 P (W), ( VAr) P (t) (A) Fgure 9: Input current sgnal Fgure 6: Power components estmated by the FF algorthm B. Power system dynamc smulaton For ths purpose, an Electromagnetc ransent Program (EP) smulaton code has been deeloped at U Kaserslautern (Germany). hrough the EP smulaton, the pure snusodal and dstorted oltage and current sgnals are generated. In Fgure 7 a sngle-lne dagram of V, 5 Hz test power transmsson system s presented. In the same fgure the power system parameters are gen. G R +R =6.5Ω G L +L =63.66mH G S=P+ Lne R L =6.5 Ω L =95.49mH L Fgure 7: est power system. R=5Ω L=mH In ths test, the fundamental harmonc of the electromote force ( E = 36.6 / V, %) the thrd (3.4%) and the ffth (7.5%) harmoncs are supermposed. he generated hgher order harmoncs caused the dstorton n the measured oltage ( t) and current () t. Both the smulated oltage () t and the current () t are depcted n Fgures 8 and 9, respectely. (t) (V) Fgure 8: Input oltage sgnal. By applyng the presented two stage NA algorthm, the acte and reacte powers depcted n Fgure are estmated. he hgh accuracy s obtaned. he relate errors were less than 4 %. P (W), (VAr) 4 3 P a Bud Fgure : Estmated power components (dynamc smulaton of power system). 5. Concluson In ths paper a new numercal measurement algorthm for dgtal meterng of power s presented and tested. It s based on the applcaton of the Newton ype Algorthm, a nonlnear estmator sutable for the spectrum estmaton of dstorted nput sgnals. he algorthm s not senste to the frequency changes of the nput sgnal, whch are typcal for power systems durng faults and abnormal operatons, or for the electrcal machnes wth the arable speed. he algorthm s tested through dynamc computer smulated tests, as well as by usng data obtaned from dynamc smulaton of power system operated under non-snusodal condtons. he obtaned results confrmed a hgh accuracy of the deeloped algorthm. he fast algorthm conergence offers the opportunty to apply the algorthm n processes wth fast/ery fast transents. Besde acte and reacte power meterng, ths algorthm also enables measurng and estmaton of other power qualty quanttes, such as: HD, apparent power, dstorton power, power factor, RS. he technque s not lmted to the measurement applcatons n power systems only. It may be used n desgnng the algorthms for some other and new applcatons n the other feld of engneerng, as well RE&PJ, Vol., No.3, arch 5

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