Power Harmonic Analysis Based on Orthogonal Trigonometric Functions Family

Transcription

1 Chna Intenatonal Confeence on Electcty Dstuton Powe Hamonc Analyss Based on Othogonal Tgonometc Functons Famly Ruoyu Wang, Yuefeng Zhang, Jnghong Guo, Huzhou Electc Powe Bueau, Huzhou, Zheang 33, Chna, State Gd Electc Powe Reseach Insttute of Chna, anng, Jangsu 3, Chna Astact: Hamonc detecton s the ass of hamonc ssues and hamonc detecton method s the ey of hamonc measuement. Based on Foue Tansfom and Wavelet Tansfom, ways and means of detectng and analyzng ae developed, ut oth of them have the polem of spectum leaage. In ths pape, a novel powe hamonc algothm ased on the famly of tgonometc functons s developed fo the chaactezaton of hamoncs. The algothm has desgned a set of dscete efeence sgnals and only estmated the ampltudes and the phases whch ae needed. Besdes t hasn t een nvolved n the nfluence of the leaage, ths algothm s also easy to cay out and has much hgh esoluton pecson than the tadtonal Foue algothm and Wavelet algothm. The algothm also has the chaactestc of eng moe senstve to samplng fequency and less senstve to samplng ponts. Usng ths chaactestc, the algothm can adust samplng paametes n ode to gve consdeaton to oth pecson and eal-tme pefomance. The smulaton esults have poved that the outstandng advantage of the algothm s that when chosen a pope samplng fequency the algothm can pc out hamoncs components wth hgh pecson n vey few fundamental peods. Compang wth Fast Foue Tansfom (), Wndowed- and Wavelet Pacage Tansfom, the algothm has ts cetan advantage. Key wods: Hamonc; Foue Tansfom; Wavelet Pacage Tansfom; Tgonometc Functons. Intoducton Wth the futhe eseach of powe Hamonc analyss, tadtonal Foue analyss exposue some defcency n tme-fequency localzaton. It can t pocess eal tme sgnal analyss pefectly. Wavelet Tansfom can e used to astact the chaactestc fequency y dvdng the sgnals n fequency-doman. But n pactce, the polem that the dyadc fequency-dvson of wavelet tansfom [] can t dvde the fequency and defntely maes the analytcal effot ad, especally to the analyss of hgh-fequency segment. R.R Cofman, Y.Y Meye and M.V Wchehause [,3] have put fowad Wavelet pacage Tansfom, whch can solve the polem of hgh-fequency contnuatve dvson effectvely. But n fact, Wavelet pacage Tansfom also has the polem of the enegy ovelappng etween neay fequency ands. Ths polem also affects the analyss pecson. In addton, Daueches has poved the esult n efeence [4] that thee have no compactly suppoted othogonal wavelets except Haa wavelet. Ths means when we choose a wavelet tansfomaton whch has a good effect n tme-fequency localzaton of the powe hamonc analyss, we have to gve up the pecse phase analyss of the hamonc. The Complex Wavelet and Bothogonal Wavelet can solve the polem of pecse phase analyss, ut the algothms s complcated whch need to e mpoved. In pactce, people lttle study the powe hamoncs whch have the fequency of moe than tmes of the fundamental

2 Chna Intenatonal Confeence on Electcty Dstuton fequency. Unde ths condton, ths pape has developed a novel algothm ased on Othogonal Tgonometc Functons Famly. The algothm only analyzes the powe hamonc sgnals y dnt of othogonalty of Tgonometc Functons. Unle Foue Tansfom and Wavelet Tansfom, t needs no pefect self-contaned othogonal ass to analyze sgnal, so t can effectvely ovecome the neluctalty spectum alas when sgnals convesed fom tme-doman to fequency-doman. Ths algothm can get hgh pecson ampltude s and phase s nfomaton of the steady-state powe hamonc. It also has the advantage of easy ealzng. The algothm has een valdated y smulatons.. Foue Tansfom and Wavelet Tansfom. Foue tansfom (FT) FT s one of the most popula tools n powe hamonc analyss. A gven dscete tme sees x, x,, x, satsfed the equement that n x n <, ths sees Dscete Foue Tansfom and Invese Dscete Foue Tansfom s as follow: X( ) xw,,,,, W e π () n / n n n xn X( ) X( ) W,,,, n (). Wavelet pacage tansfom (WPT) WPT s the esult of Wavelet Tansfom fathe development.wpt s most supeoty s that t can acheve fne su and decomposton. Suppose { V, Z} ae the seal suspaces of the gven sgnal( x() t ) space, usng mult-esoluton analyss the othogonal wavelet can decompose the sgnal space laye y laye: V V W, V V W,, V V+ W+. Fo the same scale, and V s the suspace of low fequency sgnal W s the suspace of hgh fequency sgnal. The expanson coeffcents a ( n ) of x() t n V eflect the oveall pctue of the sgnal and the expanson coeffcents d ( n ) n W eflect the specfc detals as well. Unde the scale, the ass wavelet of the suspaces can e deduced y the followng equatons: ( ) ( ) W () t how ( t ) ( ) ( ) W+ () t h W ( t ) In whch, h, h ae the coeffcents of the conugate mo flte and s the sequence nume of the suspaces unde the scale. The coespondng ecuence fomula fo the decomposton and econstucton of the wavelet pacet coeffcents s: d h d d h d +, n, n l,( l ) l +,n+, n l,( l ) l, n +,n +,n+ l,( l ),( l ) (3) (4) In whch, d h d + h d (5) 3. Powe hamonc algothm ased on the famly of tgonometc functons 3. Algothm pncple In ths secton, the algothm ased on the famly of tgonometc functons wll e ntoduced. Fst we gve the followng pepaaton theoem: Lemma 3.: suppose that mn, ae postve nteges, m n. Let e α [, π ), ased on the othogonalty of the tgonometc functons, we have: sn( πmx )sn( π nx ) dx (6) sn( πmx )cos( π nx ) dx (7) sn( πmx + α )sn( π nx) dx (8) sn( πnx )sn( π nx ) dx.5 (9)

3 Chna Intenatonal Confeence on Electcty Dstuton sn( π nx + α )sn( π nx) dx.5cos α () Defne equaton () y functon y( α ), then y( α ) s wave shape on nteval α [, π ) s shown as Fgue.. We can see fom Fgue that y( α ) has the only maxmum value and mnmum value. The aove lemma maes t possle to use the tgonometc functons to detect the powe hamonc sgnals. concluson we wll gve an ostensve algothm as follow. Suppose ut () s a sgnal contanng cetan ode hamoncs and the ode s less than 3. ut ( ) has the expesson as follow: ut ( ) [sn( π ft + α)+ Asn(π ft + α) (6) A sn(π 3 f t + α )] Its sampled value s: 3 3 u ( ) [sn( π f / fs + α) + Asn(π f / fs + α) A sn(π 3 f/ f + α )],,,,..., 3 s 3 (7) ( ) y α In whch, A,,,...,3 s the ampltude of each ode hamonc, A and we defne A. α,,,...,3 s Fgue. The wave shape of y( α ) The dscete fom of Lemma 3. wll e ntoduced as follow. Theoem 3.: suppose that m, n ae postve nteges, m n, f s the fundamental fequency and f s s the samplng fequency. fs / f, α [, π ), when s g enough, we have: α sn( π mf / fs)sn( πnf / fs) () sn( π mf / fs)cos( πnf / fs) () sn( πmf / fs + α)sn( πnf / fs) (3) Theoems 3.: suppose that m, n ae postve nteges, m n. f s the fundamental fequency and f s s the samplng fequency. fs / f, α [, π ), when s g enough, we have: sn( πnf / )sn( / ) fs πnf f s sn( πn / )sn( πn / ) sn ( π nx) dx / Futhemoe, we have: (4) sn( πnf / fs + α)sn( πnf / fs) ( cos α) / (5) Fom Theoem 3. we can see that the esults of fomula (4) and (5) have nothng to do wth hamonc ode n. Based on ths the phase of each ode hamonc, α [, π ),,,...,3. Suppose the estmated ampltude and phase of each ode s A %, % α,,,...,3.the pupose of the algothm s to estmate A%, % α fom u ( ). We defne efeence ampltude A : f f (8) A() sn ( π / ),,,...,3 s Fom fomula (4) we have: Δ () ()... (3) /, then we defne a efeence A A A A sgnal: ua(, ) sn( πf / fs),,,,..., (9) Togethe we suppose f floo( fs /( f)), floo(): R R s a ottom ntegal functon. By summng ove the esult of dsplacement multplcaton of u ( ) and u (, ), we have: S( m) u( )sn( πf / ( )/ ) fs + π m f () m,,..., f In addton, let S ( m% ) e S( m% ) max( S( m) m,,..., f),,,...,3 () Then the estmated ampltude and phase of each ode hamonc s espectvely epesented as follow: ( ) A % S ( m % )/ / S ( m % )/,,,...,3 () % α π ( m% ) / f,,,...,3 (3) As to the mplementaton of ths algothm, we can stoe A

4 Chna Intenatonal Confeence on Electcty Dstuton u (, ) A as a matx, and ths matx can e stoed n the memoy devce n advance. u (, ) A u ( ) n f Y? u ( ) 3? Y Fgue 4. The Algothm flow chat ua(, ),,,,..., ua(, ),,,,..., u A(, ),,,...,3;,,..., ;... ua(3, ),,,,..., Fgue. Refeence dscete sgnal stoed as fom of matx When dentfyng ode hamonc, we can tae out the pesoted data fom ow. ext step, we multply the data fom ow wth the sampled sgnal u ( ) coespondngly and sum the poduct. Then y the left cyclc shft of ow (altogethe shft f tmes) we can get a sees of sums. Lastly, we see the maxmum value fom the sees. Wth the left cyclc shft, we can educe the computatonal cost. u() u() u() u( ) u (,) u (,) u (,) u (, ) u (,) u (,) u (,3) u (,) A A A A A A A A Fgue 3. Multplcaton wth left cyclc shft The algothm flow chat s gven as follow: 3. Algothm smulaton Suppose a sgnal contanng 3 ode hamoncs and the fundamental fequency s 5Hz. Selected samplng fequency s 5 Hz. Samplng tme s. second and the sampled ponts ae 5. The estmated ampltude and phase value s espectvely lsted n Tale. Afte addng a unfom dstuted whte nose les etween [-,], the estmated value s also lsted n Tale. Tale. The estmated ampltude Ode Real Estmated Estmated Eo osy ampltude ampltude afte ( / ) eo addng ( / ) nose Tale. The estmated phase Ode Real Estmated Estmated Eo phase phase afte addng nose ( / )

5 Chna Intenatonal Confeence on Electcty Dstuton (ote: The estmated phase s the same efoe and afte addng whte nose.) In ths example, the ampltude of each ode hamonc (except the fundamental) s andom geneated n [,5] and the phase s andom geneated n[, π ).We can see fom Tale, the Hamonc Components ae ch and he polem of spectum leaage s vey seous, ut the estmated ampltude eo s less than.5% and the estmated phase eo s less than %.Afte addng whte nose, the estmated ampltude eo s less than % and the estmated phase eo s unchanged. The algothm has cetan nose esstng alty and almost has nothng to do wth the spectum leaage. 4. Compang wth and WPT 4. Compang wth s always a poweful tool to analyses sgnal spectum. We suppose a sgnal whch s close to pactce. The effectve fundamental ampltude s V. The fundamental fequency s 5Hz and ntal phase s degee. The followng effectve ampltude (V) fom nd to 3 th ode hamonc s espectvely:., 6.6,.88,,.66, 4.4,.44 and.. The phase (degee) s espectvely:,,, 5,, 4, 3 and 33. We compae the new algothm wth and addng foth ode Blacman-Has (B-H) wndow. The samplng fequency s 64Hz. We sample 8 ponts to the new algothm and 56 ponts to. The followng Tale3 and Tale4 s the esult of smulaton. Tale3. The esult of ampltude estmaton O d e A (V) B-H The new algoth m Eo (%) espectvely (ote: A s the shotcut fo Ampltude and P s fo Phase) Tale4. The esult of phase estmaton O d e P (de ge e) B-H The new Eo (%) espectvely Fom Tale3 and Tale4, we can daw a concluson that when we choose a pope cetan hgh samplng fequency, the new algothm can gve hgh pecse estmaton wth elatvely fewe sampled ponts and samplng tme. 4. Compang wth WPT The samplng fequency s 64Hz. We sample 8 ponts to the new algothm and 56 ponts WPT. When usng WPT, consdeng aout su-and dvson, we only analyze odd ode hamonc (Even ode hamonc s lttle analyzed n pactce). We choose d43 wavelet pacages to analyze the gven sgnal, the decomposton scale s 5. So the ottom su-and andwdth s 5 3 / Hz, whch can mae sue to detect the cetan odd ode hamonc fequency. Shannon cteon s chosen as the cteon of optmal wavelet tee. The followng Tale5 s the esult of smulaton. Tale5. The Result of ampltude estmaton de Ampltude (V) WPT The new algothm Estmated Eo (%) Estmated Eo (%) Fom Tale5, we can see that the new algothm has much moe poty n ampltude estmaton than the d43 WPT. Because the wavelets of d famly s symmety popety ae poo,

6 Chna Intenatonal Confeence on Electcty Dstuton we do not ty to compae the estmaton of phase. 4.3 Analyss of computatonal costs Suppose the sampled ponts ae, as analyzed fom the aove secton 3, the new algothm needs of ( f ) tmes multply opeatons and ( f ) tmes add opeatons to calculate the fundamental hamonc s ampltude and phase. When samplng only one fundamental peod, ( f ) [ floo( fs /( f)) ] ( / ) tmes multply opeatons and ( ) ( / ) tmes add opeatons ae needed to calculate the ode hamonc ( means ottom ntegal functon). The computatonal weght of the new algothm s, whch s equvalent wth DFT. But when samplng moe than one fundamental peod, f floo( fs /( f)) <, the computatonal weght s less than DFT. The moe we sample, the moe supeoty of computatonal costs we have. Fom Tale3 totale5 we can see that when samplng fundamental peods, the pecson of and WPT s stll lowe than the new algothm whch sampled only one fundamental peod. When added a Blacman-Has wndow, the s qute equvalent n pecson wth the new algothm. Compaed wth and WPT, the new algothm has the pecson advantage when sampled ponts ae few. 5. Analyss on unsteady sgnals In fact, thee ae a a lot of unsteady hamonc sgnals n eal powe system. They come fom extenal dstuances o some cetan devces n the powe system. The occuences and dsappeaances ae uncetan. Suppose a sgnal: t ( ) sn(π 5 t) t 8T + 3 sn(π 5t + ) T t 4T +.5 sn(π 5t + 3) 6T t 8T + 5 sn(π 35t + 5) 3T t 7T Fgue 5. The souce sgnal and ts hamonc The fundamental fequency s 5Hz and the peod T.s. We stll choose d43 WPT and as confeences wth the new algothm. When calculatng the ampltude of each ode hamonc, we detemne the nume of sampled ponts y efeence to the hamonc sgnal s occuence and dsappeaance n tme doman. The method of how to confm the tme can e found n the authos anothe pape [5]. The followng Tale6 gves the smulaton esults: Tale6. Estmaton of the unsteady hamoncs O d e A (V) WPT The new Eo (%) espectvely Fom Tale6, we can see that the pecson of the new algothm s outstandng. The d43 WPT has a seous spectum leaage etween su-and Hz to 3Hz so the 5 th hamonc s not well detected. As to the, the esoluton capalty of unsteady hamonc sgnals s vey poo. In fact, fom the deducton of the algothm n secton 3, we can daw the concluson that the algothm s not senstve to the nume of sampled ponts. If the samplng fequency s hgh enough and the samplng tme s moe than only one fundamental peod, we can get a cetan hgh pecson coespondngly.

7 Chna Intenatonal Confeence on Electcty Dstuton 6. Conclusons Bogaphy Ths pape has developed a novel algothm ased on Othogonal Tgonometc Functons Famly. The algothm analyzes the powe hamonc sgnals y dnt of othogonalty of Tgonometc Functons. Unle Foue Tansfom and Wavelet Tansfom, t needs no pefect self-contaned othogonal ass to analyze sgnal, so t can effectvely ovecome the neluctalty spectum alas when sgnals convesed fom tme-doman to fequency-doman. Ths algothm can get hgh pecse ampltude s and phase s nfomaton of the steady-state powe hamonc. It also has the advantage of easy ealzng. Also we gve two examples to vefy the advantage of ths algothm. The dsadvantage of the algothm s that the ealzng of the algothm s ased on the exact dentfcaton of the fundamental fequency. The soluton of ths polem s also mentoned n the pape [5]. Comned wth wavelet tansfom, the algothm has an deal esoluton capalty wth the unsteady hamoncs. By contast wth the tadtonal algothms ths algothm has ts cetan supeoty of hamoncs detecton. Refeences [] S Mallat, A Theoy fo Multesoluton Sgnal Decomposton: the Wavelet Tansfom, IEEE Tansacton on Patten Analyss and Machne Intellgence, Vol, o. 7, pp , 989. [] R R Cofman, M V Wchehause, Entopy-ased Algothms fo Best Bass Selecton, IEEE Tansactons on Infomaton Theoy, Vol 38, o., pp , 99. [3] M V Wchehause, Lectues on Wavelet Pacet Algothms, IRIA, USA, 99. [4] I Daueches, Othonomal Bases of Compactly Suppoted Wavelets, Comm. Pue and Appl. Math, Vol 4, o. 7, pp , 989. [5] Ruoyu Wang, Jnghong Guo, Hamoncs Detecton n Powe System Usng Wavelet Tansfom Based on Lftng Scheme, Powe System Technology, Vol 3, Supplement, pp.5-, Jun. 8. Ruoyu Wang: on n Lanyungang, Jangsu, chna n Ap Gaduated fom State Gd Electc Powe Reseach Insttute of Chna n anng n 8 and have got Maste Degee of Electc Powe System and Automaton. The mao feld of study s powe system automaton and dgtal sgnal pocessng. He woed n the eseach and development cente of the State Gd Electc Powe Reseach Insttute of Chna when he was a gaduate student and patcpated n the poect of desgnng a Real-tme Data Acquston Chp wth Hgh Pecson. Tll now he has pulshed two papes espectvely n Jounals of Powe System Technology and Compute Smulaton. ow he wos n Huzhou Electc Powe Bueau as opeatng pesonnel n a 5V tansfome sustaton. Emal: wanguoyu@39.com Yuefeng Zhang: on n Wux, Jangsu, chna n 979. Gaduated fom Chongqng Unvesty and have got Maste Degee of Electc Powe System and Automaton n 8. Hs mao feld of study s powe system automaton. ow he wos n Huzhou Electc Powe Bueau as opeatng pesonnel n a 5V tansfome sustaton. Emal: zyf73@yahoo.com.cn Jnghong Guo: on n J-an, Anhu, chna n 967. Gaduated fom atonal Mole Communcatons Reseach Laoatoy of Southeast Unvesty and have got Docto Degee n. Then he fnshed hs postdoctoal eseach wo n the school of electonc and nfomaton engneeng of Unvesty of Sydney. The mao feld of study s powe system automaton and dgtal sgnal pocessng. ow he wos n the State Gd Electc Powe Reseach Insttute of Chna, anng. Emal:.h.guo@na-chna.com