Symmetrical Components 1

Transcription

1 Symmetril Components. Introdution These notes should e red together with Setion. of your text. When performing stedy-stte nlysis of high voltge trnsmission systems, we mke use of the per-phse equivlent iruit. Also, when performing symmetril fult (three-phse fult) nlysis of high-voltge trnsmission systems, we mke use of the per-phse equivlent iruit. But for unsymmetril fults (single line to ground, two line to ground, nd line to line) nlysis, the three phses no longer see the sme impedne, whih violtes the si requirement of per-phse nlysis (phses must e lned).

2 There is very elegnt pproh ville for nlyzing unsymmetri three-phse iruits. The pproh ws developed y mn nmed Chrles Fortesue nd reported in fmous pper in 98. It is now lled the method of symmetril omponents. We will spend little time studying this method in order to understnd how to use it in unsymmetril fult nlysis. Chrles Le Geyt FORTESCUE Professor ws orn out 878 in Keewntin, Northwest Territories, Cnd. Hudson By Ftory where the Hyes River enters Hudson By He immigrted in 9 to USA. He ppered in the ensus on April 93 in Pittsurgh, USA. Chrles died on 4 Deemer 936 in Pittsurgh, USA. The Chrles LeGeyt Fortesue Sholrship ws estlished in 939 t MIT s memoril to Chrles LeGeyt in reognition of his vlule ontriutions to the field of eletril engineering Chrles LeGeyt Fortesue, orn t York Ftory, Mnito, 876, son of hief ftor of Hudson By Compny-ws the first eletril engineering grdute of Queen's University. After grdution Fortesue joined Westinghouse Eletri nd Mnufturing Compny t Est Pittsurgh nd ttined universl fme for his ontriutions to the engineering priniples nd nlysis of power trnsmission nd distriution systems. He is espeilly noted for development of polyphse systems nlysis y the symmetril omponents method. He mde his wy, evenutlly, to MIT where he eme very well known nd respeted professor. Its fsinting tht Ceil Lewis Fortesue orn 88 lso eme Professor of Eletril Engineering in London University, in the sme period. One wonders if they herd out eh other? Chrles Le Geyt FORTESCUE Professor nd Louise Cmeron WALTER were mrried out 95. Louise Cmeron WALTER3 ws orn out 885 in Pennsylvnni, USA. She ws Sulptor Chrles Le Geyt FORTESCUE Professor nd Louise Cmeron WALTER hd the following hildren: Jne Fithful FORTESCUE.

3 . Symmetril Components: Motivtion Def: A symmetril set of phsors hve equl mgnitude & re º out of phse. Gol: Deompose set of three unsymmetril phsors into One unsymmetri ut equl set of 3 Two symmetril sets of 3 3

4 Then we n nlyze eh set individully nd use superposition to otin the omposite result. In wht follows, we demonstrte tht: Step : An unsymmetril set, not summing to, n e deomposed into two unsymmetril sets: o n equl set nd n o unsymmetril set tht does sum to ; Step : An unsymmetril set tht sums to n e deomposed into two symmetril sets Step : Consider set of phsors tht do not dd to zero (euse of different mgnitudes or euse of ngulr seprtion different thn º or euse of oth). Assume tht they hve phse sequene --. Add them up, s in Fig., i.e., R () 4

5 -- -- R Fig. : Addition of Unsymmetril Phsors So we see from () tht R () Define: R 3 (3) Then: 3 (4) (5) Define: Then: A B C (6) 5

6 A B C (7) Conlusion: We otin n unsymmetril set of voltges tht sum to y sutrting from eh originl phsor, where is /3 of the resultnt phsor, illustrted in Fig.. C A -- R B - =- R /3 Fig. : Sutrting from unsymmetril phsors Step : How to deompose A, B, nd C into two symmetril sets? Cn we deompose A, B, C into -- symmetril sets? As test, try to dd ny -- symmetril sets nd see wht you get. See Fig. 3. 6

7 -- -- C -- B A Fig. 3: Adding symmetril -- sets Note tht in dding the phsor sets, we dd the two -phse phsors, the two -phse phsors, nd the two -phse phsors. One n oserve from Fig. 3 tht the resultnt phsor set, denoted y the solid lines, re in ft symmetril! 7

8 It is possile to prove mthemtilly tht the sum of ny -- symmetril sets is lwys nother symmetril set. Let s try different thing. Let s try to dd two symmetril sets, ut let s hve one e -- (lled positive sequene) nd nother e -- (lled negtive sequene). As efore, in dding the phsor sets, we dd the two -phse phsors, the two - phse phsors, nd the two -phse phsors. The result of our efforts in shown in Fig. 4. 8

9 C B A Fig. 4: Adding symmetril -- set to symmetril -- set The resultnt phsor set is unsymmetril! We n gurntee tht the three phsors in this unsymmetril phsor set sums to zero, sine we otined it y dding two phsor sets tht sum to zero, i.e., 9

10 + + = + + = (8) A + B + C = Now onsider Fig. 4 gin. Assume tht someone hnds you the unsymmetril set of phsors A, B, nd C. Cn you deompose them into the two symmetril sets? Cn you e ssured tht two suh symmetril sets exist? The nswer is yes, you n e ssured tht two suh symmetril sets exist. Fortesue s pper ontins the proof. I simply rgue tht the three phsors given in Fig. 4, A, B, nd C, re quite generl (there is nothing speil out them), with the single exeption tht they sum to zero.

11 Clim: We n represent ANY unsymmetril set of 3 phsors tht sum to s the sum of onstituent symmetril sets: A positive (--) sequene set nd A negtive (--) sequene set. Given this lim, then the following theorem holds. Theorem: We n represent ANY unsymmetril set of 3 phsors s the sum of 3 onstituent sets, eh hving 3 phsors: A positive (--) sequene set nd A negtive (--) sequene set nd An equl set These three sets we will ll, respetively, Positive,, Negtive,, zero,, sequene omponents.

12 The implition of this theorem is tht ny unsymmetril set of 3 phsors,, n e written in terms of the ove sequene omponents in the following wy: (9) We n write the equtions of (9) in more ompt fshion, ut first, we must desrie mthemtil opertor tht is essentil. 3. The α-opertor To egin on fmilir ground, we re ll onversnt with the opertor j whih is used in omplex numers. Rememer tht j is tully vetor with mgnitude nd n ngle: j 9 ()

13 In the sme wy, we re going to define the α opertor s: j.866 It is esy to show the following reltions: () 3 (3) 4 (4) We lso hve tht: 6 (5) s illustrted in Fig. 5. () +α α Fig. 5: Illustrtion of +α Note tht 4 6 (6) 3

14 Similrly, we my show tht: 6 (7) 3 3 (8) 33 (9) 35 () 3 5 () And there re mny more reltions like this tht re sometimes helpful when deling with symmetril omponents. (See the text lled Anlysis of fulted power systems y Pul Anderson, pg. 7.) 4. Symmetril omponents: the mth We repet equtions (9) elow for onveniene: (9) 4

15 We n relte the three different quntities hving the sme supersript. Zero sequene quntities: These quntities re ll equl, i.e., () Positive sequene quntities: The reltion etween these quntities n e oserved immeditely from the phsor digrm nd n e expressed using the α-opertor. -- Fig. 6: Positive sequene omponents (3) 5

16 Negtive sequene quntities: The reltion etween these quntities n e oserved immeditely from the phsor digrm nd n e expressed using the α-opertor. -- Fig. 8: Negtive sequene omponents (4) Now let s use equtions (), (3), nd (4) to express the originl phsors,, in terms of only the -phse omponents,,, i.e., we will eliminte the -phse omponents,, 6

17 7 nd the -phse omponents,, This results in (9) So we hve written the quntities (phse quntities) in terms of the +- quntities (sequene quntities) of the -phse. We n write this in mtrix form s: (5) Defining

18 8 A (6) we see tht eq. (5) n e written s: A (7) We my lso otin the +- (sequene) quntities from the (phse) quntities: A (8) where 3 A (9)

19 Equtions -9 hold for Line-to-line voltges Line-to-neutrl voltges Line urrents Phse urrents 9