Examples 4.2 and 4.3

Transcription

1 Exampe 4.2 Exampes 4.2 an 4.3 Consie a 6-Hz synchonous machine with the foowing pu paametes:.7 k k k.55 H 2.37 s.5 Obseve that this is a oun oto machine because =, ~, an >. Because the -wining an the -wining shae appoximatey the same fux path, we wi assume that =. Aso, because fux components fo both of these winings ae geneate fom ey cuents, we wi assume that =. Thus, we have =.526 an =.54. Aso, we nee the mutua tems fo the couping between the an the winings () an between the an the winings (Y). But we eca that, in pu, a mutua tems fo couping on the -axis ae eua, i.e., A m Y Since we know =.49, we aso have that =Y=.49. We esie to eveop the state-space euation i fo this system. Soution ( N) i v om (4.75 ) we have that

2 v v v whee Y Y i i i i i i i i i i i i k Without -cicuit: ωn.3 pu k k With -cicuit: ωn pu k k k Without -cicuit: k pu k k With -cicuit: Y Y fom which we may compute by atab 2

3 Without -cicuit: With -cicuit: pu pu Then we may compute pu With -cicuit: N Without -cicuit: N Exampe 4.3 Obseve that the coefficient matix is seen to contain in 2 of its 25 tems in the expession without the -wining, an 8 of its 36 tems in the expession with the -wining. This gives some iea of the compexity of the euations. epeat Exampe 4.2 fo the fux inkage moe given by (4.38), with an without the -cicuit. Soution We wok this pobem using the ata of Exampe 4. (an using =.526 an =.54). The esie state-space moe is given as: 3

4 Without -wining (see e. (4.38)): Tm j j 3 j 3 j 3 j 3 j j With -wining: (4.38) j 3 j 3 j 3 j 3 j 3 j j T m j (4.38 ) Now we obseve fom the state-space euations that many tems euie the eakage inuctances,, an. These tems ae not pesent in the cuent-state-space moe (give eason why). But when expessing the fux-inkage-state-space moe, they appea in the expessions fo λ A an λ A via the auxiiay euations when we epesent satuation, an when we o not epesent satuation, these eakage inuctances appea iecty in the state matix as in (4.38) an (4.38 ). We aso obseve the pesence of,, which we nee to compute. 4

5 These eakage inuctances ae not povie in the oigina ata (see Exampe 4.2), an so we must compute them. But they ae easy to compute via the pu es. (4.7) an (4.8) in you book (p. 3 in the peunitization notes). I have ewitten these eations hee: - = - = - = A. om these eations, we see that: whee = -( - ); = -( - ); ( ) A mu u u u Thus, we can compute an via: = - ; = - ; A simia teatment fo the -axis uantities esuts in: = - ; = - ; eca the ata given fo Exampe k k k.55 H 2.37 s.5 Cacuations given in the book (p. 3) to aess without -wining ae:.5 pu pu pu pu In aition, we nee the tems, which can be obtaine fom 5

6 A ; A (See (4.9) an (4.22), an notes on uxinkageeuations, p. 4 an p. 6). These cacuations ae povie in the book fo the case without the -wining, as foows (p. 3): pu pu pu pu The compete cacuation is one, without the -wining, is povie in the book as foows: 6

7 .5 pu pu pu pu pu pu pu pu j j an we get fo the state-space euation fo the fist six vaiabes, with = 3 j j j T m 7

8 Exampe 4.4 What is beow is exacty what is in the book. We wi moify it to accommoate the -wining. Exten Exampes 4.2 an 4.3 to incue the effect of the tansmission ine an toue euations. The ine constants ae e.4 pu, 2H a. The infinite bus votage constant K an the amping j toue coefficient ae eft unspecifie. Soution, e Then.96 ˆ 2. ˆ 2.4 e e e ˆ+ Nˆ ˆ By igita compute we fin Then ˆ ˆ ˆ Nˆ

9 an we compute Ksin.7 Ksin K sin 6.67 Kcos.7 Kcos.67 K cos ˆ.8 K sin 5.87 Theefoe the state-space cuent moe is given by i i i i i i i i i i.32i.29i.29i.3i.28i K sin K sin K sin K cos.67 K cos.559tm The fux inkage moe is of the fom T C, whee T, C, an ae given by (4.59) (4.62). Substituting, 9

10 T T The matix C is mosty the same as that given in Exampe 4.3 except that the tems ae moifie C

11 K sin K cos.559 Tm