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(143) 1*=_;X=#HFUef5"9C13o3h;*."801*90E

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(147) 13C8/-5KN5"Q:6.757EXF_;7D 8/-5"=n :?=-."905;*E05S8/-5SE8J;e:?2?:?8 D F_;9>:?=nk • :?=-."905;*E05S8/-5S4 ag •. . . X1*9X5"df;O5".7=-1*NFU:?=e>L:?=R1*;=c8:61*B= /R;18a348:6E /-5Ad98 :6BC1*91G:?8df: 157E0NcE:?e3[90:?1*2?:?F8:6578E/-.5U;=cEDefE805U5"F ;*.0/df:61*5"Q):?=#578XN^g134h34Q:685"/-BA5Ako41*=-5U1:63

(148) E8E0/-5"58 e#DP8/-58/-5"9F_;2-2?:?FO:?8

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(152)

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(170) :6E8 /-5S9J;8057N df1BC5"9a13I8/-5XF_;*.0/:?=-5g S. . %/-5AEB:?=>_5#HR;8:61*=nkB/:6.0/c:6EeR;*E057N 1*=ca5"B801*= EE057.71*=-NL2Z;7B :6E%>:?Q)5"=ce#Df Jω. B/-5"905. dω = Pm − PG dt. `g . :6E8/-5SFG57.0/R;=:6.;2df1BC5"9:?=dH8k. Pm = ωTm :6E8/-5U;*."8:?Q)5A5"2657."89:6.;2df1BC5"91*H8dH8k PG = ωTe 6 : E 8 / S 5 G F 57.0/R;=:6.;2801*9 #H-5k Tm 6 : E 8 / S 5 " 5 6 2 7 5 ."89:6.;2801*9 #H-5g Te. 2?2nQ;9:Z;e2657E:?=p8/:6E8/-57E:6E;905S:?= dng HngH=2657E0E 1*8/-5"9B:6E05A:6E E8J;8057N^g. .

(171) . . #8J;98:?=>UB:?8/O8/-5 EB:?=>A5 #HR;8:61*= `g "kFG1*8:61*=_13f8/-5%901*801*9k#.;=Gef5N57E0."9:?ef57N e#DL8/-5S8 BC1G5#HR;8:61*=-Ek-."2Z;*E0E:6.;2nFG1N5"2h `g δ˙ = ω `g 1 ω˙ = (P − P − Dω) M. m. . G. B/-5"905 9J;*N C:6E8/-5S901*801*9a;=>265k δ M> r:6E8/-5S:?=-5"98:Z;O13I8/-5S901*8J;8:?=>GdR;980Ek M. . .

(172) .

(173) D. . . . , E r:6E8/-5AN-;FOd:?=>O3h;*."801*9g . <:?>H905Ug?SE/-1B Ea;P>)5"=-5"9J;801*9a.71*==-57."8057Np801G;K=-5"8 BC1*9M. eH-Eg . <:?>H905Ug?

(174) . >)5"=-5"9J;801*9.71*==-57."8057Np801_;PeH-Eg. %/-5S5"2657."89:6.;2df1BC5"9a1*H8dH813I8/-5S>)5"=-5"9J;801*9:6E>:?Q)5"=pe#Df 0. PG =. Eq U 0. sin(δ − θ). xd U 0 QG = 0 Eq cos (δ − θ) − U xd. `g `g . B/-5"905 ;=-N 9J;*N X;905K8/-5KQ)1*2?8J;>)5G;=-N ;=>265G;8S8/-5KeH-EXB/-5"905O8/-5K>)5"=-5"9J;801*9A:6E U .71*==-57."80θ57N^k :6E8/-5X905;*."8:?Q)5U5"2657."89:6.Sdf1BC5"9a1*H8dH8k QG 6 : E 8/-5AN;!b:6E89J;=-E:65"=#8905;*."8J;=-.75k xd 6 : E 8/-5S:?=#805"9=R;2Q)1*2?8J;>)5g Eq 2?2^Q;9:Z;e2657Ea;905SN5 -=-57N ;=-NLH=-N5"90E8011Nc;*E:?= ? g. . 0. 0. . ; <. A; <. #HR;8:61*=-E `g %;=-N `g %;905X8/-5a8 BC1G5#HR;8:61*=-EH-E057N_3`1*9%8/-5X."2Z;*E0E:6.;2^FG1N5"2 13<8/-5SED=-.0/901*=-1*H-EF_;*.0/:?=-5 ? g

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(176) %/-5XND=R;FO:6.7E%13<8/-5a1*=-5+;!b:6E%FG1N5"2^.;=. . C; <.

(177) . . . . . . . . . ef5SN57E0."9:?ef57N e#DL8/-5a3`1*2?261B:?=>l5#HR;8:61*=-E δ˙ = ω 1 (Pm − Pe − Dω) ω˙ = M 0 1 xd 0 xd − x d 0 ˙ U cos(δ − θ) Eq = 0 Ef − 0 Eq + 0 Tdo xd xd. . `g ( `g `g & . B/-5"905 [E r:6E8/-5AN;!b:6E89J;=-E:65"=#8a1*df5"= .":?90."H:?88:?FG5A.71*=-E8J;=#8k Tdo :6E8/-5AN;!b:6E ED=-.0/901*=-1*H-E 905;*."8J;=-.75k xd :6E8/-5S>)5"=-5"9J;801*9 R5"26N Q)1*2?8J;>)5g E 0. ?>. . f. . '+FG67'1F !-5 H. n1);*NE.71*==-57."8057Nl:?=l8/-5df1BC5"9EDE805"F13801N-;7Dl;905a13nQ;9:61*H-EC8 Ddf57E;=-N_/R;7Q)5 N: ]^5"905"=#8P.0/R;9J;*."805"9:6E8:6.7Eg_%/-5K261);*N .0/R;9J;*."805"9:6E8:6.7EU/R;7Q)5_;LE:?>=: R .;=#8U:?FOdR;*."8. ; <. . 1*=p8/-5AND=R;FO:6.7E :?=p8/-5AEDE805"Fpg =-5S261);*NpFG1N5"2nB/:6.0/c:6E .71*FOFG1*=2?D H-E057NL:6E 8/-5o@IhFG1N5"2 ? gO 8S.71*=-E:6E80EA13C8/90575OdR;980Ekn/R;7Q:?=>c.71*=-E8J;=#8A:?FOdf57N-;=-.75k .71*=-E8J;=#8C."H9905"=#8k;=-NG.71*=-E8J;=#8rdf1BC5"9C.0/R;9J;*."805"9:6E8:6.7Eg

(178) %/-5o@IhFG1N5"2R.;=Gef5 N57E0."9:?ef57N e#DL8/-5a3`1*2?261B:?=>l5+bd9057E0E:61*=-E UL 2 UL PLk = PL0 kpz + kpi + kpp UL0 UL0 2 UL UL + kqp QLk = QL0 kqz + kqi UL0 UL0. `g? `g? . B/-5"905 :6E8/-5A;*."8:?Q)5Sdf1BC5"9261);*Nc;8eH-E k k PLk :6E8/-5S905;*."8:?Q)5Sdf1BC5"9261);*Nc;8eH-E k k QLk 6 : E8/-5S=-1*FO:?=R;2I;*."8:?Q)5Sdf1BC5"9 261);*N^k PL0 6 : E8/-5X=-1*FO:?=R;2905;*."8:?Q)5Adf1BC5"9 261);*N^k QL0 :6E8/-5A;*."8HR;2Q)1*2?8J;>)5k UL :6E8/-5S=-1*FO:?=R;2nQ)1*2?8J;>)5k UL0 ;=-N ;905C8/-5r3[9J;*."8:61*=-E413f.71*=-E8J;=#84:?FOdf57N-;=-.7513R8/-5;*."8:?Q)5 ;=-NP905;*."8:?Q)5 kpz df1BC5"9k-90k57qzEdf57."8:?Q)5"2?D)k ;=-N k ;905O8/-5K3[9J;*."8:61*=-EP13.71*=-E8J;=#8P."H9905"=#8K13%8/-5l;*."8:?Q)5_;=-N 905;*."8:?Q)5 kpi df1BC5"9k-9057Eqidf57."8:?Q)5"2?D)k.

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(180) 8/-5A261);*NEH-EHR;2?2?DpN5"df5"=-N 1*= 5;*.0/j1*8/-5"9k-3`1*9 ?: =-E8J;=-.75k#8/-5905;*."8:?Q)5adR;98%13n8/-5 261);*Nl.71*H26N_ef5 FG1N5"2657NL;*E Q = P tan φ k B/-5"905 φ :6ES8/-5K261);*N ;=>265gG FG1*905O>)5"=-5"9J;2

(181) FG1N5"2

(182) BC1*H26N ;26E0L0 1L:?=-."2?H-L0 N5GFG1*905 13X8/-5pND=R;FO:6.7El13X8/-5 261);*N^kC3`1*9O:?=-E8J;=-.75 ;26E01 8/-5pN5"df5"=-N5"=-."D 13X8/-5L3[905+ #H-5"=-."D ? g

(183) %/-57E05S8 Ddf57E 13IFG1N5"26E;905S=-1*8905 #H:?9057N :?=p8/:6E8/-57E:6Eg E:?FOd265OFG1N5"24H-E057Nj801 N5"9:?Q)5O8/-5 %(')*,+,'.-1 ' B *R+6* +6* *,')G-/)0 ' FG1N5"2hk N57E0."9:?ef57Nc:?= 57."8:61*= g -kR:6Ea.71*=-E8J;=#8X:?FOdf57N-;=-.75P261);*NE ? ga%/-5K;*NFO:?88J;=-.75 13I8/-5S.71*=-E8J;=#8:?FOdf57N-;=-.75A261);*Np;8 eH-E k .;=p8/-5"=pef5S.;26."H2Z;8057Npe#Df Y¯. ; <. ; <. `g? . kk. %/:6E:6E 5#HR;2n801OE05"88:?=>. . . . PLk − jQLk Y¯kk = . Uk2 kpz = 1. ;=-N. kqz = 1. :?= 8/-5 o@IhFG1N5"2hg. 15 + 67* 67'1F !-5 H ' + 6 -D%( 67+*" 5 $ 1$&%(/6 . . %/-5

(184) ef5"/R;7Q:61*9I13o;FUH2?8: hF_;*.0/:?=-5%89J;=-EFO:6E0E:61*=Pdf1BC5"9IEDE805"FT.;=Uef5rN57E0."9:?ef57N e#D_;SE05"8C13n5#HR;8:61*=-Eg4%/-5"905:6E

(185) df1BC5"9CeR;2Z;=-.75X;8C5;*.0/leH-Er:?=_8/-5EDE805"Fpk;=-N 8/-5Xdf1BC5"9eR;2Z;=-.75A5 #HR;8:61*=-Ea;8 =-1N5 k ;905X>:?Q)5"=pe#D ? . #; <. 0=. 0=. N X l=1 N X l=1. Bk,l Uk Ul sin(θk − θl ) − PLk + PGk. k = 1, . . . , N + n. `g? . Gk,l Uk Ul cos(θk − θl ) − QLk + QGk. k = 1, . . . , N + n. `g?" . B/-5"905 :6E8/-5S:?F_;>:?=R;9DLdR;9813I8/-5A;*NFO:?88J;=-.75AF_;89: b Y¯ ;8df1E:?8:61*= k, l k Bk,l :6E8/-5S905;2ndR;9813I8/-5A;*NFO:?88J;=-.75AF_;89: b Y¯ ;8df1E:?8:61*= k, l k Gk,l 6 : E 8/-5X=#HFUef5"9a13I261);*N eH-E057Ek N :6E8/-5S=#HFUef5"9a13I>)5"=-5"9J;801*90Ek n ;=-N Q ;905S8/-5S>)5"=-5"9J;8057Nc;*."8:?Q)5P;=-NL905;*."8:?Q)5Adf1BC5"9X;8eH-E k g P. C1*FUe:?=:?=>L8/-5Adf1BC5"9aeR;2Z;=-.75P5 #HR;8:61*=-EX;=-Np8/-5PND=R;FO:6.U5 #HR;8:61*=-E 3`1*98/-5 Gk. Gk. . .

(186) . . . . . . . . . . >)5"=-5"9J;801*90E ;UE05"8C13 -90E8C1*90N5"9%=-1*=2?:?=-5;91*90N:?=R;9D_N: ]^5"905"=#8:Z;2n5#HR;8:61*=-E;=-NL; E05"8 13I=-1*=2?:?=-5;9X;2?>)5"e9J;:6.U5#HR;8:61*=-E;905A1*e8J;:?=-57N 8/R;8N57E0."9:?ef5S8/-5Xef5"/R;7Q:61*9 13I8/-5AEDE805"F ? . ; <. `g? `g? . x˙ = f (x, y) 0 = g(x, y). B/-5"905 :6E8/-5AE8J;805SQ;9:Z;e2657EQ)57."801*9134N:?FG5"=-E:61*= n · n k x :6E8/-5S=#HFUef5"9a134E8J;8057E 13I8/-5X>)5"=-5"9J;801*9FG1N5"x2hk nx :6E8/-5A;2?>)5"e9J;:6.AQ;9:Z;e265SQ)57."801*9134N:?FG5"=-E:61*= 2(n + N ) g y. <:?>H905Gg lE/-1B EA;_FUH2?8: hF_;*.0/:?=-5G89J;=-EFO:6E0E:61*=. 5"9J;801*9eH-E057E;=-N. N. 261);*NLeH-E057Eg. EDE805"F .71*=-E:6E8:?=>p13. n. >)5"=. xd' 1. xd' 2. ' xdn. <:?>H905Ug

(187) . FUH2?8: hF_;*.0/:?=-5Pdf1BC5"9EDE805"Fpg. aE0EHFG58/-5"905S;905 >)5"=-5"9J;801*9eH-E057E%;=-N 261);*NleH-E057Eg

(188) 5"88:?=>KHdl8/-5X5#HR;! 8:61*=-ES3`1*9S8/-5GEDE805"nF 9057EH2?80EA:?= n · n -90EN8A1*90N5"9UN: ]^5"905"=#8:Z;2C5#HR;8:61*=-EP;*ES:?= 5 #HR;8:61*= `g? ;=-N ;2?>)5"e9Jx;:6.U5#HR;8:61*=-Ea;*E:?=p5#HR;8:61*= `g? "g 2(n + N ). . .

(189) . . . . &. . 5 %(05 +- 5 '1F !$4!5 %(+ %('5 . 57."8:61*=lg XE/-1BC57NG8/R;8

(190) 8/-5%ef5"/R;7Q:61*9r13f8/-5df1BC5"9rEDE805"F .71*H26NOef5N57E0."9:?ef57N e#DO5#HR;8:61*=-E `g? 4;=-N `g? "g<%/-5 %(')*,+,'.-1' B *+6* +6* *,')G-/)0 ' FG1N5"26E

(191) 8/-5 df1BC5"9lEDE805"F B:?8/ 1*=2?D ; E05"8_13 -90E8_1*90N5"9lN: ]^5"905"=#8:Z;2a5#HR;8:61*=-E ? g 8 .;= ef5 H-E057N H=-N5"9p8/-5 ;*E0EHFOd8:61*= 8/R;8 8/-5j>)5"=-5"9J;801*90Ec.;= ef5jN57E0."9:?ef57N e#D 8/-5j."2Z;*E0E:6.;2XFG1N5"2S1*9 1*=-5+;!b:6ELFG1N5"2hk8/-5 FG57.0/R;=:6.;2Adf1BC5"9 :?=dH8 :6E .71*=-E8J;=#8kX8/-5 N-;FOd:?=> D :6E 5"901 ;=-N 8/R;8 8/-5 261);*NEp.;= ef5jFG1N5"2657N ;*E .71*=-E8J;=#8 :?FOdf57N-;=-.757Egr%/-5X:?= 57."8057Np."H9905"=#8e#DL>)5"=-5"9J;801*9 k :6E. . ; <. ¯k ¯0 − U E qk. I¯gk =. k = 1, . . . , n.. 0. jxdk. dd2?D:?=>GWX:?90.0//-1] E - 90E8 2Z;7BAk8/-5X3`1*2?261B:?=>l5#HR;8:61*=-E;905A1*e8J;:?=-57N^ 0=. E; < . n X l=1. ¯l − I¯gk Y¯k,l U. k = 1, . . . , n.. `g?!( `g? . =-."2?H-N:?=>_m :?=cEDE805"Fpk8/-5U;*."8:?Q)5Sdf1BC5"9a.;=pef5S.71*=#8901*2?2657Nc8/901*H>/c8/-5 m ? g

(192) %/-5X:?= 57."8057Np."H9905"=#8a13I8/-5Am 2?:?=Ml:6E>:?Q)5"=pe#Df. B/-5"905 ;=-N I¯rec 8:?Q)5"2?D)k. . `g?& . −Prec + jQrec I¯rec = ∗ ¯rec U −Pinv + jQinv I¯inv = ¯∗ U I¯inv. . ;905 8/-5am. `g . inv. ."H9905"=#8;8C8/-5 9057."8: R5"9;=-NG:?=#Q)5"9805"9E:6N5k)9057Edf57.+. 2?:?= Mfk. :6E8/-5U;*."8:?Q)5Sdf1BC5"9.71*FOdf1*=-5"=#8a13I8/-5Am. Prec = −Pinv = PDC :6E

(193) 8/-5 905;*."8:?Q)5 df1BC5"9%.71*FOdf1*=-5"=#8%13^8/-5m 2?:?=Mfk Qrec = Qinv = QDC > 0 ; = N ; 0 9. 5 8 / 5 7 . * 1 O F d 6 2 + 5 l b 7 . * 1 = , H # > ; 0 8 5 1 n 3 8 / 5 0 9 7 5 " . 8 : R " 5 9 ; = G N ? : # = ) Q " 5 9 0 8 " 5 C 9 eH-E ∗ ¯rec ¯∗ U U inv. NN:?=>K8/-5a."H9905"=#8 138/-5Xm 2?:?=M_801P5 #HR;8:61*=`g?

(194) 8/-5 3`1*2?261B:?=>G5 #HR;! 8:61*=p:6E1*e8J;:?=-57N^ ` g Q)1*2?8J;>)57Ek9057Edf57."8:?Q)5"2?D)g. . . 0=. n X l=1. ¯l − I¯gk − Y¯k,l U. m X j=1. I¯hvdc,k,j. . k = 1, . . . , n.

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