e-issn : 0975-0 Hee-Jn Lee / Internatonal Journal of Engneerng an echnology (IJE) Dynamc oelng of a Synchronous Generator Usng -S Fuzzy Approach Hee-Jn Lee Department of Electronc Engneerng Kumoh Natonal Insttute of echnology, Gum-s, Gyeongbuk, Korea jnlee@kumoh.ac.kr Abstract he ynamc behavor of power systems s affecte by the nteractons between lnear an nonlnear components. o analyze those complcate power systems, the lnear approaches have been wely use so far. Especally, a synchronous generator has been esgne by usng lnear moels an tratonal technues. However, ue to ts we operatng range, comple ynamcs, transent performances, an ts nonlneartes, t cannot be accurately moele as lnear methos base on smallsgnal analyss. hs paper escrbes an applcaton of the akak-sugeno (-S) fuzzy metho to moel the synchronous generator n a sngle-machne nfnte bus (SIB) system. he -S fuzzy moel can prove a hghly nonlnear functonal relaton wth a comparatvely small number of fuzzy rules. he smulaton results show that the propose the -S fuzzy moelng captures all ynamc characterstcs for the synchronous generator, whch are eactly same as those by the conventonal nonlnear moelng methos. Keywors ynamc moelng; power system; synchronous generator; -S fuzzy. I. INRODUCION A synchronous generator n a power system s a nonlnear fast-actng multple-nput multple-output (IO) evce. Due to ts we operatng range, comple ynamcs, nonlnearty, an the changng system confguraton, the entre system cannot be accurately represente by a fe moel, whch s then use for the esgn of conventonal lnear system/controllers. Otherwse, by usng the IF-HEN rule, the akag-sugeno (-S) fuzzy moelng makes t possble to analyze a nonlnear system by appromatng the system as a lnear nput-output system n certan range []-[6]. hs range can be epane over the range efne by other lnearzaton technues wthout losng generalty when the system operates n we range of operatng ponts [7]. Also, the stablty of the system can be analyze by usng a lnear matr neualty (LI) metho base on the Lyapunov conton, whch can be formulate by the -S fuzzy moelng. It has two avantages; one s the convenence of analyss through lnearzaton, an the other s the accuracy of analyzng nonlnear systems. oreover, ths LI technue on the -S fuzzy moelng can prove the powerful capablty to esgn the robust controller of any nonlnear ynamc systems. In ths paper, the -S fuzzy moelng metho s frstly apple to moel an nverte penulum system. hen, t s use to moel the synchronous generator, whch can be escrbe by the fourth-orer nonlnear fferental euatons, n a sngle-machne nfnte bus (SIB) power system. By applyng the large (three-phase short crcut) an small (± 5% step changes n the reference voltage of ecter) sturbances to the SIB system, ts effectveness s evaluate to show the same ynamc behavors as gven n the eact nonlnear moel of the synchronous generator. II. -S FUZZY ODELING A. akag-sugeno Fuzzy oel here s neee new moel whch s fferent from lnear moel to analyze more accurate when large accents occur. In ths paper, the -S fuzzy metho s use to analyze nonlnear system. he -S fuzzy system can be represente accorng to Fg. structure. It s har to analyze nonlnear system whch has a lnear an a nonlnear part. he nonlnear system s converte to -S fuzzy moel to analyze easly. Fg.. Approach base on the -S fuzzy moelng p-issn : 39-863 Vol 8 No Apr-ay 06 97
e-issn : 0975-0 Hee-Jn Lee / Internatonal Journal of Engneerng an echnology (IJE) he th rule of the affne -S fuzzy moels are followng forms, where contnuous fuzzy system (CFS), respectvely. oel Rule IF z (t) s an an z p (t) s p, HEN A C,,,, r Here, j s the fuzzy set an r s the number of moel rules; (t) R n s the state vector, A R nn, C R nm ; z (t),, z p (t) are known premse varable that may be functons of the state varables, eternal sturbances, an/or tme. Each lnear euaton A C s represente n () [7]. Where n ( ) ( ), j j j r {A C } ( ){A C } () r r h h( ) ( ) r ( ) h ( ) 0, r h ( ) () B. Desgn Eample: Inverte Penulum In ths secton, an llustratve eample s prove to emonstrate the valty of the suggeste stablty analyss an synthess metho [8]. ass ( ) Control voltage DC motor 0 : Ø Ø Ø l Gear Pvot pont Fg.. Inverte penulum controlle by a DC motor L R I V v K K 0 b b m b Fg. 3. oel of an armature-controlle DC motor he plant to be controlle s an electro-mechancal system as shown n Fg.. otor s nerta s neglgble when compare wth that of penulum. he euvalent crcut of ths system s llustrate n Fg. 3. he torue s supple by motor, that s, m KmI. (3) the torue of penulum s elvere by gear (0:), that s, 0 0K I. () By usng Krchhoff s loop rule to get euaton p m m p-issn : 39-863 Vol 8 No Apr-ay 06 975
e-issn : 0975-0 Hee-Jn Lee / Internatonal Journal of Engneerng an echnology (IJE) V LI RI K 0 b. (5) he knetc torue apple penulum s sum of torue of knetc energy an torue of gravty. p l m lmg sn (6) where, (-) sgn means that the knetc torue s n opposton to torue of gravty. he ynamc euaton s gven by 3 ABu where the varable an the parameters are as follows [9]. [ ] [ I] 0 0 sn A K 0 K 0 K3 K g 0K K, m 0K K, b R K 3, K, K5 l lm L L L (, the angle measure wth respect to the vertcal as); (, the tme ervatve of ); u (the control voltage) = 0; m (the mass of the penulum) = kg; l (the stance of the center of mass m from the pvot pont) = m; g (gravtatonal constant) = 9.8 m/s ; R (euvalent resstance) = Ω; L (euvalent nuctance) = 00 mh; K b K m (constant value of generator) = 0. Vs/ra; (constant value of mass) = 0. Nm/A; 0 B 0 K 5 In ths paper, a controller oesn t be consere. So, the control nput u = 0. It s straghtforwar to compute sn the -S fuzzy system. In ths eample, the nonlnear term s only. he system matr A s separate nto matr. IF-HEN rules are follows, Inverte Penulum oel Rule IF () t s, HEN A IF () t s, HEN A 0 0 A K 0 K 0 K3 K 0 0 0 K3 K, A K 0 K Nonlnear system euaton (7) s the same as the -S fuzzy euaton (8) when t s satsfe follows by ()- (). Euaton (0) can be rewrtten by (8) an (9). hk hk K (7) (8) h( )A ha h A, h h (9) sn sn h ( h) p-issn : 39-863 Vol 8 No Apr-ay 06 976
e-issn : 0975-0 Hee-Jn Lee / Internatonal Journal of Engneerng an echnology (IJE) sn h( ) sn h Euaton (0) s always satsfe regarless of values of an. Here,, an 0 are assume. (0) 0 8 6 0 = 8.0 0 = 5.0 0 = 0.0 0 = -5.0 0 = -8.0 [ra] 0 - - -6 0 5 0 5 me [sec] 0 8 6 (a) 0 = 8.0 0 = 5.0 0 = 0.0 0 = -5.0 0 = -8.0 [ra] 0 - - -6 0 5 0 5 me [sec] (b) Fg.. Penulum poston (u=0, =.5, 3 =0) (a) Nonlnear moel (b) -S fuzzy moel Fg. shows the nverte penulum pvot angle as a tme an varous ntal angle spee ( 0 ). Fg. (a) s the nonlnear moel n (7) where the system has fferent stable ponts accorng to varous ntal angle spees. Fg. (b) s the -S fuzzy moel n (8), where ts trace s the same as nonlnear moel. hs eample proves the valty of the -S fuzzy moelng. III. SYNCHRONOUS ACHINE ODELING A. Synchronous oelng he SIB system s shown n Fg. 5. he system s one of the funamental systems an can be epane mult-machne system easly [0]. In ths paper, fourth-orer synchronous generator moel s escrbe by usng the -S fuzzy moelng metho. he state varables are -as voltage, -as voltage, rotor angle an rotor spee [], []. he synchronous generator can be epresse by usng - as n the phasor agram of Fg. 6 n steay state [3], []. p-issn : 39-863 Vol 8 No Apr-ay 06 977
e-issn : 0975-0 Hee-Jn Lee / Internatonal Journal of Engneerng an echnology (IJE) hese mathematcal moels are gven n ()-(3). Armature an transmsson resstance are neglgble compare wth reactance. Usng Park s transform, - as voltage euatons are follows. he rotor angle an rotor spee are escrbe by. E E ( ) I E t o f s t E o E ( ) I () t H E I E I ( ) II FW () t s Fg. 5. Sngle-machne nfnte bus system E as j I V t V b I ( / In () an (), (, / I, I as Fg. 6. Steay state phasor agram of the SIB system ) are the - as components of current n the armature wnngs, ) s the - as components of (steay, transent) reactance n the armature wnngs, the fel voltage, o an o are tme constants of - as transent. In (), s the rotor angle wth reference to the nfnte bus, s s the synchronous spee n steay state, m s turbne output shaft torue, FW s frctonal an wnage torue, an H s the polar moment of nerta. By usng voltage euaton n (), an swng euaton n (), the fourth-orer set of fferental euaton can be foun for the ynamc moel of the generator. he current euatons for the transmsson system (n Fg. 5) are gven n (3), whch are erve from the methos of crcut an Park s transformaton theory E Vb cos E Vbsn I, I (3) Where I an I are the - as components of transmsson current, E components of transent voltage, V b s the nfnte bus voltage, reactance. B. -S Fuzzy to the SIB System For the -S fuzzy moelng, euaton (), (5) are gven n below. AC, E E e an e E f s E are the - as e s the component of transmsson lne p-issn : 39-863 Vol 8 No Apr-ay 06 978
e-issn : 0975-0 Hee-Jn Lee / Internatonal Journal of Engneerng an echnology (IJE) Vb( ) cos ( ) 0 ( ) 0 o e o e Vb( ) sn 0 ( ) ( ) 0 A o e o e 0 0 0 s Vbsn s Vbcos ( ) ( ) 0 0 H e H e E C 0 f s m s o H he SIB system gven n () has nonlnear terms ( cos, sn, sn, cos ). atr A s ve nto 6 lnear matr by the -S fuzzy moelng. he nput-output form of the fuzzy system of () s represente as 6 () h( ){A C}, 0 h ( ), h ( ) (5) a an n values of nonlnear terms escrbe n able are nserte the poston of separate matr by the -S fuzzy rule. Each nonlnear terms have values an represente follow IF-HEN rule. 6 ABLE I. FUZZY ODEL PARAEERS OF HE SIB SYSE Nonlnear terms cos sn Inserte embershp functon values a n a n 0 0 cos sn sn 0 cos 0 3 sn cos 3 3 he SIB System oel Rule IF s a, s a, 3 s a an s a, HEN A C, IF s a, s a, 3 s a an s n, HEN A C, IF s n, s n, 3 s n an s n, HEN A6 C,... h 3 h 3 h 6 3 For eample, to fn A, nonlnear term cos / s replace by, sn / s replace by, sn s replace by, an cos s replace by. Vb( ) ( ) 0 ( ) 0 o e o e Vb( ) 0 ( ) ( ) 0 A o e o e 0 0 0 s Vb s V b ( ) ( ) 0 0 H e H e he 6 lnear matres can be foun above metho. Rotor angle ( ) of synchronous generator s boune 0 / by (5) to satsfy the -S fuzzy moelng metho. p-issn : 39-863 Vol 8 No Apr-ay 06 979
e-issn : 0975-0 Hee-Jn Lee / Internatonal Journal of Engneerng an echnology (IJE) C. Smulaton Results he -S fuzzy moel must have ynamc characterstcs the same as nonlnear moel, to confrm the valaton of the -S fuzzy moelng. Smulaton parameters are gven follows. f e s (funamental freuency) = 60 Hz; (steay state rotor spee) = 0 ra/s; (steay state -as reactance) =.8 Ω; (transent -as reactance) = 0.3 Ω; (steay state -as reactance) =.7 Ω; (transent -as reactance) = 0. Ω; e (transmsson lne reactance) = 0. Ω; o (transent -as tme constant) = 8; o (transent -as tme constant) = 0.; wo fferent types of sturbances, namely, a ± 5% step changes n the reference voltage of ecter an a 00 ms three-phase short crcut at the nfnte bus n Fg. 5, are carre out to evaluate the effectveness of the -S fuzzy moelng for the synchronous generator..05 -S fuzzy moel Nonlnear moel.0 ermnal Voltage [pu].03.0.0 0.99 0 5 0 5 0 5 30 35 0 5 me [sec]. (a) -S fuzzy moel Nonlnear moel 0.9 Rotor angle [ra] 0.8 0.7 0.6 0.5 0. 0 0.5.5.5 3 3.5.5 5 me [sec] (b) Fg. 7. Performance evaluaton of the -S fuzzy moelng: (a) ermnal voltage response when 5(%) step-change s apple to the ectaton system (b) Rotor angle response when 00 ms to the nfnte bus three-phase short crcut fault s apple Fg. 7 (a) shows termnal voltage response of synchronous generator when the etaton voltage changes. he termnal voltage ncreases as the etaton voltage ncreases at 5 sec. When the etaton voltage ecreases to ntal value at 5 sec, the termnal voltage ecreases, too. he sol lne whch ncates the nonlnear moel shows same pattern wth the otte lne that represents the -S fuzzy moel. Fg. 7 (b) shows the smulaton p-issn : 39-863 Vol 8 No Apr-ay 06 980
e-issn : 0975-0 Hee-Jn Lee / Internatonal Journal of Engneerng an echnology (IJE) result when three-phase short crcut fault occurs for 0. sec. he ampng effect can be observe because of system unstablty when accent occurs. In ths case, the same routnes are observe n both moels; nonlnear moel (sol lne) an the -S fuzzy moel (otte moel). he effectveness of the -S fuzzy moel can be checke through these smulaton. IV. CONCLUSIONS hs paper propose the new moelng of synchronous generator by applyng the -S fuzzy metho n orer to analyze stablty of the nonlnear SIB system. Conventonal lnear moel cannot represent the characterstcs of synchronous generator whch has we operatng range, complcate ynamc characterstcs, an nonlnearty. he -S fuzzy moel shows more accuracy than the lnear moel because t consers nonlnear characterstcs. It wll be more easy to analyzng system stablty an esgnng controller by usng the -S fuzzy moel because nonlnear terms can be gnore when applyng Lyapunov conton. Acknowlegment hs paper was supporte by Kumoh Natonal Insttute of echnology References [] K. anaka an. Sugeno, Stablty analyss an esgn of fuzzy control systems, Fuzzy Sets System, vol. 5, pp. 36-56, 99. [] K. anaka,. Ikea, an H. O. Wang, Fuzzy regulators an fuzzy observers: Relae Stablty Contons an LI-Base Desgns, IEEE ransactons on Fuzzy Systems, vol.6, no., pp. 50-65, 998. [3] E. Km an S. Km, Stablty analyss an synthess for an affne fuzzy system va LI an ILI: a contnuous case, IEEE ransactons on Fuzzy Systems, vol. 0, no. 3, pp. 39-00, 00. [] E. Km,. Park, an S. J, A new approach to fuzzy moelng, IEEE transactons on fuzzy systems, vol. 5, no. 3, pp. 38-337, 997. [5] F. Cuesta, F. Gorllo, J. Aracl, an A. Ollero, Stablty analyss of nonlnear multvarable akak-sugeno fuzzy control systems, IEEE ransactons on Fuzzy Systems, vol. 7, no. 5, pp. 508-50, 999. [6] C. W. Park, C. H. Hyun,. S. Park, C. H. Lee, J. Km, an. Park, Control of uncertan fleble jont manpulator usng aaptve akag-sugeno fuzzy moel base controller, Proc. Of the 00 IEEE Internatonal Conference on Robotcs & Automaton, Seoul, Korea, ay -6, 00, pp. 985-990. [7]. akag an. Sugeno, Fuzzy entfcaton of systems an ts applcatons to moelng an control, IEEE ransactons on Systems, an, an Cybernetcs, vol. 5, no., pp. 6-3, 985. [8] S. H. Zak an C. A. accarley, State-feeback control of nonlnear systems, Internatonal Journal of Control, vol. 3, no. 5, pp. 97-5, 986. [9] S. Kawamoto, Nonlnear control an rgorous stablty analyss base on fuzzy system for nverte penulum, Proc. of the Ffth IEEE Internatonal Conference, New Orleans, LA, USA, Sep. 8-, 996 pp. 7-3. [0] J. W. Park, G. K. Venayagamoorthy, an R. G. Harley, LP/RBF neural-networks-base onlne global moel entfcaton of synchronous generator, IEEE ransactons on Inustral Electroncs, vol. 5, no. 6, pp. 685-695, 005. [] P. Kunur, Power System Stablty an Control, cgraw-hll, 993. [] P. W. Sauer an. A. Pa, Power System Dynamcs an Stablty, Prentce-Hall, 998. [3] P.. Anerson an A. A. Foua, Power System Control an Stablty, IEEE Press, 99. [] Ha Saaat, Power System Analyss, cgraw-hll, 00. p-issn : 39-863 Vol 8 No Apr-ay 06 98