Application of Fuzzy C-Means Clustering in Power System Model Reduction for Controller Design

Transcription

1 Proeeding of the 5th WSEAS Int. Conf. on COMPUAIONAL INELLIGENCE, MAN-MACHINE SYSEMS AND CYBERNEICS, Venie, Italy, November 20-22, Appliation of Fuzzy C-Mean Clutering in Power Sytem Model Redution for Controller Deign SHU-CHEN WANG PEI-HWA HUANG 2 CHI-JUI WU 3 Department of Eletrial Engineering Department of Eletrial Engineering National aiwan Oean Univerity National aiwan Univerity of Siene and ehnology No. 2, Peining Rd., Keelung No. 43, Se. 4, Keelung Rd., aipei 060 AIWAN AIWAN Abtrat: -hi paper preent the appliation of fuzzy -mean (FCM) lutering in the order redution of dynami model for ontroller deign in a power ytem. Baed on the fuzzy -mean algorithm, a method i propoed for lutering the pole and zero of the original power ytem model into new luter from whih a redued-order model an be obtained. hen the redued-order model i ued to deign a proportional-integral type power ytem tabilizer to improve the damping in ytem oillation after a ytem diturbane. he redued-order model an ontain the ritial dynami harateriti of the original model, but let it eaier to deign the ontroller. Reult from a ample power ytem are preented to how the validity of the propoed method. he eletromehanial mode of the power ytem an be improved by the deigned power ytem tabilier from pole aignment. Key-Word: -Power ytem dynami, Model redution, Fuzzy -mean, Fuzzy Clutering, Pole aignment. Introdution Model order redution onern the tranformation of a higher-order model into a lower-order model through ome ort of omputation [, 2]. A ertain relationhip between thee two model i preerved and they are imilar in the harateriti under onideration. In power ytem tudie, reating a dynami model i the firt tep for ytem tability reearh, dynami behavior analyi, or other ytem funtional tet. A ytem beome larger, their omplexity inreae and power ytem analyi ha to takle high-order model analyi. However, omputation on the high-order model i highly omplex while the final analyi reult may have unneeary portion. In thi ae, having a low-order model that maintain the main harateriti of the high-order ytemt an replae the original ytem and ignifiantly implify the omputational problem [3-3]. If the tability performane of a power ytem i unable to atify the peifiation, the tabilizing ontroller an be ued to improve the dynami harateriti. Without tability diturbane ompenation, teady-tate performane and any other performane index are not poible. herefore, a tabilizing ontroller of power ytem i needed. he mot important appliation of the redued order model let it eaier to deign of a uitable ontroller for the original high-order ytem. Many method an ued to deign a power ytem tabilizier with output feedbak heme. he pole aignment deign allow the power ytem for the eletromehanial mode dynami to be plaed in deired loation. In thi paper, the method baed on fuzzy -mean lutering analyi [4-20] aim to group pole and zero of a power ytem tranfer funtion into ome luter. For eah luter, the original ytem pole (zero) an be replaed by eah luter enter that beome the new member repreentative of the luter. All new member repreenting their repetive luter jointly ontitute a tentative redued-order model of the original ytem. he redued-order model i ued to deign a proportional-integral power tabilizer to improve the dynami tability. he reult obtained from a ample power ytem model will be illutrated and the effetivene of the method i thu onfirmed by the example.

2 Proeeding of the 5th WSEAS Int. Conf. on COMPUAIONAL INELLIGENCE, MAN-MACHINE SYSEMS AND CYBERNEICS, Venie, Italy, November 20-22, Fuzzy -mean Cluter Analyi he method propoed in thi paper utilize fuzzy -mean lutering (FCM) analyi [4-6] to redue the original high-order model into a low-order model. Cluter analyi [-20], of whih the tak i to laify non-proeed data into ertain ategorie depending on variou trait, i a bai tool ommonly ued in everal ientifi field. Data in eah ategory have the mot reemblane while being very diimilar with data from other ategorie. Suppoe there are n data point{ x j }, j n, to be lutered into data luter. Let μ ij denote the degree of memberhip that x j belong to the i th luter. It i noted that 0 μ ij and i μ ij for eah j. Define the fuzzy partition matrix U [ μ ij ], i, j n. herefore, the objetive of the fuzzy -mean algorithm i to determine all the element of matrix U. he FCM algorithm i eentially an iterative proedure and an be formulated a the following ix tep in whih l denote the iteration number. (a) Set the number of luter. Initialize U () l randomly a U [ μ ij ], l, i. (b) Compute the luter enter i of eah luter: n m j μij x j i n m j μij () Note that the value of m normally fall in the range of.5 m 3. () Selet the weighting the weighted data point w j of every data point, then W j a Wj xj wj (2) (d) Compute the ditane d ij between the j th data point and the i th luter enter: dij i Wj (3) () l (e) l l+. Compute μ ij in U a μij (4) 2/( m ) k ( dij dkj ) (f) If ( l+ ) ( l) U U ε, a preet auray, then top; otherwie, return to Step (b). It i worth noting that in the above algorithm, the luter enter i of eah luter i referred to a the prototype of the luter and an be onidered a the repreentative of that luter. 3 Deign Method Given a tate pae linear model, dynami harateriti of the ytem an be bet revealed from it pole and zero. he following tep omprie the propoed model redution method and ontroller deign. Step: After onfiguring all the parameter of the power ytem and linearizing the ytem tate equation, the following ytem dynami equation are obtained a x& Ax + Bu (5) y Cx where A, BC, are the tate, input, and output matrie of the ytem; x, u and y denote the tate, input and output vetor, repetively. Step2: From Equation (5), the tranfer funtion i found to be G () CI ( A) B b0 + b+ b2 + L+ bm 2 a + a + a + L+ a n m n (6) Baed on the tranfer funtion, the pole and zero an be omputed. Step3: Uing the fuzzy -mean algorithm, it an luter eparately the pole and zero in the omplex plane to obtain the orreponding luter enter. In order to keep the ytem oillatory behavior, pole with and without imaginary part are lutered into ditint group, and zero are proeed likewie. Step4: he alulated luter enter replae the repetive group of pole and zero of the original ytem and olletively ontitute the et of pole and zero for the redued-order model. he tentative redued-order model tranfer funtion i thu et a

3 Proeeding of the 5th WSEAS Int. Conf. on COMPUAIONAL INELLIGENCE, MAN-MACHINE SYSEMS AND CYBERNEICS, Venie, Italy, November 20-22, E fd V ref P,Q V t Re X e V inf Figure. Single-mahine infinite bu power ytem Vref + Vp + Vt - - VOLAGE REGULAOR V KA + A K f + f SABILIZING RANSFORMER LIMIER EFD Figure 2. Blok diagram of tati exitation ytem able he parameter of generator X d 2pu X q.9pu X d 0.244pu X q 0.pu do 4.8e qo 0.55e able 2 he parameter of tati exitation ytem K A 400 A 0.05 Step5: K F t i r i F.0 ( + zi) R () () ( + p ) In order to make the time repone of the redued-order model ompatible with that of the original higher order model, a gain adjutment fator defined by G () k (8) R () 0 i ued to adjut the teady tate value of the redued order model. Step6: he parameter of a proportional-integral power ytem tabilizier (PSS) are to be determined. he power ytem tabilizier ha the tranfer funtion a i V PSS ki kpω + ω (9) hen the loed-loop tranfer funtion of the ytem i y G() (0) v k G() k G() 4 Example PSS P I Conider the ingle-mahine infinite bu power ytem hown in Figure. he generator an be repreented by the two axi model. he equation are obtained: E [ E ( X X ) I ] () d d q q q qo q FD q d d d do E [ E E ( X X ) I ] (2) he parameter of generator are hown in able. he blok diagram of tati exitation ytem i diplayed in Figure 2. he parameter of tati exitation ytem are hown in able 2. Baed on the above-deribed method, the redued order model and the ontroller deign for the tudy ytem i obtained a follow: Step: Chooe the tate vetor x a x ΔE d ΔEq Δω Δδ ΔEFD ΔV S he definition for eah tate variable are Δ E d diret-axi tranient voltage Δ E q quadrature-axi tranient voltage ω peed δ rotor angle E FD exiter output voltage V S tabilizier tranformer output voltage he ytem matrie are A B U ΔV ref [ ]

4 Proeeding of the 5th WSEAS Int. Conf. on COMPUAIONAL INELLIGENCE, MAN-MACHINE SYSEMS AND CYBERNEICS, Venie, Italy, November 20-22, Figure 3. Comparion the original ytem and the redued model able 3 Pole and zero of the original model Pole Zero ± j ± j able4. Pole and zero of the redued model Clutered Pole Clutered Zero ± j ± j able 5 Eletromehanial mode of the power ytem Eigenvalue without Eigenvalue with PSS PSS Step2: ± j.53-2 ± j he tranfer funtion of the original model i alulated a ( +.000)( )( ± j ) G () ( ± j.53)( )( )( )( ) he pole and zero of the original model are diplayed in able 3. Step3: Uing fuzzy -mean algorithm, the pole and zero of the original model are proeed to obtain ome luter enter to be ued for repreenting the original pole and zero. able 4 how the pole and zero after lutering. In able 4, the pole ( 28.5 ) are obtained from lutering the pole of the original model, ( ), ( ), ( ), and ( ). he pole ( ± j.53 ) of the eletromehanial mode are retained. Regarding the zero, the lutered zero are ( 4.85 ± j ). Step4: he luter enter i obtained after omputation and i ued to replae the pole and zero of the original ytem to beome the redued model. he tentative tranfer funtion for the redued model are R( ) Step5: ( ± j ) ( )( ± j.53) he gain adjutment fator i ued to adjut the ytem repone to make the redued-order model ompatible with the original model. For the tudy ytem, the gain adjutment fator are alulated a G () k R () λ 0.35 where λ i the eletromehanial mode. After the above tep, the tranfer funtion of the redued order model i R () kr() whih i given below Step6: ( ± j ) R ( ) ( )( ± j.53) he redued-order model i ued to deign a proportional-integral power ytem tabilizer. If the eletromehanial mode of the loed-loop ytem i to be aigned at λ -2 ± j, the parameter of power ytem tabilizier are obtained a [ k k ] [ ] P I From able5, the eletromehanial mode of of the original model ytem are obviouly improved. he time repone of the original ytem with

5 Proeeding of the 5th WSEAS Int. Conf. on COMPUAIONAL INELLIGENCE, MAN-MACHINE SYSEMS AND CYBERNEICS, Venie, Italy, November 20-22, and without ontroller after a mall ditrubane i hown in Figure 3. 5 Conluion A model redution method for reduing the order of power ytem dynami model in ontroller deign ha been propoed in thi paper. Baed on the fuzzy -mean algorithm, the propoed method perform lutering on the pole and the zero of the original ytem model into new luter from whih a redued-order model an be derived. he redued-order model that maintain the main harateriti of the high-order ytem an ignifiantly implify the deign of a power ytem tabilizer. Reult from applying the method to a ample power ytem have been demontrated to how the validity of the propoed method. Aknowledgment hi work wa upported in part by the National aiwan Oean Univerity and the National Siene Counil of aiwan. Referene: [] A. Bergen, Power Sytem Analyi, Prentie Hall, New York, [2] D. rudnowki, Order redution of large-ale linear oillatory ytem model, IEEE ran. on Power Sytem, Vol. 9, No., 994, pp [3] F. Saleh and M. Mahmoud, Deign of power ytem tabilizer uing redued-order model, Eletri Power Sytem Reearh, Vol. 33, 995, pp [4] A. Feliahi, X. Zhang and C. Sim, Power ytem tabilizer deign uing optimal redued order model, IEEE ran. on Power Sytem, Vol. 3, No. 4, 988, pp [5] R. Catro and J. de Jeu, A wind park redued-order model uing ingular perturbation theory, IEEE ran. on Energy Converion, Vol., No. 4, 996, pp [6] N. Nihei and. Oyama, A tudy on deompoition and model redution for wide area power ytem tability aement, Power Engineering Soiety 999 Winter Meeting of the IEEE, Vol., 999, pp [] N. Sinha and J. Pal, Simulation baed redued order modeling uing a lutering tehnique, Computer & Elet. Engineering, Vol. 6, No. 3, 990, pp [8] N. Sinha, Redued-order model for linear ytem, Pro. of the IEEE Conferene on Sytem, Man and Cyberneti, 992, pp [9] M. Duri, Z. Radojevi and E. urkovi, A redued order multimahine power ytem model uitable for mall ignal tability analyi, Eletrial Power & Energy Sytem, Vol. 20, No. 5, 998, pp [0] G. Obinata and B. Anderon, Model Redution for Control Sytem Deign, Springer, 200. [] P. Benner, R. Mayo, E. Quintana-Orti, and G. Quintana-Orti, A ervie for remote model redution of very large linear ytem, Pro. of the International Parallel and Ditributed Proeing Sympoium, 2003, pp [2] H. Ukai, H. Maubara, M. Kobayahi, and H. Kandoh, Stabilizing ontrol of erie apaitor ompenated power ytem on the bai of redued order model, Pro. of the 4 th International Conferene on Advane in Power Sytem Control, Operation and Management, 99, pp [3] F. Saleh and M. Mahmoud, Deign of power ytem tabilizer uing redued-order model, Eletri Power Sytem Reearh, Vol. 33, 995, pp [4] N. Pal, K. Pal, J. Keller, and J. Bezdek, A poibiliti fuzzy -mean lutering algorithm, IEEE ran Fuzzy Sytem, Vol. 3, No. 4, 2005, pp [5] S. Naimento, B. Mirkin and F. Moura-Pire, A fuzzy lutering model of data and fuzzy -mean, Pro. of the IEEE Conferene on Fuzzy Sytem, 2000, pp [6] J. Bezdek, J. Keller, R. Krinapuram, and M. Pal, Fuzzy Model and Algorithm for Pattern Reognition and Image Proeing, Kluwer Aademi, 999. [] F. Hoppner, F. Klawonn, R. Krue, and. Runkler, Fuzzy Cluter Analyi: Method for Claifiation, Data Analyi and Image Reognition, John Wiley & Son, 999. [8] M. Aldenderfer and R. Blahfield, Cluter analyi, Sage Publiation, Beverly Hill, 984. [9] L. Kaufman and P. Roueeuw, Finding Group in Data: An Introdution to Cluter Analyi, John Wiley & Son, 990. [20] J. Chiang and Y. Chen, Inorporating fuzzy operator in the deiion network to improve laifiation reliability, Computer & Elet. Engineering, Vol. 28, 2002, pp