ROBUST OVERLAPPING DECENTRALIZED CONTROL FOR MULTI-AREA LONGITUDINAL POWER SYSTEMS

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1 ROBUS OVERLPPG DECERLZED COROL FOR MUL-RE LOGUDL POWER SYSEMS Xaohua L Xue-Bo Che Yuawe Jg We Wag 3.Faculty of formato Scece ad Egeerg ortheaster Uversty Sheyag 0004 Cha.School of Electroc ad formato Egeerg sha Uversty of Scece ad echology sha 4044 Cha 3.Dala Uversty of echology Dala 603 Cha bstract: he problems of robust overlappg decetralzed cotrol for a class of multarea logtudal power systems are dscussed ths paper. specal overlappg decomposto method s preseted terms of cluso prcple. Based o the decomposto a ew cotrol method s formed by meas of Lear Matrx equalty (LM) approach of Orgacally-Structured Cotrol (OSC). he proposed method s appled to a three-area logtudal power system ad the smulato results show that the performaces of the cotrollers desged by proposed method are better tha the oe by OSC drectly. Copyrght 005 FC Key words: overlappg decomposto robust decetralzed cotrol logtudal power systems Lear Matrx equalty (LM). RODUCO t s well kow that the overlappg decomposto has bee wdely appled to varous cotrol desgs of complex systems (Slak 99; Che ad Stakovc 996; kar ad Ozguer 00; Kttel et al. 00). he cocept of overlappg decetralzed cotrol has bee put forward to solve the problems about the decetralzed cotrol of the tercoected systems. Geerally the overlappg meas that there s a commo part dfferet subsystems whch s called overlappg part. t may be dfferet whe the partto of the subsystems s dfferet. he choce of the overlappg or the partto of the subsystems wll affect the overlappg decetralzed cotrol of overall system. Kttel et al. (00) provded a example about choosg overlappg ad poted out the fact that cosderg the overlappg systems ca make overall system has better cotrol performaces. herefore how to use of the overlappg ad overlappg decomposto complex systems to mprove cotrol performaces s a researchable problem. he decetralzed cotrol for tercoected power systems has also attracted cosderable atteto of researchers the feld of complex ad large-scale systems (Slak 99; Che ad Stakovc 996; Che et al. 00; Che et al. 00; Stakovc 999). We kow that mult-area tercoected power systems have a complex structure. hey ofte exst etwork forms but some specal cases they ca also exst logtudal or loop or radal structure. order to smplfy the problem o cotrol desg we ca study respectvely for dfferet cases. Che et al. (00 ad 00) preseted a decetralzed cotrol method for mult-area power systems.e. t decomposed the overall system as a group of par-wse area subsystems the desged the decetralzed cotrollers for each area accordg to the decetralzed desg method for two-area overlappg power systems (Che ad Stakovc 996). ths way maybe there are may decetralzed cotrollers for each area. But the method dd ot gve us the strategy to choose optmze ad coordate the cotrollers. t also dd

2 ot cosder the tercoectos betwee the par area systems whe the par area subsystems are desged. herefore the performaces of cotrol are ot perfect. o mprove the cotrol performace dces ad cosder the tercoectos betwee subsystems more completely a specal overlappg decomposto method s studed framework of the cluso prcple ad LM method of Orgacally- Structured Cotrol (Slak ad Stpaovc 00) s adopted to add the postve effect of the tercoectos the cotrol desg for the multarea tercoected power systems. he obectve of ths paper s to preset a ew decetralzed cotroller desg dea for mult-area logtudal tercoected power systems. order to mplemet the cotrol dea a overlappg decomposto method whch decomposes the tercoected power system to a group of parwse area subsystems s preseted terms of the restrcto codtos of cluso Prcple. he decomposto mode ca separate out the tercoectos betwee each par-wse area. Whe the cotrollers are desged the tercoectos ca be cosdered as a postve factor. Based o the decomposto the decetralzed cotrollers of each subsystem par-wse area subsystems are desged by usg LM approach of OSC expaded space. he all the cotrollers ca be combed together ad cotracted back to orgal space by usg the cotracto codto of cluso prcple ad mplemeted orgal space. he proposed method s appled to a three-area logtudal power system ad the smulato results show that the performaces of the cotrollers desged by proposed method s better tha the oe by OSC drectly.. MODELS D SRUCURE OF SYSEMS We cosder a class of tercoected power systems composed of tercoected areas descrbed by the followg dfferetal equatos (Calovc 984; Slak 978) S : x x + Bu + Fξ y Cx where the state vector x() t R () cotrol put m l ut () R output yt () R ad perturbato put ξ() t R. he correspodg costat matrces wth approprate dmesos ca be represeted by B blockdag( B B B ) C blockdag( C C C ) F blockdag( F F F ) () matrx the dagoal block represets the parameter of -th area subsystem ad s the tercoecto betwee areas. he equato of -th area subsystem ca be descrbed as where x x+ Bu+ Fξ + x y Cx (3) x R s the -th area state vector x [ x v Pe ] x R s the state vector of the other areas x [ x v Pe ]. Suppose ad. he physcal meag of each varable ca be foud (Calovc 984; Slak 978). u R m s the cotrol put vector of -th area system m m. the m. B F (3) are descrbed as follows C ad 0 a t d α m 0 0 α 0 0 m b C 0 0 f B 0 C 0 0 F 0 (4) where C s the costat matrx blocks correspodg to the state m d R x ; a m R t are costat couplg vectors; R s the bas factor related to area cotrol error of GC; α / P0 P 0 s a steady-state load ormalzato factor based o area. fact the ormalzato factor α s ucerta because the loads of the areas are costat alog wth the load perturbato ad the fluctuato of geeratg power. We otce the fact from the model () ad () that there are the tercoectos oly matrx. B C ad F have bee block-dagoal matrces. f the power systems have logtudal structure the matrx ca be smplfed as follows:

3 (5) he ma characterstc of logtudal structure s that the area subsystems are coected lke a cha. Suppose -th area subsystem s the mddle of the cha. Because -th area subsystem cludes smultaeously the formato of two areas coectg wth t.e. the formato overlappg affected by the other two areas s formed -th area. herefore -th area ca be regarded as the overlappg structure of adacet areas. he overlappg decomposto of ths structure has a partcular character. 3. SPECL OVERLPPG DECOMPOSO OF LOGUDL SRUCURE Specal overlappg decomposto s dfferet from geeral overlappg decomposto t regards the whole area subsystem ot part subsystem as the overlappg part. he tercoected system wll be expaded ad decomposed to a group of par-wse area subsystems by usg the restrcto codto of cluso prcple. Suppose the expaded system of the logtudal power system () s S : x x + Bu + F ξ y Cx (6) where x () t R m ut () R l yt () R ξ () t R are the state cotrol put output ad ucerta perturbato put of the expaded system S respectvely. B C ad F are stll costat matrces wth expaded dmesos. d there are always < m< m l < l. Suppose that the pars of matrces (UV) (QR) (S) are gve ad V U R Q S ad are all full rak trasformato matrces ther dmesos are m m m m l l l l respectvely ad satsfy UV QR m S l where deotes a detty matrx. terms of cluso prcple the the matrces B C ad F ca be expressed as VU + M B VBQ + M B C CU + M C F VFQ + M F. (7) where M M B M C ad M F are the complemetary matrces wth the dmesos m l m. o let S be a expaso of S a proper choce of M M B M C ad M F s requred ad the restrcto codtos of the cluso prcple ca be used. he codtos are dvded to two classes. hey are provded by the followg theorem (.Slak 99; Stakovc 999). heorem. he system S s oe of restrctos of S f there exsts full rak matrces V R ad such that V V BR VB CV C FR VF. (8) heorem. he system S s oe of restrctos of S f there exsts full rak matrces V ad R such that MV 0 M R 0 MV 0 M R 0. (9) B Because the tercoectos betwee subsystems are oly matrx for the system () we oly cosder the overlappg structure decomposto of the state equato the system (). Frst of all the trasformato matrces should be chose. For the logtudal power systems choose the overlappg decomposto factor β 0.5 accordg to the cluso prcple method the - β 0.5. he matrces are 3 ( ) V blockdag( ) 3 ( ) ( ) ( ) U blockdag( ) m m3 m ( ) R blockdag( m m ) m m3 m ( ) m m ( ) m m m ( ) Q blockdag( m ) (0) Whe satsfes the codto V V (8) we have ( )( ) 0 ( ) ( )( ) ( ) ( ) () where dotted les mark out - par-wse area subsystems. Whe satsfes the codto MV 0 C F

4 (9) the complemetary matrx M s M ( )( ) 0.5( )( ) ( )( ) 0.5( )( ) ( ) 0.5( ) 0 () Whe B satsfes the codto MB R 0 (9) the complemetary matrx MB s MB Rewrte the model to a compact form B 0.5B S: x Dx+ BDu+ h( x) 0 0.5B 0.5B u KDx (9) B( )( ) 0.5B( )( ) 0 where KD blockdag( K K ). the compact B( )( ) 0.5B( )( ) 0 form the quadratc costrat of the tercoecto hx ( ) s (4) + h ( x) h( x) x α H H x (0) Smlarly F ad C ca be obtaed too. t should be oted that the state vector of overall systems s trasformed from x [ x x x ] to x [ x ]. x x x x x S : x x + B u + F ξ + x S : x x + B u + F ξ + x (5) where +. x ad x are the tercoectos betwee two subsystems. Here suppose there s ot selftercoecto each subsystem. Let x h( x) x h( x) where x deotes x [ x x ]. he frst oe of the equatos (5) s used to accout for the desg procedure. the same way whe B satsfes the codto BR VB (8) we have Because the load ormalzato factorα s ucerta but bouded the tercoectos we ca B suppose the tercoectos to satsfy the followg quadratc costrats. B B h ( ) ( ) (6) x h x α x H Hx 0 0 B where α >0 s the bouds of ucerta B( )( ) 0 0 tercoectos H s a costat matrx. Cosder the decetralzed state feedback cotrol laws as B( )( ) 0 follows B u ( ) (3) x Kx (7) he the closed-loop subsystems ca be descrbed as S : x x + B K x + h( x) (8) Cosderg the robust coectve stablty of the system (5) the tercoected matrx E ( e ) s added to the tercoectos of the system (Slak 978). herefore 4. ROBUS DECERLZED COROL DESG he power system () s decomposed to a group of par-wse area subsystems expaded space after overlappg decomposto. We ca use LM approach of OSC (Slak ad Stpaovc 00) to desg the robust decetralzed cotrollers for each par subsystems. he model of par-wse area subsystems ca be descrbed by the followg h( x) e x () he the sze of the tercoectos s further lmted by mposg the costrats where h( x) α e β x () e s a elemet of the fudametal tercoecto matrx E ( e ) wth

5 dmesos correspodg to the system (5) orm of the tercoecto ormalzato factor α β s a excepted ucerta each two-area system. Based o () the quadratc costrats of the tercoecto ca be deoted as where Smlarly h( x) h ( x) h( x) α e β x α x H H x (3) H blockdag{ eβ e } β (4) H blockdag{ e β e } β (5) order to obta the decetralzed cotrol law u KDx such that the closed-loop system (9) to be robust coectvely stable we use Lyapuov stablty theory ad Schur complemetary formula to educe followg LM optmzato problem (Slak ad Stpaovc 00). + Mmze γ Subect to Y D > 0 Φ YDH YDH < 0 HY D 0 γ 0 H+ YD 0 0 γ + Φ Y D D + YDD + BDLD + LDBD α. d the ga matrx K D where / be computed as (6) γ of the cotroller ca herefore we ca arrve at K blockdag( K K K ) (30) 5. SMULO RESULS Cosder a three-area tercoected power system wth logtudal structure. he parameters of the two areas three areas ca be foud (Calovc 984; Slak 978) ad the other oe s chose accordg to the oe of the parameters of two areas. he parameters are 33 a t3 [ ] d 3 [ ] m3 m3 [ ] B 33 [ ] f3 a t 3 C33 C (3) Let 3 the expadg matrces are selected by (0). fter the procedure of expaso ad cotracto the decetralzed cotrollers are desged terms of above proposed method. pply them to the threearea tercoected power system the output respose curves to the load dsturbace cludg devato of frequecy varatos ad the te le power exchage varatos f 3 Pe 3 are show Fg. he comparso s doe betwee the results obtaed by proposed method ad the oe by usg OSC approach drectly (Slak ad Stpaovc 00). he real les deote proposed method ad the dotted les deote OSC approach. f Output Respose Curve K L Y (7) D D D fter the decetralzed cotrollers are desged for each par-wse area subsystem the ga matrx K D ca be formed expaded space as follows K blockdag( K K K K K ) (8) D 3 Pe Output Respose Curve o mplemet the decetralzed cotrol the orgal system ad form overlappg decetralzed cotrol laws K D should be cotracted back to the orgal space based o the restrcto codtos of the cluso prcple (Slak 99). he robust decetralzed cotrollers for system () satsfy K QKV (9)

6 f Output Respose Curve Output Respose Curve Pe decetralzed cotrol method for mult-area logtudal tercoected power systems. Because OSC approach suffcetly cosders the tercoectos betwee subsystems ad the effect brought by ucerta structure perturbato the cotrol s robust ad t guaratees the coectve stablty of the systems. t the same tme the smulato results of proposed method prove the fact that cosderg the overlappg systems properly ca mprove cotrol performaces of overall systems. he proposed method ca be also appled to the other cotrol systems. CKOWLEDGEME hs research reported here was supported by the SFC of Cha uder grat o ad by the USRP of Laog Educato Departmet uder grat o Output Respose Curve f 3 Output Respose Curve Pe 3 Fg. he output respose curves of the ma varatos to the load dsturbace From Fg. we ca kow that the performaces of the cotrollers desged by the former are better tha the latter. he reaso s that the two cotrollers cosderg dfferet tercoectos are combed together ad the tercoectos are used suffcetly. he smulato results llumate that the proposed method are feasble. 6. COCLUSOS Makg use of specal overlappg decomposto of the systems the paper presets a ew robust REFERECE kar M. ad Ozguer U. (00). Decetralzed sldg mode cotrol desg usg overlappg decompostos. utomatca vol. 38 pp Calovc M.S(984). utomatc geerato cotrol: decetralzed areawse optmal soluto. Electrc Power Systems Research vol. 7 pp Che X.-B. ad Stakovc S. S(996). Overlappg decomposto ad decetralzed LQG cotrol for tercoected power systems. : Proc. of EEE SMC 96 vol.3 pp Che X B Slak D D Stakovc S S(00). Decetralzed Cotrol of Mult-area Multmache Overlappg tercoected Power Systems. : Proc. of CC 0 Pretora South frca Dec. -4 pp Che X B Slak D D Stakovc S S(00). Decetralzed H desg of automatc geerato cotrol. : Proc. of FC 5th World Cogress Barceloa Spa July -6 vol.g pp Kttel D. Gga D. Laroche E. ad Mathel M.de (00). Decetralzed H cotrol of large scale web trasport systems. : Proc. of FC 5th real World Cogress Barceloa Spa. Slak D. D. ad D. M. Stpaovc (00). Orgacally-structured cotrol. : Proc. of the merca Cotrol Coferece rlgto V pp Slak D.D(978). Large-Scale Dyamc Systems: Stablty ad Structure. ew York: orth-hollad. Slak D.D(99). Decetralzed cotrol of complex systems cademc Press ew York. Stakovc S.S. Che X.B. Matausek M.R. ad Slak D.D(999). Stochastc cluso prcple appled to decetralzed automatc geerato cotrol. teratoal Joural of Cotrol vol. 7 3 pp