Reachability Analysis Based Minimal Load Shedding Determination

Transcription

1 Reachability Aalyi Baed Miimal Load Sheddig Detemiatio Haifeg Liu, Studet Membe, IEEE, Licheg Ji, Studet Membe, IEEE, Vekataamaa Ajjaapu, Seio Membe, IEEE, Rateh Kuma, Seio Membe, IEEE, Jame D. McCalley, Fellow, IEEE, Nicola Elia, Membe, IEEE, Vijay Vittal, Fellow, IEEE Abtact-- Thi pape popoe a method to detemie the miimal amout of load heddig baed o backwad eachability aalyi. The backwad eachable et i calculated uig level et method to compute the egio of tability of a table equilibium poit. The miimal amout of load heddig a a fuctio of time elaped ice the pot-ditubace coditio ca be ytematically deived a the hoizotal ditace betwee the pot-ditubace tajectoy ad the tability bouday of the table equilibium poit. Cae tudie with a ytem coitig of a ifiite bu feedig a dyamic load though a tap-chagig tafome ae peeted to illutate the appoach. Idex Tem-- Level et method, load heddig, eachability aalyi, voltage tability. V I. INTRODUCTION OLTAE itability i oe of the majo theat to powe ytem opeatio []. I thi pape, we focu o logtem lage-ditubace voltage tability. Accodig to the ecet epot of the IEEE/CIRE joit tak foce [], logtem lage-ditubace voltage tability i the ability of a powe ytem ivolvig low actig equipmet uch a tapchagig tafome to maitai teady voltage at all bue afte a lage ditubace fom a give iitial opeatig coditio. I face of log-tem lage-ditubace voltage itability, load heddig i geeally a effective coective cotol cadidate [3]. It dicoect a factio of load i ode to ihibit futhe ytem deteioatio. It i deied that the pecetage of load to be hed i miimized whe deigig a udevoltage load heddig cheme. The peviou tudie dealig with load heddig agait voltage collape ca be claified ito two categoie: the tatic ad the dyamic method. The tatic method [4, 5, 6, 7] cocetate o detemiig the miimal actio eeded i ode to etoe a olutio to the powe flow equatio o to iceae tability magi. Howeve, i pite of thei elative advatage i tem Thi wok wa uppoted i pat by fudig fom the Natioal Sciece Foudatio ad fom the Office of Naval Reeach ude the Electic Powe Netwok Efficiecy ad Secuity (EPNES) pogam, awad ECS33734 ad ECS33379, by the Natioal Sciece Foudatio ude the gat NSF-ECS- 87, NSF-ECS-4473, ad NSF-ECS-4448, ad by a DoD-EPSCoR gat though the Office of Naval Reeach ude the gat N46. The autho ae with the Depatmet of Electical ad Compute Egieeig, Iowa State Uiveity, Ame, IA 5 USA ( hfliu@iatate.edu;lcji@iatate.edu;vajjaap@iatate.edu;kuma@iatate.edu; jdm@iatate.edu; elia@iatate.edu; vvittal@iatate.edu). of computig time, oe of the tatic method ae able to accout fo tempoal ifluece. I paticula the ifluece of the heddig delay, which ha bee how i [8, 9] to be cloely elated to the miimal load to hed, i ot take ito accout. Thi poblem equie model icopoatig ytem dyamic [9, ]. Howeve, the miimal amout of load heddig a a fuctio of time i uually obtaied by imulatio uig tial ad eo [9] which i time coumig. I thi pape, a ytematic appoach baed o the egio of tability i popoed to detemie the miimal amout of load heddig a a fuctio of time elaped ice the potditubace coditio. The egio of tability of a table equilibium poit efe to the egio of opeatig coditio i tate pace that evetually ted to a table equilibium poit. Thee have bee eveal effot made to etimate the egio of tability []. Vaiou method uig Lyapuov fuctio [] ae popoed to etimate the egio of tability. I additio to the difficulty to cotuct good Lyapuov fuctio fo geeal powe ytem model, the method baed o Lyapuov fuctio ca oly fid ome ubet of the exact egio of tability, which i foud to be ovely coevative, with a upedictably vayig degee of coevativee [3]. O the othe had, Chiag et. al. i [4] aalyzed the topological ad dyamical chaacteizatio of the tability boudaie fo a cla of oliea autoomou dyamic ytem. Ude cetai aumptio, the tability bouday of a table equilibium poit wa how to coit of the table maifold of all the utable equilibium poit o the tability bouday. The method i ot applicable if the ytem oly ha oe table equilibium poit uch a the evee Va de Pol ytem [5]. What i moe, fidig the table maifold of a equilibium poit i a otivial poblem fo thid ad highe ode ytem. I thi pape, the exact egio of tability of a table equilibium poit i computed baed o backwad eachability aalyi uig level et method, ad illutated fo a ytem icludig a dyamic load ad a tap-chagig tafome. The popoed method doe ot equie cotuctio of Lyapuov fuctio. It i applicable fo geeal oliea ytem, ad it doe ot equie idetificatio of the utable equilibium poit. The pape i ogaized a follow. Some fudametal cocept of the eachability aalyi ad level et method ae itoduced i Sectio II. Sectio III decibe a algoithm to detemie the egio of tability. I Sectio IV we calculate

2 miimal load heddig baed o the popoed method. Sectio V peet dicuio ad cocluio. II. REACHABILITY ANALYSIS AND LEVEL SET METHOD A. Reachability Aalyi We coide the followig oliea autoomou ytem: dx f( x) dt = () whee x R i the tate vecto, ad f(x) i the vecto field. The tability poblem i powe ytem i mathematically fomulated a oe of euig that the tate of the powe ytem at the itat of cleaig the fault i iide the egio of tability (ROS) of the pot-fault table equilibium poit []. We ue the backwad eachable et aalyi to obtai the ROS. ive ytem dyamic ad a taget et of tate, the backwad eachable et i the et of tate fom whee tajectoie ca each the taget et. If the taget et i choe a a mall eough -ball aoud the table equilibium poit uch that the -ball i withi the egio of tability of the table equilibium poit, the coepodig backwad eachable et ca be ued to obtai the ROS of the table equilibium poit. Oe way to decibe a et of cotiuou tate i kow a the implicit uface fuctio epeetatio. Coide a cloed et S R. A implicit uface epeetatio of S would defie a fuctio φ( x): R R uch that φ( x) if x S ad φ ( x) > if x S. Oe way to tack the backwad eachable et of a taget et i to olve the ytem equatio () with t eplaced by t fo all x o the bouday of the taget et i.e., fo all x with φ ( xt, = ) =, whee φ ( xt, ) i the implicit uface epeetatio of the backwad eachable et at time t, meaig the collectio of tate that ca each the taget et i time t o le. Becaue φ ( xt, ) = ca have ifiitely may olutio, we dicetize it ito a fiite umbe of ubegio ad calculate the evolutio of all ubegio imultaeouly. Howeve, the accuacy of the method ca deteioate quickly alog the evolutio of the implicit uface [6]. I ode to avoid the above poblem, we ca ue the implicit fuctio φ ( xt, ) both to epeet the bouday of the backwad eachable et ad to evolve the bouday. Fig. illutate a implicit uface popagatig with vecto field f(x). Outide φ ( xt, ) > Iide φ( x, t) f( x) Fig.. Implicit uface popagatig with vecto field f(x). The evolutio of the implicit fuctio φ ( xt, ) i defied by [6]: T φt + ( φ) f( x) =, () whee the t ubcipt epeet a patial deivative with epect to the time vaiable t, φ i the gadiet of φ ( x). We ca ewite () a: φt + fi( x) =. (3) i= xi The patial diffeetial equatio () defie the motio of the bouday whee φ ( xt, ) =. It i kow a Eleia fomulatio of the iteface evolutio. Equatio () i alo efeed to a the level et equatio which wa itoduced by Ohe ad Sethia fo umeical iteface evolutio of bouday [7]. B. Level Set Method Actually, () i the pecial fom of the followig geeal Hamilto-Jacobi-Iaac equatio: φt + H ( φ) =, (4) whee the Hamiltoia H ca be a fuctio of both pace ad time. Fo the autoomou oliea ytem give by (), T H( φ) = ( φ) f( x). (5) Ulike ay othe fomulatio of the backwad eachable et, the HJI PDE fomulatio ca be olved vey accuately uig umeical method baed o the level et techique. Level et method ae a collectio of umeical techique that ca tack the implicit uface evolvig accodig to the aociated vecto field [8]. They have bee ued i a vaiety of image poceig ad compute viio, computatioal phyic, fluid mechaic, ad combutio applicatio. I level et method, a et of data poit (kow a a gid) i defied fit. The mot popula gid ae Cateia gid defied a: {( xi, yj) i m, j } (6) Oce φ (,) x ad f( x) ae defied at each Cateia gid poit, umeical method ca be applied to evolve φ i time movig the bouday aco the gid. If the egio of tability i bouded, the equece of evolvig boudaie will evetually covege. Let φ ( x) = φ( x, t ) (7) epeet the value of φ at time t. I ode to update φ i time, we eed to fid ew value of φ at evey gid poit afte ome time icemet t. Thee ew value of φ ae deoted a + φ ( x) = φ( x, t + ) (8) whee t+ = t + t. We ca dicetize () a follow: + φ ( x) φ ( x) T + ( φ ) f ( x) = (9) t whee φ i the gadiet of φ. Equatio (9) ca be ewitte a:

3 3 + φ ( x) φ ( x) + φx f ( ) i i x =, () t i= whee φx i the patial deivative of φ with epect to x i i. Oe might evaluate thee patial deivative a: φ ( xi) φ ( xi ) φx () i xi Highe-ode accuate diffeecig method uch the Hamilto-Jacobi oocillatoy method [7] ca alo be ued to iceae the accuacy of the appoximatio fo φ. III. AN ALORITHM TO DETERMINE REION OF STABILITY I thi ectio we decibe the algoithm to detemie the egio of tability of a SEP. Step : Fid a table equilibium poit of a autoomou oliea ytem by olvig f(x)=, ad let x xi R be a table equilibium poit. Step : Specify a ε -ball ceteed at the table equilibium poit with adiu ε a the taget et. Defie a implicit uface fuctio at time t= a φ( x,) x x The the taget et i the zeo ublevel et of the fuctio φ ( x,), i.e., it i give by, = ε () { x R φ( x,) } = { x R x x ε} (3) Theefoe, a poit x i iide the taget et if φ ( x,) i egative, outide the taget et if φ ( x,) i poitive, ad o the bouday of the taget et if φ ( x,) =. Step 3: Popagate i time the bouday of the backwad eachable et of the taget et by olvig the followig HJI PDE: φt + fi( x) = (4) i= xi with temial coditio φ( x,) = x x ε (5) The zeo ublevel et of the vicoity olutio φ ( xt, ) to (4), (5) i the backwad eachable et at time t: { x R φ( x, t) } (6) Step 4: The backwad eachable et of the ε -ball aoud the table equilibium poit i computed uig the toolbox of level et method [9]. It i alway cotaied i the egio of tability of the table equilibium poit. Ad if t goe to ifiity, the backwad eachable et appoache the tue egio of tability. If the egio of tability i bouded, the level-et method baed umeical computatio of the backwad eachability et evetually covege to the egio of tability withi a fiite computatio time. IV. APPLICATION TO MINIMAL LOAD SHEDDIN Becaue the dyamic load etoatio mechaim play a impotat ole i voltage itability, we coide the ytem of Fig. coitig of a ifiite bu feedig a dyamic load though a lole tamiio lie ad a ideal tap-chagig tafome. I the model, we ue the cotiuou tapchagig tafome model [] to appoximate the dicete oe fo the pupoe of a moe coveiet aalytic aalyi. E V V Fig.. A igle-load model. The ytem model i give a follow: L jx : T = P V (7) T = V V (8) ef ( V ) E ( V ) + X ( V ) = (9) 4 Equatio (7) epeet the dyamic behavio of a cotat load whee V i load voltage P i powe et poit, ad i the cotext of load heddig, P i the load powe demad afte the heddig actio. i cotollable vaiable load coductace which i adjuted to maitai cotat powe T L i load ecovey time cotat. Note that we aume that the dyamic load ha uity powe facto. Equatio (8) i the appoximated cotiuou tap-chagig tafome model whee V i voltage et poit i tap atio of the tap-chagig tafome T i the time cotat of the appoximated cotiuou tap-chagig tafome which ha bee deived i []. I Equatio (9), E i the voltage of the ifiite bu, X i the eactace of the tamiio lie. Fom (9), we ca get E V = () 4 + X Theefoe, the et of equatio (7)-(9) ca be expeed a a et of fit ode odiay diffeetial equatio ()-(): E TL = P () 4 X + E T = V () 4 + X Equatio () ad () ca be ewitte a: E = ( P 4 ) = f (, ) (3) TL + X E = ( V ) 4 = f (, ) (4) T + X The equilibium coditio fo the oigial model (7)-(9) ae a follow: V = V (5)

4 4 = P/ V = (6) 4 ( V) E ( V) + X P = (7) Equatio (7), fo value of P lowe tha the maximum powe give by Pmax = E /( X) (8) ha two olutio ad u, whee coepod to the table ad u to the utable equilibium poit, ad > u I the followig example, the paamete ae choe a: X=.5, V =., P =.9, E =.6 =.956, TL= ec., T=5 ec. The we get the table equilibium poit a ( =.9, =.9 ). A. Regio of Stability Followig the algoithm of ectio III, we compute the egio of tability of the table equilibium poit ( =.9, =.9 ) a follow: ): Specify a ε -ball ceteed at the table equilibium poit ( =.9, =.9 ) with adiu ε =. a the taget et. The implicit uface fuctio i defied a φ ((, ),) = (.9) + (.9). (9) The the taget et i the zeo ublevel et of the fuctio φ ((, ),). ): Calculate the backwad eachable et of the taget et by olvig the followig HJI PDE: φt + ( f) + ( f) = (3) with temial coditio φ ((, ), ) give by (9). Fo thi example, the backwad eachable et computatio covege i 93. ecod. Fig. 3 how the computed egio of tability of the table equilibium poit of the pot-ditubace ytem. I the figue, the tability bouday i idicated by the olid lie. We have validated ou eult by plottig the phae potait i the ame figue computed uig time domai imulatio of ample tajectoie. The dahed lie with aow epeet ample tajectoie. It i clea that the computed egio of tability i accuate. (p.u.) (p.u.) Phae Potait Computed Stability Bouday Fig. 3 Computed egio of tability ad phae potait. We ue tate tajectoie to veify ou eult. Fig. 4 ad Fig. 5 how the tate tajectoie fo two diffeet iitial coditio. Fom ou computed egio of tability, we coclude that the tate tajectoie tatig fom the iitial coditio ( () =.3, () =.5) covege to the table equilibium poit, wheea the tate tajectoie tatig fom the iitial coditio ( () =.3, () =.5) divege. Fig. 4 how coveget tajectoie ad o the othe had, Fig. 5 how that the voltage begi to collape at about 35 ecod Active Powe t (ecod) Fig. 4 State tajectoie fo the iitial coditio ( () =.3, () =.5) Active Powe t (ecod) Fig. 5 State tajectoie fo the iitial coditio ( () =.3, () =.5). B. Miimal Load Sheddig Detemiatio Oce the egio of tability of the table equilibium poit of the pot-ditubace ytem i calculated, thi ifomatio ca be ued to deive the miimal amout of load heddig a a fuctio of the time elaped ice the iitial potditubace coditio equied to tabilize the ytem. Fo example, whe the iitial coditio of the pot-ditubace ytem i ( () =.4, () =.9), the tajectoy i diveget which i idicated by the dahed cuve i Fig. 6. Fig. 7 how the tajectoie with the iitial coditio ( () =.4, () =.9), whee we ca ee that the voltage begi to deceae at about 8 ecod. Let P () be the load befoe heddig ad () the coepodig admittace, uch that the ytem i at equilibium pio to the ditubace:

5 5 P() = () V (3) If a pecetage λ of load i hed immediately afte the ditubace, the pot-heddig load coductace i: = ( λ) (3) () The pot-heddig load demad i: P = V (33) If the pot-heddig iitial coditio ae withi the egio of tability, the tajectoie will covege to the table equilibium poit. Theefoe, the miimal amout of load heddig a a fuctio of time elaped ice the iitial potditubace coditio ca be deived a the hoizotal ditace betwee the pot-ditubace tajectoy ad the tability bouday of the table equilibium poit. Hee, we ue the hoizotal ditace becaue the tap atio of the tapchagig tafome i uchaged at the itace of load heddig. Fo the choe iitial coditio, the miimal amout of load coductace heddig (i. p.u.) a a fuctio of time elaped ice pot-ditubace coditio i how i Fig (p.u.).5 miimal load heddig (p.u.).5 Phae Potait pot-ditubace table equilibium poit pot-ditubace iitial coditio pot-ditubace tajectoy Computed Stability Bouday Fig. 6. Pot-ditubace tajectoy ad egio of tability. Active Powe hed (p.u.) t (ecod) Fig. 8. Miimal amout of load heddig a a fuctio of time elaped ice pot-ditubace coditio. Fig. 8 how that the equied miimal amout of load heddig iceae damatically afte ecod. The eult ae validated though time domai imulatio fo diffeet time delay. Fig. 8 idicate that the miimal amout of load heddig at t=5. i.3 p.u. It mea that if the amout of load which i geate tha o equal to.3 p.u. i hed at time 5., we ca guaatee the ytem i table. The time domai imulatio how i Fig. 9 cofim thi eult. O the othe had, if a le amout of load i hed at the time of 5., the ytem may loe tability. Fig. give the time domai imulatio fo the ituatio that. p.u. load i hed at the time of 5.. It i evidet fom time domai imulatio that the voltage collape evetually. If the load heddig actio i take ooe, le load eed to be hed to tabilize the ytem. Fig. 8 how that oly.5 p.u. load eed to be hed at t=3.3 i ode to tabilize the ytem. The coepodig time domai imulatio eult i peeted i Fig...4. Active Powe t (ecod) Fig. 7. The tajectoie with the iitial coditio ( () =.4, () =.9) t (ecod) Fig. 9. Time domai imulatio fo.3 p.u. load heddig at t=5..

6 Active Powe t (ecod) Fig.. Time domai imulatio fo. p.u. load heddig at t= Active Powe t (ecod) Fig. Time domai imulatio fo.5 p.u. load heddig at t=3.3. V. CONCLUSION AND DISCUSSION Thi pape peet a appoach fo computig the miimal amout of load heddig equied to tabilize a ytem that othewie will divege. The computatio utilize egio of tability computatio. The egio of tability of a table opeatig poit of the pot-ditubace ytem i computed a the backwad eachable et of a mall eighbohood of the table equilibium poit, ad i computed uig level et method. The miimal amout of load heddig a a fuctio of time elaped ice pot-dutubace coditio i ytematically deived a the hoizotal ditace betwee the pot-ditubace tajectoy ad the tability bouday of the table opeatig poit. The umeical tet coducted o a model coitig of a ifiite bu feedig a dyamic load though a tap-chagig tafome how the effectivee of the popoed appoach. The peeted method i applicable to moe complex ytem. The computatio time of backwad eachable et i acceptable fo ytem with up to five tate pace dimeio. Howeve, the method may ot be efficiet fo lage powe ytem. Futhe wok i eeded to iceae the efficiecy of the backwad eachable et computatio algoithm. VI. REFERENCES [] C. W. Taylo, Powe Sytem Stability. EPRI Powe Sytem Egieeig Seie. Mcaw Hill, 994. [] IEEE/CIRE joit tak foce o tability tem ad defiitio, Defiitio ad claificatio of powe ytem tability, IEEE Ta. o Powe Syt., vol. 9, pp , Aug. 4. [3] C. W. Taylo. Cocept of udevoltage load heddig fo voltage tability. IEEE Ta Powe Delivey, vol. 7, pp , Ap. 99. [4] Z, Feg, V. Ajjaapu, D. J. Maatukulam, A compeheive appoach fo pevetive ad coective cotol to mitigate voltage collape, IEEE Ta Powe Syt., vol. 5, pp May. [5] S. eee, I. Dobo, ad F. L. Alvaado, Cotigecy akig fo voltage collape via eitivitie fom a igle oe cuve, IEEE Ta. Powe Syt., vol. 4, pp.3-4, Feb [6] M. Ni, J. D. McCalley, V. Vittal, S. eee et al., Softwae implemetatio of olie ik-baed ecuity aemet, IEEE Ta. Powe Syt., Vol. 8, pp. 65-7, Aug. 3. [7] T. J. Ovebye, Computatio of a pactical method to etoe powe flow olvability, IEEE Ta. o Powe Syt., vol., pp. 8-87, Feb [8] T. Va Cutem. A appoach to coective cotol of voltage itability uig imulatio ad eitivity. IEEE Ta. Powe Syt., Vol., pp. 66-6, May 995. [9] T. Va Cutem, C. Moo, ad D. Lefebve, Deig of load heddig cheme agait voltage itability uig combiatoial optimizatio. i Poc. IEEE Powe Eg. Soc. Wite Meetig, vol.,, pp [] S. Abog,. Adeo, D. J. Hill, ad I. A. Hike. O ifluece of load modellig fo udevoltage load heddig tudie. IEEE Ta. Powe Syt., vol. 3, pp , May 998. [] M. A. Pai, Eegy Fuctio Aalyi fo Powe Sytem Stability. Boto/Dodecht/Lodo: Kluwe Academic Publihe, 989. [] R. eeio, M. Tataglia ad A. Vicio, O the etimatio of aymptotic egio of tability: tage of the at ad ew popoal, IEEE Ta. Automatic Cotol, vol. 3, pp , Aug [3] M. Pavella, D. Et, ad D. Ruiz-Vega, Taiet Stability of Powe Aytem: A Uified Appoach to Aemet ad Cotol. Boto/Dodecht/Lodo: Kluwe Academic Publihe,. [4] H. D. Chiag, M. W. Hich ad F. F. Wu, Regio of tability of oliea autoomou dyamical ytem, IEEE Ta. Automatic Cotol, vol.33, pp. 6-7, Ja [5] H. K. Khalil, Noliea Sytem, 3d ed. Uppe Saddle Rive, NJ: Petice- Hall, [6] S. Ohe, ad R. Fedkiw, Level Set Method ad Dyamic Implicit Suface. New Yok: Spige-Velag Ic., 3. [7] S. Ohe, ad J. Sethia, Fot popagatig with cuvatue depedet peed: algoithm baed o Hamilto-Jacobi fomulatio, Joual of Computatioal Phyic, vol. 79, pp. -49, 988. [8] J. A. Sethia, Level Set Method ad Fat Machig Method Evolvig Iteface i Computatioal eomety, Fluid Mechaic, Compute Viio, ad Mateial Sciece. Cambidge, Uited Kigdom: Cambidge Uiveity Pe, 999. [9] Ia Mitchell. A Toolbox of Level Set Method Veio.. [Olie]. Available: [] Thiey Va Cutem, Cota Voua, Stability of Electic Powe Sytem. Boto/Dodecht/Lodo: Kluwe Academic Publihe, 998. [] P. W. Saue ad M. A. Pai, A compaio of dicete v. cotiuou dyamic model of tap-chagig-ude-load tafome, i Poc. Bulk Powe Sytem Pheomea III Stability ad Secuity, 994, pp VII. BIORAPHIES Haifeg Liu eceived the B.S. ad M.S. degee i electical egieeig fom Zhejiag Uiveity, Hagzhou, Chia, i ad 3, epectively. He i cuetly puig the Ph.D. degee i the Depatmet of Electical ad Compute Egieeig at Iowa State Uiveity, Ame. Hi eeach iteet i powe ytem hybid cotol. Licheg Ji eceived the B.S. ad M.S. degee i electical egieeig fom Zhejiag Uiveity, Hagzhou, Chia, i ad 3, epectively. She i

7 7 cuetly puig the Ph.D. degee i the Depatmet of Electical ad Compute Egieeig at Iowa State Uiveity, Ame. He eeach iteet ae powe ytem hybid cotol ad dicete evet ytem. Vekataamaa Ajjaapu eceived hi Ph.D. degee i Electical Egieeig fom the Uiveity of Wateloo, Otaio, Caada, 986. Cuetly, he i a Aociate Pofeo i the Depatmet of Electical ad Compute Egieeig at Iowa State Uiveity, Ame. Hi peet eeach i i the aea of eactive powe plaig, voltage tability aalyi, ad oliea voltage pheomea. Rateh Kuma eceived the B.Tech. degee i electical egieeig fom the Idia Ititute of Techology, Kapu, Idia, i 987 ad the M.S. ad Ph.D. degee i electical ad compute egieeig fom the Uiveity of Texa at Auti, i 989 ad 99, epectively. He i cuetly a Aociate Pofeo i the Depatmet of Electical ad Compute Egieeig, Iowa State Uiveity, Ame. Hi pimay eeach iteet ae dicete-evet ytem ad thei applicatio. Jame D. McCalley eceived the B.S., M.S., ad Ph.D. degee fom eogia Ititute of Techology, Atlata, i 98, 986, ad 99, epectively. Cuetly he i a Pofeo i the Electical ad Compute Egieeig Depatmet at Iowa State Uiveity, Ame, whee he ha bee ice 99. He wa with Pacific a ad Electic Compay fom 986 to 99. Nicola Elia eceived the Lauea degee i electical egieeig fom Politecico di Toio, i 987 ad the Ph.D. degee i electical egieeig ad compute ciece fom the Maachuett Ititute of Techology i 996. Peetly he i a Aitat Pofeo at the Depatmet of Electical ad Compute Egieeig at Iowa State Uiveity. Hi eeach iteet iclude computatioal method fo cotolle deig ad hybid ytem. Vijay Vittal eceived the B.E. degee i electical egieeig fom Bagaloe, Idia, i 977, the M.Tech. degee fom the Idia Ititute of Techology, Kapu, Idia, i 979, ad the Ph.D. degee fom Iowa State Uiveity i 98. Cuetly, he i a Pofeo i the Electical ad Compute Egieeig Depatmet at Iowa State Uiveity, Ame. D. Vittal i the ecipiet of the 985 Peidetial Youg Ivetigato Awad.