Reducing the Order of Very Large Power System

8/22/2019 Reducing the Order of Very Large Power System

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IEEE Transactions on Power Systems, Vol. 3, No . 1 , February 1988 127

G Troul l i nos , J . DorseySchool of E lectr i cal Engineer i ngGeorgi a I nsti t ute of Technol ogyAt l ant a, Geor gi a 30332-0250

ABSTRACTResul t s are present ed f or r educed or der model s ofpower syst ems as l ar ge as 254 gener at or s, 2500 buses,based on t he modal coherency met hod of model r educt i on.Extensi on of order est i mati on, based on bal ancedreal i zati ons, t o l arger systems i s di scussed. Anal t ernati ve order est i mati on approach based on thei ntergenerator coherency r anki ng t abl e i s i ntr oducedand compared t o t he bal anced r eal i zat i on approach. Theal ternat i ve approach i s consi st ent w th t he bal anci ngr esul t s and more t han an or der of magni t ude f aster .

INTRODUCTIONA topi c of ongoi ng i nter est i n t he area of powersystem anal ysi s has been t he determ nat i on of usef ulr educed order power syst emmodel s. The i ni t i al moti va-t i on f or r educi ng t he order of t he power syst em modelwas t o r educe t he comput ati on t i me for t r ansi entstabi l i ty studi es . A decade ago, a t r ans i ent s t abi l i t yst udy coul d easi l y t ake many hour s of computer t i me.The evol ut i on of comput er hardwar e and sof t war e hasamel i or at ed t hi s probl em and t he speed and memor ycapaci t y of pres ent computer s has made tr ansi ents tabi l i t y anal ysi s a manageabl e t ask. Furt heri ncreases i n comput er capaci t y w l l be wel comed becausethey w l l al l ow power system analyst s t o study l argersystems more ef f i c i ent l y , but they w l l not spel l thedi f f erence bet ween doi ng or not doi ng an ef f ecti vetr ans i ent s tabi l i ty s tudy.The i mpr ovement s i n the speed and memor y of compu-t ers t hat have made tr ansi ent stabi l i t y anal ysi s ar out i ne busi ness, have necessari l y opened up t he possi -bi l i t y of sol vi ng other more compl ex probl ems, l ong ofi nterest . Two of t he more prom nent pr obl ems at t hemoment are secur i t y assessment and on- l i ne syst emi denti f i cati on. These are pr obl ems wher e the comput a-t i onal bur den i s ver y hi gh and accur ate r educed ordermodel s may be of vi t al i mpor t ance. Thi s paperdi scuss es some r educed order model i ng t echni ques thatwere ori gi nal l y desi gned for use i n off - l i ne t ransi entstabi l i t y s tudi es . I n the course of events , themodel i ng t echni ques have been i ncor por ated i nt o asof t ware package, and test ed i n a prel i m nary way onsome f ai r l y l arge power systems. The resul t s of that

37 IJM 097-9by the I EEE Power Syst emEngi neer i ng Comm t t ee oft he I E E E Power Engi neeri ng Soci et y f o r present at i onat the IEEE/PES 1987 W nt er Meeti ng, New Or l eans,Loui si ana, February 1 - 6, 1987.August 25, 1986; sade avai l abl e f o r pr i nt i ngNovember 17, 1986.

A paper r ecommended and appr oved

Manuscri pt suhmt t ed

H. Wong and 3. Myer sSout hern Company Servi ces, I nc.P. O Box 2625Bi r m ngham Al abama 35202

test i ng are cont ai ned i n t hi s paper. I n addit i on, t hi spaper exam nes, i n a prel i m nar y way, how t hesemodel i ng t echni ques can be appl i ed to t he probl em ofon- l i ne sys tem i dent i f i cat i on. A c ruc i al i s sue fo rth i s probl em i s determ ni ng the m ni mum order of themodel of t he exter nal system The order est i mati ont echni ques devel oped i n thi s paper pr ovi de an answer t othat questi on.The r educed or der model s di scussed i n the sequelare based on an anal ysi s t echni que, modal - coherency,f or whi ch an order est i mat i on scheme has beenprevi ousl y proposed [1,2]. The anal yti cal j usti f i ca-t i on for t hi s model i ng techni que has been previ ousl ychroni cl ed, and t he present paper concentr ates on f i r stt esti ng t he vi abi l i t y of t he model s and second on

di scuss i ng means by whi ch t he comput at i onal speed oft he model i ng and order est i mat i on process es can bei mproved. Comput ati onal resul t s are presented f ort hree power syst em model s, one w t h 86 gener ator s and600 buses, and t wo w t h approxi matel y 250 gener ator sand 2500 buses. The vi abi l i t y of esti mati ng t hef easi bl e amount of order r educti on i s al so assessed.Previ ous r esul t s [1,21 i ndi cated t hat t he concept ofbal anced real i zat i ons coul d be successf ul l y appl i ed t osystems w th 100 gener ator s or l ess. The appl i cabi l i tyof t hi s approach t o l arger power syst ems i s di scussed,and some resul t s provi ded for a system w th 240generat or s and 2600 buses.These resul t s ar e t hen used as a basel i ne agai nstuhi ch t o compare an al t ernat i ve means of esti mati ngorder r educti on. Thi s al t ernat i ve measure is approx-i mate, but has t he advantages of bei ng comput ati onal l ymore ef f i ci ent and appl i cabl e t o syst ems w t h up t o 700generat ors.

1. TESTS ay LARGE SYSTEWSThe t est i ng of t he modal - coherency met hod ofgenerat i ng reduced or der model s has been done l ar gel yon power syst em model s of t he Sout her n Company Syst emI ni t i al l y, an 86 gener ator model was used [21.Subsequent t est i ng has been done on a 254 gener at ormodel deri ved fr om the base case condi t i ons i n thesummer of 1984, and a 240 gener at or model deri ved f r omt he base case condi t i ons in 1986. The sour ce of botht hese model s i s t he f ul l Sout hern Company Model whi chhas j ust over 400 generat ors. The i ni t i al model r educ-t i on was done usi ng engi neeri ng j udgement , t o pr ovi de anodel l arge enough to t est t he order est i mati on t ech-ni ques, but m ni m ze the devel opment costs. Both oft hese l arger model s have roughl y 2500 buses and 4000l i nes. Even w th t hese model s, i t was not possi bl e t odi r ectl y appl y t he order est i mati on based on bal anced

r eal i zat i ons [21 because of t he si ze of t he syst em TOsee why, consi der t he power sys t emmodel used i n modal -coherency anal ysi s [ l - 51 whi ch consi st s of a dynam cequat i on of t he f ormd2gi d6dt 2M - PM - PGi - Di

0885-8950/88/02OO-O 127$0 .OO0 988 IEEE

Authorized licensed use limited to: King Fahd University of Petroleum and Minerals. Downloaded on May 17, 2009 at 16:02 from IEEE Xplore. Restrictions apply.


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f or each generat or, i , i n an N- gener ator syst em pl us aset of al gebr ai c net work equat i ons t o repr esent t hepower f l ows, bot h real and r eact i ve, of t he systemPM , Xi , , and Di represent , respecti vel y, t hemechani cal power i nput, t he el ectr i cal power out put ,t he i ner t i a and dampi ng for gener ator i .I f t he power syst em i s l i neari zed about a st eadystat e operat i ng poi nt , and i f addi t i onal l y i t i s

assumed that t he r ati o Di / M = y i s t he same f or al lgener ator s the st ate model ' represent ati on i s of t hef o rm(1)(t) =A x(t) + B u(t)

A Aw t h x = [ A6, Aw] ' , U = [ APUAPL]' , wher e

1 1A Aand 86, Au, Am, and APL are vector s r epresent i ng,r especti vel y, angl e and speed devi ati ons of gener ator sand changes i n mechani cal i nput and l oad power. Thepr e f i x A i ndi cates t hat a vari abl e i s a smal l devi ati onf r oma st eady st ate operat i ng poi nt. C i s the i dent i tymatr i x.Thi s model i s t he si mpl est possi bl e repr esentati onof t he gener ator s of a power s ystem t hat i s asympt ot-i cal l y stabl e and, therefore , sui t abl e f or use i n thebal anci ng approach. Even at t hat , t he di mensi on of t hepl ant mat r i x i s 2N- 2, wher e N i s t he number of gener a-t ors i n t he model . Bal anci ng r equi r es t he comput at i onof both t he cont r ol l abi l i t y and observabi l i ty gram ans,bot h of t he same order as the pl ant matr i x. I npracti ce i t has not been possi bl e t o rel i abl y anal yzesyst ems w t h more t han 200 gener ator s. That i s, oncet he contr ol l abi l i t y and observabi l i t y gramm ans reach

di mensi on 400, convergence is unl i kel y. Two s ol ut i onmet hods have been t r i ed, t he or i gi nal approach whi chcomput es the i nvers e of t he pl ant matr i x [2], and ani t erati ve sol uti on of t he Lyapunov equati ons 161. Bothmethods f ai l at about t he same syst emsi ze.Syst ems w t h more t han 200 gener at or s can beanal yzed by empl oyi ng some of t he propert i es of modal -coherency model s establ i shed i n ear l i er work [l -41.Speci f i cal l y, t he modal - coherency approach i s capabl eof det erm ni ng a nest ed set of ei t her l ocal or gl obalmodel s of t he power syst em based on an rms coher encymeasur e

and usi ng st ep di st urbances [4, 51. The gl obal model sr eveal t he macroscopi c i nter area behavi or of t hesystem t he l ocal model s are used for area speci f i cdi st urbances such as a f aul t at a par t i cul ar gener ator.For ei t her t ype of model , a sequence of di st urbancesare appl i ed to t he power syst em t o produce a r anki ngt abl e t hat contai ns the rel ati ve i nt ergeneratorcoher ency measure bet ween ever y pai r of gener at or s i nt he power syst em An exampl e of such a t abl e i s showni n Tabl e 1 f or a gl obal di st urbance of t he 39-BusNew Engl and Syst em Thi s t abl e r anks gener ator s f r om

Table 1 Global Ranking Table for 39-Bus N e w EnglandSystem.Rnk i n p Ge mr a to f

1.2.3.4.5.6.7.8.9.10.11.1213.14.15.16.17.18.19.20.21.2223.24.25.26.27.28.29.30.31.32.33.34.35.36.37.38.39.40.41.42.43.44.45.

t he most coher ent

Pair(6.71(1.8)(4.7)(4.6)(4.8)(2.3)(3.8)(2.8)(7.8)(1.2)(1.3)(1.4)(4.5)(1.7)(6.8)(3.41(3.7)(1.6)(2.4)(2.7)(3.6)(8.9)(4.91(2.61(7.9)(5.71(1.91(6.9)(5.6)(3.9112.9)(5.8)(3.5)(1.5)(5.9)(2.5)(1.10)(8.10)(2.10)(2.10)14.10)(7.10)(6.10)(9.10)(5.10)

c&muq Aweeat ionMuwn LWE I2.74 1275 23.123.30 33.823.89 43.913.923.933.984.w 54.024.084.134.164.194.284.364.384.484.494.584.654.694.714.844.874.905.335.465.755.935.976.006.148.678.929.229.W10.6710.7710.9611.171241

4806

7

8

pair t o the l east coher ent pai r.Gener at or s can t hen be aggr egated i nt o groups byappl yi ng a speci f i c gr oupi ng rul e [ 4, 51 to the r anki ngt abl e. Each aggr egati on ei t her combi nes t wo unaggre-gated generat or s i nt o a new gr oup, adds an unaggr egat edgener ator t o an exi st i ng group, or combi nes t woexi sti ng groups i nt o a si ngl e group. No mat t er whi chevent OCCUKS, each aggr egati on reduces t he number ofgener ator s in t he model by one. Carr i ed t o i t sext reme, t hi s process wi l l i n N-1 st eps r educe t hewhol e power s yst em t o one equi val ent generator . Theprocess has t he val uabl e att r i but e t hat i t pr oduces aconsi st ent set of model s. That i s, a part i cul ar gener-ator w l l never bel ong to one gr oup at some poi nt i nt he aggr egati on process and then move t o a di f f er entgroup at s ome poi nt f ur t her al ong i n the aggr egati onprocess. Thi s i s somet hi ng t hat cannot be guaranteedw t h other aggregat i on t echni ques. Furt her more, si ncet he met hod i s based on gener at or coher ency, even thought he anal ysi s techni que uses a l i near model , t he act ualr educed or der power syst em equi val ent can be nonl i near[ 4, 51-I n t he aggr egati on process, t he most coherentgener ator s are combi ned f i rs t . Previ ous r esul t s havemai ntai ned t hat t he modes of t he l i near model di scardedat each aggr egat i on are t he most uncont r ol l abl e.

I n a syst em w t h 250 odd generat ors, t he most coherentgener ator s ar e very coher ent. The modes associ at edw th t hese generat ors can be el i mnated w th l i t t l eeff ect on the Or der est i mati on process because t hei rel i m nati on does not appreci abl y ef f ect t he cont rol l -abl e and margi nal l y contr ol l abl e modes. That i s, t hel ower order model w l l pr eser ve the contr ol l abl e andmargi nal l y cont r ol l abl e modes of t he hi gher ordermode 1


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129I n t erms of the l i near model , aggregati on ofgener ator s i and j cor r esponds t o addi ng rows i and jand col umns i and j of t he matr i x -KC. Thi s i ni t i alaggr egati on i s based on t he gl obal r anki ng t abl e,created by di sturbi ng al l t he generat ors of the systemThe ef f ect i s t o make the matr i ces used i n thebal anci ng computat i ons numeri cal l y more st abl e.I n pract i ce thi s method seems to work f ai r l y wel l .F i gures l a and l b are gl obal order esti mati on curves

f or t he 254 gener at or model where t he model has beeni ni t i al l y aggregated down to 1 8 0 and 140 gener at or s,r espect i vel y. These cur ves ar e based on a measur e of

wher e t he U . are t he si ngul ar val ues of t he bal ancedsyst em [1,2f. Thi s measure has been normal i zed t o ascal e of 100. The l arger p(i ), t he l ess accurat e t her educed model . The si ze of the r educed model i s t he(a)

AGGREGATIONLEVEL1 I

AGGREGATIONLEVELFig. la L lb Global Order ESt iMt iOn Curve for 254Generator System Initially Reduced t o 180and 140 Generators.di f f erence between the number of gener ator s i n theunr educed model and t he aggr egati on l evel . Thesegl obal order r educti on curves have a char acteri st i cshape. They r i se l i nearl y w t h a f ai r l y l ow sl ope andt hen bend sharpl y upwards. Experi ment al r esul t s i n ai n a subsequent sect i on w l l show t hat t he bend i n thecurve matc hes qui t e wel l t he l argest j umps bet weenaggregati ons i n t he associ ated ranki ng t abl e. Thecurves can be compared by deter m ni ng t he poi nt on eachcurve at whi ch the ri se ceases to be l i near and thepoi nt where t he tangent of t he knee i s 45 degrees.Note that f or both curves, subtr acti ng the model si zef rom the aggregati on l evel at whi ch the cur ve ceases i t

l i near r i se, y i e l ds about 90 equi val ent generat ors.Subtr act i ng t he aggregati on l evel at whi ch t he t angentt o the knee is 45 degrees, f rom t he model si ze yi el dsabout 30 equi val ent generators. Thi s rei nforces t hei dea that i f t he f ul l 254 gener ator model coul d beanal yzed, t he l i near port i on of the cur ve woul d bel onger by ( 254-m aggregati ons wher e m i s t he order oft he reduced model , movi ng t he nonl i near par t of t hecur ve t o the ri ght by t hat same number of aggr egati ons,so t hat measuri ng t he poi nt where t he l i near r i se endsand t he t angent r eaches 45 degr ees on t he unreducedmodel woul d yi el d essenti al l y t he same i nfor mati on,namel y 90 and 30 equi val ent gener ators .

Thi s approach was appl i ed to t he 240 gener at ormodel t o deter m ne reduced order model s f or a part i c-ul ar di st urbance, namel y a three phase f aul t at Pl antScherer . Pl ant Scher er i s a good choi ce f or theappl i cat i on of a d is t urbance because, e l ectr i cal l y , i ti s nearl y i n the heart of the 500 kV t r ansm ssi onsyst em and str ongl y coupl ed to other maj or gener ati onuni ts.The pr ocess of deter m ni ng a r educed order modelf or a di st urbance at Pl ant Scher er r equi res t wo st eps.F i r s t , t he g lobal rank ing tabl e i s used to f i nd t hef i rst group to whi ch Scherer bel ongs. The si ze of t hi sgroup w l l depend upon whet her Scherer j oi ns an exi st -i ng group or i s pai red w th another generator t o form a

new group. I n t hi s case, t he f ormer event occurs , andScherer pai rs w t h Pl ant s Arkwri ght and Si ncl ai r t of orm a new group. Pl ant s Arkwri ght and Si ncl ai r aregeogr aphi cal l y cl ose t o Scherer, but are on the 115 kVr ather t han the 500 kV t r ansm ssi on syst emThe order est i mati on cur ve is obtai ned by bal anc-i ng a modi f i ed l i near model. The modi f i cati on consi st sof el i m nat i ng al l d i agonal t erms, i n the f i rs t N-1col umns, f rom t he matr i x M except t hose corr espondi ngt o the generat ors of t hat gr oup to whi ch Schererbel ongs. I n t he present case, al l di agonal el ement s of

U w l l be e l i m nated except those correspondi ng t oScherer and Ar kwri ght and Si ncl ai r. The corr espondi ngorder esti mati on curve i s shown i n Fi g. 2. The curveshows a sl ow st eady ri se to aggr egat i on l evel 1 5 0 , atwhi ch the curve r i ses very sharpl y. Thi s curve i ssomewhat i n contr ast to that f or a s i m l ar ana l ys i s of

100

80

hX 304201 fl o o o

AGGREGATION L N E LPig. 2 Order Bstimation Curve for Disturbance atScherer, 240 Generator System, InitiallyReduced to 180 Generators.

Authorzed censeduse mted to: Kn FahdUnverst o fPetroeum and Mneras. DownoadedonMa 17 2009at16:02 fromIEEEX ore . Restrc tons a .


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100-90-a0-70 -60-50 -40

Pl ant Scherer f or t he 25 4 gener at or model , shown i nFi g. 3. Here, t he cur ve i s near l y zero unti l aggrega-t i on l evel 90, r i ses sl ow y to aggregati on l evel 150,and t hen t urns upward very r api dl y. Fi gur e 3 i ssomewhat easi er t o i nterpret t han Fig. 2. I n thef ormer case, i t appear s t hat a reduct i on t o 45-50generat or s coul d be done w t hout i ncur r i ng much er r or.I n Fi g. 2, however, t he i nterpret ati on i s not so cl ear.

Izw0LTaz 3ol0 I

30 60 90 120 150 180AGGREGATION 1.EVEL

Fig. 3 Order Estimation Curve for Disturbance atScherer, 254 Generator System, InitiallyReduced t o 180 Generators.The shape of t he cur ve r esul t s f rom t he f act t hat t he24 0 gener ator model has t wo si ngul ar val ues whi ch aret hree orders of magni t ude l arger t han al l othersi ngul ar val ues. The t wo very l arge si ngul ar val uesdom nat e the denomnat or P( i ) so t hat i t r emai ns essen-t i al l y unchanged. The numerat or i s al ways smal l bycompar i son. On ot her model s st udi ed, ther e have been agroup of l arge si ngul ar val ues, so t hat t he pl ot ofP(i ) st art s out ver y l ow and t hen sw ngs upwardrapi dl y once t he l arge si ngul ar val ues begi n toi nfl uence t he numerator of P (i ).

The reduced order model i s f ound by f or m ng ar anki ng tabl e based on the di st urbance of onl y Scherer ,Ar kwr i ght , and Si ncl ai r , and t hen proceedi ng throught he r anki ng t abl e unt i l t he appropri ate number ofequi val ent gener ator s i s r eached. Gi ven the shape oft he cur ve i n Fi g. 2, t he f ol l ow ng test procedure wasused. Sever al model s were devel oped, of decreasi ngsi ze, and t hen test ed agai nst t he unr educed model t osee i f a det ectabl e degradati on i n accur acy coul d bedi scerned. Fi gur es 4, 5, and 6 show some of t her esul t s. I t i s cl ear t hat down t hrough model s of s i ze7 5 gener ator s, t he response of t he reduced order modeli s very c l ose to t hat of t he unr educed model f or oneand a hal f s econds. Af t er t hat , t her e are some angl edevi ati ons on t he order of t wo t o f i ve degr ees. I naddi t i on to t he generator aggregati ons, a generalnet work r educt i on was perf ormed on t he equi val ent s.The or i gi nal net work had 2500 buses and 4000 l i nes .I t can be seen t hat qui t e a bi t of network reduct i on i spossibl e w t h very l i t t l e change in accur acy, and thatt here i s l i t t l e to choose between the 1000 bus, 1900l i ne equi val ents o f F ig . 4, and t he 50 0 bus and 1300l i ne equi val ent s of Fi g. 5 Fi gure 6 shows t hedi st urbance of Pl ant Sowen whi ch i s on t he 500 kVtr ansm ssi on system and el ectr i cal l y cl ose to Scherer.

- _ I . -L- L-1 .-L U L L - - -0.6 1.2 1.8 2.4TIME IN SECONDS0 .0

Fig. 4 Comparison of F u l l Model (-) to 150, 100, and75 Generator Equivalents, 1003 Buses, 1919Lines, Three Cycle Fault at Scherer, Angle ofScherer.~. ,-I-.1 --T-r---

TIME IN SECONDSFig. 5 Comparison of Unreduced Model (-) to 150, 100.and 75 Generator Equivalents, 505 Buses, 1353Lines, Three Cycle Fault at Scherer, Angle ofScherer.

A l egi t i mate quest i on i s why t her e are no testr esul t s f or model s of l ess t han 75 generators. I nf a c t , a 50 gener ator case was part of t he or i gi nal t estprocedur e. However, i t was i mpossi bl e t o achi eve asati sf actory l oad f l ow for t he 50 generator case, w t hnet work r educti on. A s a resul t , i t was not possi bl e tocompar e t he 50 gener ator model i n a consi st ent way t ot he ot her model s.An i nt erest i ng f eat ure of t he t est r esul t s i s thatt he r educed model s ar e f ai r l y i mper vi ous to thesever i ty of the faul t . F i gure 7 compar es t he unreducedsyst em t o r educed model s of order 150, 100, and 75genert ors or a ni ne cycl e faul t. A s can be seen, t her esul t s are very good for over t wo seconds.


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-A L L - l - - . L L - l -0.6 1.2 1.8 2. 4TIME IN SECONDS0. 0

Pig. 6 Coqac ison of Unreduced l w e l (-1 to 150, 100,and 75 Generator Equivalents, 503 Buses, 1353Lines, Three Cycle Fault at Scherer, Angle atRowen.

0 1 2 3 4 5TIME IN SECONDS

Pig. 7 Omparison of Unreduced System l w e l with 240Generators to Reduced Order Uodels with 150,100, and 75 Generators for a Nine Cycle, ThreePhase Fault at Plant Scherer, Angle at Scherer.A poi nt t hat per haps meri t s emphasi s i s t hat t het est resul t s r eport ed her e are f or detai l ed nonl i nearmodel s t hat i ncl ude f ul l gover nor and exci t or cont r olr epr esent ati ons. For t he r educed model s, t he gener -at or s wer e aggregat ed and r epl aced by an equi val entmachi ne, t he l oad buses aggr egat ed and t hen a new l oadf l ow establ i shed.

3. EXTENDING ORDER ESTIPIATION To URGER SYsTEplsThe resul t s pr esent ed i n Sect i on 2 i ndi cat e t hatt he bal anci ng appr oach gi ves a reasonabl e est i mat e oft he proper order of a r educed or der model f or moder at esi zed power syst ems w t h 200 t o 300 gener ators. I t maybe possi bl e to extend t hi s t echni que t o syst ems of 4 00t o 5 0 0 generators. The resul t s i n Sect i on 2 showedthat f or a sys temw th 250 gener ators, i t was possi bl e

t o r educe the l i near model by 30 per cent , say t o 180gener ator s, and st i l l appl y t he bal anci ng approach.However, t he 2 5 0 gener ator model was i t sel f a reducedversi on of a 4 00 generat or model . Whether i t woul d beposs i bl e to s ta r t w th t he 4 0 0 gener ator model , r educei t t o 180 gener ators , perf orm t he bal anci ng, and thenesti mate t he Order r educt i on f or t he or i gi nal 40 0generator model i s a possi bi l i ty that i s curr ent l ybei ng i nvesti gat ed.However , even i f t hi s appr oach does work, t hecomput ati onal burden i s qui t e heavy. Even w t h a l argecomput er t he t ot al CPU t i me i s nontr i vi al . For a l argeuti l i t y engagi ng i n a l arge number of t r ansi entstabi l i ty studi es on a s i ngl e base case, t hi s t i me w l l

be i nconsequent i al compared t o t he ti me saved by usi ngthe equi val ent s, but t he approach i s cert ai nl y notcompat i bl e w t h t he Curr ent r esearch pr obl ems ofsecur i t y assessment and on- l i ne i dent i f i cat i on.I f r educed Or der model s are t o be appl i cabl e t ot hese l att er pr obl ems, t hen t hey must be comput ed veryqui ckl y. To achi eve a mor e rapi d est i mator f or veryl ar ge syst ems, use can be made of t he ranki ng t abl e ofrel ati ve coher ency. For an N gener ator power syst emt h i s t abl e w l l have N x (N-1)/2 entr i es. Tabl e 1 i sthe ranki ng tabl e f or a gl obal di st urbance of t he 39-Bus New Engl and sys t em whi ch has 1 0 generator s. Thet abl e has onl y 4 5 ent ri es, maki ng i t a sui t abl e candi -

dat e for demonst r ati on. The groupi ng r ul e appl i ed t ot hi s t abl e i s t he so- cal l ed commut ati ve rul e [ 2 , 5 1 .Basi cal l y thi s r ul e says that a generator can be addedt o a gr oup onl y when i t has est abl i shed a coherencyconnecti on t o al l exi st i ng member s of t hat group. I nt' erm of t he r anki ng t abl e, generator i i s added t o agroup at t hat l evel of t he tabl e where t he r el ati vecoher ency measures l i nki ng gener ator i t o al l member sof t he exi st i ng group are at or above t hat l evel i n t heranki ng t abl e. For exampl e, i n Tabl e 1 , t he t wo mostcoherent gener ator s are 6 and 7 whi ch i mmedi atel y f or ma gr oup at l evel 1 of t he t abl e. The next t wo mostcoherent generat ors are 1 and 8 whi ch agai n f orm agroup at l evel 2 of t he t abl e. The next aggregat i on i snot f ormed unt i l l evel 4 at whi ch poi nt gener at or 4 hasestabl i shed coher ency w t h both gener ator s 6 and 7 andj oi ns that group.I ni t i a l l y , the groupi ngs occur very rapi dl y , that

i s, very f ew l evel s of t he r anki ng t abl e are tr aversedbet ween consecut i ve aggr egati ons. A s t he aggregat i onproceeds, however, more l evel s of t he coherency tabl ehave t o be t r aversed to est abl i sh t he next aggr egati on.Thi s i s a nat ural consequence of t he f act t hat each newgener ator pai r t hat i s encounter ed has t o be consi der edi n terms of i ts r el ati onshi p t o al l the groups t hathave been f or med hi gher up i n t he t abl e. I n t er ms oft he bal anced r eal i zati on methodol ogy, t he si ze of t hej umps ref l ect s the re l at i ve uncontr ol l abi l i ty o f themodes bei ng el i m nated. At t he t op of t he t abl e,aggregati ons occur vi rt ual l y at each l evel , so t hati ni t i al l y a l arge number of smal l groups of t wo ort hree gener ator s f orm very rapi dl y. These aggregat i onscorr espond t o t he el i m nat i on of t he most uncontr ol l -abl e modes. Fur t her i nto the t abl e t he groups begi nt o grow in si ze, and many l evel s of t he tabl e aret r aversed bet ween aggregati ons. Thi s ref l ects the f actt hat t he cohernecy r el ati onshi ps are weaker, andconsequent l y, t he modes t hat are el i m nat ed are morecontr ol l abl e. Thi s behav ior i s c l ear l y re f l ected i nTabl e 1 .One of t he f eat ures of t he commutat i ve r ul e i st hat t hi s f i nal aggregati on i nt o a si ngl e generatordoes not occur unt i l t he f i nal ent ry i n the ranki ngt abl e i s consi der ed. Thi s f eat ure of t he commutat i verul e can be used t o est abl i sh t he f ol l ow ng Or der


6/7

132est i mati on measur e. Gi ven t he r anki ng t abl e f or aparti cul ar di st urbance, start at the t op of t he ranki ngt abl e and det erm ne al l t he reduced order model s,keepi ng t r ack of t he number of ent r i es of t he ranki ngt abl e tr aver sed bet ween successi ve aggregat i ons.Nor mal i ze t hese j umps by t he l argest j ump, and pl ot t henor mal i zed numbers vers us the aggregat i on l evel s atwhi ch t hey occur. F i gure 8 shows such a pl ot f or t hedi st urbance at Pl ant Scherer. The bal anci ng orderest i mati on cur ve i s shown f or compar i son. Note t hatt he l arge j umps occur at t he knee of t he cur ve, andt hat t he two measures ar e gi vi ng about t he same

I100

70 Iaz 40

~

WWII : 50-a 40zc 30-a 20 -v

30 10. .'"110 20 30 40 1"1111150 60 70 80AGGREGATION L E E L

Fig. 8 Coqarison of Ranking Table and BalancingOrder Estimation, Disturbance at PlantScherer, 86 Generator System.est i mate of order r educti on. F i gure 9 shows t her anki ng t abl e est i mate for a di st urbance at Pl antScher er f or t he 2 5 4 gener at or syst em Based uponFi g. 3 , t he reduced Or der model w l l be i n the r ange30- 60 gener at or s, dependi ng upon wher e one st ops on t hesl ow ri se that st art s about aggregati on l evel 1 0 5 . Theesti mate f rom Fi g. 9 woul d be on the order of 3 5 - 4 0gener ators. The answers ar e not i denti cal , but t heyar e i n t he same range, and by compar i ng t he t wo, onewoul d pr obabl y concl ude t hat a r educed or der model of4 0 gener ator s woul d be ver y accur ate.

1 0 0

70

6oi

AGGREGATION I - N ELFig. 9 Ranking Tabl e Order Estimation Curves

Disturbance at Scherer, 254 Generator Model.

4. IUNIMOM ORDER HIR G I D W W E L SThere i s curr ent l y a hi gh degr ee of i nt erest i nprobl ems r el ati ng t o t he assessment of t he st abi l i t y ofl arge power systems. To per f orm t hi s assessment i nr eal t i me, OK near real t i me, r equi r es t he si mpl estpossi bl e model of t he over al l power system that wi l lst i l l y i el d a val i d resul t . The power syst emmodel sdi scussed i n thi s paper are al l deri ved f r om t he samemast er model of t he Sout her n Company, r educed by engi -

neeri ng j udgement t o var i ous l evel s of compl exi t y.These model s pr ovi de an oppor t uni t y to make a r eason-abl e assessment of t he m ni mum Or der gl obal model t hatmay be r equi r ed f or securi t y assessment probl ems.F i gure 1 0 i s t he order est i mati on curve based onbal anci ng for t he 2 5 4 gener ator model . Her e t he l i nearpar t of t he cur ve ends around aggr egat i on l evel 1 1 5 ,yi el di ng a gl obal model of or der 65. Fi gure 1 1 i s t heorder est i mati on cur ve based on the ranki ng t abl e f ort he same model. The f i r st si gni f i cant j umps occuraround aggr egati on l evel s 1 8 5 and 2 0 0 . I n bet ween ar eSome r el ati vel y smal l j umps. Thi s i ndi cat es t hat oncet he l arge j ump at aggregat i on l evel 1 8 5 i s accepted,t he model accuracy w l l not decl i ne si gni f i cant l y unti laround aggr egati on l evel 2 0 0 . Thi s yi el ds a mni mumgl obal model i n the range of 50 t o 7 5 gener ator s.

100

70z 60-

AGGREGATION LEVELFig. 10

100-90-ao-7 0 -

2 60-II : 50-

40z= 30-20 -10-

W0W

v

Global Order Estimation Curve for 254Generator System.

0 40 80 IZ U I O U LW W LTVAGGREGATI0N I I V EL

Fig. 1 1 Ranking Table Global Order Estimation Plot for254 Generator nodel.


7/7

13 3

Fi gure 12 i s t he order est i mati on cur ve based ont he r anki ng t abl e for t he 240 generat or model . Asi m l ar anal ysi s here yi el ds a m ni mum gl obal model ont he order of 50 t o 70 gener at or s, dependi ng uponwhet her t he f i r st maj or j ump at aggregat i on l evel 170i s accept ed.

10090

70

AGGREGATION LEVEL

Pig. 12 Ranking Table Global Order Estimation Plot for240 Generator Model.The si ze of t hese m ni mum gl obal model s cl ear l yhas some i mpact on t he anal ysi s t echni qugs t hat w l l beappropri ate t o securi t y assessment . For i nstance,consi der t he probl em of i dent i f yi ng a useful model oft he power s yst em exter nal t o t he boundar i es of t heSout her n Company. Under t he most opti m st i c est i mat e,t he gl obal model of t he over al l syst emwoul d have about10 ar eas w t hi n Sout hern Company, and 40 areas, orequi val ent gener ator s, exter nal t o Sout her n Company.Thi s pr esents a f orm dabl e i dent i f i cat i on probl em onet hat may not be sol veabl e w t hout t he shari ng of aconsi derabl e amount of dat a between ut i l i t i es.

5. SUmlARYTest r esul t s on l arge power syst ems have been

present ed whi ch i ndi cate t hat or der r educti on based ont he modal coher ency met hod can deter m ne accurat e l oworder model s of a power system Or der est i mat i on basedupon bal anced r eal i zati ons appears t o gi ve r easonabl yaccurat e esti mates of the f easi bl e order r educt i on,but more extensi ve test i ng i s needed to compl et el yval i dat e t hi s approach. An al t ernati ve measure oforder est i mat i on based upon t he modal - coher ency ranki ngt abl e has been i ntr oduced and shown t o be consi st entw t h t he order r educti on est i mates based on bal ancedreal i zat i ons. I t has the att ract i ve f eature of bei ngconsi derabl y f ast er t han the bal anci ng method andappl i cabl e t o much l arger syst ems.

Or der est i mat i on based on t he coher ency r anki ngtabl e has some very at t ract i ve f eatures. F i rst , i t i snumeri cal l y st abl e and it i s possi bl e t o appl y thi st echni que t o syst ems w t h 700 t o 800 gener ator s.Second, t he r educt i on can be done qui t e rapi dl y. For asys t em w t h 250 gener ator s and 4000 buses, a gl obalr educed or der model can be obt ai ned i n about 60 C. P.Useconds on an I BM 5860. A l ocal model t akes about30 C. P. U seconds. Over hal f of t hi s ti me goes to thecomput at i on of t he i nter gener ator coherency measures.Si nce t he sof t war e that comput es t he coherency measureshas not been opti m zed, t he comput at i onal speeds can bei mproved f ur t her. Thus, i f t he evol uti on of comput erscont i nues as expected, i t shoul cf be possi bl e i n thef oreseeabl e f uture to uti l i ze model reducti on tech-ni ques as a part of a l arge oper ati onal cont r ol andassess ment scheme. Whether t he r educed model s ar ebased on modal coherency r emai ns t o be seen. Ther e maybe bett er approaches. What i s cl ear i s t hat vi abl e,r educed order nonl i near model s can be deter m ned, anddet erm ned qui t e rapi dl y.

1. G. Troul l i nos , J. Dors ey, " Appl i cati on of Bal ancedReal i zati ons to Power SystemDynam c Equi val ent s, "I EEE Transacti ons on Automati c Cont r ol , V o l . AC-30, NO 4, pp. 414-416, Apr i l 1985.2. G TKOUl l i nOS, J . Dorsey, H Wong, J . Myer s,

S. Goodw n, "Est i mat i ng Or der Reducti on for PowerSyst emDynamc Equi val ent s, " I EEE Transacti ons onPower Apparat us and Syst ems, Vo l . PAS-104 , No. 12,pp. 3475-3481, December 1985.3. J . DOKSey, R. A. schi uet er, "Gl obal and LocalDynam c Equi val ent s Based on Str uctur al Archet ypesf or Coher ency, " I EEE Tr anSaCt i OnS on PowerAppar at us and Syst ems, Vol . PAS- 102, NO 6,pp. 1793-1801, J une 1983.4. J. Dorsey, R. A Schl ueter , "Str uctura l Archet ypesf or Coher ency, A Fr amewor k f or Compar i ng PowerSyst em Equi val ent s, " Automati ca, V o l . 20, No. 3,pp. 349-353, May 1984.5. R A. Schl uet er, U Ahn, and H Modi r , "Modal

Anal ysi s Equi val ent s Deri ved Based on the RMSCoher ency Measure, " I EEE Transact i ons on PowerAppar at us and Syst ems, Vol . PAS-98, No. 1143,J ul y/ Auqust 1979, ( Abst r act) and Text of Abstr act-Papers , 1979 W nter Power Meet i ng, I EEE Publ i ca-t i on 79 CH 1418-C.6. R H BaKt el S, G . W Stewart , "Al gor i t hm 432Sol ut i on of t he Matr i x Equat i on .Ax + xB = C, "Communi cat i ons of t he ACM Vol . 15, No. 9,pp. 820-823.

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Reducing the Order of Very Large Power System