New Concepts for Automatic Generation Control in Electric Power Systems Using Parametric Quadratic Programming

Copyrlrh, fJ I.·AC Rnl Timt Di,;ul Con,roI Appl;"a'ioru c.. adalajarl. Mu i(o 198'

NEW CONCEPTS FOR AUTOMATIC GENERATION CONTROL IN ELECTRIC POWER SYSTEMS USING PARAMETRIC QUADRATIC PROGRAMMING J.

L. Carpentier*, G . Cono** and P. L. N icderlander***

· Electricite de France , 2 rue Lo uis Murat , ]5008 , PorU , France "Elecfricife de France, 22 avenue de Wag ram , 7j 008 , Paris, Fra'lce " ' £ POS, J8 ru ede I'Yvette , 7j016 , Paris. Fran ce

Abstract. New concePts for Automa tic Generation Control in E'ectric Power Systems a r e presented , whe r e the two components of Automatic Generation Control , Load Frequency Contr ol and Economic Dispatch are perforned at the same rate . i . e . a few seconds , and where Economic Dispatch takes network security into account . This gives network secu r ity and iOOd transients, avoidin~ contradictory actions of Load Frequency Control and Economic Dispatch on the generating uni ts . The corner stone of the SOll:tion is the use of a new fast on- line Optimal Power Flow , using a new oarametric quadratic programming method . which is presented in details, ~eywords . Digital control : load dispatching : on- line operation : power system control : quadratic progranming : automatic generation control .

INTRODUCTION The Automatic Generation Control (AGC) o f an electric power system has two main objectives : first adjust power generation to load , second minimize the sum of the genera t ion costs. In the present state of the a rt! 1 1 [2] [3J (41. the first objective is met throuah the so-called "Load Frequency Contro l ' (LFC) , the second one using an "Economic Di spatch" func t lon (EO) , Usua lly, these two functions are cOr:lpletely separated in time , LFC taking the system dyn8r:lics closely into account , with A cycle rate between 2 and 10 seconds , whereas EO only changes the set points of LFC eVery fifth minute . E x c~pt a few exceptions. among which New Yo r k ~ower Pool may be mentionned (5' , no car e is t aken of power transmiSSion network security in the EconomiC Dispatch function . In these conditions , as mentionned in (1], the usua l Automatic Generation Control process has two main dra wbacks : LFC and EO control actions may of ten be contradictory , which results in undesi r able oscillations for the gene r ated powe rs. the power t ransmisSion networ k secu ri ty is not met . ElectriCite de France is now studying ne w concepts for AGC : avoiding these t wo dra wbacks was included in the objectives ,

UDCA-T

PRINCIPLES OF A NEW AaC

SYSTf'~.

Respective Functions of Load Frequency Control and Economic Dispatch The objective of the new Aac system project is to adjust generation to load in the most economical way, meetin~ generating unit constraints and transmission ~ecurity const r aints . If the latter constraints are well defined (transmission line and transformer currents within given limits under various conditions), it Is not the sa"'e ~or all the generatina unit constraints: o f course unit power as well as Dower ranps versus tine must remain within given limits: but also freql;er1: and large pO'lIer osci llati ons on a un!: must be avoided, which is not so clearly clefined ; if the latter constraint did not exist , the most economical AGC would only include EO :a few units , the "incremental units" , in economic balance with the syster:l , would adjust power generation to load in the ~ost economical way . But , as load variations versus time have an oscillatory corr_poner.t , large osc i llations might occu r on the powers of these i ncremen ta l un i ts . wh ich is unces i ra bE, To avoid such l arge oscillations on a few units , the load variations a r e partitionned into t wo components : the "Oscillato r " " component , for which LFC shares the oscill
'96

J . L. Carpen lie r, G. Co Cto and P . L. Nied e d a nde r

the average available powe r s of these un i ts a nd is costly . but this cost corr esponds t o spa r ins inc r emental unit "'e a r . Un it regul at-

i ng bandwidths are determined by schedul i ng one day in advance and i ncide ntall y ad justed eve ry hou r ,de f ini ng t he LFe possi b i li ties . Then , f or on - line cont r ol , the probl e m le ft is to apply Lf e to t he oscil la tory component of load var iations a nd EO to t he trend

e. ~ .

component .

Basic Pr inciples o f the New AGe System PrQject

The basic typical properties of the pr oposed system consist of performi n ~ ; Lfe and EO f unctions at th e s a ;ne r a t e , a few seconds , e . g . 10 seconds ( which , so f ar , seems sufficient for the F r ench system ) , ED with t r ansmission securi t y constraints (l.fC need not meet these constraints , due to the osc illatory character of phenorena and to the line thernal tire laj.'!sl .

In these conditions , the essenti a l functions related with AGC include: Every day , power scheduling with basic de termination of the unit regulating band.... idths . Every hour , final optlr.>al alloca t ion of re Ilulating band .... idths to the variOUS uni tS fo r the neKt hour lassociate d wi t h an opti l'1al po wer rlow) . Eve r y fifth minute , act i ve lonly) optimal po we r flow incl~dlnR elec t r i cal ne t work security analysis and spec i al ly p r ov i ding a " reduced r.>odel " of the product i on trans lIIission system , to be used eve r y 10th second . Eve r y 10th second , the on - l i ne AGC function itself . Including 1 . Decomposition of inout si~nals i n t o oscil latory and trend components (these input signals , which are the measurements of the variations of load to provide and of the currents to adjust in crit i cal lines , .... i l l be defined Mo r e accurately belo .... ) . 2 . Aoplication of secure Economic Dispatch to the trend co~ponents of input signals : this gives va r iations LIP to apply to the generating unit powe r s , 3 . Application of Load frequency Control to the osci llatory components of input signals this p.ives variations lIpf to ap p ly t o the generatina unit powers Cof course , LI P then LIP · • LIP • lIpf rreet the generat i ng unit constraints conct"rning the limits of POWB"'S and po .... e r ramps) , d . Sending of the orde r {)p " IJP · IJP f t o eac h Ilenerati ng unit , In p ut Signals

16

It was shown i n a p r ev i ous paper J tha t the i np ut signals ma y be tak en equ al t o

6Pe • I t . 6b , = -~ 0 ( lI F • --,--) d t

(1)

f o r th e me a su r emen t of t he var i a t i o n o f lo a d t o p r ovide

. -ot )

fo r

0

M!i.. dt ), 1.

(2(

t he me asurement of th e var i a t i on o f a

critical cu rrent to adjust , wi t h t he f ollowing notations t : time lIf " : frequency deviati.on

.

LI P; : e Kported power deviation M1 : cu rr ent deviation i.n 1 ine or t ra ns f o r mer L

L .I. . ), L : constants (possibl e tobe a dju s ted) giv i ng t he dynanlc prope r ties o f the cont r ol systen . Tt .... as denonstra,ed that using thes e inpu t signals allo ws to iet for ~he co~plete AGC the sal"e t r ansients as fo r the LfC i ntegra l control used n., .... in France , ''' hich 1S satisfacto r y , speciall', very stable . ~,

Load FrequenCl Control function

Th. Load freQ;.len cy con tr'J l lIPt:' J b.Pj

'J Secure

(31. '''i th

CBj/ I:B j'lIb; co~ponent

of

i. very Slnple

"r

relative

<0

the j th unit

regulating band .... idth of the j th unl t. Econo~ic

Dispatch function

The Secure Econonic Jispatch must use th e trend co~ponents of the input signa ls li b, ... lIbL ' " in order to provide the corrections LIP. This prcblen is not simple to solve within a very lit t le time and need ne .... techniques we shall call " fast On - Line Optina \ Po we r fl o w" ( " fOLOPF " ) and presen t be 10101 . Th e fe a s i b i1 ity of these new techniques is the co r ne r stone of the process and was the f i r s t s t ud y pe rformed after the process was Imag i ned . FOLOPf teChniques are nerived from t he tech niques of the usual Optimal Power Flo .... performed every fifth minute , and we sha ll fi r st briefly recall the latter .

5 r~INUTE RECURRENCE OPTIMAl. POWER fLOW Problem to be s'Jlved for this program , the data are the equip~ent on service for generation and trans~ission, the f!Jel costs , the l'Jads and the voltage magn I tudes Capprox imate if not we 11 known) a t every b>.Js . The results are the acti'/e powers P of the generatinll uni ts minimizing the total ruel cost and neeting gene r ating un i t constraints and trans~ission security constraints , for the Intact systeM and '_mde r contingency . Notice that the fuel costs ma y be take n a s they are o r biassed in order to ta ke a n a ccount of const r ai n ts (as po wer ra mp const r aints) like l y to appea r u p t o o ne or t .... o hours late r . Diffe r e nti a l Injections Met hod The problem is solved usi ng the "Dif f ere n t ial Injec t ions " method r 7] r 81 whi. c h bel ongs to t he f a mi.l y of " compact" Op t ima l Power fl o w Me t hods [9). These compa ct methods are c ha ra c t e r ized by the building of a "red uced mode l" of the problem , val uable i n a mo re o r l e ss Io'ide re gion , wh e re t he us eful co ns t r aints o f the p r obl e m ar e e Kp re s sed o nl y ve r sus the co n t r o l v ar i a bles P , i . e . t he

'97

New Concepts fo r Automa ti c Gene ration Control active powers of the generating uni ts "'hereas the compl ete proble~ is usually also expressed ve rsus the state variables e , voltage phase angles a t every bus of the network) . In the Diffe re ntial Injections I:lethod . the r educ ed model is non-linear, the losses being a quadratic function of p . and the fuel cost may be a quadratic function of P . In these conditions , the r educed model remains valuable on a wide region . Two main steps are performed ite ratively : reduced model build_ ing . r educed problem optimization . till the reduced problem is equivalent to the complete problem . The process nay be sunmarized as follo ws: 1. Start : find an initial solution pO(us",al_ Iy optimal but not feasible) . 2 . 'iith a physical load fIo ...· (Newton r.-ethod) comp u t e the electrical syste~ state e " f(P) . 3 . Analyse and se l ect the useful cons traints: this step includes contingency analysis if security under contingency is des ired ; usually t he useful const raints are not nume r ous , less than 12 fo r a pr oblem wi th 100000 potential constra ints. 4 . Build the coefficients o f the reduced problem t hrough sensitivity techniques (dOL/dP for the currents . dp/dP , d'p/dP' , p being the system 108ses) . At this point the reduced model is built . 5 . Test if the Just built reduced orobler!1 is iden t ical t o the previous one . If yes , the optimum is reached: i f no , go to 6 . 6 . Opti mi ze the r educed problem : Using the reduced model . one finds : 1 ) a feasibl e solution f o r P , 2) a feaSib l e and optimal solu ti on fo r P. Then go to 2 . Step I is initializati on , steps 2.3 .4 reduced model building , step S test, step 6 reduced problem opti~ization .

to 40 seconds with contingency in the security analysis . SECURE ECONOMIC DISPTACH TO BE SOLVED EVERY 10TH SECOND IN AUTOMAT IC GENERATIO N CONTROL Basic Ideas The basic idea to Ret a fast solut ion of the secure econom i c dispatch to be solved every 10th second in the AGe algorithm consists o f noticing that : 1) the reduced model built every fifth minute by the Different ial Injections method remains valuable to describe the system operation during the S following minutes (exceot the case when a sudden event occurs: then the corresponding addition to the reduced ~odel may be computed ve r y q·l!ckly . ·... hich keeps this proposition valuable) . 2) at the beginning of a 10 second cycle , we have the opt i mal solution P corresponding to the previous load: it is the solution of the previous 10 second cycle o r of the Differential !njectiorsal"ori t hm at the beginning of a S minute cycle : in these conditions . it is sufficient to trac k the optirr.a l solution ..,hen load changes, L e . to perform a parametric optimization . instead of solving an optimiz atio n problem with an initia l flat start . Pr oblem State me n t followlng the a bove basic ideas , t he problem to be solved ma.y be writ ten :

lA, J

IIPj

~A!.j

liP,

J

m

lIPj

~

"

" "tPI!'

1:.... fbL (L ~2 )

IIPj ~

J

Proble~

(S)

VJ

(a)

(6)

liP')

Reduced

(d )

X.I.'lP i Dj J liP J • Ob,

mlnimale

(7)

Optimization th lIPj va riat ion of the powe r Pj of t he jth unit (or Dart of un i t beinit defined by a quadratic cost) . IIb l • IIb L : trend component of t he input signals de f ined by (1) and (2) : they represent the variations of the conditions of the proble~ due to load changes since the las t ten second cycle . b L : current margin for current in line L at the end of the l ast 10 second cycle ; for those lines exactly on their limi ts, b L E 0 ap ap Alj • 1 - ~ • with ~ " lrst differen'oil

The rpduced problem optimiZat i on is performed using the Gene ra lized Reduced Gradien t (i.RG) technique [10!. In this algorithm , a t each step the variables are moved , when possible , in the opposi te d irection o f the gradient projected on the linearized constraints ; the length in this direction is given by the mi nimur!1 of the objective function o r by the fact a new constra int is rlet before ; then , at each step. non- linearities o f the const ra ints are take n into account thanks to Newton's me thod . A very il"'po r tant feature ~u !t; be noticed in this method : the gradient,i . e . the first derivative o f the objective func _ tion . o nly gives the steepest descent direction , but not the opt i mum point d i rection nor its posit ion . GRG is used here t o solve a quadratic program but it is a gene ral conve x programming method without special pr ope r ties fo r quadratic programs . Usually , 3 iterations ( reduced model building . reduced problem opt imization ) are sufficient. The nu~erical perfo rmances are as follows In IBM 30- 81 : f o r a system with SOO busses . 800 lines o r t ransformers, 65 the rmal olants , it takes about 20 seconds without cont ingency and up

tial losses (cons tant )

o

a 'p

.

.

Dij" apiOaPjo " 2nd dlfferental losses (constant ) ALj : sensitivity of current in line L versus IIPJ . inci denta l l y after trip o f a line or unit d . IIPj , IIP~J : bounds for IIP j . due to physica l bounds and ramp limits du ring 10 seconds. c J ' e J : coeffiCients of the ob jec tive func-

J . L. Car pen ti er , C . COt t o and P . L. Niede riander

598 tion I'"j

~O ,

ej

~ O) .

r ;. 'j t.f' j

( ll l represents rhe halarlCe h ... · ~,(>.~~ ;::"'r,"r,,'\ I.

IS

load and losses. represents ~ he - ra:'s-issi .

(6

constrai:l >s , r epresents the

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r l'.

r ''',

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•. -' .>:' ; '1:':'1 _.,.,

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act'ml val'le : In q;8.1ra"\·· I."_;'::·"-,~.:,': !. s I"'ethod does n':lt "Xlst : s , s L·· ,. problel'l , ... e had t' I",a~ine a ;;"-.' i rl:'gra:-C-i!h!. ",> · n,:.-: . side . '.
SEt'UHf: f." ;11. '; ! ' T'lI',-,U~:".

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First , two lt1"a,; al !,,', c'jl t:1 (of 'b' pr .hl ... :'· 1 ,. ,I '/'. "l'an inter,..r·diStl" : Hi,lr'l' i 'r' ;Jr-,· ~ l' '. 11 ,''',I' ronstrllinls : ',s" . ,'~-l'l:~'-"l ,!'<', ";' ' S ; in th ..se con{\i~i ,ns , ~.,. it,>_',",' 1; 1'- 'I ('~r'l r1etriC" 'l'Ja'jr't" j, pr-,>!r,,- ',:,'! \:1.' -,;' .5train~s and s,..all :i~"';SI .-"s . ,;j" " '" " 11tions neressar::' r,!;i 0>01'1::; ".' '\ r (8) , For the para-<:,tr.,· 1'''~ ;j'R.' ,., : , • .t:' ,itself , Wl' firs' r~,r;1 ,'. q,,, "'11'\,,'\ ''''hich wO'J11 be op~l~al j,!1 .. ,.- .'Itl, "0;'; " . ; ne'''' c'1nslrai:lt is -e' ; 'ht,:~ ,,' c ';." di r er ' i'm tj"!lnec b-; :,V '"I'S ',,:. 'I'> ,~ ,_ traint ra«es 1':: p'~ssir-·lo(' : .,~< .. ".. -"1"_' ',1ther s ~ ef' taklnl! in', 9 '<-,,tra!nt -",t . (,'c ,., T 'I"': 1:-' 6,." .... . . .. . ' ", ~~~.q',es 'lseri in :.inl"ar ;:'r' >o1ra--l." .,\:, .~, ,5":; GR"; cann'~t '0 ... jse~ f-i · :. .. r : 1'; :""-: , '.
!I,

Use of an

In,;er~eciate

"

",

!:-

;:; ' -: 'f.S'r!oll~ .

'l.f"!lera' ,'\I!

This pr.,ble-. is (I ,':flira " j< r~' '::-"- , ";" . " quadratic ')bje(' t i'.'" : "'" '.l n' ;;r.'i·" ;';8dratic (''l!1st:'ain' la '; ,••' ....'.':' , ~', . ,{" 'h' 'I \' if llb ' , tJ.r ~ is ' ; - '1. b' c.,,,,rs,," ft'8slb'>: 5' i . ~ .

5')1 • t \">n

_,

, 9'

t.b:

O~adra ' lc

?roO!ra-

~.'-_

linear Constraints It ::lay be shown :10' ~hat. :'.r '''_'' :~:j-alit-: oroperties, (6) is equivalent ~ o (11

r - r

t. P _

'. "'·__ :r,i!. -1'".1' ~ ;al va riable assoria 't"d ""ith .., • " , S ,';1" ' !,;", ~roble- lA , i.t is s ·· , 1"1", ' s",· .. {:: '.hen',:,correctc.P :' :~.:,' - l ( " " : ' as ~erf,:,r."er. i.!; the r,R~ "I'~':'i·r.'(, ~

"5'-

,

~f'laxa·I.,"

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'~s

':. Cons 'rainls

j.

.:'1 a:j-;an'ageJ'.. s t, ':-onsi'ler at 'r.VS" relati mships IfI'hirh are ,c':",, ' i'!c , W'J.,pin~ !Qr "r.,laxlng" the others ',"'" . !:,. '-U,,'Uli,m . O!" ro"rse this involves • h' 'asks ; ',;;]' -1-,\ :,I! • '.' . ' j j •• 'S ! t~t ~ar.i11I\S ,r the relax.·tj 'r.!;I'ralr. ' s ; If a -argin oec~~es ~t,r· . 'r.'· 'ns'rai:'.'s bec(>"',!!' f'ffecti·le . ~a'·r.::·,1I 'r'" ('!"'ec'\ve cons.raints '"hich s'. ,_~ '0'" I'f"iaxe; : 'his hap!)ens "'he:'. t.bl. l~ 2" ! ' tl::''j ':a> co)rresP'JtlC!ing d:.Jal variable :L h,,<" '· .... 5 P'Js i t i vc , or when the t.!' are not n~; f' ."'," a va!-iation 6bL which wou l d ~~"1
•

-", • : S' , I ' " , 5 "',. d,

",·;i tj "l1s , (111

!

'I

~.a~{

be restrictf'd

( ') 1

:. I

.I ( 12)

!. } 2

(; r ;,. t:.t

'~"'.:

,,

.{ t:.V'

(13)

is 'h" prc,b;e,.. rerair,lnFt t .... be solved 'I s ·pp ','here ~he set o:;~ constraints 'l~,"'S ~.?t change.

;;.sj·:" .::::

Fri~.C'if'e

.! 301,~tion

~!'a' ~c ~r~sra-

,f a Parare-ric O'~a ­

t:'~nstraineo

bj a Set of

:'ir.e;)r El,u'_ions T!':e gl::'... ral

fo r - o!" the croira- (:3) -ay be

tlri-:e~

).

~

;- "cx

x ,=,(;P ,Q b

el7)

x :; 8 ~XT JII

. C.P1'1 , 8

r (;'o l l '" AI DiJ It/j '" i , DU

'99

Nc,",' Co nc e p ts f o r Aut oma ti c Generati o n Co ntro l 'IJ e know for b .. 0 , x " 0 is the oPtlnal sible) sobtlon .

l'ca-

Reduced gradient. The variable...se!, ; x : is oartitionned into 2 $ubse~s I X, x ; ; :'~ll,:, ­ "" inJl, the sane oaftlt.!.on , A • fA, A'. The partition is s'.Ich that ;, 1S sq'~ar" ;;C'n sin"!';lar ; ~ is the basic co"'ponent of x. ;- ' he n':l!~-ba 5 i c co~ponen t . (161 gives rif . (~ , xT ~l d ! ~(c

...

(:4 Of

• X

- -, t -,;

gives A d ;

. • d;

•r.; •

,~

•

0 " Jx .. ('

7:.

(.9,

...cA' "-

1 ,,...

'T or "

~

-""~·ln.l! f"s ~

x . x' • 6x at the

0

and R ' x' , ( -,

122

,': 'r.

,

OJ

; OJ and a' >

-

.j

and

, 'j' j , 'j

'he reG·.cel .u·!es

15

and

a

.

gT

r.

· "P:T • r.

tile.

0

being a oarti tion 01

(2<1 ) 'j

such that.

(;oSI PrincipiI.' of the paranetric optil'liza·i'm. Th(' principle consists of tryilll~ ~o sol'le 'I': and (24) at a time . If x '" x' , .... lth g ~ ~' and llb ,. b - Ax' , .'~ '.·an ~

r.

Cl x '

.ob

.

-;

(25) ,

1If'(27) .

"r'''~si'-

III

I ilt '1- ~

Ril\~ , 7

8.

U,.dot(' P (p' . !\P) , '11 .... l-~la)(€'d constraint nar "Ith,' 11rS' ro:;,,,,, A, =>A, - llPTU·) . y

~!"

•...

n

·p' .• el,- '

,

I

YA- '

'ht""k 11 a '"rl\nll"'15sion line constraint -'Ill ' b,. dr :,p~d (6bl ., 0 and u L <: 0 or i··j ......sslbili'"l t, piv:H and 6b l tending to 10 "s'.·n 'he ,...,..,n!>.raintt . Ir so , redefi.ne 'A, -:-"""O."€, aRa1n "A- ' , '1 " y r;:- :, If p <: 1 and rlvr'inA: n~' n"r:'orMed . go to 9 ; lfp '" 1 ~ r p <:. al.'1 I'i"'':l~in-'l '"as JUSt perfo r red , f< ' ') 2 . !I • ir." . t, .. e~155ar:; because a cons t raint : :'('·:e,.teo P fr')~ being eq'Jal to 1 . This is j"."rl,r:-:"';"'It.h ~he sa~e selection cri t eria ,~ ir. :"'~a: 3i"'~lex "e~hod . Then go to 7 .

"r

'l-L 'llE

O?T:'~/,:'

P',)'IJER fLO'6'

P!'5'JLF

}

;':xlJ,:,:,j-en·, o:;,rganisation

iT

'lie chanp;e only ~ and x (6 x -

Exp':':'i-er.~s

which Iti'!es ~ he funda~en t al relations h ios to -ee ~ : A 6~ ' • Cl b

G 6; '

-:.,

,~.

,'n

fAST

0,

.: ~

J't '

1).,

A(x· · l l x · . b

A ll x '

';'i

i,,'h q,a~ ~ Cl ~ 1 and , : ! ' I ,'-1i:,5 ·('as:o:" . Cl ",a:: be 11m " , t· :"\ ."..- 67 , '. -,.'l."!'; ..... ne ~f i t5 bounds r· -, .!!( it ,;.. iax"d cr·ns~ralnt becol:les . ;-.. : the rlp'illi'i'le ·/al;.;(' 0 1 Cl nust •. ,~.,. ;,.' ••• I" .1~· a s,...,all correction to -."., 0 #. " RO t, 6 ; if Cl = 0 'I' " ;,,·i!!!!)l .. C. l 1;l'\ited Cl , g':l to 2 aft a',,1- 1: .•1 1 ,t'il! ','1"11 lIPj to:;, belong to t he • _"X ,. s('. c.F i.l Cl • t) for another reason , ' >,

At the point x ' , .Iet 1S oa r titjo'l x " Ix. : : ; such that ~j <: 'X J <: SJ or X'J ~ o.j and gj <: 0 ':lr X'J _ SJ anrl aJ > 0 , The op t i""H 11 t.'1 ",ay be r eached • r:ti ng to ge t & _ 0, or

'.. !lP_ ~

.:. .

.'

[0

:'p'

,l,

0'

j

y - . A.

~

ine llP s'lCh that llf-'j and gj , 0 > .;, :' .\f. '0.;" ;: ne . : Illb be il~g ,":'1 :'- ',: '1;1.' sec'md "'e-berl , ',". -j ", . ' i' .' iN ,. ...',·hp-1 ; ,.'11 E"~ g':"' to ~ ,

"

<0

rr-:'lble- constraints .

,.,

-:

Opt! Mill I ty cond i t ions . Thl"~; cor;:; ~I'''nrl '~ df ). 0 ror alll possibl .. ehanjZe ~ X, ... hich -'1b(' '.. rl tten :

'j

~ Cl -{ 1

Opti",al Po .... er Flow basic

~n - I..ir.e

,' ,

1?3

--,,

'~\e

'Ill

0

f~!.-_~PF aljZ')ri'h", fled'/ts directly fr~"l ' h'" ;:'''':i .s a:.al::s1s . I ' ~,a:1 be brie!!:. "'s,'rlb"'c as : ;.)11-:.0""5 ; '::'('1,:" :~')'" ll~' .. O. Use the part i tion t:.'F , r ,- the last ;":10",n solution ( which :""'!i; : ~9 ':"1 c.P " Q opti~al \ ......Co~P"J t e '"'' ,,,, ~ ~ cf, 'riP , rhe basis A , it s 11::":'';'' ~-' , "he ·'d·.;al vecto r " u z 1'A-' ,

12'

o~i.n.

""axli"-'''' such that

Cl

T~; ..

ro'" vector , is '11(' "rerLc,..1 p.ra:ii"~:'" f versus '; ; (? 1)\ sh,',",'s 1. is a 1 i :-.ear .- .:;!""'.i">n ,1 x . t!

6x' ,

a:!" :'i .!~~

i,

gradier.~

Cl

!l"F :

·,,1 th

A

C.x _

-Ilt

I)', .

(27)

,

This gives the variation Cl x . '" [Cl x • • Clx " 0 .... hich .... ould be opt1mal and ~eet the complete va riation llb of the second membe r if no ne'" const ra i n t appe ar ed . As a ma t t er of f act , the variation Axo is not a l .... a ys pos sibl e to perform , and the final variation for one itera tion will be defin e d by

... e r e carried o·~t :-o:;,r 29 case s or :-or ~ 1pica 1 1980 . 81 hours , ·... i t ~ ab':llt '}"" '::i;.lsses , 8QO lines , 65 t he r mal '~ni's , ... 1 th data cor-,in", fro~ s~'ste r:l me a S'J r e r>ents . Starting from an actua l case ( ab out 2-:>000 to 25000 therr:lal ;.j\v) l oads were inc reased or decreased by abou t 1200 MW fo r 15 c a ses and abo'~t 600 Woi' for 14 cases . On one ha nd , startina f~e OPtimal solution P of the actual case , the FOLOPf algorithm '..as "Ioolied ("fOLO?f solution") ; on the othe r ha nd , the optiC!lal so l ution .... as computed d irec tl y by the usual Optimal Po .... er Flo.... for the fi na l loa d conditions ("direct solution" ) . Th e o b j e c ti ve .... as to test ho .... siC!li l ar .... ere t he t ....o s o l u ti o m .~,'"

frf'r,o:~

s1s . e~ ,

600

J. 1... Carpentier , C. Cono and P. L. Niederlander

and how much CO~Dute~ time was required to get the FOLOPf solution .

Solution Accuracy Exper iments were carried out in two cases: with quadratic fuel cos t s and with linear fuel costs ....·ith quadratic fuel costs . the differences bet·... een the FOLOPF and direct solutions P were less than 3 MW . This proves 2 things: 1) fOLOP. algor! thn runs cor :,ectl~'

and accurately ; 2) the use of the reduced mode l al one remains valuable even for fairly large load changes . With linear fuel cos ts, a difficulty appeared : the fOLO?F solution

gave the same optimal cost as the direct solution , but the differences between the gene rated powers P could !'each :.Ip ~o 300 '.''11 . This is due to the existence of an infinit'l of equivalent optimal solutions with linear f'lel cos ts ; as the proposed AGC ~ay have to use solu t ions coming from both types of co::"putations , it wa s recolrl'lendeci to use only quadra tic fuel cosu .

Computation Speed for the 1200 MW load changes , the nur:ber of FOLOPF iterations where between 2 and :2 . the IBM 30- 81 computation time between 8 and 90 !fIi 11 iseconds ; for the 600 MW load chanp,es, the number of FOLOPF iterations were between 2 and 9 , the computati on times between 7 and 51 milliseconds . The nUl'lber of Iterations and computati on times ma inly depended on the nUMber of effective security constraints . These computations are very fast: always less than 90 (fIS f or 1200 MW load chan~e ; due to possib le additional computa ti ons ( e . g . in case of sudden system change) , t.he ma x imum e xpected computation time for FOLOPF may be estimated between 0 . 2 and 0 . 4 second : comDared with the 20 to 40 seconds necessary for a complete Optimal Power Flow , with FOLOPF the computation time is divided by a ratio lOO •

No w, there e x ist minicomputers us able in an Enerps Control Center with half the sDeed of the IBM 30- 81 : with such a minicomputer, the ma xi mum FOLOP F computation tine for a 1200 MW load change would be 0 . 4 x 2 • 0 . 8 second kno wi ng that in France 1200 Wi represents a ma x imum load change seldom reached within a 10 second internal , FOLOPF would need at most 8 % of the cycle time .

CONCLUSION New concepts f o r AGC were presented, 'IIhere on one hand transmission security Is net . and on the othe r hand LFC and EO are performed at the same rate, e . p' . 10 seconds . The feasibility o f these conce pts is due to the us e of a new tool , "Fast On- Line Optimal Power Flow" ("FOLOP F" ) , which divides by 100 the computer t i me of secure EO problems . This FOLOP F uses a new parametric quadratic Dro4 r amming algorithm ; it may be noticed this algo r ithm Is general and not limited to electric power s yste m appli cations ; wi th particular addi-

tions !'Iirroilar to those oresented in the pape r, it could even certainly in SOr1e cases be used AS an approxination for no re ~ene ral convex progra:nmin 4 · REFERENCES 1 . raavitsch H. ann Stoffel J ., "Automatic r.eneration Cor.trol , a seJrvey", International ,lo'.1rna1 of Electrical Power and Energy Systerr,s , Vol . 2 , No 1 . January 1980, po 21 - 28. 2 . Of' ~:ello LP" ~alls R. . T. ami B'Rells W.F., "Autoratic r,ene rati ~n Control . Part I . Process '~ odellin~ ", 'f.E!': rrans. PAS 92 . 1973 , pp 710_715 . 3 . De ··:el10 F.P .• ";ills R. J . and B'Re11s w.f., "A'.. to-atic r.eneration Control . Part :~ . uigital Control Techniques", IEEE Trans . PAS 92, 1973 , op 716-724 , 4 . Schells~ede n . and Wagner H., "Design aspects of a so!t war e package for Automatic ,';eneration Cont r o l 'II! th instantaneous economic dispatch and load forecasting functions" . IFAC Synoosium on A.utomatic Control in Power Generation , Distribution and Protection , Pretoria 1980 , OP 61 - 69 . 5 . Elacqua A. J . and Corey S . L . • "Security constrained disp at ch at the New York Power Pool" , IEEE paper No 82 WM Of\