OPF_IEEE.pdf

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IEEE T r a n s a c t i o n on P o w e r A p p a r a t u s a n d S y s t e m s , V o l . P A S - 1 0 3 , N o . 1 0 , O c t o b e r 1 9 8 4OPTIMAL POWER FLOW BY NEWTON APPROACH

D a v i d I . Su nMember Bruce A s h l e yMember Brian BrewerMember Art HughesS r . M e m b e rESCA Corporation 1 3 0 1 0 N o r t h u p W a y Bellevue WA 98005

A b s t r a c t - T h e classical o p t i m a l power flowproblem with a n o n s e p a r a b l e o b j e c t i v e function can b esolved by an explicit Newton approach. E f f i c i e n t ,robust solutions can b e obtained for problems of anypractical s i z e or k i n d . Solution effort i sapproximately p r o p o r t i o n a l t o network s i z e , a n d i srelatively independent o f the number o f controls orbinding inequalities. Th e k e y idea i s a directsimultaneous solution f o r a l l o f th e unknowns i n t h eLagrangian f u n c t i o n on each i t e r a t i o n . E a c h i t e r a t i o nminimizes a quadratic approximation o f the L a g r a n g i a n .F o r any given set o f binding constraints th e processconverges t o the Kuhn-Tucker conditions i n a f e wi t e r a t i o n s . Th e challenge in algorithm d e v e l o p m e n t i st o efficiently i d e n t i f y the binding i n e q u a l i t i e s .INTRODUCTION

T h i s paper describes a new approach t o t h es o l u t i o n o f th e classical o p t i m a l power flow ( O P F )problem based on an explicit Newton f o r m u l a t i o n . Testsp e r f o r m e d on l a r g e p r o b l e m s with prototype a l g o r i t h m sshow t h a t th e approach f u l f i l l s a l l o f the r e q u i r e m e n t sf o r p r a c t i c a l OP F programs. T h e Newton f o r m u l a t i o nappears t o b e a s fundamental a n d e f f e c t i v e f o r OPF asi t , i s f o r power flow ( P F ) .T h e c l a s s i c a l OP F i s a P F p r o b l e m i n which certaincontrollable variables a r e t o b e adjusted t o m i n i m i z ean objective function such a s t h e c o s t o f active p owergeneration or losses, while satisfying physical a n doperating limits on various controls, d e p e n d e n tvariables, an d functions o f v a r i a b l e s . Because th e

o b j e c t i v e must include l o s s e s , a n d the controls includereactive d e v i c e s , the problem i s characterized by an o n - s e p a r a b l e o b j e c t i v e function. T h i s c h a r a c t e r i s t i c ,which sets th e classical OPF apart from similaroptimization problems, also makes i t more difficult t osolve.T h e t y p e s of controls t h a t a n OPF m u s t b e able t oaccommodate include a c t i v e an d reactive poweri n j e c t i o n s , generator voltages, transformer ta p r a t i o s ,and phase shift a ng le s. In a given OP F study, activepower controls, reactive power controls, o e i acombination of both may b e optimized. Because o f thenon-separability, o p ti m iz i ng r ea ct iv e power control i smore difficult than a c t i v e - p o w e r control. I n g e n e r a l ,any method t h a t ca n solve t h e reactive problem ca n a l s osolve the c o m b i n e d active-reactive problem. T h e r e f o r e ,

the emphasis of t h i s paper i s on the r e ac ti v e p r ob le m.T o b e practical, OPF programs need t o have

8 4 WM 0 4 4 - 4 A paper recommended and approvedb y t h e IEEE Power S y s t e m E n g i n e e r i n g Committee o ft h e IEEE Power E n g i n e e r i n g S o c i e t y f o r p r e s e n t a -tion at t h e I E E E / P E S 1 9 8 4 Winter M e e t i n g , Dallas,Texas, January 2 9 - F e b r u a r y 3 , 1 9 8 4 . M a n u s c r i p ts u b m i t t e d S e p t e m b e r 1 , 1 9 8 3 ; m ad e a va il ab le f o rp r i n t i n g November 1 8 , 1 9 8 3 .

William F . TinneyFellowC o n s u l t a n t

performance requirements that i n c l u d e : a s o l u l t i o n timethat varies approximately in proportion t o network sizean d i s r e l ati v el y i n d ep e n d en t o f the number of controlsor inequality constraints; rapid a n d consistentconvergence t o t h e Kuhn-Tucker ( K - T ) optimalityconditions; absence of use r suppli ed tuning andscaling f a c t o r s f o r the optimization process; n ocompromises i n OPF problem definitions; a n d problemsof any practical size o r k i n d should b e a s easilysolvable b y an OPF a s by a P F . Newton-based OPFprograms can satisfy these criteria for practicality.The k e y idea o f th e approach i s asparsity-oriented, simultaneous solution for a l l o f theu n k n o w n s o f a quadratic approximation o f the Lagrangianon each iteration. General large-scale nonlinearoptimization problems are usually solved byquasi-Newton rather than explicit N ew to n m e th od s sincethe l a t t e r would b e t o o burdensome o r even completelyintractable. T h e OP F i s a notable e x c e p t i o n ; thesparse Hessian matrix of the Lagrangian f un ct io n ca n b eexplicitly evaluated a n d operated o n efficiently.Efficiency ca n b e further enhanced by decoupledf o r m u l a t i o n s f o r which th e convergence remainssuperlinear. F o r a given s e t o f equalities, a NewtonOP F converges t o t h e K - T conditions i n a f e witerations. The m a j o r challenge in algorithmdevelopment i s t o identify t h e binding inequalitiesefficiently.A s d e s c r i b e d here, the Newton approach i s af l e x i b l e formulation that ca n b e u s e d t o developdifferent O P F a l go r it hm s suited t o the requirements o f

different applications. Although t h e Newton approachexists as a concept entirely apart from any specificmethod o f implementation, i t would not b e possible t odevelop practical OP F programs without employingspecial sparsity techniques. T h e concept a n d thetechniques together comprise t h e given approach. OtherNewton-based approaches ar e possible.Th e paper describes t h e inte rim re sult s o f anongoing r e s ea r c h p r oj e ct , R P 1 7 2 4 - 1 , being p er fo rm ed b yESCA Corporation under contract with t h e Electric PowerResearch I n s t i t u t e ( E P R I ) . Th e purpose of th e projecti s t o explore O PF s ol ut io n methods. More work h a s beendone than i s reported here a n d more w or k r em ai ns t o b ed on e b efo re t h e project i s completed. Other papersdescribing p r o j e c t results a r e planned.

NOTATIONTh e f o l l o w i n g symbol definitions a r e usedthroughout t h e t e x t . Most symbols a r e also defined inthe t e x t where they f i r s t appear. Symbols u s e d only int h e APPENDIX are d e f i n e d there.

k , mAwr tt

S u b s c r i p t s denoting b u s e s ( n o d e s ) orb u s pairs.P r e f i x on scalars and vectorsdenoting incremental correctionof quantity."with respect t o " .S u p e r s c r i p t meaning transpose.0 0 1 8 - 9 5 1 0 / 8 4 / 1 0 0 0 - 2 8 6 4 $ 0 1 . 0 0 1 9 8 4 IEEE

2 8 6 4


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2 8 6 5

Matrix entry denoting a l a r g e number.S c a l a r s

P k Active power i n j e c t i o n a t b u s k .Q k R e a c t i v e power i n j e c t i o n a t b u s k .v k M a g n i t u d e of c o m p l e x v o l t a g e at b u s k .E ) k A n g l e o f c o m p l e x v o l t a g e at b u s k .t ; km Transformer tap r at io b et we en buses k a n d m .0k m P h a s e shifter a n g l e b et we e n b us es k a n d m .Yk S y m b o l f o r any sta te o r control variable.F O b j e c t i v e functionL Lagrangian function.X p k L a g r a n g e m u lt i p l i e r f o r P k .X q k Lagrange multiplier f o r QPi L a g r a n g e m u l t i p l i e r f o r active i n e q u a l i t yconstraint i .N Number of b u s e s i n network.S Q u a d r a t i c penalty weighting f a c t o r .o l i Quadratic penalty function f o ri n e q u a l i t y constraint i .Vectorsy A l l variables v , E , 0 , t .X A l l Lagrange m u l t i p l i e r s X - p a n d X q" p S ub ve ct or o f X .I q Subvector of X.z C o m p o s i t e o f subvectors y a n d X, u Lagrange multipliers f o r bindinginequality c o n s t r a i n t s .g G r a d i e n t o f L wrt z .z S u b v e c t o r of z f o r real power variables.z " l Subvector of z f o r reactive power variables.g Subvector of g f o r r e a l power variables.g " f Subvector of g f o r reactive power variables.M a t r i c e sH Hessian o f the Lagrangian.J Jacobian for O P F .W B o r d e r e d H e s s i a n . Composite o f H , J p j t .W ' Bordered Hessian f o r P O s u b s y s t e m .W " l Bordered Hessian f o r Qv s u b s y s t e m .H ' Hessian submatrix o f W ' .J Jacobian submatrix of W ' .H " Hessian submatrix o f W".d " l Jacobian submatrix of W " .

B A C K G R O U N DP r o g r e s s on OPF a n a l y s i s has been reviewedperiodically [ 1 - 3 ] , and i t was reviewed f o r thisproject. A t t e m p t s t o solve th e OP F problem date backover twenty years. Practical s o l u t i o n s f o r OP Fp r o b l e m s w i t h s e p a r a b l e objective functions have b e e no b t a i n e d with special l i n e a r programming methods [ 4 ] ,b u t the classical OPF ha s d e f i e d practical s o l u t i o n s .O f th e many proposed methods, only a few have beentested on p r o b l e m s large enough t o evaluate theirp e r f o r m a n c e . I t seems t h a t a l l o f t he se m et ho ds f a l lshort of being practical because o f limitations i ns p e e d , problem size or robustness, or because o fc o m p r o m i s e s with the problem d e f i n i t i o n . T h i sconclusion i s supported b y th e observation that t h ec l a s s i c a l OPF h a s no t become a standard a p p l i c a t i o n .R a t h e r than attempt t o evaluate many d i f f e r e n tmethods, only three are discussed. They ar e relevantt o t h e Newton approach and are r e p r e s e n t a t i v e o fothers.T h e reduced gr ad ie nt m ethod of D om me l and Tinney[ 5 ] h a s b e e n f r e q u e n t l y cited as a b e n c h m a r k . . Several

o t h e r reduced g r a d i e n t method s have also b e e r n

published. Although s om e s uc ce ss has been claimed f o rthese m e t h o d s , recent f i n d i n g s show that g r a d i e n tmethods c an no t s ol ve the O P F . This wa s convincinglyshown i n a recent paper b y Burchett, Happ a n d Wirgau[ 6 ] i n which t h e y c o m p a r e d a reduced gradient methodwith a quasi-Newton method which i s strong e n o u g h t osolve t h e p r o b l e m .Sasson, Viloria a n d Aboytes [ 7 ] were t h e first t o

show that a sparse factorization o f a n e xp li ci t H es si a nmatrix f o r the OPF could be performed. However, theyattempted t o solve the PF equations a s well a s t om in im i z e t h e objective function b y using o n l y t h eH es si an m at ri x in an augmented Lagrangian formulation.Evidently, this does not work well. W i th theirformulation, the Hessian matrix had second-neighborfill-in, making it considerably less spar se than theH es si an m at ri x of the approach described i n this paper.T h e a fo re me nt io ne d q ua si -N ewt on method ofBurchett, e t a l [ 6 ) utilizes second o rd er i nf or m at io ncontained in an iteratively c o ns t ru c te d r e du ce d ' H e s s i a nmatrix. I t i s strong enough t o solve the O P F . I neach iteration th e q u a s i - N e w t o n method obtains adescent direction by operating o n the reduced gradientwith an approximation o f the factors of the symmetrical

b u t dense reduced Hessian. ( T h e r ed uc ed H es s ia n h a st h e dimension o f th e superbasic v a ri ab l es ) . U p d a ti n g thedense approximate factors in each iteration, and t h enumber o f i t e r a t i o n s r e q u i r e d f o r the approximateHessian t o generate a good descent d i r e c t i o n ,contribute t o a l a r g e computation a n d storagerequirement f o r p r o b l e m s o f p r a c t i c a l size.PRELIMINARIES

T h i s section b r i e f l y describes how th e basicaspects o f nonlinear optimization a r e a p p l i e d i n th egiven approach.E x a m p l e Problem

T h e five bus network o f F i g . 1 i s u s e d throughoutas a s p e c i f i c example.

2

T 2 4

/ 0 - l3

T 3 5

4 5F i g . 1 Example Network

G e n e r a t o r s at buses 1 and 3 are d i s p a t c h a b l esources o f a c t i v e a n d reactive power. Load buses 2 , 4 ,a n d 5 have scheduled active a n d reactive power. Th ep h a s e shifter h a s controllable angle 0 4 . 5 and thetransformers have controllable ta p ratios t 2 4 a n d t 3 5 .Th e reference angle o f the c o m p l e x b u s v o l t a g e s i s e j .The p r o b l e m i s t o minimize t h e total cost of P 1an d P 3 while s a t i s f y i n g the scheduled loads and limitsof different v a r i a b l e s . P 1 y Q l P 3 Y Q 3 , 0 4 5 , t 2 4 andt 3 s a r e controllable.

V a r i a b l e sI n an o p t i m i z a t i o n p r o c e d u r e th e variablescan c h a n g e roles. Variables are classified a s

1

j


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2 8 6 6c o n t G r o l ( s u p e r b a s i c ) , dependent state ( b a s i c ) o r , ifthey have reached their l i m i t s , constant ( n o n b a s i c ) .I n t h e Newton a p p r o a c h these distinctions areunnecessary. Basic a n d superbasic variables a r ep r o c e s s e d i d e n t i c a l l y . V ariables t h a t become nonbasica r e constrained a t their limiting values withoutactually c h a n g i n g their roles a s variables. Th eoriginal variables establish the dimension o f t h evector of v a r i a b l e s , y , and i t i s n e v e r c h a n g e d .

In t h e e x a m p l e , y consists of thirteen v a r i a b l e s ;0 4 5 , t 2 4 , t 3 5 g , 1 , vl , e 2 , v 2 , 0 3 , v 3 , e4 , v 4 , 8 5 , an dv 5 . Although O 1 , t h e reference angle, rema insconstant, i t i s included in y for completeness.I n e q u a l i t y Constraints

T h e f o l l o w i n g q u a n t i t i e s have upper a n d / o r lowerl i m i t s ;1 . Dispatchable sources of P an d Q .2 . Variables; voltages, tap ratios, phase shifterangles.3 . Functions such as line f l o w s , m e a s u r e s o f s e c u r i t y ,e t c c .Active E q u a l i t i e s

T h e s e t o f e q u a l i t i e s , A , a l w a y s includes t h e PFequations f o r s ch ed ul ed l oa d a n d generation. I t alsoincludes t h e following set o f b i n d i n g i n e q u a l i t i e s :1 . Th e P F e q u a t i o n s of an y dispatchable sources o f Por Q that are constrained a t their l i m i t s .2 . Th e e q u a t i o n s of any other inequality functionsconstrained a t t he ir l im it s.3 . Th e trivial e q u a t i o n s o f variables constrained a ttheir l i m i t s .O b j e c t i v e F u n c t i o n

T h e o b j e c t i v e f u n c t i o n , F ca n a s s u m e severald i f f e r e n t f o r m s , b u t th e d i f f e r e n c e s have nos i g n i f i c a n t i m p a c t on the approach. F f o r th e e x a m p l ei s g i v e n i n ( 1 ) .F=CP+CI11 33 ( 1 )

C I and C 3 a r e th e slopes of piecewise linear segmentsof the active power cost c u r v e s o f dispatchableg e n e r a t o r s 1 an d 3 . In an actual program suitablelogic would be needed t o change C 1 a n d C 3 as P1 a n d P3change i n t he s ol ut io n process. T h i s l o g i c i s omittedhere because i t has no impact o n the approach. Ifdesired, quadratic approximations of the cost curvescould also b e used in F .

Lagrangian F u n c t i o nTh e Lagrangian, L , for the Newton OPF is s h own in

(2).

L CP+CP-.1 1 3 3 l j k X p ' P k ' ' q k Q ~ k (2 )Th e s u m m a t i o n s a r e over all bu s e s f r o m 1 to N .However, the Lagrange multipliers, Xql an d X q 3 , b e c o m en o n z e r o only when thei r r e sp e ct i ve e q ua t io ns fo r QL an d

a r e in A . Th e values of the Lagrange multipliersare determined b y the solution p r o c e s s . In thefive bus example, X l, and X3 are zero because theycorrespond t o dispatVhable s % d r c e s operating within

t he ir l im it s.I n ( 2 ) , L i s a l i n e a rIn the Newton approachvariables a n d functionspenalty functions a i which

combination o f PF equations.inequality constraints o na r e enforced b y quadratica u g m e n t L a s shown i n ( 3 ) .L=CP 3 C - XP- 2 2Q-i i 1 3 3 pk kqkk i ( 3 )

Th e summation for i i s over a l l binding inequalities.L for the example, with n o active penalties a n domitting Q 1 and Q 3 f o r d i s p a t c h a b l e V AR sources, i s ,

l I l 33 p11 p2 2 Ap3 3p 4 P 4 p 5 P 5 X q 2 Q 2 - X q 4 Q 4 - X q 5 Q 5 ( 4 )

I n some optimization methods L i s augmented b yother equations t o e nha nce i ts positive definiteness.Such augmentation, which adversely affects sparsity, i sunnecessary in t h e Newton O P F .Solution Conditions

A minimum of the o b j e c t i v e f u n c t i o n occursthe K - T optimality conditions a r e satisfied.quantitative indicators for evaluating theconditions require n o additional c o m p u t a t i o n .following conditions must exist f o r a minimum:

whenT h eK-TT h e

1 . T h e mismatches of a l l P F e qu at io ns in set A a r ewithin tolerance.2 . Th e inequality c on st r ai n ts a re all satisfied.3 . T h e p r o j e c t e d g r a d i e n t i s zero ( e x c e p t f o rr o u n d o f f ) .4 . Th e s e n s i t i v i t y , P i t b e t w e e n each b i n d i n g

i n e q u a l i t y c o n s t r a i n t a n d th e o b j e c t i v e function i ss u c h t h a t further c o s t reduction c a n b e achievedo n l y i f t h e constraint i s v i o l a t e d .5 . Th e p r o j e c t i o n o f the Hessian i n t h e feasibleregion i s positive d e f i n i t e . Conceptually, thismeans t h a t th e m u l ti - d i me n si o na l o b j ec tiv e functioni s b o w l shaped; therefore, t h e stationary point i sa true m i n i m u m , no t a saddle point.

A l t h o u g h l o c a l , p o s i t i v e definiteness e n s u r e s t h a ta stationary p o i n t i s a m i n i m u m , i t d o e s n ot e n s u r et h a t i t i s a g l o b a l minimum.SUBMATRICES

T h i s section describes t h e J a c o b i a n a n d Hessianmatrices, J a n d H , an d displays them fo r the exampleproblem. J an d H a r e submatrices i n the Newtonformulation. Their arrangements i n the displays i nthis section are conceptual only. In an actuali m p l e m e n t a t i o n t h e y are combined an d rearranged i n t o asingle m a t r i x or matrix pair. H owe ve r, t he arrangementshown f o r H would b e correct if its separatef a c t o r i z a t i o n were needed.Jacobian Matrix

J i s a m a t r i x o f f i r s t p a r t i a l derivatives o f PFe q u a t i o n s w r t y . I n t h e N ew to n a p pr oa ch t h e n u m b e r ofrows - i n J i s a l w a y s 2 N a n d t h e nu m b e r o f columns i se q u a l t o the dimension o f y . J fo r t h e example i sshown i s ( 5 ) .


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+ 4 5 1 2 4 T 3 s 0 1 u n 0 2 V 2 0 3 5 / 3 0 4 5 / 4 0 5 " 5 5i J j i

j j i i i ' I

The rows f o r c u r r e n t l y i n a c t i v e PF equations canb e o m i t t e d from J . Each symbol j indicates a nonzeroelement and its c o o r d i n a t e s i n d i c a t e a specificderivative. For e x a m p l e , j w i t h c o o r d i n a t e s Q4 , v5 i sa Q 4/ a v 5 . A nonsingular J a co b ia n m a tr i x for th e N e w t o nPF could be formed b y d e l e t i n g c e r t a i n c o l u m n s o f J .+ 4 5 T 2 4 T 3 5 0 1 V I 0 2 V 2 0 3 V3 0 4 5 4 0 5 V5

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I h I h h h hh h h h h h

h h h h hh h h h

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h h

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( 6 )

H e s s i a n MatrixH i s a symmetric matrix o f second partiald e r i v a t i v e s o f L wr t y . Its dimension i s t h e s a m e a sthat of y . Each element o f H i s a linear combinationof second p a r t i a l derivatives of PF e q u a t i o n s . Details

on f o r m i n g H a r e g i v e n in the APPENDIX. H for t h ee x a m p l e i s shown i n ( 6 ) .Each s y m b o l h indicates a n o n z e r o element an d i t scoord inat es ind icate a s p e c i f i c derivative. Fo re x a m p l e , h with coordinates t24,v4 i s 8 2 L / 8 t E ) vO n l y the u p p e r t r i a n g l e i s shown.The s p a r s i t y of th e 2x 2 b l o c k structure of themain submatrix ( h e a v y border) i s t h e s a m e a s that ofthe network i nc id e nc e m at ri x. Therefore, it can b efactorized i n the u s u a l sparsity-directed way.P r o c e s s i n g o f elements outside o f the h e a v y border ca nb e e f f e c t i v e l y s e p a r a t e d f r o m t h e p r o c e s s i n g o f themain s u b m a t r i x . T h i s w i l l b e c l a r i f i e d i n the nextsection. H b y i t s e l f , however, i s not factorized inthe Newton approach.Elements of H representing the c o u p l i n g s betweenvar iab le p a i r s e an d v , G a n d t , a n d a n d v a r e v e r y

small c o m p a r e d t o the a v e r a g e m a g n i t u d e s of otherelements. T h i s favorable property i s exploited in thedecoupled formulations.NEWTON OPF FORMULATIONS

I n this s ec ti on t he c o u p l e d a n d d e c o u p l e d NewtonOPF formulations a r e explained b y t em p or ar il y i g n o r i n gthe complications o f identifying a n d enforcing t h ebinding inequalities. Th e only a ct iv e co ns tr ai nt sassumed at t h i s point a r e the PF equations for th egiven loads. ( W i t h o u t some additional constraints thep r ob l em w ou l d be unsolvable since the op ti ma l sol ut ionwould normally be f a r outside any normal o p e r a t i n grange.)

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2 8 6 8C o u p l e d Formulation

L e t , z b e a vector c o m p o s e d of subvectors y a n d X .Differentiating L twice wrt z leads t o t h e symmetricmatrix W symbolized in ( 7 ) .H J t ]W = l l - J

7 . Return t o step 2 with updated z .

Decoupled FormulationA d ec ou pl ed f or mu la ti on o f t h e Newton OPF i ss y m b o l i z e d i n ( 1 0 ) ,

W I x A z , =-'t( 7 )W " t x A z l = _g .

This matrix i s well k n o w n ( p g . 2 3 9 of r e f . ( 1 2 ) b u tapparently unnamed. Here i t i s r e f e r r e d t o s i m p l y a sW . W provides t h e basis f o r the Newton OPF formulationshown i n ( 8 ) ,W x A z = - g ( 8 )

where g i s th e g r a d i e n t vector o f f i r s t p a r t i a lderivatives o f L wrt z , and Az i s a v ector o f Newtoncorrections in z .

( l O a )( 1 0 b )

where the symbols ar e a s defined i n the NOTATIONsection. T h e primes o n symbols a r e u s e d t o suggest ananalogy with the decoupled PF [ 9 ] . T h e sparsityarrangement o f ( 1 0 ) for the example i s s h o w n i n ( 1 1 ) .1 2 3 4 5

h h j h j

h h j

h_ h h h

o o

h j h h j

h

hjjhhhh

A ) 2A k P 2

A S 3A O , 3A S 4A S 5

A T , 4A T 3 5Av s

A V 2

A x3A s 4

A A q 4A s 5A A q 5

- a L l a + , ,

- a L l a o 2- a L/ a X 5- a u a e 3

a l

- a u a l , 4- a L / a o s- a L / a k

- a L U a T 2 4- a L / a T 3 5I - a L l a v- a L / a v 2- a L a l q 2- a L I a v 3- a L / a s 1- L / a A q 4I - O u a x V

( l l a )

( ( l i b )

2F o r sparse factorization i t i s necessary t orearrange W . Equation ( 9 ) o n the previous p a g e showsthe sparsity a r r a n g e m e n t f o r the example.

Equation ( 9 ) also shows i n detail t h e elementst h a t comprise t h e s y s t e l b % First derivatives o f L wr t Xi n vector g are the familiar r e s i d u a l s o f th e PFequations, i.e., the differences between the actual a n dscheduled injections o f Pk a n d Q k .This arrangement c r e a t e s a main subma tri x ( h e a v yb o r d e r s ) whose 4 x 4 block structure i s t h e same as t h a tof th e network incidence matrix. P r o c e s s i n g of thee xt er na l s ub ma tr ice s ca n b e effectively separated fromprocessing o f the main s u bm a tr i x b e ca u se elimination o fthe external variables only modifies certain nonzerob l o c k s within the m ain submatrix. Furthermore, theordering o f t h e external variables has no e f f e c t onsparsity. ( T h e s e same remarks also a p p l y t o t h e

arrangement o f H i n ( 6 ) . )F a c t o r i z a t i o n a n d repeat solution o f W r e q u i r e sf o u r t i m e s a s much c o m p u t a t i o n a l e f f o r t as the s a m eoperations with t h e p owe r flow J a c o b i a n , a n d the matrixstorage requirement i s d o u b l e . E f f i c i e n c y a n d s t o r a g erequirements can b e further improved b y d e v e l o p i n gdecoupled formulations.

C o u p l e d SolutionT h e coupled O P F , subject only t o equalityconstraints, ca n b e solved by t h e following s i m p l i f i e dalgorithm;

1 . M ak e st ar ti ng e st im at es f o r z = ( y , X ) .y can b e the s a m e a s the starting values f o r a P F .X can b e zero or any reasonable guess.

2 . Evaluate g a s a f u n c t i o n of z .3 . Check f o r solution.

I f K - T conditions are s a t i s f i e d , OPF i s s o l v e d .Else

4 . E v a l u a t e W a s a function o f z .5 . F a c t o r i z e W a n d solve f o r Az.

3

4

5

2

3

4

- 5

The decoupling s ho wn d iv id es the problem i n t o P ea n d Qv sysbystems. Th e elements of H t h a t a r e omittedi n the decoupling a r e negligibly small; the elementso f J t h a t a r e omitted, w h i c h a r e t h e same a s thoseomitted in the decoupled PF [ 9 ] , m a y no t b e negligiblei n some problems. Any defects that might arise fromt h i s particular decoupling would b e due t o th eapproximation f o r J , not H . T h e r e a r e remedies f o rdefects due t o decoupling of J . Other versions o fdecoupling ar e possible. For e xa mp le , versions basedon decoupling H b u t no t J .

Th e sparsity o f the 2x2 block structure o f t h emain s u b m a t r i x o f W ' a n d W " ( h e a v y b o r d e r s ) i s the sameas t h a t o f the network incidence matrix. T h e remarksa b o u t processing of t h e e x t e r n a l submatrices f o r thecoupled version also apply t o this decoupled version.Factorization o f W a n d W " together requiresapproximately the same comp uta ti ona l e ff or t a sf a c t o r i z a t i o n o f th e Newton P F . T h e matrix storage i salso similar.

6 . U p d a t e z b y Az.

1


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2 8 6 9Decoupled Solution

The d e c o u p l e d O P F , s u b j e c t only t o equalityconstraints, can be solved b y t h e f o l l o w i n g a l g o r i t h m :

1 . Make starting estimates f o r z .2 .3 .4 .

E v a l u a t e g t a n d W ' as functions o f z .Factorize W ' and solve f o r Az'.U p d a t e z b y Az'.

5 . E v a l u a t e g " as function o f u p d a t e d z .6 . Check f o r s o l u t i o n .

I f K-T c o n d i t i o n s ar e s a t i s f i e d , OPF i s s o l v e d .E l s e

7 . E v a l u a t e W " as f u n c t i o n o f u p d a t e d z .8 . Factorize W I ' a n d s o l v e fo r Az".9 . U p d a t e z b y A z " .

1 0 . Return t o step 2 with updated z .

ENFORCING INEQUALITY CONSTRAINTSThis section discusses methods for enforcinginequality constraints. Methods f o r determining whichinequalities t o enforce a r e discussed i n the nextsection. Th e f o ll o wi n g a t tr i bu te s are advantageous ininequality constraint e n fo r ce m e nt m e th od s :

1 . Th e method should b e efficient for enforcinginequalities singly o r a few at a time withoutrefactori z i n g .2 . Th e method should provide for automatic control ofthe h ar dn es s of constraint enforcement. There aresituations where a certain amount of freedom a round

an inequality limit i s beneficial. Ce r t a i nstrategies f o r identifying the b i nd i ng i n eq u al i ti e srequire control of the h ar dn es s of enforcement.Soft limits may also b e needed to obtain usefulnonfeasible solutions w h e n feasible ones d o notexist.3. Th e me thod should not make uncoordinated s t e pchanges in the variables o r functions in order t ose t them to desired values. A l l such changesshould b e m a d e in the s i mu l ta n eo us s o lu t io n of thecorrection vector A z . Uncoordinated changesdisrupt convergence.

Inequality c o n s t r a i n t s are consi dered in t h r e ecategories: ( 1 ) D ispatchable active a n d reactivep o w e r , ( 2 ) variables, ( 3 ) functions of variables.Enforcing Limits on Dispatchable Power Sources

Th e PF equations for dispatchable V A R sourcesa r e omitted from A a s long a s they a r e feasible,an d restored t o A when necessary to p r e v e n t th e Q kfrom becoming infeasible. A d d in g and removingthese equations causes no s tr uc tu ra l c ha ng es inW' and W". As an example, s u p p o s e that at somei tera tion dispatchable r e a c t i v e s o u r c e at bus k e x c e e d sits maximum. Then the equation for A X q k w o u l d b eactivated by entering the current evaluation of itselements in its previously dummied row/column in W".Since Q k had b e en feasible u p to this p o i n t , Xqk w o u l db e currently zero, and t h i s i s the value that would beused i n th e initial evaluation. After the nexti t e r a t i o n , Xqk would assume a nonzero value. The sign

o f X q k indicates whether Q k would return t o itsfeasible range if it s equation were again removed.The PF equations themselves provide only hardenforcement. Methods f o r soft enforcement o f limits onpower sources have been developed but their explanationi s b e y o n d the s c o p e of this paper.

Enforcing Limits on VariablesAny variable can b e made i n t o a constant b y simplyeliminating the equation for i t s correction a n d settingthe var iab le t o t he d esi re d v a l u e . Fo r e xa mp le , a ss um evoltage a t l o a d bus k i s below i t s minimum. Th e limitcould be enforced b y s et ti ng v k t o it s l i m i t and makingroW/column f o r v k i n W " a dummy. Th e equation fo r Qkremains i n W " t o enforce i t s scheduled value.Th e change o f variables t o constants, or v ic e v er sa ,can b e made i n this manner whenever W ' or W I ' i srefactorized, b u t more efficient methods are needed f o renforcing o n e o r a few inequalities a t a time. Also,a s p o i n t e d out, uncoordinated changes i n variablesshould not b e made. A var iab le should be moved t o itsrelevant limit in c o or d i na t i on w i th t h e simultaneouscorrection o f a l l o th er v ar i ab l es .Quadratic penalty functions ideally fulfill therequirements for inequality c o ns t ra i n t e nf or c em e nt i ns e co n d- o rd e r m e th od s. T h i s i s i n marked contrast t othe erratic a n d generally unsuccessful b eh av io r oflinearized quadratic penalties used i n gradient-basedO P F 'methods. I n a second-order method t h e effect o f aquadratic penalty i s accurately controllable. I tprovides a two-sided constraint that ca n clamp avariable a t an exact target value o r allow acontrollable amount o f freedom for variation around t h etarget value. I t also provides t h e means f o r moving avariable or f u n c t i o n t o the target value i nc oo rd i na t io n w i th t h e simultaneous c o r r e c t i o n o f allo th er v ar ia bl es. Equation ( 1 2 ) shows the quadraticpenalty function used t o constrain variable y i .

i . ( Y = - Y ) ( 1 2 )where Y i i s t h e target value, y i i s t h e current va luea n d S i i s a weighting factor that i s automaticallycontrolled t o give t h e a p p ro p ri a te a mo un t o f hardnesso f enforcement. Th e first a n d second d e r i v a t i v e s o f d ia r e :

1 - S. ( y . - Y i )y .21d y i S

( 1 3 )

( 1 4 )Application of t h e penalty will force y i t oc oi nc id e w it h or remain as close t o y i as desired. T h e

q u a d r a t i c penalty augments the L . Its firstderivative i s added t o a L / a y i o f g a n d its secondderivative i s added t o a 2 L / a y i o f W .The proper value f o r S i i s a u t o m a t i c a l l ycontrolled. I f S i i s l a r g e , Yi acts a s a h ar d li mi t a n dy i moves a r b i t r a r i l y close t o the l i m i t . If S i i ss m a l l , Y i acts as a s o f t l i m i t . Th e corrections i n y iar e coordinated with all other corrections on eachi t e r a t i o n . Th e v a l u e of a 2 / a y i i n W " o r W " i s u s e di n a u t o m a t i c a l l y adjusting S i t o the proper value f o r asoft l i m i t . An y S i larger than a c er ta i n v a lu e willeffectively produce a h ar d l im it .After a p e n a l t y ha s been i m p o s e d to constrain avariable, the sign of its derivative can b e used t odecide whether the penalty i s still needed. - T h e


7/17

2 8 7 0d e r i v a t i v e o f a penalty on a variable i s t h e negativeof i t s Lagrange m u l t i p l i e r , p i , for the enforced limitingvalue. T h e derivatives of i m p o s e d penalties are alsou s e d t o determine whether t h e K-T conditions a r es a t i s f i e d .

Quadratic penalties e l i m i n a t e uncoordinatedc h a n g e s i n v a r i a b l e s , p e rm it c on tr ol o f the h ar dn es s ofconstraint enforcement, indicate whether imposedconstraints a r e still binding, a n d provide t h eincremental costs o f the imposed constraints.Enfor Limits on S p e c i a l Functions

I t i s necessary t o b e able t o enforce l i m i t s ons p e c i a l f u n c t i o n s . O n e example i s t h e functiondefining t h e flow of p owe r thr ough a line. Limits o nsuch f u n c t i o n s could b e enforced, when needed, b yadding their equations explicitly t o t h e Lagrangian.B u t t h i s would b e undesirable because i t wouldradically c h a n g e t h e favorable a n d ot he rwi se cons ta ntsparsity structures o f W ' a n d W" . I t i s m uc h s im pl erand e qua ll y e ff ect iv e t o enforce such functionali n e q u a l i t i e s with penalties. T h e following example o fAn i n eq u a li ty c on s tr a i nt o n t h e flow of p owe r through al i n e i l l u s t r a t e s the general c a s e . Other functionsd e fi ni ng i ne qu al it ie s c an b e handl ed in a similar way.

The power flow through l i n e ( k m ) can b e helda p p ro x im a te l y w it h in a specified limit b y requiring thepower angle ( - 0 k O m ) t o b e close t o som e p re com pute dv a l u e 9 k m . I f t h i s i s n ot a cc ur at e e n o u g h , Okm ca n b ei t e r a t i v e l y adjusted t o achieve any desired degree o faccuracy. The quadratic p ena lt y f unct ion i f o re n f o r c i n g an a ng le l im it 8 k m i s :.~~~~~~~.S ; .1 = 2 ( 8 k e )2

where Si i s a weighting factor.Thein ( 1 6 ) .

( 1 5 )

first and second derivatives of i a r e given

d u i S ( 8 - e )dk , i k mk

i - S ( e _e )i k mm

2d ao s1.

2gd ki S.2 1d 0 m

d 2 a .1d 8 dBmk

d 2 u I -sd_ d 8 = ik m

( 1 6 a )

( 1 6 b )

( 1 6 c )

( 1 6 d )

( 1 6 e )

T o a ct iv at e a penalty for ( O k 0 m ) ev a lua te thefi rst derivatives of i an d add them t o th ecorresDondina d er iv at iv es i n g. A d d +S, t o a 2 L / 3 ) k a n da 2 L I / a 9 , a n d a d d - S i to a 2 L / a H k 3 6 m of W'. Note t h a t th epenalty only a u g m e n t s e x i s t i n g non-zero terms in W'.An y n u m b e r of penalties s u c h a s t his can b e imposedwith n o burden o n the m a t r i x . Si can b e adjustedautomatically t o suit requirements. Th e f i r s td e r i v a t i v e ( t h e two are e qua l e xce pt for sign) is th e

Lagrange m u l t i p l i e r t 1 l f o r th e enforcedinequality. I t indicates whether i t i s necessary t ocontinue enforcing th e inequality, and i t i s also usedi n d et er mi ni ng w he th er t h e K-T conditions aresatisfied.I n d iv i d ua l E n fo r ce m e nt o f Inequalities

Th e enforcement o r removal o f a s i n g l e i n e q u a l i t yconstraint involves only a few small changes in W ' orW " a n d t h e s e changes a ff ec t o nl y a f e w rows o f t h efactorization. T h e r e a r e tw o sparsity methods f o robtaining repeat solutions for small changes i n afa ctoriz ed m atr ix wi th out h av ing t o perform a completerefactoriz ation: partial refactoriz ation andcompensation. Either o r both can b e u s e d t o impose o rremove inequality c on st r ai nt s s i ng l y o r a few a t a timewithout refactorizing W ' o r W " I .

Partial refactorization exploits t h e property thats m a l l changes i n a sparse matrix a f f e c t o n l y a few rowso f its sparse f a c t o r i z a t i o n . Fo r simple c h a n g e s , whicho cc ur m os t f r e q u e n t l y , a special f a c t o r u p d a t i n g s c h e m e[ 1 0 ] i s most e f f i c i e n t . For more c o m p l i c a t e d changest h e normal f a c t o r i z a t i o n i s used but a p p l i e d o n l y t ot h e affected rows. To impose o r remove a PF equationo r a penalty f o r an i n e q u a l i t y , proceed as f o l l o w s :1 . Modify th e affected rows o f the factors of W or W "b y partial refactorization.2 . Modify g ' or g " by adding or removing thederivatives f o r th e penalty o r equation change.3 . P e r f o r m a f a s t f o r w a r d solution [ 1 1 ] using o n l y th erows o f W ' o r W t affected by t h e updating.4 . Back solve t o obtain the updated solution.

Small matrix c h a n g e s followed by updated solutionscan b e p e r f o r m e d b y t h i s scheme repeatedly. Theupdates i n the matrix a n d solutions a r e c u m u l a t i v e .Th e e f f i c i e n c y o f each ne w update i s u n a f f e c t e d by thepreceding updates.

With compensation [ 1 1 1 ] , which i s similar t o t h eSchur c o m p l e m e n t t e c h n i q u e [ 1 3 ) used i n o p t i m i z a t i o nl i t e r a t u r e , the f a c t o r s are not altered. T h e e f f e c t o fmatrix c h a n g e s on the solution i s c o m p u t e d directlyfrom sparse c o m p e n s a t i o n vectors. ' S u c c e s s i v e matrixchanges a r e no t c u m u l a t i v e ; each new solution bycompensation i s a change f r o m t h e original basesolution. H o w e v e r , successive changes can b eaccumulated i n t h e compensation vectors t o makesolutions for cumulative changes more efficient. Theefficiency o f t he m et hod declines a s t h e number o fsuccessive changes i ncr ea se s. C om pe ns at ion i s moreefficient than p a r t i a l refactorization up t o acrossover p o i n t that d e p e n d s on th e t o t a l number a n dt y p e of changes.DETERMINING THE B I N D I N G I N E Q U A L I T Y SE T

I n d e c o u p l e d schemes t h e PO and Q v s u b s y s t e m s aresolved i n alternating c y c l e s . Some o f t h e moste f f e c t i v e algorithms tested t h u s f a r perform f a s t t r i a li t e r a t i o n s within e ach cycl e t o determine t h e currentlyb i n d i n g inequalities. Depending o n the application,t h e scheme o f t r i a l iteration used in one cycle mayd i f f e r from t h e o n e u s e d i n th e alternate cycle.S c h e m e s may also differ d e p e n d i n g on t h e s t a g e o f thesolution process. In th e f o ll o w i n g s i m p li f i e dd e s c r i p t i o n s th e p a r t i c u l a r c y c l e i s not s p e c i f i e d .Each a l g o r i t h m could b e used f o r either th e P O or Q vc y c l e . T h u s , instead o f s p e c i f y i n g W " or W " , thes y m b o l W i s u s e d to m e a n either or b o t h . The


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2 8 7 1 .descriptions ar e i n c o m p l e t e ; t h e intent i s o n l y t os u g g e s t s t r a t e g i e s f o r f i n d i n g t h e bindinginequalities.

T h e a lg or it hm s d es cr ib ed employ t w o kinds ofs o l u t i o n s o f t h e d e c o u p l ed q u a d r a t i c a p p r o x i m a t i o n s ofL : main iterations a n d t ri a l i te r at i on s.A m ai n i te ra ti on f o r e it he r c yc le i s a s f o l l o w s :

1 . Evaluate W a n d g a s functions o f t h e latest updateof z a n d A .2 . Factorize W and solve f o r Az .

A trial iteration i s p e r f o r m e d either b y p a r t i a lrefactorization o r b y c o m p e n s a t i o n . T h e aim o f triali t e r a t i o n s i s t o i d e n t i f y t h e c u r r e n t l y b i n d i n gi n e q u a l i t i e s . I n e a c h t r i a l t h e e f f e c t s o f d i f f e r e n tc o m b i n a t i o n s o f c o ns tr ai n t e n fo r ce m e nt and release at atentative s o l u t i o n p o i n t are e x a m i n e d , V a r i o u sstrategies a r e employed t o minimize t h e number o ftrials needed t o f i n d t h e binding s e t . When t h ec ur r en tl y b in di ng i n e q u a l i t i e s have b een s a t i s f a c t o r i l yidentified, t h e solution i s a d v a n c e d t o t h e next s t e pwith t h e b i n d i n g i n e q u a l i t i e s e n f o r c e d . A trialiteration i s always f a s t compared t o a main i t e r a t i o n ,b u t t h e r a t i o o f t h e i r s pe ed s d ep en ds on t h e number a n dkinds of matrix changes involved i n t h e t r i a l and t h ea m o u n t o f m o n i t o r i n g r e q u i r e d t o e x a m i n e t h e tentatives o l u t i o n .Example A l g o r i t h m s

T h e aim o f t h i s s e c t i o n i s t o s u g g e s t howa l g o r i t h m s b a s e d on t h e a p p r o a c h can b e d e v e l o p e d .Three r a t h e r fundamental a l g o r i t h m s , each o f w h i c hcould have many variations, are outlined. Otherd i f f e r e n t a l g o r i t h m s are a l s o p o s s i b l e .A l g o r i t h m I1 . C he c k f o r s o l u t i o n . I f n o t s o l v e d , continue.2 . C h e c k inequalities.

a . C h a n g e A for enforcement o f all violatedi n e q u a l i t i e s .b . Change A t o release a l l p r e v i o u s l y enforcedi n e q u a l i t i e s where enforcement i s no l o n g e rn e e d e d .

3 . P e r f o r m main i t e r a t i o n t o o b t a i n A z .4 . U p d a t e z b y Az.5 . P r o c e e d t o alternate c y c l e .

Algorithm I i s analogous t o t h e s c h e m e s used - i nconventional PF programs t o i d e n t i f y a n d enforcei n e q u a l i t y c o n s t r a i n t s . I t does n o t maintain strictf e a s i b i l i t y b u t a l l o w s c o n s t r a i n t s t o b e , violatedb e f o r e enforcing them on t h e following iteration.C o n s t r a i n t s are enforced a n d released s i m u l t a n e o u s l y i ng r o u p s . A s o b s e r v e d i n P F programs, s i m u l t a n e o u sschemes s u c c e e d i n i d e n t i f y i n g t h e binding inequalit iesb u t t h e y t a k e many i t e r a t i o n s . Algorithm I could b ei m p r o v e d b y s t a r t i n g w i t h s o f t p e n a l t i e s ands y s t e m a t i c a ll y h a r d en i n g t h e m as t h e b i n d i n g s e tb e c o m e s c l a r i f i e d . M a n y d i f f e r e n t s i m u l t a n e o u se n f o r c e m e n t s t r a t e g i e s are p o s s i b l e .

Algorithm I I1 . - 3 . Same a s i n Algorithm I .4 . Find s c a l a r K ( K i 1 . O ) such that a t z + K ( A z ) onlyon e inequality has moved t o i t s limit.5 . U p d a t e z by K ( A z ) .6 . Proceed t o alternate half c y c l e .

Except f o r t h e e f f e c t of t h e alternate decoupledc y c l e , A l g o r i t h m I I would maintain strict feasibility.I t w o u l d , however, b e prohibitively s l o w . I t could b egreatly improved b y adjusting K t o allow f o r acontrolled a mount of violation on each iteration. Ame ri t funct ion [ 1 2 ] could b e developed t o determine ag o o d value f o r K b y t a k i n g into account t h e k i n d ,number and magnitude o f t h e v i o l a t i o n s . M a n yvariatiQns o f t h e m e r i t function a n d t h e strategy f o ri t s application are p o s s i b l e .

A l g o r i t h m I I I1 . - 3 . Same a s i n Algorithm I .4 . T e n t a t i v e l y u p d a t e z b y Az.5 . Check i n e q u a l i t i e s a t z + Az .

a . I f t h e i n e q u a l i t i e s s a t i s f y th e criteria f o r ana c c e p t a b l e i t e r a t i o n , a d v a n c e t h e solution t ot h e t e n t a t i v e . s t e p ; return t o step ( 1 ) andproceed t o alternate half cycle.E l s e

b . According to th e trial strategy select a set o fviolated i n e q u a l i t i e s f o r enfQrcement a n d a s e tof p r e v i o u s l y enforced i n e q u a l i t i e s f o rrelease. M o d i f y A a c c o r d i n g t o t h e s e l e c t i o n .C o n t i n u e .

6 . Enforce and release the s e l e c t e d i n e q u a l i t i e s b y at r i a l iteration t o obtain A z ( t r i a l ) .7 . Return t o s t e p ( 5 ) w i t h z = A z ( t r i a l ) .

A l g o r i t h m III maintains a s p e c i f i e d amount o ff e a s i b i l i t y . A t t h e end o f each series of t r i a li t e r a t i o n s f o l l o w i n g a m a i n i t e r a t i o n t h e enforcementof i n e q u a l i t i e s i s e x a c t l y or a p p r o x i m a t e l y resolvedd e p e n d i n g on t h e criteria f o r an a c c e p t a b l e iteration.In e f f e c t , t h e t r i a l i t e r a t i o n s test differentc o m b i n a t i o n s of enforcement i n order t o f i n d t h e b e s tone. Many v a r i a t i o n s of Algorithm III are p o s s i b l e .Criteria Fo r E n f o r c i n g I n e q u a l i t i e s

T h e enforcement o f different t y p e s of i n e q u a l i t i e shas d i f f e r e n t e f f e c t s . T h e e f f e c t s a r e a p p r o x i m a t e l yp r e d i c t a b l e from t h e k n o w n behavior o f power s y s t e m sand t h e s o l u t i o n process. T h i s k n o w l e d g e can b e u s e di n e s t a b l i s h i n g c r i t e r i a f o r e n f o r c i n g i n e q u a l i t i e s andi n d e v e l o p i n g m e ri t f un ct io ns .When a violated v o l t a g e limit a t a l o a d b u s i s


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2 8 7 2enforced, i t tends t o pr oduce cha nge s in the aamedirection i n other voltages in its area of the network.Simultaneous enforcement of ha rd limits on sev era lv io la te d v ol ta ge inequalities in one area c a n crea tedisruptive reacti v e power- flows. E nforcing vol ta geinequalities one a t a time, or sev era l a t a time withsoft c on s tr a in t s, a l l ev i a te s this difficulty. Cr i t e r i acan b e developed for s e l e c t i n g the inequality mostlikely to b e binding out of a group o f voltageinequality violations.

Enforcing an inequality o n a dispatchable sourceof active or reacti v e power forces the remainingdispatchable sources to ma ke up for the deficiency orsurp lus. This action can f o r c e other dispatchablesources o f the same kind to exceed t he ir l im it s. Thesecascading effects shopld b e resolved by trialiterations before proceeding . Violations ofinequalities in this class shoul4 b e enforcedsimultaneously because t h e y g e n e r a l l y tend to c r e a t emore violations of t h e sam e kind.

Exact enforcement of all inequalities on eachiteration i s generally wasteful. T h i s waste can b ereduced b y d ev el op in g c ri te ri a f o r a c c e p t a n c e of ani t f e r a t i o n that ta kes into account the number, kind andmagnitude of violations a s well a s bus mismatches.These f e w examples o f criteria a r e i l l u s t r a t i v e , b u tnot exhaustive.Selective Monitoring

Th e forward solutions for trial iterations inAlgorithm III can b e p e rf or me d e f fi ci e nt l y because onlya few columns of the f a c t o r s a r e directly involved.The back solutions can b e speeded u p in a similar wayb y s ol vi ng only for those quantities that need t o b emonitored. After each trial iteration i t i s onlynecessary to check two groups of inequalities; ( 1 )those that were nonfeasible on the last t r i a l ; a n d ( 2 )those t h a t were marginally feasible b u t t hr ea te ne d w it hviolation by th e enforcement of o th e r i n eq u al i ti e s.The first group would normally consist of b u s vQltagesfor which i t would b e necessary to check t he e ff ec t o fenforcing one on reducing the violations of the others .This group would rapidly diminish on e ac h s uc ce ss iv etrial i t e r a t i o p . Since most of the e f f o r t o f trialiterations i s in the b ac k solution, large gains can b emade by s e le c ti v e m o ni t or i ng .

TEST RESULTSAt t h e time of this writing the development a n dtesting o f algorithms i s continuing. Th e intent of theinvestigation i s to determine how d if fe re nt a lg or it hm sperform on d if fe re nt p ro bl em s, no t to produce adefinitive solution m ethod . All t ; e s t i n g i s being done

on t h e V AX 1 1 / 7 8 0 c o m p u t e r . To give some e vi de nc e ofh ow the a pp r oa ch works, t he r e s u l t s of one experimentala l g Q r i t h m on one te s t problem are shown. T h i s examplei s r ep re se nt at iv e of res ult s b eing obtained withdifferent probl ems and different e x p e r i m e n t - a lalgorithms. I t i s expected t h a t much better algorithmswill b e developed.Test P r o b l e m

The network of the test problem i s a portion o fthe p o w e r -system of the N o r t h e a s t e r n U n i t e d S t a t e s .The problem is r e a c t i v e cpntrol optimization - o n l y .Nor ma ll y, m ul ti pl e sources o f a c t i v e power c a n b edispatched. In this particular example, h o w e v e r , asingle sl ack is us e d . T h e main a t t r i b u t e s of t h e t e s tproblem a r e :

- N e t w o r k : 9 1 2 buses. 1 6 3 7 branches.- T o t a l M W generation: 1 2 0 0 p.u. ( 1 0 0 MV A base)- Starting C o n d i t i o n : unconverged network solution.- Objective F u n c t i o n : Minimum active p ow er l os se s- Controls: 255 d i s pa t ch a bl e r e ac ti v e s o u r c e s and 4 1controllable transformers.- I ne qua li ty C ons tr ai nt s: U pp er a n d l ow er l im it s ondispatchable reactive sources, controllablet r a n s f o r m e r tap ratios and all bus voltages.

A version of Algorithm III f o r a decoupledf o r m u l a t i o n wa s u s e d f o r t he t es t . Th e strategy o finequality constraint enforcement was as f o l l o w s : Onth e f i r s t iteration only t h e l i m i t s on generatorvoltages and transformer t ap r at ios were enforced. Onthe second iteration l i m i t s o n reactive sources w e r ealso enforced. On t h e third a n d s ub se qu en t i te ra ti on sa l l inequalities w e r e enforced. A l l enforeements w e r ewith hard limits. Each Qv cycle was mad e comp letelyfeasible b y t r i a l i te r at i on s b e fo r e t h e a l go ri th m w asadvanced t o the n e x t cycle.

Results o f T e s t ProblemT h e h i s t o r y o f i t e r a t i o n s f o r th e t e s t p r o b l e m i ss h o w n i n TABLE I . Each i te ra ti on a ct ua ll y r ep r es en tsboth a P O a n d a Qy c y c l e . H o w e v e r , the t r i a literations a n d i ne q ua l it y c on st r ai nt e n fo r ce m en t s a p p l yonly t o the Qv cycle.

TABLE I .O p t i m i z a t i o n S u m m a r y F o r A 9 1 2 - B u s S y s t e m .( 1 ) ( 2 ) ( 3 ) ( 4 ) ( 5 ) * ( 6 ) * ( 7 ) * ( 8 )N o . o f N o . of A c t i v e R M S of Ma xM a i n T r i a l B i n d ' g P o w e r M a x P Ma x Q P r o j e c ' d A v o r

I t e r . I t o r . L i m i t s L o s s M a m t c h M sm tc h G ra di en t A t a p1 1 1 1 4 2 1 1 . 4 4 8 0 . 2 8 6 6 . 1 3 6 . 0 0 5 5 5 4 . 1 8 2 42 8 1 4 3 1 0 . 3 2 5 0 . 4 1 8 0 . 8 2 0 . 0 0 1 1 6 7 . 0 6 8 93 2 3 1 3 1 1 0 . 2 7 1 0 . 3 0 3 0 . 3 4 4 . 0 0 1 4 1 7 . 0 4 6 24 2 1 3 3 1 0 . 3 0 2 0 . 0 9 7 0 . 1 2 6 . 0 0 1 3 6 7 . 0 1 8 65 1 1 3 2 1 0 . 3 0 6 0 . 0 2 4 0 . 0 2 8 . 0 0 0 2 5 3 . 0 1 9 36 2 1 3 1 1 0 . 3 1 6 0 . 0 4 8 0 . 0 3 6 . 0 0 0 3 6 6 . 0 2 2 57 0 1 3 1 1 0 . 3 2 7 0 . 0 6 6 0 . 0 4 1 . 0 0 0 3 1 8 . 0 1 2 88 0 1 3 1 1 0 . 3 3 2 0 . 0 6 2 0 . 0 1 4 . 0 0 0 0 9 4 . 0 0 2 79 0 1 3 1 1 0 . 3 3 4 0 . 0 4 8 0 . 0 0 4 . 0 0 0 0 4 7 . 0 0 0 6

1 0 0 1 3 1 1 0 . 3 3 1 0 . 0 3 0 0 . 0 0 4 . 0 0 0 0 4 6 . 0 0 0 21 1 0 1 3 1 1 0 . 3 3 3 o . o 1 4 0 . 0 0 4 . 0 0 0 0 4 3 . 0 0 0 11 2 0 1 3 1 1 0 . 3 3 4 0 . 0 0 2 0 . 0 0 2 . 0 0 0 0 3 4 . 0 0 0 1

* 1 0 0 MV A b a s e .Legend For T a b l e I

C o l u m n 1 i s t h e count o f i t e r a t i o n s . A n iterationc o n s i s t s o f on e m a i n iteration for the P O cycle a n d onemain iteration p l u s a variable n q m b e r of triali t e r a t i o n s for th e Qv c yc le . Th e system i s updated a tt h e e n d o f t he m ai n i te ra ti on o f the P O c O y c l e . F o r theQ v cycle t h e system i s n ot u pd at ed until the e n d of thet r i a l iterations.


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Column 2 i s t h e count o f trial iterations n e e d e dt o identity t h e currently b i nd i n g i n eq u al i tie s f o r a nacceptable i t e r a t i o n . I f the main i t er a ti on i ts el f i sfeasible, no trials are needed a n d t h e count i s z e r o .Column 3 i s t h e number of binding inequalities a tt h e e n d o f the i t e r a t i o n .Column 4 i s t h e value o f the o b j e c t i v e function a t

t h e start o f the i t e r a t i o n .Column 5 a n d 6 are t h e l ar ge st r es id u al s o f the PFequations a t t h e start o f the iteration.C o l u m n 7 i s th e R M S of the g r a d i e n t o f L f o r t h ebasic a n d superbasic variables at t h e beginning o f theiteration. T h e R M S value i s t h e s q u a r e root o f the s u mof t h e squared g r a d i e n t s , divided b y t h e number o f thebasic and superbasic variables. T h i s n u r d b e r i s am ea sur e of the steepness of the objective f u n c t i o n . Att h e solution i t should b e z e r o except f o r r o u n d o f f .Column 8 i s th e maximum per unit iterativecorrection i n a v o l t a g e or transformer tap r a t i o . Atthe solution i t should b e zero except f o r r o u n d o f f .

DISCUSSION OF R E S U L T ST h e t o t a l c o m p u t a t i o n a l effort f o r t h e solutioncan b e e s t i m a t e d f r o m the total of main a n d t r i a li t e r a t i o n s . F o r a well-written code each maini te ra ti on w ou ld r e q u i r e about 1 . 2 times as much effortas a N e w t o h PF i t e r a t i o n . T h e e f f o r t o f t r i a literations i s v a r i a b l e . With g o o d s p a r s i t y t e c h n i q u e st h e cost o f a t r i a l should a v e r a g e about on e t e n t h o f af u l l iteration. T h i s makes t h e total effort f o r th etest problem e q u i v a l e n t ; t o about 1 8 Newton PFiterations.Most o f the t r i a l i t e r a t i d n s we r e p e r f o r m e d at anearly s t a g e . Eleven trials were needed i n t h e f i r s titeration t o f i n d the currently b i n d i n g t a p s and

generator v o l t a g e s . E i g h t t r i a l s were needed i n thesecond i t e r a t i o n when reactive source l i m i t s were alsoenforced.Twenty three trials were needed i n th e thirditeration when l o a d v o l t a g e s were i n c l u d e d i n thee n f o r c e d i n e q u a l i t i e s . A f t e r the t h i r d iteration o n l yf i v e more t r i a l s were n e e d e d t o i d e n t i f y t h e b i n d i n gs e t 1 f o r t h e s o l u t i o n . T h e 1 3 1 b i n d i n g i n e q u a l i t yconstraints consisted o f 1 4 V A R l i m i t s and 1 1 7 busV o l t a g e limits.I t e r a t i o n s s e v e n t h r o u g h t w e l v v e w e r e needed o n l yt o solve the power flow e q u a t i o n s while e n f o r c i n g af i x e d s e t o f ' b i n d i n g i n e q u a l i t i e s . C o n v e r g e n c e wa sslow near t h e e n d b e c a u s e o f t h e w e a k h e s s o f thed e c o u p l e d PF formulation when t h e PF r e s i d u a l s become

s m a l l . A c h a n g e in s o l u t i o n scheme a t t h i s p o i n t wouldb e a d v a n t a g e o u s .I t i s also i n t e r e s t i n g to note that the o b j e c t i v ef u n c t i o n r e a c h e d i t s l o w e s t ; v a l u e on the t h i r di t e r a t i o n a r i d then gradually increased as t h e powerf l o w m i s m a t c h e s were d i m n i n i s h e d . More d e t ' a i l e dm o n i t o r i n g a n d a n a l y s i s o f w h a t i s happening i n th ec o u r s e o f t h e s o l u t i o n i s expected t o provide i n s i g h tf o r i m p r o v i n g a l g o r i t h m s .T h e p o s s i b i l i t y t h a t O PF s O l u t i o n s m ay n o t b eu n i q u e i s o f c o n s i d e r a b l e concern. Several O P Fp r o b l e m s h a v e b e e n s o l v e d with W i d e l - y d i f f e r e n ts t a r t i n g c o n d i t i o n s a n d with e n t i r e l y differenta l g o r i t h m s . I n a l l s u c h tests identical solutions wereo b t a i n e d f o r th e s a m e problems. While t h i s d oes not

2 8 7 3prove t h a t all OPF problems have unique solutions, itstrongly suggests that a t least some of them d o .

SUMMARY AN D CONCLUSIONST h e Newton approach i s suitable fo r development;ofpractical O P F p r o g r a m s . This c o n c l u s i o n i s supportedby tests o f Newton-based algorithms W i t h prototypec o d e s . Although efficiency has not b een fully

exploited in t h e prototype codes u s e d i n t est in g, i t i spossible t o estimate the performances o f efficientcodes f o r the same operations.T h e r e i s always some risk in d rawing conclusionsabout a new approach that h a s not been e x t en s ive lytested i n production U s e . I t seems unlikely, however,t h a t a n y difficulties ca n a ri se that would i n v a l i d a t ethe m a i n outline of the given approach. If an explicitNewton method cannot b e m a d e t o solve a p r o b l e m , anyexisting quasi-Newton m et ho d wo ul d be n o better. T h eonly known alternatives a r e weaker, more approximatemethods that involve more computation.Th e k e y ideas o f the g i ve n N ew to n approach are,

1 . A n e x pl ic it N ew t on f o r m u l a t i o n ' .2 . Decoupling.3 . Quadratic penalties f o r enforcing inequalityconstraints.4 . S p e c i a l strategies f o r finding the bindingi n e q u a l i t i e s .5 . S p e c i a l sparse m a t r i x / v e c t o r t e c h n i q u e s .

Some o f t h e attributes o f a l g o r i t h m s b a s e d on theapproach are,1 . Superlinear convergence t o th e K-T conditions.2 . Solution time c o m p a r a b l e t o a fe w conventionalPF s .3 . S o l u t i o n t i m e a n d matrix storage no t adverselyaffected by n um be r of c ontr ol s o r i n e q u a l i t i e s .4 . N o need f o r user-supplied tuning f a c t o r s o ri n t e r a c t i o n t o obtain a solution.5 . A d a p t a b i l i t y t o a l l three subdivisions o f O P F ;active/reactive d isp at ch, a cti ve only, reactiveonly.

A solution a l g o r i t h m b a s e d on the g i v e n Newtonapproach ca n b e roughly d i v i d e d i n t o ,1 . P r o b l e m d e f i n i t i o n : objective f u n c t i o n , c o n t r o l s ,c o n s t r a i n t s , e t c .2 . Solution f o r m u l a t i o n : d e c o u p l i n g , a p p r o x i m a t i o n s ,etc.3 . Methods f o r e n f o r c i n g i n e q u a l i t y c o n s t r a i n t s ,penalties, trial iterationsi etc.4 . S t r a t e g i e s f o r finding the binding inequalities.5 . S p a r s i t y t e c h n i q u e s .

I n d e v e l o p i n g an a l g o r i t h m , choices must b e m a d eamong t h e s e v e r a l possibilities i n each part. The


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2 8 7 4choices in any one p a r t will b e affected b y the choicesm a d e i n others. T h e number o f c om b in a ti o ns o f choicesi s l a r g e a n d each a p p l i c a t i o n a rea ma y r e q u i r ed i f f e r e n t choices. As more k n o w l e d g e and e x p e r i e n c ea r e gained, the b e s t choices fo r each a p p l i c a t i o ns h o u l d b e co me e vi de nt . P r a c t i c a l , p r o d u c t i o n - g r a d e OP Fp ro gr am s c an b e d e v e l o p e d fr om wh at i s p r e s e n t l y k n o w n .

ACKNOWLEDGEMENTST h i s research i s s p o n s o r e d b y E P R I under contractR P 1 7 2 4 - P h a s e I . ESCA C o r p o r a t i o n i s the p r i m econtractor. Hermann D o m m e l , Philip G i l l , Walter Murray,Michael Saunders, Larry S t a n f e l , W illiam Tinney, andMargaret Wright are the consultants t o the project. D r .John Lamont i s the E P R I project manager.Specific a p p r e c i a t i o n i s e x p r e s s e d to D r . J o h nL a m o n t f o r i n i t i a t i n g the p r o j e c t a n d f o r h i sc o n t i n u i n g g u i d a n c e ; t o Walter M u r r a y , M a r g a r e tW r i g h t , P h i l i p G i l l , a n d Michael Saunders f o r theire x p e r t advice o n t h e o v e r a l l a p p l i c a t i o n ofoptimization m e t h o d s ; t o D r . Hermann D o m m e l f o rs h a r i n g h i s broad p e r s p e c t i v e a n d e x p e r i e n c e o n powersystem a n a l y s i s ; and t o D r . L a r r y Stanfel f o r h i si n v e s t i g a t i o n of t h e r e l a t i o n s h i p between th e Newtonm e t h o d a n d Linear P r o g r a m i n g i n a real t i m e o p e r a t i o n senvironment.T h e a u t h o r s a l s o wish t o thank t h e project'si n d u s t r y a d v i s e r s f o r t h e utility perspective t h e yp r o v i d e d .

APPENDIXE x p r e s s i o n s Fo r M at ri x E le me nt s

T h e m a t r i x W i s c o m p o s e d of submatrices J a n d H .Expressions f o r J are well kn own f ro m t h e Newton P F ;expressions f o r H a r e less f a m i l i a r . More than f i f t yexpressions a r e r e q u i r e d f o r H , b u t many fewer arerequired fo r H a n d H " .Each element o f H ' or H " i s o f the f o r m 3 L / ! y k , y 1 Y .S i n c e L i s j u s t a combination o f P F e q u a t i o n s , eachterm of H ' o r H " i s a sum of second partial derivativesof t h e PF equations w r t , Yk and y m . F o r any p a i r o fvariables Yk and Y m , - nonzero second partialderivatives e x i s t only for those PF e q u a t i o n s thatcontain Yk an d y m . T h e n um be r of n on ze ro terms in thesum f o r any element o f H ' an d H " i s small. T h eexpressions f o r representative elements needed f o r W 'and W ' a r e developed in the f o l l o w i n g . Expressionsf o r the f i r s t p a r t i a l derivatives of the PF equationsare needed f o r J i and J " . E x p r e s s i o n s f o r secondpartial d e r i v a t i v e s o f PF evaluations a r e needed f o r H 'and H " .

Power Flow E q u a t i o n sT h e p o l a r f o r m of the PF e q u a t i o n s i s shown below.

2, ~ ~ ~ 2P = v ( C - E t G ) +( v Y , v t Y c o s 4 ' )k k kk km km k m km km kmZ . 2k k ( k k E t k m B k m ) + ( v k X m t k m Y k m s i n km

( C k + j B ) Transfer admittance o f branch k m .km kmY = G 2+ B 2km k m k m 7 = arc tan B / Gkm km km

G ? k k j B k k Driving point admittance a t busk exclusive o f contributionsfrom any controllable transfor-mers with their tappers at b u s k .t k n m

0 k m

I f branch ( k , m ) i s a controlla-ble t r a n s f o r m e r , tk m i s its tapratio. Otherwise, t k m = 1 . O .I f branch ( k , m ) i s a phaseshifter, 0 k m i s its shi ft angle.Otherwise, , k m = 0 .

I n the following d e r i v a t i v e s ,= ( - a + 40 - 7 )km k m km km

First Partial Derivatives o f P for J a n d ' .a P ka P k = -v Iv v Y s i n 4 'a O k k m k m k m k mkap? r

v V t Y s i n ;a O r e l k m k m km k ma p l ,- - - ' V I ' v t Y s i n 4 'ao k 1 a k m km km km

First Partial Derivatives of Q for J " a n d g

a v k 2 v k ( B k r + t k m B k m ) + V m k m k m sink~~~~~~~~~~~~ka o k_ v t y s i n ;a v ~ k k m k m k m

= 2 v - t B + v v Y sin;a t k k m km k m k m k mk m !

R e p r e s e n t a t i v e Second P a r t i a l Derivatives o f P f o r H2a p ,k = - v z v t Y c o s 42 a o 2 k m km k m kma k2 a Qcok 'Q-kkma - k V m t k m Ykm k k m al ; m m

R e p r e s e n t a t i v e S e c o n d P a r t i a l Derivatives o f fo r H "

Q k _22 = 2 ( B k k + t k B k )a v 2kSumma tions a r e over a l l b r a n c h e s ( k , m ) c o n n e c t e d t o b usk .

2a Q k = t Y s i n p ka v a v km k mk m


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Example of H e s s i a n ElementsFollowing ar e some o f t he e xp re ss io ns f o r elementsof H ' a n d H" for this example problem. R e f e r to theexpression for L of the example in ( 2 ) .

a 2 Laoj

a 2 L. a v 4 , a v 5

I _ _ X 2- I - P 22 . 2 XQ2 22

2 2

- x P 4 4 _ __4 4a o 2 a o 2. 2

a _ _ 4 _ _ _ a __4X)4________- x Q 5a v 4 a v 5 a v 4 a v 5

Expressions for o t h e r e l e m e n t s of H' a nd H" f o l l o wdirectly from these examples.REFERENCES

1 . B . Stott, 0. Alsac and J.L. Marinho, " T h eOptimal Power Flow Problem", in E l e c t r i c PowerP r o b l e m s ; Th e Mathematical Challenge, B o o k , S I A M ,pp 3 2 7 - 3 5 1 , P h i l a d e l p h i a , 1 9 8 0 .

2 . A . M. Sa sson and H . M. Merril, " S o m eApplications of Optimization Techniques to PowerSystem Problems ", I E E E P r o c e e d i n g s , v ol . 6 2 , p p.9 5 9 - 9 7 2 , July 1 9 7 4 .3 . H . H . H a p p , "Optimal P o w e r Dispatch - AComprehensive S u r v e y " , I EE E T r an sa c ti o ns on PowerApparatus and S y s t e m s , vol. P A S - 9 6 , p p . 8 4 1 - 8 5 4 ,

May/June 1 9 7 7 .4 . B . Stott, J . L . Marinho a nd 0. A l s a c , " R e v i e wof Linear Programming Applied t o Power SystemRescheduling", IEEE PICA C onference P r o c e e d i n g s ,Cleveland, p p . 1 4 2 - 1 5 4 , Ma y 1 9 7 9 .5 . H . W. Dommel and W . F . T i n n e y , "Optimal P o w e rFlow Solutions". IEEE Transactions on PowerApparatus an d Systems, v o l . PAS-87 , p p.1 8 6 6 - 1 8 7 6 , October 1 9 6 8 .6 . R . C . B u r c h e t t , H . H . H a p p a nd K. A . Wirgau," L a r g e Scale Optimal Power Flow" , I b i d , vol.P A S - 1 0 l , N o . 1 0 , pp. 3 7 2 2 - 3 7 3 2 , October 1 9 8 2 .7 . A . M. S a s s o n , F . V i lo r ia and F . A b o y t e s ,

" O p t i m a l Load Flow Solution Using the He s s i a nM a t r i x " , Ibid, vol. P A S - 9 2 , p p . 31-41,J a n u a r y / F e b r u a r y , 1 9 7 3 .8 .

1 1 . 0. A l s a c , B . Stott a n d W ."Sparsity-Oriented C o m p e n s a t i o nModified Network S o l u t i o n s " , I b i d ,N o . 5 , pp . 1 0 5 0 - 1 0 6 0 , May 1 9 8 3 .F . T i n n e y ,Methods f o rv o l . P A S - 1 0 2 ,

1 2 . P . E . G i l l , W . Murray and M . H . W r i g h t ,Practical O p t i m i z a t i o n , B o o k , Academic P r e s s , 1 9 8 1 .1 3 . P . E . G i l l , W . Murray, M . A . Saunders, M. H .Wright, Sparse Mat rix Techniques i n Optimization,Report SO L 8 2 - 7 1 , Department o f OperationsR e s e a r c h , Stanford U n i v e r s i t y , California, December1 9 8 2 .

BiographyB r u c e Ashley received h i s BSEE from the Universityof Washington i n 1 9 8 0 and attended Purdue U n ive r s it yunder a Purdue Electric Power Committee Scholarship.He j o i n e d ESCA i n December, 1 9 8 1 , and i s a m e m b e r ofth e IEEE P E S , I A S , and t h e Computer S o c i e t y .Brian Brewer received h i s BSEE degree in 1 9 8 0 andMSEE degree in 1 9 8 2 from the University of Washington,a n d i s a member of IEEE PE S an d Tau B eta P i . He joined

ESCA i n J u l y 1 9 8 2 .Ar t H u g h e s received BSEE degree f r o m t h eUniversity o f Manitoba in 1 9 7 2 a n d MS a n d P h D degreesfrom the University o f Texa s a t Arlington in 1 9 7 6 and1 9 7 8 . H e i s presently employed by the ESCA Corp.David I . S u n received BSEE and MSEE degrees fromResselaer Polytechnic Institute i n 1 9 7 4 and 1 9 7 6 , andP h D f r o m t h e University o f Texas at Arlington i n 1 9 8 0 .He i s p r e s e n t l y e m p l o y e d by th e E S C A C o r p .William F . Tinney received BS and MS degrees f r o mStanford University in 1 9 4 8 and 1 9 4 9 . He worked f o rthe Bonneville Power Administration from 1 9 5 0 to 1 9 7 9and w a s head of the System Analysis Section a t th e timeof h i s r e t i r e m e n t . Most o f h i s experience ha s been in

p o w e r system computation. He i s presently a consultantin this f i e l d .

F . L . Al v a ra do and M . K. E n n s , " Blocked SparseMatrices in Electric Power Systems", Paper A 7 63 6 2 - 4 , I E E E PE S Summer Meeting, P o r t l a n d , July1 9 7 6 .9 . B . Stott and 0 . Al s a c, "Fast Decoupled L o a dF l o w s " , I E E E Transactions o n Power A p p a r a t u s andS y s t e m s , vol. P A S - 9 3 , p p . 8 5 9 - 8 6 9 , May/June 1 9 7 4 .

1 0 . E . C . Housos a nd G. D . I r i s a r r i , " A SparseVariable Metric Optimization M e t h o d Applied t o t h eSolution o f Power S y s t ; e m Problems", I b i d , vol.P A S - 1 0 1 , no. 1 , p p . 1 9 5 - 2 0 2 , January 1 9 8 2 .

2 8 7 5


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2 8 7 6D i s c u s s i o nS . K . C h a n g a n d J . L e q u a r r e ( S y s t e m s C o n t r o l , I n c . , P a l o A l t o , C A ) :T h i s p a p e r h a s p r e s en t ed a r e f r e s h i n g n e w a p p r o a c h t o t h e s o l u t i o n o fo p t i m a l p o w e r f l o w p r o b l e m . A t S y s t e m s C o n t r o l , w e h a v e d e v e l o p e da L o s s M i n i m i z a t i o n P r o g r a m ( R e a c t i v e C o n t r o l O p t i m i z a t i o n ) e m p l o y i n ga s i m i l a r N e w t o n - b a s e d a p p r o a c h e d a n d t h e t e s t r e s u l t s i n d i c a t e t h a t t h i sm e t h o d i s v e r y p r o m i s i n g f o r p r a c t i c a l a p p l i c a t i o n s . W e w o u l d t h e r e f o r el i k e t o s h a r e o u r e x p e r i e n c e b y o f f e r i n g t h e f o l l o w i n g c o m m e n t s :1 . A l t h o u g h n o t s t r e s s e d i n t h e p a p e r , w e f e e l t h a t a g o o d i n i t i a l e s t i -

m a t e o f z i s e s s e n t i a l f o r i m p r o v i n g t h e r o b u s t n e s s o f t h e a l g o r i t h m .T h i s i s d u e t o t h e f a c t t h a t N e w t o n ' s m e t h o d i s i d e a l o n l y i n t e r m so f i t s l o c a l c o n v e r g e n c e p r o p e r t i e s , b u t n o t i n t e r m s o f g l o b a l c o n -v e r g e n c e . An i n i t i a l s t a r t w i t h f l a t v o l t a g e a n d z e r o X w i l l g e n e r a l l yc a u s e d i f f i c u l t o r e r r a t i c c o n v e r g e n c e . A t y p i c a l s c h e m e w o u l d b ea s f o l l o w s : P e r f o r m o n e l o a d f l o w i t e r a t i o n t o o b t a i n i n i t i a l s t a t ev a r i a b l e s c o r r e s p o n d i n g t o a r e a s s i g n e d s e t o f c o n t r o l v a r i a b l e s . T h e nc o m p u t e t h e i n i t i a l X f r o m t h e r e l a t i o n [ J v J T h = v , w h e r e [ J v ] i st h e n o n s i n g u l a r p o w e r f lo w J a c o b i a n m a t r i x , a n d g v i s t h e g r a d i e n tv e c t o r o f t h e o b j e c t i v e f u n c t i o n w i t h r e s p e c t t o t h e s t a t e v a r i a b l e s .F o r t h e L o s s M i n i m i z a t i o n p r o b l e m , t h e i n i t i a l a s s i g n m e n t o f m a x -imum v o l t a g e s t o t h e g e n e r a t o r c o n t r o l v a r i a b l e s w a s f o u n d e x -p e r i m e n t a l l y t o g i v e b e s t c o n v e r g e n c e c h a r a c t e r i s t i c s .2 . T h e p a r t i t i o n i n t h e p a p e r o f t h e W m a t r i x i n t o a n e x t e r n a l s u b -m a t r i x a n d a m a i n s u b m a t r i x s u g g e s t s t h e p o s s i b i l i t y o f i l l -c o n d i t i o n i n g i f a c o n t r o l l a b l e t r a n s f o r m e r h a s n e g l i g i b l e c o n d u c -t a n c e . F o r e x a m p l e , t h e d i a g o n a l t e r m ( A t , A t ) c o r r e s p o n d i n g t oa c o n t r o l l a b l e t r a n s f o r m e r i n t h e L o s s M i n i mi z a t i o n p r o b l e m c a nb e w r i t t e n a s ( A t , A t ) - t e r m = 2 V k 2 [ g ( l - X p K ) + b X Q k l =2 V k 2 b . X Q k g = o w h e r eVk i s t h e v o l t a g e m a g n i t u d e a t t h e t a p - s i d e b u s kg , b t h e t r a n s f o r m e r c o n d u c t a n c e a n d s u s c e p t a n c e ,r e s p e c t i v e l yX P k , XQk t h e L a g r a n g e m u l t i p l i e r f o r r e a l a n d r e a c t i v e p o w e rf l o w e q u a t i o n s a t b u s k , r e s p e c t i v e l y .I f b u s k i s o f P V- t y p e a n d XQk i s z e r o , t h e t e r m ( A t , A t ) i s z e r oa n d c a n c a u s e n u me r i c a l d i f f i c u l t y i f t h e p a r t i t i o n s t r a t e g y i s a p -p l i e d . A m o d i f i e d o r d e r i n g o r p r o c e s s i n g s c h e m e , e . g . , p l a c i n g t h et a p v a r i a b l e i n t h e m a i n s u b m a t r i x , i s t h e r e f o r e r e q u i r e d t o c i r c u m -v e n t t h i s p r o b l e m .3 . I n s p i t e o f i t s h e u r i s t i c n a t u r e , t h e s t r a t e g y o f e n f o r c i n g / r e l e a s i n gm u l t i p l e c o n s t r a i n t s a t a t i m e s e e m s t o b e e s s e n t i a l f o r e f f i c i e n t i d e n -t i f i c a t i o n o f t h e b i n d i n g i n e q u a l i t y s e t i n l a r g e s c a l e OPF p r o b l e m s .To a p p l y t h i s s t r a t e g y , h o w e v e r , t h e p o s s i b l e p h e n o m e n o n o f" z i g z a g g i n g " s h o u l d b e p r o p e r l y g u a r d e d a g a i n s t s o t h a t a c o n -s t r a i n t i s n o t r e p e a t e d l y e n f o r c e d a n d r e l e a s e d d u r i n g t h e t r i a l i t e r a -t i o n s . A l t h o u g h f i n d i n g a m e t h o d f o r i d e n t i f y i n g t h e b i n d i n g i n e -q u a l i t i e s s t i l l i s a m a j o r c h a l l e n g e , i t s e e m s t h a t a s t r a t e g y t h a t u t i l i z e sz i g z a g g i n g - g u a r d e d m u l t i p l e c o n s t r a i n t e n f o r c e m e n t / r e l e a s e c o u p l e dw i t h d e t e r m i n i s t i c s i n g l e c o n s t r a i n t e n f o r d e m e n t o r r e l e a s e w o u l db e a p r a c t i c a l c h o i c e . T h e a u t h o r s ' c o m m e n t s c o n c e r n i n g t h i s p o i n tw i l l b e a p p r e c i a t e d .F i n a l l y , w e w o u l d l i k e t o commend t h e a u t h o r s o n a w e l l - w r i t t e n a n di m p o r t a n t p a p e r .

M a n u s c r i p t r e c e i v e d F e b r u a r y 1 3 , 1 9 8 4B . K . J o h n s o n ( P o w e r T e c h n o l o g i e s , I n c . , S c h e n e c t a d y , N Y ) : T h e a u t h o r sh a v e p r e s e n t e d a v e r y i n t e r e s t i n g p a p e r o n m e t h o d s o f o p t i ma l p o w e rf l o w s o l u t i o n s . H a v i n g d e v e l o p e d a t e c h n i q u e s i m i l a r t o t h e i r c o u p l e ds o l u t i o n s e v e r a l y e a r s a g o , [ 1 J I r e a d t h e p a p e r w i t h p a r t i c u l a r i n t e r e s t .A p o i n t t o n o t e i s t h a t i f t h e o b j e c t i v e f u n c t i o n , F i s t o t a l c o s t o f g e n e r a -t i o n t h e n t h e L a g r a n g e m u l t i p l i e r s X p k am d X p a e q u a l t h e i n c r e m e n t a lc o s t a s s o c i a t e d w i t h t h e i r p a r t i c u l a r c o n s t r a i n t e q u a t i o n , i . e . , t h e i n -c r e m e n t a l c o s t o f p o w e r o r v a r s a t s y s t e m b u s e s w h e r e p o w e r o r v a r sa r e c o n s t r a i n e d . T h i s s u g g e s t s t h a t t h i s i n f o r m a t i o n m i g h t b e u s e f u l f o rd e t e r m i n i n g t h e c h a r g e s w h i c h s h o u l d b e a s s e s s e d f o r i n t e r c h a n g e s b e -t w e e n u t i l i t i e s o r t h e m o s t e c o n o m i c l o c a t i o n f o r c a p a c i t o r b a n k s . H a v et h e a u t h o r s p u r s u e d s u c h a n e x t e n s i o n o f t h e i r m e t h o d ?

R E F E R E N C E[ 1 ] V. D . A l b e r t s o n , B . K . J o h n s o n , a n d W. S c o t t M e y e r , " E x a c t E c o -n o m i c D i s p a t c h W i t h Q u a d r a t i c C o n v e r g e n c e " , I E E E C o n f e r e n c eP a p e r N o . 6 8 CP 671-PWR p r e s e n t e d a t t h e I E E E S u m m e r M e e t i n g ,J u n e 2 3 - 2 8 , 1 9 6 8 .M a n u s c r i p t r e c e i v e d F e b r u a r y 2 1 , 1 9 8 4

W a l t e r L . S n y d e r , J r . ( L e e d s a n d N o r t h r u p C o . , N o r t h W a l e s , P A ) : T h ea u t h o r s a r e t o b e c o m p l i m e n t e d on a w e l l d e v e l o p e d , p r a c t i c a l , a n dw o r ka b l e a p p r o a c h t o t h e o p t i m a l p o w e r f l o w p r o b l e m . T h i s d i s c u s s i o np r e s e n t s a m o r e c o m p a c t a n d g e n e r a l i z e d m a t h e m a t i c a l s t a t e m e n t o f t h ep r o b l e m o b j e c t i v e , a n d e x p l o r e s t h e t r e a t m e n t o f a c t i v e g e n e r a t i o n f o rb o t h b i n d i n g a n d n o n b i n d i n g c o n s t r a i n t s .T h e a u t h o r s ' a p p r o a c h m i n i m i z e s t h e f o l l o w i n g L a g r a n g i a n t y p ef u n c t i o n :L ( z ) = L ( y , X ) = [ C * P ] - R i * d P ] + ( 2 ) * [ d x * [ S ] * d x ] ( e q u a t i o n d i )w h e r e : [ a n d ] d e n o t e r o w a n d c o l u m n v e c t o r s , r e s p e c t i v e l yT h e d " o p e r a t o r " d e n o t e s a f i n i t e c h a n g e , n o t d i f f e r e n t i a t i o np ] = P ( y ) ] , a n d i n c l u d e s b o t h r e a l a n d r e a c t i v e p o w e r ( o r e i t h e r on ei n a d e c o u p l e d f o r m u l a t i o n )d p ] = P ( y ) ] - P ( s c h ) ]d x ] = [ A ] * Y ] - x ( s c h ) ]E q . ( d i ) s ho u ld b e i d e n t i c a l t o t h e a u t h o r s ' f o r m u l a t i o n a n d l e a d t ot h e s a m e s o l u t i o n e q u a t i o n , b u t e x p r e s s e s t h e p r o b l e m i n m a t r i x f o r m ,e x p l i c i t l y i n c l u d i n g t h e q u a d r a t i c p e n a l t y f u n c t i o n s . T h e a d v a n t a g e o ft h e f o r m g i v e n i n ( d i ) i s t h a t i t a l l o w s o n e t o m o r e c l e a r l y r e v i e w a n de v a l u a t e t h e c o m p l e t e o b j e c t i v e .I t s h ou ld b e p o i n t e d o u t t h a t t h e l a m b d a t e r ms o f ( d i ) c o n t a i n d P r a t h e rt h a n P a s i n t h e a u t h o r s ' p a p e r . C o u l d t h e a u t h o r s p l e a s e v e r i f y t h a tP i n t h e i r l a m b d a t e r m i s a c t u a l l y a m i s m a t c h , d P , w h i l e P i n t h e i r c o s tt e r m r e p r e s e n t s t h e a c t u a l i n j e c t i o n a n d n o t a m i s m a t c h . I f s u c h i s n o tc o r r e c t , w o u l d t h e a u t h o r s p l e a s e c l a r i f y t h e s e e x p r e s s i o n s .

T h e d i s c u s s e r v i e w s C a n d X a s p l a y i n g t h e s a m e r o l e , w h e r e b o t h a r en e ve r n o n - ze r o a t t h e s a m e t i m e f o r a g i v e n e le me n t o f P . I f C i s z e r o ,X i s a v a r i a b l e w h i c h i s s o l v e d s u c h t h a t P i s c o n s t r a i n e d t o a s c h e d u l e dv a l u e b y e n f o r c e m e n t o f t h e K u h n - T u c k e r c o n d i t i o n s . A n o n - z e r o C ,h o e v e r , i s v i e w e d a s a p r e s p e c i f i e d , f i x e d i n c r e m e n t a l c o s t t h e r e b y t a k -i n g t h e p l a c e o f l a m b d a . F o r n o n - z e r o C , P i s n o l o n g e r c o n s t r a i n e d t oa s c h e d u l e d v a l u e s i n c e C i s n o t a v a r i a b l e , r a t h e r t h e C*P t e r m i s m i n i m i z -e d i n c o n j u n c t i o n w i t h t h e r e s t o f t h e c o s t t e r m s . F o r c o n s i s t e n c y , as c h e d u l e d v a l u e o f 0 c o u l d b e a s s u m e d i n t h e C*P t e r m s , a l t h o u g h t h i ss c h e d u l e d v a l u e i s n e v e r e n f o r c e d . T h e a p p r o a c h w o u l d t e n d t o r e d u c et h e d i s p a t c h e d e l e m e n t s o f P t o z e r o w e r e i t n o t f o r m a i n t e n a n c e o f ap o w e r b a l a n c e v i a t h e J a n d H s e n s i t i v i t i e s w h i c h r e l a t e d i s p a t c h e d Pt o s c h e d u l e d P .T h e a u t h o r s ' c o m m e n t s o n t h e p r e c e d i n g p a r a g r a p h w o u l d b e a p -p r e c i a t e d . I n p a r t i c u l a r , t h e t r e a t me n t o f r e a l p o w e r C a n d l a mb d a c o e f -f i c i e n t s w i t h r e s p e c t t o b i n d i n g a n d n o n - b i n d i n g c o n s t r a i n t s i s n o t c l e a r -l y e x p l a i n e d i n t h e p a p e r . A r e t h e r e a l p o w e r l a m b d a s h e l d a t z e r o w h i l eP i s d i s p a t c h e d w i t h i n l i m i t s , a n d a r e t h e a s s o c i a t e d c o s t s s e t t o z e r o w h e nP i s a t a l i m i t ? I f C i s l e f t n o n - z e r o w h e n P i s b i n d i n g , d o e s t h e n o n - z e r ol a m b d a e f f e c t i v e l y o v e r r i d e t h e C*P t e r m ? I n E q . ( 4 ) o f t h e p a p e r , t h ei n c l u s i o n o f r e a l p o w e r b u t n o t r e a c t i v e p o w e r l a mb d a t e r m s f o r b u s e sa n d 3 l e n d s t o t h e c o n f u s i o n , s i n c e b o t h r e a l a n d r e a c t i v e p o w e r a r ed e f i n e d a s d i s p a t c h a b l e o n t h e t w o b u s e s .S i m i l a r c o n f u s i o n i s e v i d e n t i n E q s . ( 6 ) a n d ( 9 ) o f t h e p a p e r w h e r er e a c t i v e l a m b d a t e r m s a r e ' d u m m i e d o u t ' ( s e t t o z e r o ) f o r b u s e s 1 a n d3 w h i l e r e a l p o w e r l a mb d a t e r m s a r e n o t . A g a i n , t h i s s u g g e s t s t h a t r e a lp o w e r o n b u s e s 1 a n d 3 i s s c h e d u l e d w h i l e t h e p a p e r s t a t e s t h a t i t i sd i s p a t c h a b l e .I t s h o u l d b e p o i n t e d o u t t h a t t h e z e r o d i a g o n a l t e r m s i n ( 6 ) a n d ( 9 )w i l l p r e s e n t n o p r o bl em s i n c e f i l l i n w i l l o c c u r w h e n t h e m a t r i x i s a c t u a l -l y f a c t o r e d d u e t o t h e j a co b i a n t e r m s o f t h e a s s o c i a t e d d i a g o n a l m a t r i xp a c k e t . I t i s a s s u m e d t h a t t h e o n l y p u r p o s e s o f t h e a r b i t r a r i l y l a r g ed i a g o n a l t e r m s i s t o e n s u r e a z e r o c o r r e c t i o n t o l a m b d a a n d v o n e x e c u -t i o n o f t h e ' d i v i d e b y d i a g o n a l ' s t e p o f t h e f o r w a r d - b a c k w a r d s u b s t i t u -t i o n p r o c e s s .F i n a l l y , t h e l a s t t e r m o f E q . ( d i ) a t t h e b e g i n n i n g o f t h i s d i s c u s s i o ni s s e e n a s a m o r e g e n e r a l e x p r e s s i o n o f t h e q u a d r a t i c p e n a l t y t e r m s u s e dt o e n f o r c e l i m i t s o n t h e v a r i a b l e s t h e m s e l v e s a s w e l l a s o n l i n e a r f u n c -t i o n s o f t h e v a r i a b l e s . [ A ] i s s i m p l y a n i n c i d e n c e t y p e m a t r i x ( n o t t o b ec o n f u s e d w i t h t h e s e t , A , o f b i n d i n g c o n s t r a i n t s ) , a n d i s a t r i v i a l i d e n t i -t y f o r l i m i t s o n t h e v a r i a b l e s t h e m s e l v e s . Was t h i s m e t h o d c o n s i d e r e db y t h e a u t h o r s f o r " b e y o n d t h e s c o p e o f t h i s p a p e r " s o f t e n f o r c e m e n to f l i m i t s o n p o w e r s o u r c e s ?I n E q s . ( 1 6 a ) a n d ( 1 6 b ) o f t h e p a p e r , s h o u l d a s c h e d u l e d p o w e r a n g l et e r m b e i n c l u d e d a s w a s d o n e i n E q . ( 1 5 ) ?O v e r a l l , t h e p r e s e n t a t i o n o f t h i s p a p e r w a s s t i l l e x c e l l e n t a n d w a s e x -t r e me ly m o t i v a t i n g t o t h e d i s c u s s e r , i n j e c t i n g n e w l i f e i n t o t h e o p t i m a lp o w e r f l o w p r o b l e m .

M a n u s c r i p t r e c e i v e d F e b r u a r y 2 3 , 1 9 8 4


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2 8 7 7B. S t o t t , 0 . A l s a c a n d A . B o s e ( E l e c t r i c a l & C o m p u t e r E n g i n e e r i n g ,A r i z o n a S t a t e U n i v . , T e m p e , A Z ) : F o r ov er t w e n t y years, r e s e a r c h e r sh a ve h a d l i m i t e d s u c c e s s i n d e v e l o p i n g e f f i c i e n t , r e l i a b l e s o l u t i o n m e t h o d sf o r l a r g e " c l a s s i c a l " O PF p r o b l e m s w i t h n o n s e p a r a b l e o b j e c t i v e f u n c -t i o n s . T h e s e p r o b l e m s a r e e n c o u n t e r e d m a i n l y w h e n s c h e d u l i n g r e a c t i v ep o w e r f o r t h e m i n i m i z a t i o n o f o b j e c t i v e s s u c h a s p r o d u c t i o n c o s t or l o s s e s .T h e p r e s e n t p a p e r i n t r o d u c e s a n a p p r o a c h w i t h r e a l p o t e n t i a l f o r ab r e a k t h r o u g h . I n d o i n g s o i t r e v e r s e s a r e c e n t t r e n d t o w a r d s t h e us e o fs o p h i s t i c a t e d m o d e r n g e n e r a l - p u r p o s e c o n s t r a i n e d - o p t i m i z a t i o n m e t h o d sa n d / o r c o d e s , w h i c h a p p e a r t o s o l v e O PF p r o b l e m s c o r r e c t l y b u t r e l a t i v e l yv e r y s l o w l y . H i g h s p e e d i s a i m e d f o r b y r e v e r t i n g t o o p t i m i z a t i o n f i r s tp r i n c i p l e s a n d f o c u s s i n g t h e d e v e l o p m e n t e f f o r t on s t r u c t u r e - s u i t e d i m -p l e m e n t a t i o n , r e s u l t i n g i n an a p p r o a c h w i t h g o o d c o n v e r g e n c e p o w e ra n d e x c e l l e n t n e t w o r k s p a r s i t y p r e s e r v a t i o n .T h e a t t r a c t i o n o f t he a p p r o a c h c e n t e r s on t he f a c i l i t y w i t h w h i c h t h ee q u a l i t y - c o n s t r a i n e d OPF p r o b l e m ca n b e s o l v e d . T h e N e w t o n s o l u t i o no f i t s L a g r a n g i a n n e c e s s a r y - c o n d i t i o n e q u a t i o n s i s c o n c e p t u a l l y s i m p l e ,p e r f o r m s s e c o n d o r de r m i n i m i z a t i o n w i t h no t u n i n g , a n d i s a m e n a b l et o e f f i c i e n t e x p l o i t a t i o n o f p r o b l e m s t r u c t u r e . U n f o r t u n a t e l y , t h i s ap -p r o a c h h a s n o a c c o m p a n y i n g p o w e r f u l w e l l - e s t a b l i s h e d means f o r h a n d l -i n g i n e q u a l i t i e s . T h e r e f o r e h e u r i s t i c s c h e m e s , t a i l o r e d t o p a r t i c u l a r OPFf o r m u l a t i o n s , n e e d t o b e d e v e l o p e d f o r s w i t c h i n g l i m i t c o n s t r a i n t s i n t oa n d o u t o f t h e b i n d i n g s e t v i a e q u a t i o n a d d i t i o n / r e m o v a l or q u a d r a t i cp e n a l t i e s .T h e s a m e g e n e r a l a p p r o a c h h a s b e e n t r i e d i n s m a l l - s c a l e r e s e a r c h an u m b e r o f t i m e s b e f o r e ( s e e e s p e c i a l l y [ A ] ) . A m o n g t h e m a j o r i m -p r o v e m e n t s i n t r o d u c e d b y t h e p r e s e n t work i s a superior way o f o r g a n i z i n gt h e n e c e s s a r y - c o n d i t i o n e q u a t i o n s f o r e f f i c i e n t d e c o u p l e d s p a r s e s o l u -t i o n . I t i s s h o w n t h a t q u a d r a t i c p e n a l t i e s can b e u s e d e f f e c t i v e l y f o r s t a t e -v a r i a b l e l i m i t s , a s w e l l a s f o r f u n c t i o n a l i n e q u a l i t i e s . I m p r o v e m e n t s i ne f f i c i e n c y t h r o u g h t he u s e o f a d v a n c e d s p a r s e m a t r i x t e c h n i q u e s ar ee m p h a s i z e d .T h e m o s t c r i t i c a l f a c t o r f o r s u c c e s s i n t h e a p p r o a c h i s t o f i n d s c h e m e st h a t d e t e r m i n e t h e b i n d i n g l i m i t s e t r e l i a b l y a n d r a p i d l y , o n a w i d e rangeo f p o w e r s y s t e m s , c o n s t r a i n t t y p e s a n d O P F f o r m u l a t i o n s . T h i s i s p r e c i s e -l y t h e m a j o r p r o b l e m e n c o u n t e r e d i n many p r e v i o u s O P F m e t h o d s a n d ,a s f a r a s i s k n o w n , n e v e r y e t s a t i s f a c t o r i l y r e s o l v e d . T h e p r e s e n t p a p e re m p h a s i z e s t h i s a s t h e k e y p r o b l e m t h a t must b e o v e r c o m e b y f u r t h e rd e v e l o p m e n t . I t g i v e s v e r y r o u g h o u t l i n e s o f s e v e r a l c a n d i d a t e t r i a l -i t e r a t i o n s c h e m e s .A s t h e p a p e r i t s e l f s t a t e s , i t i s i n t r o d u c i n g a m e t h o d o l o g y as mucha s a d e v e l o p e d m e t h o d . Many q u e s t i o n s o f d e t a i l r e m a i n t o b e a n s w e r e d .A f e w o f t h e s e q u e s t i o n s a r e a s f o l l o w s . I t i s s t a te d t ha t e n h a n c e m e n to f p o s i t i v e - d e f i n i t e n e s s o f t h e L a g r a n g i a n i s unneces s a ry, i m p l y i n g t h ea b s e n c e o f c o n ve r ge n ce p r o b l e m s . Has s u f f i c i e n t e x p e r i e n c e w i t h d i f -f e r e n t s y s t e m s a n d s i t u a t i o n s b e en a c c u m u l a t ed t o v e r i f y t h i s ? I s d e c o u p l -i n g s e n s i t i v e t o b r a n c h R/X r a t i o s ? Wh a t i s t h e e f f e c t o f d ec o u p li n g w h e nv o l t a g e m a g n i t u d e l i m i t s n e e d t o b e e n f o r c ed b y a c t i v e power s c h e d u l -i n g . W h a t i s t h e i n f e a s i b i l i t y b e h a v i o r o f t h e a p p r o a c h , a n d w h a t i n -f e a s i b i l i t y s t r a t e g i e s a r e m o s t a p p r o p r i a t e w i t h i t ? P e n a l t i e s ar e a d v o c a t e df o r f u n c t i o n a l i n e q u a l i t i e s , b u t o t he r t h a n f o r s i

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